<ol>
+
+ <li> New: A new element FE_Q_iso_Q1 has been implemented that is defined by
+ a subdivision of the element into smaller Q1 elements. An element of order
+ @p p is similar to FE_Q of degree @p p with the same numbering of degrees of
+ freedom. The element is useful e.g. for defining a sparser preconditioner
+ matrix for AMG at higher order FE_Q elements or for representing a component
+ of a system of PDEs where higher resolution is preferred over high order.
+ <br>
+ (Martin Kronbichler, 2013/05/14)
+ </li>
+
<li> Fixed: setting values in TrilinosWrappers::SparseMatrix
in parallel was adding the values instead.
<br>
*/
/**
- * A namespace in which classes relating to the description of
- * 1d polynomial spaces are declared.
+ * A namespace in which classes relating to the description of 1d polynomial
+ * spaces are declared.
*/
namespace Polynomials
{
{
public:
/**
- * Constructor. The
- * coefficients of the
- * polynomial are passed as
- * arguments, and denote the
- * polynomial $\sum_i a[i]
- * x^i$, i.e. the first element
- * of the array denotes the
- * constant term, the second
- * the linear one, and so
- * on. The degree of the
- * polynomial represented by
- * this object is thus the
- * number of elements in the
- * <tt>coefficient</tt> array
- * minus one.
+ * Constructor. The coefficients of the polynomial are passed as
+ * arguments, and denote the polynomial $\sum_i a[i] x^i$, i.e. the first
+ * element of the array denotes the constant term, the second the linear
+ * one, and so on. The degree of the polynomial represented by this object
+ * is thus the number of elements in the <tt>coefficient</tt> array minus
+ * one.
*/
Polynomial (const std::vector<number> &coefficients);
/**
- * Constructor creating a zero
- * polynomial of degree @p n.
+ * Constructor creating a zero polynomial of degree @p n.
*/
Polynomial (const unsigned int n);
/**
- * Constructor for Lagrange polynomial and its
- * point of evaluation. The idea is to
- * construct $\prod_{i\neq j}
- * \frac{x-x_i}{x_j-x_i}$, where j is the
- * evaluation point specified as argument and
- * the support points contain all points
- * (including x_j, which will internally not
- * be stored).
+ * Constructor for Lagrange polynomial and its point of evaluation. The
+ * idea is to construct $\prod_{i\neq j} \frac{x-x_i}{x_j-x_i}$, where j
+ * is the evaluation point specified as argument and the support points
+ * contain all points (including x_j, which will internally not be
+ * stored).
*/
Polynomial (const std::vector<Point<1> > &lagrange_support_points,
const unsigned int evaluation_point);
/**
- * Default constructor creating
- * an illegal object.
+ * Default constructor creating an illegal object.
*/
Polynomial ();
/**
- * Return the value of this
- * polynomial at the given point.
+ * Return the value of this polynomial at the given point.
*
- * This function uses the Horner
- * scheme for numerical stability
- * of the evaluation.
+ * This function uses the Horner scheme for numerical stability of the
+ * evaluation.
*/
number value (const number x) const;
/**
- * Return the values and the
- * derivatives of the
- * Polynomial at point <tt>x</tt>.
- * <tt>values[i],
- * i=0,...,values.size()-1</tt>
- * includes the <tt>i</tt>th
- * derivative. The number of
- * derivatives to be computed is
- * thus determined by the size of
- * the array passed.
+ * Return the values and the derivatives of the Polynomial at point
+ * <tt>x</tt>. <tt>values[i], i=0,...,values.size()-1</tt> includes the
+ * <tt>i</tt>th derivative. The number of derivatives to be computed is
+ * thus determined by the size of the array passed.
*
- * This function uses the Horner
- * scheme for numerical stability
- * of the evaluation.
+ * This function uses the Horner scheme for numerical stability of the
+ * evaluation.
*/
void value (const number x,
std::vector<number> &values) const;
/**
- * Degree of the polynomial. This
- * is the degree reflected by the
- * number of coefficients
- * provided by the
- * constructor. Leading non-zero
- * coefficients are not treated
- * separately.
+ * Degree of the polynomial. This is the degree reflected by the number of
+ * coefficients provided by the constructor. Leading non-zero coefficients
+ * are not treated separately.
*/
unsigned int degree () const;
/**
- * Scale the abscissa of the
- * polynomial. Given the
- * polynomial <i>p(t)</i> and the
- * scaling <i>t = ax</i>, then the
- * result of this operation is
- * the polynomial <i>q</i>, such that
- * <i>q(x) = p(t)</i>.
+ * Scale the abscissa of the polynomial. Given the polynomial <i>p(t)</i>
+ * and the scaling <i>t = ax</i>, then the result of this operation is the
+ * polynomial <i>q</i>, such that <i>q(x) = p(t)</i>.
*
- * The operation is performed in
- * place.
+ * The operation is performed in place.
*/
void scale (const number factor);
/**
- * Shift the abscissa oft the
- * polynomial. Given the
- * polynomial <i>p(t)</i> and the
- * shift <i>t = x + a</i>, then the
- * result of this operation is
- * the polynomial <i>q</i>, such that
- * <i>q(x) = p(t)</i>.
+ * Shift the abscissa oft the polynomial. Given the polynomial
+ * <i>p(t)</i> and the shift <i>t = x + a</i>, then the result of this
+ * operation is the polynomial <i>q</i>, such that <i>q(x) = p(t)</i>.
*
- * The template parameter allows
- * to compute the new
- * coefficients with higher
- * accuracy, since all
- * computations are performed
- * with type <tt>number2</tt>. This
- * may be necessary, since this
- * operation involves a big
- * number of additions. On a Sun
- * Sparc Ultra with Solaris 2.8,
- * the difference between
- * <tt>double</tt> and <tt>long double</tt>
- * was not significant, though.
+ * The template parameter allows to compute the new coefficients with
+ * higher accuracy, since all computations are performed with type
+ * <tt>number2</tt>. This may be necessary, since this operation involves
+ * a big number of additions. On a Sun Sparc Ultra with Solaris 2.8, the
+ * difference between <tt>double</tt> and <tt>long double</tt> was not
+ * significant, though.
*
- * The operation is performed in
- * place, i.e. the coefficients
- * of the present object are
- * changed.
+ * The operation is performed in place, i.e. the coefficients of the
+ * present object are changed.
*/
template <typename number2>
void shift (const number2 offset);
/**
- * Compute the derivative of a
- * polynomial.
+ * Compute the derivative of a polynomial.
*/
Polynomial<number> derivative () const;
/**
- * Compute the primitive of a
- * polynomial. the coefficient
- * of the zero order term of
- * the polynomial is zero.
+ * Compute the primitive of a polynomial. the coefficient of the zero
+ * order term of the polynomial is zero.
*/
Polynomial<number> primitive () const;
Polynomial<number> &operator -= (const Polynomial<number> &p);
/**
- * Test for equality of two polynomials.
- */
+ * Test for equality of two polynomials.
+ */
bool operator == (const Polynomial<number> &p) const;
/**
void print(std::ostream &out) const;
/**
- * Write or read the data of this object to or
- * from a stream for the purpose of serialization.
+ * Write or read the data of this object to or from a stream for the
+ * purpose of serialization.
*/
template <class Archive>
void serialize (Archive &ar, const unsigned int version);
protected:
/**
- * This function performs the
- * actual scaling.
+ * This function performs the actual scaling.
*/
static void scale (std::vector<number> &coefficients,
const number factor);
/**
- * This function performs the
- * actual shift
+ * This function performs the actual shift
*/
template <typename number2>
static void shift (std::vector<number> &coefficients,
const number factor);
/**
- * Transforms polynomial form of
- * product of linear factors into
- * standard form, $\sum_i a_i
- * x^i$. Deletes all data structures
- * related to the product form.
+ * Transforms polynomial form of product of linear factors into standard
+ * form, $\sum_i a_i x^i$. Deletes all data structures related to the
+ * product form.
*/
void transform_into_standard_form ();
/**
- * Coefficients of the polynomial
- * $\sum_i a_i x^i$. This vector
- * is filled by the constructor
- * of this class and may be
- * passed down by derived
+ * Coefficients of the polynomial $\sum_i a_i x^i$. This vector is filled
+ * by the constructor of this class and may be passed down by derived
* classes.
*
- * This vector cannot be constant
- * since we want to allow copying
- * of polynomials.
+ * This vector cannot be constant since we want to allow copying of
+ * polynomials.
*/
std::vector<number> coefficients;
/**
- * Stores whether the polynomial is in
- * Lagrange product form, i.e.,
- * constructed as a product $(x-x_0)
- * (x-x_1) \ldots (x-x_n)/c$, or not.
+ * Stores whether the polynomial is in Lagrange product form, i.e.,
+ * constructed as a product $(x-x_0) (x-x_1) \ldots (x-x_n)/c$, or not.
*/
bool in_lagrange_product_form;
/**
- * If the polynomial is in Lagrange
- * product form, i.e., constructed as a
- * product $(x-x_0) (x-x_1) \ldots
- * (x-x_n)/c$, store the shifts $x_i$.
+ * If the polynomial is in Lagrange product form, i.e., constructed as a
+ * product $(x-x_0) (x-x_1) \ldots (x-x_n)/c$, store the shifts $x_i$.
*/
std::vector<number> lagrange_support_points;
/**
- * If the polynomial is in Lagrange
- * product form, i.e., constructed as a
- * product $(x-x_0) (x-x_1) \ldots
- * (x-x_n)/c$, store the weight c.
+ * If the polynomial is in Lagrange product form, i.e., constructed as a
+ * product $(x-x_0) (x-x_1) \ldots (x-x_n)/c$, store the weight c.
*/
number lagrange_weight;
};
{
public:
/**
- * Constructor. Takes the degree
- * <tt>n</tt> of the Lagrangian
- * polynom and the index
- * <tt>support_point</tt> of the
- * support point. Fills the
- * <tt>coefficients</tt> of the base
- * class Polynomial.
+ * Constructor. Takes the degree <tt>n</tt> of the Lagrangian polynom and
+ * the index <tt>support_point</tt> of the support point. Fills the
+ * <tt>coefficients</tt> of the base class Polynomial.
*/
LagrangeEquidistant (const unsigned int n,
const unsigned int support_point);
/**
- * Return a vector of polynomial
- * objects of degree <tt>degree</tt>,
- * which then spans the full
- * space of polynomials up to the
- * given degree. The polynomials
- * are generated by calling the
- * constructor of this class with
- * the same degree but support
- * point running from zero to
- * <tt>degree</tt>. This function may
- * be used to initialize the
- * TensorProductPolynomials
- * and PolynomialSpace
- * classes.
+ * Return a vector of polynomial objects of degree <tt>degree</tt>, which
+ * then spans the full space of polynomials up to the given degree. The
+ * polynomials are generated by calling the constructor of this class with
+ * the same degree but support point running from zero to
+ * <tt>degree</tt>. This function may be used to initialize the
+ * TensorProductPolynomials and PolynomialSpace classes.
*/
static
std::vector<Polynomial<double> >
private:
/**
- * Computes the <tt>coefficients</tt>
- * of the base class
- * Polynomial. This function
- * is <tt>static</tt> to allow to be
- * called in the
- * constructor.
+ * Computes the <tt>coefficients</tt> of the base class Polynomial. This
+ * function is <tt>static</tt> to allow to be called in the constructor.
*/
static
void
};
+
/**
- * Given a set of points along the
- * real axis, this function returns
- * all Lagrange polynomials for
- * interpolation of these
- * points. The number of
- * polynomials is equal to the
- * number of points and the maximum
- * degree is one less.
+ * Given a set of points along the real axis, this function returns all
+ * Lagrange polynomials for interpolation of these points. The number of
+ * polynomials is equal to the number of points and the maximum degree is
+ * one less.
*/
std::vector<Polynomial<double> >
generate_complete_Lagrange_basis (const std::vector<Point<1> > &points);
void
Polynomial<number>::serialize (Archive &ar, const unsigned int)
{
- // forward to serialization
- // function in the base class.
+ // forward to serialization function in the base class.
ar &static_cast<Subscriptor &>(*this);
ar &coefficients;
ar &in_lagrange_product_form;
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2009, 2010, 2011, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+#ifndef __deal2__polynomials_piecewise_h
+#define __deal2__polynomials_piecewise_h
+
+
+
+#include <deal.II/base/config.h>
+#include <deal.II/base/exceptions.h>
+#include <deal.II/base/subscriptor.h>
+#include <deal.II/base/polynomial.h>
+#include <deal.II/base/point.h>
+
+#include <vector>
+
+DEAL_II_NAMESPACE_OPEN
+
+/**
+ * @addtogroup Polynomials
+ * @{
+ */
+
+/**
+ * A namespace in which classes relating to the description of 1d polynomial
+ * spaces are declared.
+ */
+namespace Polynomials
+{
+
+ /**
+ * Definition of piecewise 1D polynomials for the unit interval. This space
+ * allows the description of interpolating polynomials on parts of the unit
+ * interval, similarly to the definition of finite element basis functions
+ * on the subdivided elements. This primary purpose of this class is to
+ * allow constructing FE_Q_iso_Q1 elements that put additional degrees of
+ * freedom into an equivalent of a refined mesh instead of higher order
+ * polynomials, which is useful when using mixed finite elements.
+ *
+ * @author Martin Kronbichler, 2013
+ */
+ template <typename number>
+ class PiecewisePolynomial : public Subscriptor
+ {
+ public:
+ /**
+ * Constructor for Lagrange polynomial on an interval that is a subset of
+ * the unit interval. It uses a polynomial description that is scaled to
+ * the size of the subinterval compared to the unit interval, the total
+ * number of intervals (subdivisions), the current index of the interval
+ * as well as if the polynomial spans onto the next interval (e.g., if it
+ * lives on two neighboring intervals).
+ *
+ * If the number of intervals is one, the piecewise polynomial behaves
+ * exactly like a usual polynomial.
+ */
+ PiecewisePolynomial (const Polynomial<number> &coefficients_on_interval,
+ const unsigned int n_intervals,
+ const unsigned int interval,
+ const bool spans_next_interval);
+
+ /**
+ * Default constructor creating an illegal object.
+ */
+ PiecewisePolynomial ();
+
+ /**
+ * Return the value of this polynomial at the given point, evaluating the
+ * underlying polynomial. The polynomial evaluates to zero when outside of
+ * the given interval (and possible the next one to the right when it
+ * spans over that range).
+ */
+ number value (const number x) const;
+
+ /**
+ * Return the values and the derivatives of the Polynomial at point
+ * <tt>x</tt>. <tt>values[i], i=0,...,values.size()-1</tt> includes the
+ * <tt>i</tt>th derivative. The number of derivatives to be computed is
+ * thus determined by the size of the array passed.
+ *
+ * Note that all the derivatives evaluate to zero at the border between
+ * intervals (assuming exact arithmetics) in the interior of the unit
+ * interval, as there is no unique gradient value in that case for a
+ * piecewise polynomial. This is not always desired (e.g., when evaluating
+ * jumps of gradients on the element boundary), but it is the user's
+ * responsibility to avoid evaluation at these points when it does not
+ * make sense.
+ */
+ void value (const number x,
+ std::vector<number> &values) const;
+
+ /**
+ * Degree of the polynomial. This is the degree of the underlying base
+ * polynomial.
+ */
+ unsigned int degree () const;
+
+ /**
+ * Write or read the data of this object to or from a stream for the
+ * purpose of serialization.
+ */
+ template <class Archive>
+ void serialize (Archive &ar, const unsigned int version);
+
+ protected:
+
+ /**
+ * Underlying polynomial object that is scaled to a subinterval and
+ * concatenated accordingly.
+ */
+ Polynomial<number> polynomial;
+
+ /**
+ * Stores the number of intervals that the unit interval is divided into.
+ */
+ const unsigned int n_intervals;
+
+ /**
+ * Stores the index of the current polynomial in the range of
+ * intervals.
+ */
+ const unsigned int interval;
+
+ /**
+ * Store if the polynomial spans over two adjacent intervals, i.e., the
+ * one given in subinterval and the next one.
+ */
+ bool spans_two_intervals;
+ };
+
+
+
+ /**
+ * Generates a complete Lagrange basis on a subdivision of the unit interval
+ * in smaller intervals for a given degree on the subintervals and number of
+ * intervals.
+ */
+ std::vector<PiecewisePolynomial<double> >
+ generate_complete_Lagrange_basis_on_subdivisions (const unsigned int n_subdivisions,
+ const unsigned int base_degree);
+
+}
+
+
+/** @} */
+
+/* -------------------------- inline functions --------------------- */
+
+namespace Polynomials
+{
+ template <typename number>
+ inline
+ unsigned int
+ PiecewisePolynomial<number>::degree () const
+ {
+ return polynomial.degree();
+ }
+
+
+
+ template <typename number>
+ inline
+ number
+ PiecewisePolynomial<number>::value (const number x) const
+ {
+ AssertIndexRange (interval, n_intervals);
+ number y = x;
+ // shift polynomial if necessary
+ if (n_intervals > 1)
+ {
+ const number step = 1./n_intervals;
+
+ // polynomial spans over two intervals
+ if (spans_two_intervals == true)
+ {
+ const number offset = step * interval;
+ if (x<offset)
+ return 0;
+ else if (x>offset+step+step)
+ return 0;
+ else if (x<offset+step)
+ y = x-offset;
+ else
+ y = offset+step+step-x;
+ }
+ else
+ {
+ const number offset = step * interval;
+ if (x<offset || x>offset+step)
+ return 0;
+ else
+ y = x-offset;
+ }
+
+ return polynomial.value(y);
+ }
+ else
+ return polynomial.value(x);
+ }
+
+
+
+ template <typename number>
+ template <class Archive>
+ inline
+ void
+ PiecewisePolynomial<number>::serialize (Archive &ar, const unsigned int)
+ {
+ // forward to serialization function in the base class.
+ ar &static_cast<Subscriptor &>(*this);
+ ar &polynomial;
+ ar &n_intervals;
+ ar &interval;
+ ar &spans_two_intervals;
+ }
+
+}
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
*
* @author Ralf Hartmann, 2000, 2004, Guido Kanschat, 2000, Wolfgang Bangerth 2003
*/
-template <int dim>
+template <int dim, typename POLY=Polynomials::Polynomial<double> >
class TensorProductPolynomials
{
public:
/**
- * Access to the dimension of
- * this object, for checking and
- * automatic setting of dimension
- * in other classes.
+ * Access to the dimension of this object, for checking and automatic
+ * setting of dimension in other classes.
*/
static const unsigned int dimension = dim;
/**
- * Constructor. <tt>pols</tt> is
- * a vector of objects that
- * should be derived or otherwise
- * convertible to one-dimensional
- * polynomial objects. It will be
- * copied element by element into
- * a private variable.
+ * Constructor. <tt>pols</tt> is a vector of objects that should be derived
+ * or otherwise convertible to one-dimensional polynomial objects of type @p
+ * POLY (template argument of class). It will be copied element by element
+ * into a private variable.
*/
template <class Pol>
TensorProductPolynomials (const std::vector<Pol> &pols);
/**
- * Prints the list of the indices
- * to <tt>out</tt>.
+ * Prints the list of the indices to <tt>out</tt>.
*/
void output_indices(std::ostream &out) const;
/**
- * Sets the ordering of the
- * polynomials. Requires
- * <tt>renumber.size()==n()</tt>.
- * Stores a copy of
- * <tt>renumber</tt>.
+ * Sets the ordering of the polynomials. Requires
+ * <tt>renumber.size()==n()</tt>. Stores a copy of <tt>renumber</tt>.
*/
void set_numbering(const std::vector<unsigned int> &renumber);
/**
- * Gives read access to the
- * renumber vector.
+ * Gives read access to the renumber vector.
*/
const std::vector<unsigned int> &get_numbering() const;
/**
- * Gives read access to the
- * inverse renumber vector.
+ * Gives read access to the inverse renumber vector.
*/
const std::vector<unsigned int> &get_numbering_inverse() const;
/**
- * Computes the value and the
- * first and second derivatives
- * of each tensor product
- * polynomial at <tt>unit_point</tt>.
+ * Computes the value and the first and second derivatives of each tensor
+ * product polynomial at <tt>unit_point</tt>.
*
- * The size of the vectors must
- * either be equal 0 or equal
- * n(). In the first case, the
- * function will not compute
- * these values.
+ * The size of the vectors must either be equal 0 or equal n(). In the first
+ * case, the function will not compute these values.
*
- * If you need values or
- * derivatives of all tensor
- * product polynomials then use
- * this function, rather than
- * using any of the
- * compute_value(),
- * compute_grad() or
- * compute_grad_grad()
- * functions, see below, in a
- * loop over all tensor product
- * polynomials.
+ * If you need values or derivatives of all tensor product polynomials then
+ * use this function, rather than using any of the compute_value(),
+ * compute_grad() or compute_grad_grad() functions, see below, in a loop
+ * over all tensor product polynomials.
*/
void compute (const Point<dim> &unit_point,
std::vector<double> &values,
std::vector<Tensor<2,dim> > &grad_grads) const;
/**
- * Computes the value of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the value of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * point value of the underlying
- * (one-dimensional) polynomials
- * is (unnecessarily) computed
- * several times. Instead use
- * the compute() function with
- * <tt>values.size()==</tt>n()
- * to get the point values of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each point value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function with
+ * <tt>values.size()==</tt>n() to get the point values of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
double compute_value (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the grad of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the grad of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the compute()
- * function, see above, with
- * <tt>grads.size()==</tt>n()
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function, see above, with
+ * <tt>grads.size()==</tt>n() to get the point value of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
Tensor<1,dim> compute_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the second
- * derivative (grad_grad) of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
- * numbering.
+ * Computes the second derivative (grad_grad) of the <tt>i</tt>th tensor
+ * product polynomial at <tt>unit_point</tt>. Here <tt>i</tt> is given in
+ * tensor product numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the compute()
- * function, see above, with
- * <tt>grad_grads.size()==</tt>n()
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function, see above, with
+ * <tt>grad_grads.size()==</tt>n() to get the point value of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
Tensor<2,dim> compute_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Returns the number of tensor
- * product polynomials. For <i>n</i>
- * 1d polynomials this is <i>n<sup>dim</sup></i>.
+ * Returns the number of tensor product polynomials. For <i>n</i> 1d
+ * polynomials this is <i>n<sup>dim</sup></i>.
*/
unsigned int n () const;
-private:
+protected:
/**
- * Copy of the vector <tt>pols</tt> of
- * polynomials given to the
- * constructor.
+ * Copy of the vector <tt>pols</tt> of polynomials given to the constructor.
*/
- std::vector<Polynomials::Polynomial<double> > polynomials;
+ std::vector<POLY> polynomials;
/**
- * Number of tensor product
- * polynomials. See n().
+ * Number of tensor product polynomials. See n().
*/
unsigned int n_tensor_pols;
/**
- * Index map for reordering the
- * polynomials.
+ * Index map for reordering the polynomials.
*/
std::vector<unsigned int> index_map;
/**
- * Index map for reordering the
- * polynomials.
+ * Index map for reordering the polynomials.
*/
std::vector<unsigned int> index_map_inverse;
/**
- * Each tensor product polynomial
- * <i>i</i> is a product of
- * one-dimensional polynomials in
- * each space direction. Compute
- * the indices of these
- * one-dimensional polynomials
- * for each space direction,
- * given the index <i>i</i>.
+ * Each tensor product polynomial <i>i</i> is a product of one-dimensional
+ * polynomials in each space direction. Compute the indices of these
+ * one-dimensional polynomials for each space direction, given the index
+ * <i>i</i>.
*/
- // fix to avoid compiler warnings about zero
- // length arrays
+ // fix to avoid compiler warnings about zero length arrays
void compute_index (const unsigned int i,
unsigned int (&indices)[(dim>0?dim:1)]) const;
};
-#ifndef DOXYGEN
-
-template <int dim>
-inline
-const std::vector<unsigned int> &
-TensorProductPolynomials<dim>::get_numbering() const
-{
- return index_map;
-}
-
-
-template <int dim>
-inline
-const std::vector<unsigned int> &
-TensorProductPolynomials<dim>::get_numbering_inverse() const
-{
- return index_map_inverse;
-}
-
-
-#endif // DOXYGEN
/**
{
public:
/**
- * Constructor. <tt>pols</tt> is a
- * table of one-dimensional
- * polynomials. The number of
- * rows in this table should be
- * equal to the space dimension,
- * with the elements of each row
- * giving the polynomials that
- * shall be used in this
- * particular coordinate
- * direction. These polynomials
- * may vary between coordinates,
- * as well as their number.
+ * Constructor. <tt>pols</tt> is a table of one-dimensional polynomials. The
+ * number of rows in this table should be equal to the space dimension, with
+ * the elements of each row giving the polynomials that shall be used in
+ * this particular coordinate direction. These polynomials may vary between
+ * coordinates, as well as their number.
*/
AnisotropicPolynomials (const std::vector<std::vector<Polynomials::Polynomial<double> > > &pols);
/**
- * Computes the value and the
- * first and second derivatives
- * of each tensor product
- * polynomial at <tt>unit_point</tt>.
+ * Computes the value and the first and second derivatives of each tensor
+ * product polynomial at <tt>unit_point</tt>.
*
- * The size of the vectors must
- * either be equal <tt>0</tt> or equal
- * <tt>n_tensor_pols</tt>. In the
- * first case, the function will
- * not compute these values.
+ * The size of the vectors must either be equal <tt>0</tt> or equal
+ * <tt>n_tensor_pols</tt>. In the first case, the function will not compute
+ * these values.
*
- * If you need values or
- * derivatives of all tensor
- * product polynomials then use
- * this function, rather than
- * using any of the
- * <tt>compute_value</tt>,
- * <tt>compute_grad</tt> or
- * <tt>compute_grad_grad</tt>
- * functions, see below, in a
- * loop over all tensor product
- * polynomials.
+ * If you need values or derivatives of all tensor product polynomials then
+ * use this function, rather than using any of the <tt>compute_value</tt>,
+ * <tt>compute_grad</tt> or <tt>compute_grad_grad</tt> functions, see below,
+ * in a loop over all tensor product polynomials.
*/
void compute (const Point<dim> &unit_point,
std::vector<double> &values,
std::vector<Tensor<2,dim> > &grad_grads) const;
/**
- * Computes the value of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the value of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * point value of the underlying
- * (one-dimensional) polynomials
- * is (unnecessarily) computed
- * several times. Instead use
- * the <tt>compute</tt> function, see
- * above, with
- * <tt>values.size()==n_tensor_pols</tt>
- * to get the point values of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each point value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the <tt>compute</tt> function, see above,
+ * with <tt>values.size()==n_tensor_pols</tt> to get the point values of all
+ * tensor polynomials all at once and in a much more efficient way.
*/
double compute_value (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the grad of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the grad of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the <tt>compute</tt>
- * function, see above, with
- * <tt>grads.size()==n_tensor_pols</tt>
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the <tt>compute</tt> function, see above,
+ * with <tt>grads.size()==n_tensor_pols</tt> to get the point value of all
+ * tensor polynomials all at once and in a much more efficient way.
*/
Tensor<1,dim> compute_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the second
- * derivative (grad_grad) of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
- * numbering.
+ * Computes the second derivative (grad_grad) of the <tt>i</tt>th tensor
+ * product polynomial at <tt>unit_point</tt>. Here <tt>i</tt> is given in
+ * tensor product numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the <tt>compute</tt>
- * function, see above, with
- * <tt>grad_grads.size()==n_tensor_pols</tt>
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the <tt>compute</tt> function, see above,
+ * with <tt>grad_grads.size()==n_tensor_pols</tt> to get the point value of
+ * all tensor polynomials all at once and in a much more efficient way.
*/
Tensor<2,dim> compute_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Returns the number of tensor
- * product polynomials. It is the
- * product of the number of
- * polynomials in each coordinate
- * direction.
+ * Returns the number of tensor product polynomials. It is the product of
+ * the number of polynomials in each coordinate direction.
*/
unsigned int n () const;
private:
/**
- * Copy of the vector <tt>pols</tt> of
- * polynomials given to the
- * constructor.
+ * Copy of the vector <tt>pols</tt> of polynomials given to the constructor.
*/
std::vector<std::vector<Polynomials::Polynomial<double> > > polynomials;
/**
- * Number of tensor product
- * polynomials. This is
- * <tt>Nx*Ny*Nz</tt>, or with terms
- * dropped if the number of space
- * dimensions is less than 3.
+ * Number of tensor product polynomials. This is <tt>Nx*Ny*Nz</tt>, or with
+ * terms dropped if the number of space dimensions is less than 3.
*/
unsigned int n_tensor_pols;
/**
- * Each tensor product polynomial
- * @þ{i} is a product of
- * one-dimensional polynomials in
- * each space direction. Compute
- * the indices of these
- * one-dimensional polynomials
- * for each space direction,
- * given the index <tt>i</tt>.
+ * Each tensor product polynomial @þ{i} is a product of one-dimensional
+ * polynomials in each space direction. Compute the indices of these
+ * one-dimensional polynomials for each space direction, given the index
+ * <tt>i</tt>.
*/
void compute_index (const unsigned int i,
unsigned int (&indices)[dim]) const;
/**
- * Given the input to the
- * constructor, compute
- * <tt>n_tensor_pols</tt>.
+ * Given the input to the constructor, compute <tt>n_tensor_pols</tt>.
*/
static
unsigned int
#ifndef DOXYGEN
-/* -------------- declaration of explicit specializations --- */
-
-template <>
-void
-TensorProductPolynomials<1>::compute_index(const unsigned int n,
- unsigned int (&index)[1]) const;
-template <>
-void
-TensorProductPolynomials<2>::compute_index(const unsigned int n,
- unsigned int (&index)[2]) const;
-template <>
-void
-TensorProductPolynomials<3>::compute_index(const unsigned int n,
- unsigned int (&index)[3]) const;
-
/* ---------------- template and inline functions ---------- */
-template <int dim>
+template <int dim, typename POLY>
template <class Pol>
-TensorProductPolynomials<dim>::
+inline
+TensorProductPolynomials<dim,POLY>::
TensorProductPolynomials(const std::vector<Pol> &pols)
:
polynomials (pols.begin(), pols.end()),
index_map(n_tensor_pols),
index_map_inverse(n_tensor_pols)
{
- // per default set this index map
- // to identity. This map can be
- // changed by the user through the
- // set_numbering() function
+ // per default set this index map to identity. This map can be changed by
+ // the user through the set_numbering() function
for (unsigned int i=0; i<n_tensor_pols; ++i)
{
index_map[i]=i;
-template <>
-void
-AnisotropicPolynomials<1>::compute_index(const unsigned int n,
- unsigned int (&index)[1]) const;
-template <>
-void
-AnisotropicPolynomials<2>::compute_index(const unsigned int n,
- unsigned int (&index)[2]) const;
-template <>
-void
-AnisotropicPolynomials<3>::compute_index(const unsigned int n,
- unsigned int (&index)[3]) const;
+template <int dim, typename POLY>
+inline
+unsigned int
+TensorProductPolynomials<dim,POLY>::n() const
+{
+ if (dim == 0)
+ return numbers::invalid_unsigned_int;
+ else
+ return n_tensor_pols;
+}
+
+
+
+template <int dim, typename POLY>
+inline
+const std::vector<unsigned int> &
+TensorProductPolynomials<dim,POLY>::get_numbering() const
+{
+ return index_map;
+}
+
+
+template <int dim, typename POLY>
+inline
+const std::vector<unsigned int> &
+TensorProductPolynomials<dim,POLY>::get_numbering_inverse() const
+{
+ return index_map_inverse;
+}
+
+
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#include <deal.II/base/tensor.h>
#include <deal.II/base/point.h>
#include <deal.II/base/polynomial.h>
+#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/base/utilities.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
-//TODO: This class is an almost exact copy of the TensorProductPolynomials class. They should be unified. A simple way to do that would be to derive this one from the other and in the implemented member functions simply defer to the corresponding function in the base class whenever appropriate.
-
/**
* @addtogroup Polynomials
* @{
*/
/**
- * Tensor product of given polynomials.
- *
- * Given a vector of <i>n</i> one-dimensional polynomials
- * <i>P<sub>1</sub></i> to <i>P<sub>n</sub></i>, this class generates
- * <i>n<sup>dim</sup></i> polynomials of the form
- * <i>Q<sub>ijk</sub>(x,y,z) =
- * P<sub>i</sub>(x)P<sub>j</sub>(y)P<sub>k</sub>(z)</i>and a locally
- * constant function. If the base polynomials are mutually orthogonal
- * on the interval [-1,1] or [0,1], then the tensor product
- * polynomials are orthogonal on [-1,1]<sup>dim</sup> or [0,1]
- * <sup>dim</sup>, respectively.
- *
- * Indexing is as follows: the order of dim-dimensional polynomials is
- * x-coordinates running fastest, then y-coordinate, etc. The first
- * few polynomials are thus <i>P<sub>1</sub>(x)P<sub>1</sub>(y),
- * P<sub>2</sub>(x)P<sub>1</sub>(y), P<sub>3</sub>(x)P<sub>1</sub>(y),
- * ..., P<sub>1</sub>(x)P<sub>2</sub>(y),
- * P<sub>2</sub>(x)P<sub>2</sub>(y), P<sub>3</sub>(x)P<sub>2</sub>(y),
- * ...</i> and likewise in 3d. The locally constant function has the last index
- *
- * The output_indices() function prints the ordering of the
- * dim-dimensional polynomials, i.e. for each polynomial in the
- * polynomial space it gives the indices i,j,k of the one-dimensional
- * polynomials in x,y and z direction. The ordering of the
- * dim-dimensional polynomials can be changed by using the
- * set_numbering() function.
+ * Tensor product of given polynomials and a locally constant function. This
+ * class inherits most of its functionality from TensorProductPolynomials. It
+ * works similarly to that class but adds a constant function for the last
+ * index.
*
- * @author Ralf Hartmann, 2000, 2004, Guido Kanschat, 2000, Wolfgang Bangerth 2003
+ * @author Timo Heister, 2012
*/
template <int dim>
-class TensorProductPolynomialsConst
+class TensorProductPolynomialsConst : public TensorProductPolynomials<dim>
{
public:
/**
static const unsigned int dimension = dim;
/**
- * Constructor. <tt>pols</tt> is
- * a vector of objects that
- * should be derived or otherwise
- * convertible to one-dimensional
- * polynomial objects. It will be
- * copied element by element into
- * a private variable.
+ * Constructor. <tt>pols</tt> is a vector of objects that should be derived
+ * or otherwise convertible to one-dimensional polynomial objects. It will
+ * be copied element by element into a private variable.
*/
template <class Pol>
TensorProductPolynomialsConst (const std::vector<Pol> &pols);
/**
- * Prints the list of the indices
- * to <tt>out</tt>.
- */
- void output_indices(std::ostream &out) const;
-
- /**
- * Sets the ordering of the
- * polynomials. Requires
- * <tt>renumber.size()==n()</tt>.
- * Stores a copy of
- * <tt>renumber</tt>.
- */
- void set_numbering(const std::vector<unsigned int> &renumber);
-
- /**
- * Gives read access to the
- * renumber vector.
- */
- const std::vector<unsigned int> &get_numbering() const;
-
- /**
- * Gives read access to the
- * inverse renumber vector.
- */
- const std::vector<unsigned int> &get_numbering_inverse() const;
-
- /**
- * Computes the value and the
- * first and second derivatives
- * of each tensor product
- * polynomial at <tt>unit_point</tt>.
+ * Computes the value and the first and second derivatives of each tensor
+ * product polynomial at <tt>unit_point</tt>.
*
- * The size of the vectors must
- * either be equal 0 or equal
- * n(). In the first case, the
- * function will not compute
- * these values.
+ * The size of the vectors must either be equal 0 or equal n(). In the first
+ * case, the function will not compute these values.
*
- * If you need values or
- * derivatives of all tensor
- * product polynomials then use
- * this function, rather than
- * using any of the
- * compute_value(),
- * compute_grad() or
- * compute_grad_grad()
- * functions, see below, in a
- * loop over all tensor product
- * polynomials.
+ * If you need values or derivatives of all tensor product polynomials then
+ * use this function, rather than using any of the compute_value(),
+ * compute_grad() or compute_grad_grad() functions, see below, in a loop
+ * over all tensor product polynomials.
*/
void compute (const Point<dim> &unit_point,
std::vector<double> &values,
std::vector<Tensor<2,dim> > &grad_grads) const;
/**
- * Computes the value of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the value of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * point value of the underlying
- * (one-dimensional) polynomials
- * is (unnecessarily) computed
- * several times. Instead use
- * the compute() function with
- * <tt>values.size()==</tt>n()
- * to get the point values of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each point value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function with
+ * <tt>values.size()==</tt>n() to get the point values of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
double compute_value (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the grad of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
+ * Computes the grad of the <tt>i</tt>th tensor product polynomial at
+ * <tt>unit_point</tt>. Here <tt>i</tt> is given in tensor product
* numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the compute()
- * function, see above, with
- * <tt>grads.size()==</tt>n()
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function, see above, with
+ * <tt>grads.size()==</tt>n() to get the point value of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
Tensor<1,dim> compute_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Computes the second
- * derivative (grad_grad) of the
- * <tt>i</tt>th tensor product
- * polynomial at
- * <tt>unit_point</tt>. Here <tt>i</tt> is
- * given in tensor product
- * numbering.
+ * Computes the second derivative (grad_grad) of the <tt>i</tt>th tensor
+ * product polynomial at <tt>unit_point</tt>. Here <tt>i</tt> is given in
+ * tensor product numbering.
*
- * Note, that using this function
- * within a loop over all tensor
- * product polynomials is not
- * efficient, because then each
- * derivative value of the
- * underlying (one-dimensional)
- * polynomials is (unnecessarily)
- * computed several times.
- * Instead use the compute()
- * function, see above, with
- * <tt>grad_grads.size()==</tt>n()
- * to get the point value of all
- * tensor polynomials all at once
- * and in a much more efficient
- * way.
+ * Note, that using this function within a loop over all tensor product
+ * polynomials is not efficient, because then each derivative value of the
+ * underlying (one-dimensional) polynomials is (unnecessarily) computed
+ * several times. Instead use the compute() function, see above, with
+ * <tt>grad_grads.size()==</tt>n() to get the point value of all tensor
+ * polynomials all at once and in a much more efficient way.
*/
Tensor<2,dim> compute_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Returns the number of tensor
- * product polynomials plus the constant
- * function. For <i>n</i> 1d polynomials
- * this is <i>n<sup>dim</sup>+1</i>.
+ * Returns the number of tensor product polynomials plus the constant
+ * function. For <i>n</i> 1d polynomials this is <i>n<sup>dim</sup>+1</i>.
*/
unsigned int n () const;
-
-
-private:
- /**
- * Copy of the vector <tt>pols</tt> of
- * polynomials given to the
- * constructor.
- */
- std::vector<Polynomials::Polynomial<double> > polynomials;
-
- /**
- * Number of tensor product
- * polynomials. See n().
- */
- unsigned int n_tensor_pols;
-
- /**
- * Index map for reordering the
- * polynomials.
- */
- std::vector<unsigned int> index_map;
-
- /**
- * Index map for reordering the
- * polynomials.
- */
- std::vector<unsigned int> index_map_inverse;
-
- /**
- * Each tensor product polynomial
- * <i>i</i> is a product of
- * one-dimensional polynomials in
- * each space direction. Compute
- * the indices of these
- * one-dimensional polynomials
- * for each space direction,
- * given the index <i>i</i>.
- */
- // fix to avoid compiler warnings about zero
- // length arrays
- void compute_index (const unsigned int i,
- unsigned int (&indices)[(dim>0?dim:1)]) const;
};
/** @} */
+
+/* ---------------- template and inline functions ---------- */
+
#ifndef DOXYGEN
template <int dim>
+template <class Pol>
inline
-const std::vector<unsigned int> &
-TensorProductPolynomialsConst<dim>::get_numbering() const
+TensorProductPolynomialsConst<dim>::
+TensorProductPolynomialsConst(const std::vector<Pol> &pols)
+ :
+ TensorProductPolynomials<dim>(pols)
{
- return index_map;
+ // append index for renumbering
+ this->index_map.push_back(this->n_tensor_pols);
+ this->index_map_inverse.push_back(this->n_tensor_pols);
}
+
template <int dim>
inline
-const std::vector<unsigned int> &
-TensorProductPolynomialsConst<dim>::get_numbering_inverse() const
+unsigned int
+TensorProductPolynomialsConst<dim>::n() const
{
- return index_map_inverse;
+ return this->n_tensor_pols+1;
}
-#endif // DOXYGEN
-
-#ifndef DOXYGEN
-/* -------------- declaration of explicit specializations --- */
-
-template <>
-void
-TensorProductPolynomialsConst<1>::compute_index(const unsigned int n,
- unsigned int (&index)[1]) const;
template <>
-void
-TensorProductPolynomialsConst<2>::compute_index(const unsigned int n,
- unsigned int (&index)[2]) const;
-template <>
-void
-TensorProductPolynomialsConst<3>::compute_index(const unsigned int n,
- unsigned int (&index)[3]) const;
-
-
-/* ---------------- template and inline functions ---------- */
-
-template <int dim>
-template <class Pol>
-TensorProductPolynomialsConst<dim>::
-TensorProductPolynomialsConst(const std::vector<Pol> &pols)
- :
- polynomials (pols.begin(), pols.end()),
- n_tensor_pols(Utilities::fixed_power<dim>(pols.size())+1),
- index_map(n_tensor_pols),
- index_map_inverse(n_tensor_pols)
+inline
+unsigned int
+TensorProductPolynomialsConst<0>::n() const
{
- // per default set this index map
- // to identity. This map can be
- // changed by the user through the
- // set_numbering() function
- for (unsigned int i=0; i<n_tensor_pols; ++i)
- {
- index_map[i]=i;
- index_map_inverse[i]=i;
- }
+ return numbers::invalid_unsigned_int;
}
+
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
{
public:
/**
- * Constructor for tensor product
- * polynomials of degree
- * <tt>p</tt>. The shape
- * functions created using this
- * constructor correspond to
- * Legendre polynomials in each
- * coordinate direction.
+ * Constructor for tensor product polynomials of degree <tt>p</tt>. The
+ * shape functions created using this constructor correspond to Lagrange
+ * polynomials in each coordinate direction.
*/
FE_FaceQ(unsigned int p);
virtual FiniteElement<dim,spacedim> *clone() const;
/**
- * Return a string that uniquely
- * identifies a finite
- * element. This class returns
- * <tt>FE_FaceQ<dim>(degree)</tt>, with
- * <tt>dim</tt> and <tt>degree</tt>
- * replaced by appropriate
- * values.
+ * Return a string that uniquely identifies a finite element. This class
+ * returns <tt>FE_FaceQ<dim>(degree)</tt>, with <tt>dim</tt> and
+ * <tt>degree</tt> replaced by appropriate values.
*/
virtual std::string get_name () const;
/**
* Check for non-zero values on a face.
*
- * This function returns
- * @p true, if the shape
- * function @p shape_index has
- * non-zero values on the face
- * @p face_index.
+ * This function returns @p true, if the shape function @p shape_index has
+ * non-zero values on the face @p face_index.
*
- * Implementation of the
- * interface in
- * FiniteElement
+ * Implementation of the interface in FiniteElement
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
private:
/**
- * Return vector with dofs per
- * vertex, line, quad, hex.
+ * Return vector with dofs per vertex, line, quad, hex.
*/
static std::vector<unsigned int> get_dpo_vector (const unsigned int deg);
};
const std::vector<ComponentMask> &nonzero_components);
/**
- * Return the polynomial degree
- * of this finite element,
- * i.e. the value passed to the
- * constructor.
+ * Return the polynomial degree of this finite element, i.e. the value
+ * passed to the constructor.
*/
unsigned int get_degree () const;
/**
- * Return the numbering of the underlying
- * polynomial space compared to
- * lexicographic ordering of the basis
- * functions. Returns
+ * Return the numbering of the underlying polynomial space compared to
+ * lexicographic ordering of the basis functions. Returns
* POLY::get_numbering().
*/
std::vector<unsigned int> get_poly_space_numbering() const;
/**
- * Return the inverse numbering of the
- * underlying polynomial space. Returns
+ * Return the inverse numbering of the underlying polynomial space. Returns
* POLY::get_numbering_inverse().
*/
std::vector<unsigned int> get_poly_space_numbering_inverse() const;
/**
- * Return the value of the
- * <tt>i</tt>th shape function at
- * the point <tt>p</tt>. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the value of the <tt>i</tt>th shape function at the point
+ * <tt>p</tt>. See the FiniteElement base class for more information about
+ * the semantics of this function.
*/
virtual double shape_value (const unsigned int i,
const Point<dim> &p) const;
/**
- * Return the value of the
- * <tt>component</tt>th vector
- * component of the <tt>i</tt>th
- * shape function at the point
- * <tt>p</tt>. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the value of the <tt>component</tt>th vector component of the
+ * <tt>i</tt>th shape function at the point <tt>p</tt>. See the
+ * FiniteElement base class for more information about the semantics of this
+ * function.
*
- * Since this element is scalar,
- * the returned value is the same
- * as if the function without the
- * <tt>_component</tt> suffix
- * were called, provided that the
- * specified component is zero.
+ * Since this element is scalar, the returned value is the same as if the
+ * function without the <tt>_component</tt> suffix were called, provided
+ * that the specified component is zero.
*/
virtual double shape_value_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
- * Return the gradient of the
- * <tt>i</tt>th shape function at
- * the point <tt>p</tt>. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the gradient of the <tt>i</tt>th shape function at the point
+ * <tt>p</tt>. See the FiniteElement base class for more information about
+ * the semantics of this function.
*/
virtual Tensor<1,dim> shape_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Return the gradient of the
- * <tt>component</tt>th vector
- * component of the <tt>i</tt>th
- * shape function at the point
- * <tt>p</tt>. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the gradient of the <tt>component</tt>th vector component of the
+ * <tt>i</tt>th shape function at the point <tt>p</tt>. See the
+ * FiniteElement base class for more information about the semantics of this
+ * function.
*
- * Since this element is scalar,
- * the returned value is the same
- * as if the function without the
- * <tt>_component</tt> suffix
- * were called, provided that the
- * specified component is zero.
+ * Since this element is scalar, the returned value is the same as if the
+ * function without the <tt>_component</tt> suffix were called, provided
+ * that the specified component is zero.
*/
virtual Tensor<1,dim> shape_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
- * Return the tensor of second
- * derivatives of the
- * <tt>i</tt>th shape function at
- * point <tt>p</tt> on the unit
- * cell. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the tensor of second derivatives of the <tt>i</tt>th shape
+ * function at point <tt>p</tt> on the unit cell. See the FiniteElement base
+ * class for more information about the semantics of this function.
*/
virtual Tensor<2,dim> shape_grad_grad (const unsigned int i,
const Point<dim> &p) const;
/**
- * Return the second derivative
- * of the <tt>component</tt>th
- * vector component of the
- * <tt>i</tt>th shape function at
- * the point <tt>p</tt>. See the
- * FiniteElement base class
- * for more information about the
- * semantics of this function.
+ * Return the second derivative of the <tt>component</tt>th vector component
+ * of the <tt>i</tt>th shape function at the point <tt>p</tt>. See the
+ * FiniteElement base class for more information about the semantics of this
+ * function.
*
- * Since this element is scalar,
- * the returned value is the same
- * as if the function without the
- * <tt>_component</tt> suffix
- * were called, provided that the
- * specified component is zero.
+ * Since this element is scalar, the returned value is the same as if the
+ * function without the <tt>_component</tt> suffix were called, provided
+ * that the specified component is zero.
*/
virtual Tensor<2,dim> shape_grad_grad_component (const unsigned int i,
const Point<dim> &p,
/**
- * Determine the values that need
- * to be computed on the unit
- * cell to be able to compute all
- * values required by
- * <tt>flags</tt>.
+ * Determine the values that need to be computed on the unit cell to be able
+ * to compute all values required by <tt>flags</tt>.
*
- * For the purpuse of this
- * function, refer to the
- * documentation in
+ * For the purpuse of this function, refer to the documentation in
* FiniteElement.
*
- * This class assumes that shape
- * functions of this
- * FiniteElement do <em>not</em>
- * depend on the actual shape of
- * the cells in real
- * space. Therefore, the effect
- * in this element is as follows:
- * if <tt>update_values</tt> is
- * set in <tt>flags</tt>, copy it
- * to the result. All other flags
- * of the result are cleared,
- * since everything else must be
- * computed for each cell.
+ * This class assumes that shape functions of this FiniteElement do
+ * <em>not</em> depend on the actual shape of the cells in real
+ * space. Therefore, the effect in this element is as follows: if
+ * <tt>update_values</tt> is set in <tt>flags</tt>, copy it to the
+ * result. All other flags of the result are cleared, since everything else
+ * must be computed for each cell.
*/
virtual UpdateFlags update_once (const UpdateFlags flags) const;
/**
- * Determine the values that need
- * to be computed on every cell
- * to be able to compute all
- * values required by
- * <tt>flags</tt>.
+ * Determine the values that need to be computed on every cell to be able to
+ * compute all values required by <tt>flags</tt>.
*
- * For the purpuse of this
- * function, refer to the
- * documentation in
+ * For the purpuse of this function, refer to the documentation in
* FiniteElement.
*
- * This class assumes that shape
- * functions of this
- * FiniteElement do <em>not</em>
- * depend on the actual shape of
- * the cells in real
- * space.
+ * This class assumes that shape functions of this FiniteElement do
+ * <em>not</em> depend on the actual shape of the cells in real space.
*
- * The effect in this element is
- * as follows:
+ * The effect in this element is as follows:
* <ul>
-
- * <li> if
- * <tt>update_gradients</tt> is
- * set, the result will contain
- * <tt>update_gradients</tt> and
- * <tt>update_covariant_transformation</tt>.
- * The latter is required to
- * transform the gradient on the
- * unit cell to the real
- * cell. Remark, that the action
- * required by
- * <tt>update_covariant_transformation</tt>
- * is actually performed by the
- * Mapping object used in
- * conjunction with this finite
- * element. <li> if
- * <tt>update_hessians</tt>
- * is set, the result will
- * contain
- * <tt>update_hessians</tt>
- * and
- * <tt>update_covariant_transformation</tt>.
- * The rationale is the same as
- * above and no higher
- * derivatives of the
- * transformation are required,
- * since we use difference
- * quotients for the actual
- * computation.
+ *
+ * <li> if <tt>update_gradients</tt> is set, the result will contain
+ * <tt>update_gradients</tt> and <tt>update_covariant_transformation</tt>.
+ * The latter is required to transform the gradient on the unit cell to the
+ * real cell. Remark, that the action required by
+ * <tt>update_covariant_transformation</tt> is actually performed by the
+ * Mapping object used in conjunction with this finite element.
+ *
+ * <li> if <tt>update_hessians</tt> is set, the result will contain
+ * <tt>update_hessians</tt> and <tt>update_covariant_transformation</tt>.
+ * The rationale is the same as above and no higher derivatives of the
+ * transformation are required, since we use difference quotients for the
+ * actual computation.
*
* </ul>
*/
/**
* Fields of cell-independent data.
*
- * For information about the
- * general purpose of this class,
- * see the documentation of the
- * base class.
+ * For information about the general purpose of this class, see the
+ * documentation of the base class.
*/
class InternalData : public FiniteElement<dim,spacedim>::InternalDataBase
{
public:
/**
- * Array with shape function
- * values in quadrature
- * points. There is one
- * row for each shape
- * function, containing
- * values for each quadrature
- * point.
+ * Array with shape function values in quadrature points. There is one row
+ * for each shape function, containing values for each quadrature point.
*
- * In this array, we store
- * the values of the shape
- * function in the quadrature
- * points on the unit
- * cell. Since these values
- * do not change under
- * transformation to the real
- * cell, we only need to copy
- * them over when visiting a
- * concrete cell.
+ * In this array, we store the values of the shape function in the
+ * quadrature points on the unit cell. Since these values do not change
+ * under transformation to the real cell, we only need to copy them over
+ * when visiting a concrete cell.
*/
std::vector<std::vector<double> > shape_values;
/**
- * Array with shape function
- * gradients in quadrature
- * points. There is one
- * row for each shape
- * function, containing
- * values for each quadrature
+ * Array with shape function gradients in quadrature points. There is one
+ * row for each shape function, containing values for each quadrature
* point.
*
- * We store the gradients in
- * the quadrature points on
- * the unit cell. We then
- * only have to apply the
- * transformation (which is a
- * matrix-vector
- * multiplication) when
- * visiting an actual cell.
+ * We store the gradients in the quadrature points on the unit cell. We
+ * then only have to apply the transformation (which is a matrix-vector
+ * multiplication) when visiting an actual cell.
*/
std::vector<std::vector<Tensor<1,dim> > > shape_gradients;
};
/**
- * The polynomial space. Its type
- * is given by the template
- * parameter POLY.
+ * The polynomial space. Its type is given by the template parameter POLY.
*/
POLY poly_space;
};
return std::vector<unsigned int>();
}
- template <int dim>
+ template <int dim, typename POLY>
inline
std::vector<unsigned int>
- get_poly_space_numbering (const TensorProductPolynomials<dim> &poly)
+ get_poly_space_numbering (const TensorProductPolynomials<dim,POLY> &poly)
{
return poly.get_numbering();
}
- template <int dim>
+ template <int dim, typename POLY>
inline
std::vector<unsigned int>
- get_poly_space_numbering_inverse (const TensorProductPolynomials<dim> &poly)
+ get_poly_space_numbering_inverse (const TensorProductPolynomials<dim,POLY> &poly)
{
return poly.get_numbering_inverse();
}
#include <deal.II/base/config.h>
#include <deal.II/base/tensor_product_polynomials.h>
-#include <deal.II/fe/fe_poly.h>
+#include <deal.II/fe/fe_q_base.h>
DEAL_II_NAMESPACE_OPEN
* @author Wolfgang Bangerth, 1998, 2003; Guido Kanschat, 2001; Ralf Hartmann, 2001, 2004, 2005; Oliver Kayser-Herold, 2004; Katharina Kormann, 2008; Martin Kronbichler, 2008
*/
template <int dim, int spacedim=dim>
-class FE_Q : public FE_Poly<TensorProductPolynomials<dim>,dim,spacedim>
+class FE_Q : public FE_Q_Base<TensorProductPolynomials<dim>,dim,spacedim>
{
public:
/**
- * Constructor for tensor product
- * polynomials of degree @p p.
+ * Constructor for tensor product polynomials of degree @p p.
*/
FE_Q (const unsigned int p);
/**
- * Constructor for tensor product
- * polynomials with support
- * points @p points based on a
- * one-dimensional quadrature
- * formula. The degree of the
- * finite element is
- * <tt>points.size()-1</tt>.
- * Note that the first point has
- * to be 0 and the last one 1. If
- * <tt>FE_Q<dim>(QGaussLobatto<1>(fe_degree+1))</tt>
- * is specified, so-called
- * Gauss-Lobatto elements are
- * obtained which can give a
- * diagonal mass matrix if
- * combined with Gauss-Lobatto
- * quadrature on the same
- * points. Their use is shown in
- * step-48.
+ * Constructor for tensor product polynomials with support points @p points
+ * based on a one-dimensional quadrature formula. The degree of the finite
+ * element is <tt>points.size()-1</tt>. Note that the first point has to be
+ * 0 and the last one 1. If
+ * <tt>FE_Q<dim>(QGaussLobatto<1>(fe_degree+1))</tt> is specified, so-called
+ * Gauss-Lobatto elements are obtained which can give a diagonal mass matrix
+ * if combined with Gauss-Lobatto quadrature on the same points. Their use
+ * is shown in step-48.
*/
FE_Q (const Quadrature<1> &points);
/**
- * Return a string that uniquely
- * identifies a finite
- * element. This class returns
- * <tt>FE_Q<dim>(degree)</tt>, with
- * @p dim and @p degree
- * replaced by appropriate
- * values.
+ * Constructs a FE_Q_isoQ1 element. That element shares large parts of code
+ * with FE_Q so most of the construction work is done in this routine,
+ * whereas the public constructor is in the class FE_Q_isoQ1.
*/
- virtual std::string get_name () const;
-
- /**
- * Return the matrix
- * interpolating from the given
- * finite element to the present
- * one. The size of the matrix is
- * then @p dofs_per_cell times
- * <tt>source.dofs_per_cell</tt>.
- *
- * These matrices are only
- * available if the source
- * element is also a @p FE_Q
- * element. Otherwise, an
- * exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented
- * is thrown.
- */
- virtual void
- get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
- FullMatrix<double> &matrix) const;
-
-
- /**
- * Return the matrix
- * interpolating from a face of
- * of one element to the face of
- * the neighboring element.
- * The size of the matrix is
- * then <tt>source.dofs_per_face</tt> times
- * <tt>this->dofs_per_face</tt>.
- *
- * Derived elements will have to
- * implement this function. They
- * may only provide interpolation
- * matrices for certain source
- * finite elements, for example
- * those from the same family. If
- * they don't implement
- * interpolation from a given
- * element, then they must throw
- * an exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
- */
- virtual void
- get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
- FullMatrix<double> &matrix) const;
-
- /**
- * Return the matrix
- * interpolating from a face of
- * of one element to the face of
- * the neighboring element.
- * The size of the matrix is
- * then <tt>source.dofs_per_face</tt> times
- * <tt>this->dofs_per_face</tt>.
- *
- * Derived elements will have to
- * implement this function. They
- * may only provide interpolation
- * matrices for certain source
- * finite elements, for example
- * those from the same family. If
- * they don't implement
- * interpolation from a given
- * element, then they must throw
- * an exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
- */
- virtual void
- get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
- const unsigned int subface,
- FullMatrix<double> &matrix) const;
-
- /**
- * Check for non-zero values on a face.
- *
- * This function returns
- * @p true, if the shape
- * function @p shape_index has
- * non-zero values on the face
- * @p face_index.
- *
- * Implementation of the
- * interface in
- * FiniteElement
- */
- virtual bool has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const;
-
- /**
- * @name Functions to support hp
- * @{
- */
-
- /**
- * Return whether this element
- * implements its hanging node
- * constraints in the new way,
- * which has to be used to make
- * elements "hp compatible".
- *
- * For the FE_Q class the result is
- * always true (independent of the degree
- * of the element), as it implements the
- * complete set of functions necessary
- * for hp capability.
- */
- virtual bool hp_constraints_are_implemented () const;
-
- /**
- * If, on a vertex, several finite elements are active, the hp code
- * first assigns the degrees of freedom of each of these FEs
- * different global indices. It then calls this function to find out
- * which of them should get identical values, and consequently can
- * receive the same global DoF index. This function therefore
- * returns a list of identities between DoFs of the present finite
- * element object with the DoFs of @p fe_other, which is a reference
- * to a finite element object representing one of the other finite
- * elements active on this particular vertex. The function computes
- * which of the degrees of freedom of the two finite element objects
- * are equivalent, both numbered between zero and the corresponding
- * value of dofs_per_vertex of the two finite elements. The first
- * index of each pair denotes one of the vertex dofs of the present
- * element, whereas the second is the corresponding index of the
- * other finite element.
- */
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
+ FE_Q(const unsigned int subdivisions_per_dimension,
+ const unsigned int base_degree);
/**
- * Same as hp_vertex_dof_indices(), except that the function treats
- * degrees of freedom on lines.
+ * Return a string that uniquely identifies a finite element. This class
+ * returns <tt>FE_Q<dim>(degree)</tt>, with @p dim and @p degree replaced by
+ * appropriate values.
*/
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
-
- /**
- * Same as hp_vertex_dof_indices(), except that the function treats
- * degrees of freedom on quads.
- */
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
-
- /**
- * Return whether this element dominates
- * the one given as argument when they
- * meet at a common face,
- * whether it is the other way around,
- * whether neither dominates, or if
- * either could dominate.
- *
- * For a definition of domination, see
- * FiniteElementBase::Domination and in
- * particular the @ref hp_paper "hp paper".
- */
- virtual
- FiniteElementDomination::Domination
- compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
- //@}
-
- /**
- * Determine an estimate for the
- * memory consumption (in bytes)
- * of this object.
- *
- * This function is made virtual,
- * since finite element objects
- * are usually accessed through
- * pointers to their base class,
- * rather than the class itself.
- */
- virtual std::size_t memory_consumption () const;
+ virtual std::string get_name () const;
protected:
- /**
- * @p clone function instead of
- * a copy constructor.
- *
- * This function is needed by the
- * constructors of @p FESystem.
- */
- virtual FiniteElement<dim,spacedim> *clone() const;
-
-private:
/**
- * Only for internal use. Its
- * full name is
- * @p get_dofs_per_object_vector
- * function and it creates the
- * @p dofs_per_object vector that is
- * needed within the constructor to
- * be passed to the constructor of
- * @p FiniteElementData.
- */
- static std::vector<unsigned int> get_dpo_vector(const unsigned int degree);
-
- /**
- * This is an analogon to the
- * FETools::lexicographic_to_hierarchic_numbering
- * function, but working on
- * faces. Called from the
- * constructor.
- */
- static
- std::vector<unsigned int>
- face_lexicographic_to_hierarchic_numbering (const unsigned int degree);
-
- /**
- * Initialize the hanging node
- * constraints matrices. Called
- * from the constructor.
- */
- void initialize_constraints ();
-
- /**
- * Initialize the embedding
- * matrices. Called from the
- * constructor.
- */
- void initialize_embedding ();
-
- /**
- * Initialize the restriction
- * matrices. Called from the
- * constructor.
- */
- void initialize_restriction ();
-
- /**
- * Initialize the
- * @p unit_support_points field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_unit_support_points ();
-
- /**
- * Initialize the @p unit_support_points
- * field of the FiniteElement
- * class. Called from the constructor in
- * case the finite element is based on
- * quadrature points.
- */
- void initialize_unit_support_points (const Quadrature<1> &points);
-
- /**
- * Initialize the
- * @p unit_face_support_points field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_unit_face_support_points ();
-
- /**
- * Initialize the @p
- * unit_face_support_points field of the
- * FiniteElement class. Called from the
- * constructor in case the finite element
- * is based on quadrature points.
- */
- void initialize_unit_face_support_points (const Quadrature<1> &points);
-
- /**
- * Initialize the
- * @p adjust_quad_dof_index_for_face_orientation_table field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_quad_dof_index_permutation ();
-
- /**
- * Mapping from hierarchic to
- * lexicographic numbering on
- * first face. Hierarchic is the
- * numbering of the shape
- * functions.
- */
- const std::vector<unsigned int> face_index_map;
-
-
- /**
- * Forward declaration of a class
- * into which we put significant
- * parts of the implementation.
+ * @p clone function instead of a copy constructor.
*
- * See the .cc file for more
- * information.
- */
- struct Implementation;
-
- /**
- * Allow access from other
- * dimensions. We need this since
- * we want to call the functions
- * @p get_dpo_vector and
- * @p lexicographic_to_hierarchic_numbering
- * for the faces of the finite
- * element of dimension dim+1.
+ * This function is needed by the constructors of @p FESystem.
*/
- template <int, int> friend class FE_Q;
-
- friend struct FE_Q<dim,spacedim>::Implementation;
+ virtual FiniteElement<dim,spacedim> *clone() const;
};
/*@}*/
-/* -------------- declaration of explicit specializations ------------- */
-
-template <>
-void FE_Q<1>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q<1>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
-
-
-template <>
-void FE_Q<1,2>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q<1,2>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
-
-template <>
-void FE_Q<1,3>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q<1,3>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
-
DEAL_II_NAMESPACE_CLOSE
#endif
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+#ifndef __deal2__fe_q_base_h
+#define __deal2__fe_q_base_h
+
+#include <deal.II/base/config.h>
+#include <deal.II/fe/fe_poly.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+/*!@addtogroup fe */
+/*@{*/
+
+/**
+ * This class collects the basic methods used in FE_Q and FE_Q_DG0. There is
+ * no public constructor for this class as it is not functional as a
+ * stand-alone. The completion of definitions is left to the derived classes.
+ *
+ * @author Wolfgang Bangerth, 1998, 2003; Guido Kanschat, 2001; Ralf Hartmann, 2001, 2004, 2005; Oliver Kayser-Herold, 2004; Katharina Kormann, 2008; Martin Kronbichler, 2008, 2013
+ */
+template <class POLY, int dim=POLY::dimension, int spacedim=dim>
+class FE_Q_Base : public FE_Poly<POLY,dim,spacedim>
+{
+public:
+ /**
+ * Constructor.
+ */
+ FE_Q_Base (const POLY &poly_space,
+ const FiniteElementData<dim> &fe_data,
+ const std::vector<bool> &restriction_is_additive_flags);
+
+ /**
+ * Return the matrix interpolating from the given finite element to the
+ * present one. The size of the matrix is then @p dofs_per_cell times
+ * <tt>source.dofs_per_cell</tt>.
+ *
+ * These matrices are only available if the source element is also a @p FE_Q
+ * element. Otherwise, an exception of type
+ * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
+ */
+ virtual void
+ get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
+ FullMatrix<double> &matrix) const;
+
+
+ /**
+ * Return the matrix interpolating from a face of of one element to the face
+ * of the neighboring element. The size of the matrix is then
+ * <tt>source.dofs_per_face</tt> times <tt>this->dofs_per_face</tt>.
+ *
+ * Derived elements will have to implement this function. They may only
+ * provide interpolation matrices for certain source finite elements, for
+ * example those from the same family. If they don't implement interpolation
+ * from a given element, then they must throw an exception of type
+ * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
+ */
+ virtual void
+ get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
+ FullMatrix<double> &matrix) const;
+
+ /**
+ * Return the matrix interpolating from a face of of one element to the face
+ * of the neighboring element. The size of the matrix is then
+ * <tt>source.dofs_per_face</tt> times <tt>this->dofs_per_face</tt>.
+ *
+ * Derived elements will have to implement this function. They may only
+ * provide interpolation matrices for certain source finite elements, for
+ * example those from the same family. If they don't implement interpolation
+ * from a given element, then they must throw an exception of type
+ * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
+ */
+ virtual void
+ get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
+ const unsigned int subface,
+ FullMatrix<double> &matrix) const;
+
+ /**
+ * Check for non-zero values on a face.
+ *
+ * This function returns @p true, if the shape function @p shape_index has
+ * non-zero values on the face @p face_index.
+ *
+ * Implementation of the interface in FiniteElement
+ */
+ virtual bool has_support_on_face (const unsigned int shape_index,
+ const unsigned int face_index) const;
+
+ /**
+ * @name Functions to support hp
+ * @{
+ */
+
+ /**
+ * Return whether this element implements its hanging node constraints in
+ * the new way, which has to be used to make elements "hp compatible".
+ *
+ * For the FE_Q class the result is always true (independent of the degree
+ * of the element), as it implements the complete set of functions necessary
+ * for hp capability.
+ */
+ virtual bool hp_constraints_are_implemented () const;
+
+ /**
+ * If, on a vertex, several finite elements are active, the hp code
+ * first assigns the degrees of freedom of each of these FEs
+ * different global indices. It then calls this function to find out
+ * which of them should get identical values, and consequently can
+ * receive the same global DoF index. This function therefore
+ * returns a list of identities between DoFs of the present finite
+ * element object with the DoFs of @p fe_other, which is a reference
+ * to a finite element object representing one of the other finite
+ * elements active on this particular vertex. The function computes
+ * which of the degrees of freedom of the two finite element objects
+ * are equivalent, both numbered between zero and the corresponding
+ * value of dofs_per_vertex of the two finite elements. The first
+ * index of each pair denotes one of the vertex dofs of the present
+ * element, whereas the second is the corresponding index of the
+ * other finite element.
+ */
+ virtual
+ std::vector<std::pair<unsigned int, unsigned int> >
+ hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
+
+ /**
+ * Same as hp_vertex_dof_indices(), except that the function treats
+ * degrees of freedom on lines.
+ */
+ virtual
+ std::vector<std::pair<unsigned int, unsigned int> >
+ hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
+
+ /**
+ * Same as hp_vertex_dof_indices(), except that the function treats
+ * degrees of freedom on quads.
+ */
+ virtual
+ std::vector<std::pair<unsigned int, unsigned int> >
+ hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
+
+ /**
+ * Return whether this element dominates the one given as argument when they
+ * meet at a common face, whether it is the other way around, whether
+ * neither dominates, or if either could dominate.
+ *
+ * For a definition of domination, see FiniteElementBase::Domination and in
+ * particular the @ref hp_paper "hp paper".
+ */
+ virtual
+ FiniteElementDomination::Domination
+ compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
+ //@}
+
+protected:
+ /**
+ * Only for internal use. Its full name is @p get_dofs_per_object_vector
+ * function and it creates the @p dofs_per_object vector that is needed
+ * within the constructor to be passed to the constructor of @p
+ * FiniteElementData.
+ */
+ static std::vector<unsigned int> get_dpo_vector(const unsigned int degree);
+
+ /**
+ * Performs the initialization of the element based on 1D support points,
+ * i.e., sets renumbering, initializes unit support points, initializes
+ * constraints as well as restriction and prolongation matrices.
+ */
+ void initialize (const std::vector<Point<1> > &support_points_1d);
+
+ /**
+ * Initialize the hanging node constraints matrices. Called from
+ * initialize().
+ */
+ void initialize_constraints (const std::vector<Point<1> > &points);
+
+ /**
+ * Initialize the embedding matrices. Called from initialize().
+ */
+ void initialize_embedding ();
+
+ /**
+ * Initialize the restriction matrices. Called from initialize().
+ */
+ void initialize_restriction ();
+
+ /**
+ * Initialize the @p unit_support_points field of the FiniteElement
+ * class. Called from initialize().
+ */
+ void initialize_unit_support_points (const std::vector<Point<1> > &points);
+
+ /**
+ * Initialize the @p unit_face_support_points field of the FiniteElement
+ * class. Called from initialize().
+ */
+ void initialize_unit_face_support_points (const std::vector<Point<1> > &points);
+
+ /**
+ * Initialize the @p adjust_quad_dof_index_for_face_orientation_table field
+ * of the FiniteElement class. Called from initialize().
+ */
+ void initialize_quad_dof_index_permutation ();
+
+ /**
+ * Forward declaration of a class into which we put significant parts of the
+ * implementation.
+ *
+ * See the .cc file for more information.
+ */
+ struct Implementation;
+
+ /*
+ * Declare implementation friend.
+ */
+ friend struct FE_Q_Base<POLY,dim,spacedim>::Implementation;
+};
+
+
+/*@}*/
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
+//---------------------------------------------------------------------------
+// $Id: fe_q.h 29043 2013-03-26 19:51:19Z bangerth $
+//
+// Copyright (C) 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
#ifndef __deal2__fe_q_dg0_h
#define __deal2__fe_q_dg0_h
#include <deal.II/base/config.h>
#include <deal.II/base/tensor_product_polynomials_const.h>
-#include <deal.II/fe/fe_poly.h>
+#include <deal.II/fe/fe_q_base.h>
DEAL_II_NAMESPACE_OPEN
*
*/
template <int dim, int spacedim=dim>
-class FE_Q_DG0 : public FE_Poly<TensorProductPolynomialsConst<dim>,dim,spacedim>
+class FE_Q_DG0 : public FE_Q_Base<TensorProductPolynomialsConst<dim>,dim,spacedim>
{
public:
/**
- * Constructor for tensor product
- * polynomials of degree @p p plus locally
+ * Constructor for tensor product polynomials of degree @p p plus locally
* constant functions.
*/
FE_Q_DG0 (const unsigned int p);
/**
- * Constructor for tensor product
- * polynomials with support
- * points @p points plus locally constant
- * functions based on a one-dimensional
- * quadrature formula. The degree of the
- * finite element is
- * <tt>points.size()-1</tt>.
- * Note that the first point has
- * to be 0 and the last one 1.
+ * Constructor for tensor product polynomials with support points @p points
+ * plus locally constant functions based on a one-dimensional quadrature
+ * formula. The degree of the finite element is <tt>points.size()-1</tt>.
+ * Note that the first point has to be 0 and the last one 1.
*/
FE_Q_DG0 (const Quadrature<1> &points);
/**
- * Return a string that uniquely
- * identifies a finite
- * element. This class returns
- * <tt>FE_Q_DG0<dim>(degree)</tt>, with
- * @p dim and @p degree
- * replaced by appropriate
- * values.
+ * Return a string that uniquely identifies a finite element. This class
+ * returns <tt>FE_Q_DG0<dim>(degree)</tt>, with @p dim and @p degree
+ * replaced by appropriate values.
*/
virtual std::string get_name () const;
/**
- * Interpolate a set of scalar
- * values, computed in the
- * generalized support points.
+ * Interpolate a set of scalar values, computed in the generalized support
+ * points.
*/
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<double> &values) const;
/**
- * Interpolate a set of vector
- * values, computed in the
- * generalized support points.
+ * Interpolate a set of vector values, computed in the generalized support
+ * points.
*
- * Since a finite element often
- * only interpolates part of a
- * vector, <tt>offset</tt> is
- * used to determine the first
- * component of the vector to be
- * interpolated. Maybe consider
- * changing your data structures
- * to use the next function.
+ * Since a finite element often only interpolates part of a vector,
+ * <tt>offset</tt> is used to determine the first component of the vector to
+ * be interpolated. Maybe consider changing your data structures to use the
+ * next function.
*/
virtual void interpolate(std::vector<double> &local_dofs,
const std::vector<Vector<double> > &values,
unsigned int offset = 0) const;
/**
- * Interpolate a set of vector
- * values, computed in the
- * generalized support points.
+ * Interpolate a set of vector values, computed in the generalized support
+ * points.
*/
virtual void interpolate(
std::vector<double> &local_dofs,
const VectorSlice<const std::vector<std::vector<double> > > &values) const;
/**
- * Return the matrix interpolating from the
- * given finite element to the present one.
- * The size of the matrix is then @p
- * dofs_per_cell times
+ * Return the matrix interpolating from the given finite element to the
+ * present one. The size of the matrix is then @p dofs_per_cell times
* <tt>source.dofs_per_cell</tt>.
*
- * These matrices are only available if the
- * source element is also a @p FE_Q_DG0
- * element. Otherwise, an exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented
- * is thrown.
+ * These matrices are only available if the source element is also a @p
+ * FE_Q_DG0 element. Otherwise, an exception of type
+ * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
*/
virtual void
get_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
FullMatrix<double> &matrix) const;
- /**
- * Return the matrix
- * interpolating from a face of
- * of one element to the face of
- * the neighboring element.
- * The size of the matrix is
- * then <tt>source.dofs_per_face</tt> times
- * <tt>this->dofs_per_face</tt>.
- *
- * Derived elements will have to
- * implement this function. They
- * may only provide interpolation
- * matrices for certain source
- * finite elements, for example
- * those from the same family. If
- * they don't implement
- * interpolation from a given
- * element, then they must throw
- * an exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
- */
- virtual void
- get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
- FullMatrix<double> &matrix) const;
-
- /**
- * Return the matrix
- * interpolating from a face of
- * of one element to the face of
- * the neighboring element.
- * The size of the matrix is
- * then <tt>source.dofs_per_face</tt> times
- * <tt>this->dofs_per_face</tt>.
- *
- * Derived elements will have to
- * implement this function. They
- * may only provide interpolation
- * matrices for certain source
- * finite elements, for example
- * those from the same family. If
- * they don't implement
- * interpolation from a given
- * element, then they must throw
- * an exception of type
- * FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented.
- */
- virtual void
- get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &source,
- const unsigned int subface,
- FullMatrix<double> &matrix) const;
-
/**
* Check for non-zero values on a face.
*
- * This function returns
- * @p true, if the shape
- * function @p shape_index has
- * non-zero values on the face
- * @p face_index.
+ * This function returns @p true, if the shape function @p shape_index has
+ * non-zero values on the face @p face_index.
*
- * Implementation of the
- * interface in
- * FiniteElement
+ * Implementation of the interface in FiniteElement
*/
virtual bool has_support_on_face (const unsigned int shape_index,
const unsigned int face_index) const;
- /**
- * @name Functions to support hp
- * @{
- */
-
- /**
- * Return whether this element
- * implements its hanging node
- * constraints in the new way,
- * which has to be used to make
- * elements "hp compatible".
- *
- * For the FE_Q_DG0 class the result is
- * always true (independent of the degree
- * of the element), as it implements the
- * complete set of functions necessary
- * for hp capability.
- */
- virtual bool hp_constraints_are_implemented () const;
-
- /**
- * If, on a vertex, several finite elements are active, the hp code
- * first assigns the degrees of freedom of each of these FEs
- * different global indices. It then calls this function to find out
- * which of them should get identical values, and consequently can
- * receive the same global DoF index. This function therefore
- * returns a list of identities between DoFs of the present finite
- * element object with the DoFs of @p fe_other, which is a reference
- * to a finite element object representing one of the other finite
- * elements active on this particular vertex. The function computes
- * which of the degrees of freedom of the two finite element objects
- * are equivalent, both numbered between zero and the corresponding
- * value of dofs_per_vertex of the two finite elements. The first
- * index of each pair denotes one of the vertex dofs of the present
- * element, whereas the second is the corresponding index of the
- * other finite element.
- */
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
-
- /**
- * Same as
- * hp_vertex_dof_indices(),
- * except that the function
- * treats degrees of freedom on
- * lines.
- */
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
-
- /**
- * Same as
- * hp_vertex_dof_indices(),
- * except that the function
- * treats degrees of freedom on
- * quads.
- */
- virtual
- std::vector<std::pair<unsigned int, unsigned int> >
- hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const;
-
- /**
- * Return whether this element dominates
- * the one given as argument when they
- * meet at a common face,
- * whether it is the other way around,
- * whether neither dominates, or if
- * either could dominate.
- *
- * For a definition of domination, see
- * FiniteElementBase::Domination and in
- * particular the @ref hp_paper "hp paper".
- */
- virtual
- FiniteElementDomination::Domination
- compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
- //@}
-
- /**
- * Determine an estimate for the
- * memory consumption (in bytes)
- * of this object.
- *
- * This function is made virtual,
- * since finite element objects
- * are usually accessed through
- * pointers to their base class,
- * rather than the class itself.
- */
- virtual std::size_t memory_consumption () const;
-
protected:
/**
- * @p clone function instead of
- * a copy constructor.
+ * @p clone function instead of a copy constructor.
*
- * This function is needed by the
- * constructors of @p FESystem.
+ * This function is needed by the constructors of @p FESystem.
*/
virtual FiniteElement<dim,spacedim> *clone() const;
private:
+
/**
* Returns the restriction_is_additive flags.
* Only the last component is true.
static std::vector<bool> get_riaf_vector(const unsigned int degree);
/**
- * Only for internal use. Its
- * full name is
- * @p get_dofs_per_object_vector
- * function and it creates the
- * @p dofs_per_object vector that is
- * needed within the constructor to
- * be passed to the constructor of
- * @p FiniteElementData.
+ * Only for internal use. Its full name is @p get_dofs_per_object_vector
+ * function and it creates the @p dofs_per_object vector that is needed
+ * within the constructor to be passed to the constructor of @p
+ * FiniteElementData.
*/
static std::vector<unsigned int> get_dpo_vector(const unsigned int degree);
-
- /**
- * Only for internal use. Its
- * full name is
- * @p get_dofs_per_object_vector_q
- * function and it creates the
- * @p dofs_per_object vector that is
- * needed within the constructor to
- * be passed to the constructor of
- * @p FiniteElementData for FE_Q_DG0
- * objects.
- */
- static std::vector<unsigned int> get_dpo_vector_q(const unsigned int degree);
-
- /**
- * This is an analogon to the
- * FETools::lexicographic_to_hierarchic_numbering
- * function, but working on
- * faces. Called from the
- * constructor.
- */
- static
- std::vector<unsigned int>
- face_lexicographic_to_hierarchic_numbering (const unsigned int degree);
-
- /**
- * Initialize the hanging node
- * constraints matrices. Called
- * from the constructor.
- */
- void initialize_constraints ();
-
- /**
- * Initialize the embedding
- * matrices. Called from the
- * constructor.
- */
- void initialize_embedding ();
-
- /**
- * Initialize the restriction
- * matrices. Called from the
- * constructor.
- */
- void initialize_restriction ();
-
- /**
- * Initialize the
- * @p unit_support_points field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_unit_support_points ();
-
- /**
- * Initialize the @p unit_support_points
- * field of the FiniteElement
- * class. Called from the constructor in
- * case the finite element is based on
- * quadrature points.
- */
- void initialize_unit_support_points (const Quadrature<1> &points);
-
- /**
- * Initialize the
- * @p unit_face_support_points field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_unit_face_support_points ();
-
- /**
- * Initialize the @p
- * unit_face_support_points field of the
- * FiniteElement class. Called from the
- * constructor in case the finite element
- * is based on quadrature points.
- */
- void initialize_unit_face_support_points (const Quadrature<1> &points);
-
- /**
- * Initialize the
- * @p adjust_quad_dof_index_for_face_orientation_table field
- * of the FiniteElement
- * class. Called from the
- * constructor.
- */
- void initialize_quad_dof_index_permutation ();
-
- /**
- * Mapping from hierarchic to
- * lexicographic numbering on
- * first face. Hierarchic is the
- * numbering of the shape
- * functions.
- */
- const std::vector<unsigned int> face_index_map;
-
-
- /**
- * Forward declaration of a class
- * into which we put significant
- * parts of the implementation.
- *
- * See the .cc file for more
- * information.
- */
- struct Implementation;
-
- /**
- * Allow access from other
- * dimensions. We need this since
- * we want to call the functions
- * @p get_dpo_vector and
- * @p lexicographic_to_hierarchic_numbering
- * for the faces of the finite
- * element of dimension dim+1.
- */
- template <int, int> friend class FE_Q_DG0;
-
- friend struct FE_Q_DG0<dim,spacedim>::Implementation;
};
/*@}*/
-/* -------------- declaration of explicit specializations ------------- */
-
-template <>
-void FE_Q_DG0<1>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
-
-
-template <>
-void FE_Q_DG0<1,2>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1,2>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
-
-template <>
-void FE_Q_DG0<1,3>::initialize_unit_face_support_points ();
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1,3>::face_lexicographic_to_hierarchic_numbering (const unsigned int);
DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+#ifndef __deal2__fe_q_iso_q1_h
+#define __deal2__fe_q_iso_q1_h
+
+#include <deal.II/base/config.h>
+#include <deal.II/base/tensor_product_polynomials.h>
+#include <deal.II/base/polynomials_piecewise.h>
+#include <deal.II/fe/fe_q_base.h>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+/*!@addtogroup fe */
+/*@{*/
+
+/**
+ * Implementation of a scalar Lagrange finite element @p Qp-iso-Q1 that
+ * defines the finite element space of continuous, piecewise linear elements
+ * with @p p subdivisions in each coordinate direction. It yields an element
+ * with the same number of degrees of freedom as the @p Qp elements but using
+ * linear interpolation instead of higher order one. This type of element is
+ * also called macro element in the literature as it really consists of several
+ * smaller elements, namely <i>p</i><tt><sup>dim</sup></tt>.
+ *
+ * The numbering of degrees of freedom is done in exactly the same way as in
+ * FE_Q of degree @p p. See there for a detailed description on how degrees of
+ * freedom are numbered within one element.
+ *
+ * This element represents a Q-linear finite element space on a reduced mesh
+ * size <i>h/p</i>. Its effect is equivalent to using FE_Q of degree one on a
+ * finer mesh by a factor @p p if an equivalent quadrature is used. However,
+ * this element reduces the flexibility in the choice of (adaptive) mesh size
+ * by exactly this factor @p p, which typically reduces efficiency. On the
+ * other hand, comparing this element with @p p subdivisions to the FE_Q
+ * element of degree @p p on the same mesh shows that the convergence is
+ * typically much worse for smooth problems. In particular, @p Qp elements
+ * achieve interpolation orders of <i>h<sup>p+1</sup></i> in the L2 norm,
+ * whereas these elements reach only <i>(h/p)<sup>2</sup></i>. For these two
+ * reasons, this element is usually not very useful as a standalone. In
+ * addition, any evaluation of face terms on the boundaries within the
+ * elements becomes impossible with this element.
+ *
+ * Nonetheless, there are a few use cases where this element actually is
+ * useful:
+ * <ol>
+ *
+ * <li> Systems of PDEs where certain variables demand for higher resolutions
+ * than the others and the additional degrees of freedom should be spend on
+ * increasing the resolution of linears instead of higher order
+ * polynomials. This can be the case when irregularities (shocks) appear in
+ * the solution and stabilization techniques are used that work for linears
+ * but not higher order elements. </li>
+ *
+ * <li> Stokes/Navier Stokes systems as the one discussed in step-22 could be
+ * solved with Q2-iso-Q1 elements for velocities instead of Q2
+ * elements. Combined with Q1 pressures they give a stable mixed element
+ * pair. However, they perform worse than the standard approach in most
+ * situations. </li>
+ *
+ * <li> Preconditioning systems of FE_Q systems of higher order @p p with a
+ * preconditioner based on @p Qp-iso-Q1 elements: Some preconditioners like
+ * algebraic multigrid perform much better with linear elements than with higher
+ * order elements because they often implicitly assume a sparse connectivity
+ * between entries. Then, creating a preconditioner matrix based on these
+ * elements yields the same number of degrees of freedom (and a spectrally
+ * equivalent linear system), which can be combined with a (high order) system
+ * matrix in an iterative solver like CG. </li>
+ * </ol>
+ *
+ * <h3>Appropriate integration</h3>
+ *
+ * Due to the nature of these elements as a concatenation of linears, care
+ * must be taken when selecting quadrature formulas for this element. The
+ * standard choice for an element of @p p subelements is a formula
+ * <tt>QIterated<dim>(QGauss<1>(2), p)</tt>, which corresponds to the formula
+ * that would be used for integrating functions on a finer mesh. This is in
+ * contrast with FE_Q(p) where QGauss<dim>(p+1) is the default choice. In
+ * particular, care must be taken to not use a quadrature formula that
+ * evaluates the basis functions (and their derivatives) on sub-element
+ * boundaries as the gradients of piecewiese functions on internal boundaries
+ * are set to zero. No checks are performed internally to ensure that this is
+ * not the case - it is the user's responsibility to avoid these situations.
+ *
+ * Also note that the usual deal.II routines for setting up sparsity patterns
+ * and assembling matrices do not make use of the increased sparsity in this
+ * element compared to FE_Q. This is because DoFTools::make_sparsity_pattern
+ * assumes coupling between all degrees of freedom within the element, whereas
+ * FE_Q_iso_Q1 with more than one subdivision does have less coupling.
+ *
+ * @author Martin Kronbichler, 2013
+ */
+template <int dim, int spacedim=dim>
+class FE_Q_iso_Q1 : public FE_Q_Base<TensorProductPolynomials<dim, Polynomials::PiecewisePolynomial<double> >,dim,spacedim>
+{
+public:
+ /**
+ * Constructs a FE_Q_iso_Q1 element with a given number of subdivisions. The
+ * number of subdivision is similar to the degree in FE_Q in the sense that
+ * both elements produce the same number of degrees of freedom.
+ */
+ FE_Q_iso_Q1(const unsigned int n_subdivisions);
+
+ /**
+ * Return a string that uniquely identifies a finite element. This class
+ * returns <tt>FE_Q_iso_q1<dim>(equivalent_degree)</tt>, with @p dim and @p
+ * equivalent_degree replaced by appropriate values.
+ */
+ virtual std::string get_name () const;
+
+ /**
+ * @name Functions to support hp
+ * @{
+ */
+
+ /**
+ * Return whether this element dominates the one given as argument when they
+ * meet at a common face, whether it is the other way around, whether
+ * neither dominates, or if either could dominate.
+ *
+ * For a definition of domination, see FiniteElementBase::Domination and in
+ * particular the @ref hp_paper "hp paper".
+ */
+ virtual
+ FiniteElementDomination::Domination
+ compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const;
+ //@}
+
+protected:
+
+ /**
+ * @p clone function instead of a copy constructor.
+ *
+ * This function is needed by the constructors of @p FESystem.
+ */
+ virtual FiniteElement<dim,spacedim> *clone() const;
+};
+
+
+
+/*@}*/
+
+DEAL_II_NAMESPACE_CLOSE
+
+#endif
DEAL_II_NAMESPACE_OPEN
-template <int dim> class TensorProductPolynomials;
+template <int dim, typename POLY> class TensorProductPolynomials;
/*!@addtogroup mapping */
}
template<typename T>
- void copy_subrange (const unsigned int begin,
- const unsigned int end,
+ void copy_subrange (const unsigned int begin,
+ const unsigned int end,
const dealii::Vector<T> &src,
- dealii::Vector<T> &dst)
+ dealii::Vector<T> &dst)
{
memcpy(&*(dst.begin()+begin), &*(src.begin()+begin),
(end-begin)*sizeof(T));
}
template<typename T, typename U>
- void copy_subrange (const unsigned int begin,
- const unsigned int end,
+ void copy_subrange (const unsigned int begin,
+ const unsigned int end,
const dealii::Vector<T> &src,
- dealii::Vector<U> &dst)
+ dealii::Vector<U> &dst)
{
const T *q = src.begin()+begin;
const T *const end_q = src.begin()+end;
}
template<typename T, typename U>
- void copy_subrange_wrap (const unsigned int begin,
- const unsigned int end,
+ void copy_subrange_wrap (const unsigned int begin,
+ const unsigned int end,
const dealii::Vector<T> &src,
- dealii::Vector<U> &dst)
+ dealii::Vector<U> &dst)
{
copy_subrange (begin, end, src, dst);
}
template <typename T, typename U>
void copy_vector (const dealii::Vector<T> &src,
- dealii::Vector<U> &dst)
+ dealii::Vector<U> &dst)
{
+ if (PointerComparison::equal(&src, &dst))
+ return;
+
const unsigned int vec_size = src.size();
const unsigned int dst_size = dst.size();
if (dst_size != vec_size)
#include <deal.II/base/utilities.h>
#include <deal.II/base/memory_consumption.h>
#include <deal.II/base/tensor_product_polynomials.h>
+#include <deal.II/base/polynomials_piecewise.h>
#include <deal.II/base/mpi.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/fe/fe_poly.h>
const FE_Poly<TensorProductPolynomials<dim>,dim,dim> *fe_poly =
dynamic_cast<const FE_Poly<TensorProductPolynomials<dim>,dim,dim>*>
(&fe.base_element(0));
+ const FE_Poly<TensorProductPolynomials<dim,Polynomials::
+ PiecewisePolynomial<double> >,dim,dim> *fe_poly_piece =
+ dynamic_cast<const FE_Poly<TensorProductPolynomials<dim,
+ Polynomials::PiecewisePolynomial<double> >,dim,dim>*>
+ (&fe.base_element(0));
// This class currently only works for elements derived from
- // FE_Poly<TensorProductPolynomials<dim>,dim,dim>. For any other
- // element, the dynamic cast above will fail and give fe_poly == 0.
- Assert (fe_poly != 0, ExcNotImplemented());
+ // FE_Poly<TensorProductPolynomials<dim>,dim,dim> or piecewise
+ // polynomials. For any other element, the dynamic casts above will
+ // fail and give fe_poly == 0.
+ Assert (fe_poly != 0 || fe_poly_piece != 0, ExcNotImplemented());
if (n_fe_components == 1)
{
- lexicographic_inv[no][fe_index] =
- fe_poly->get_poly_space_numbering_inverse();
+ if (fe_poly != 0)
+ lexicographic_inv[no][fe_index] =
+ fe_poly->get_poly_space_numbering_inverse();
+ else
+ lexicographic_inv[no][fe_index] =
+ fe_poly_piece->get_poly_space_numbering_inverse();
AssertDimension (lexicographic_inv[no][fe_index].size(),
dof_info[no].dofs_per_cell[fe_index]);
}
// ok, we have more than one component
Assert (n_fe_components > 1, ExcInternalError());
std::vector<unsigned int> scalar_lex =
- fe_poly->get_poly_space_numbering();
+ fe_poly != 0 ? fe_poly->get_poly_space_numbering() :
+ fe_poly_piece->get_poly_space_numbering();
AssertDimension (scalar_lex.size() * n_fe_components,
dof_info[no].dofs_per_cell[fe_index]);
std::vector<unsigned int> lexicographic (dof_info[no].dofs_per_cell[fe_index]);
{
const FE_Poly<TensorProductPolynomials<dim>,dim,dim> *fe_poly =
dynamic_cast<const FE_Poly<TensorProductPolynomials<dim>,dim,dim>*>(&fe);
- Assert (fe_poly != 0, ExcNotImplemented());
- lexicographic = fe_poly->get_poly_space_numbering_inverse();
+ const FE_Poly<TensorProductPolynomials<dim,Polynomials::
+ PiecewisePolynomial<double> >,dim,dim> *fe_poly_piece =
+ dynamic_cast<const FE_Poly<TensorProductPolynomials<dim,
+ Polynomials::PiecewisePolynomial<double> >,dim,dim>*> (&fe);
+ Assert (fe_poly != 0 || fe_poly_piece, ExcNotImplemented());
+ lexicographic = fe_poly != 0 ?
+ fe_poly->get_poly_space_numbering_inverse() :
+ fe_poly_piece->get_poly_space_numbering_inverse();
// to evaluate 1D polynomials, evaluate along the line where y=z=0,
// assuming that shape_value(0,Point<dim>()) == 1. otherwise, need
polynomials_adini.cc
polynomials_bdm.cc
polynomials_nedelec.cc
+ polynomials_piecewise.cc
polynomial_space.cc
polynomials_p.cc
polynomials_raviart_thomas.cc
Assert (values.size() > 0, ExcZero());
const unsigned int values_size=values.size();
- // evaluate Lagrange polynomial and
- // derivatives
+ // evaluate Lagrange polynomial and derivatives
if (in_lagrange_product_form == true)
{
- // to compute the value and all derivatives of
- // a polynomial of the form
- // (x-x_1)*(x-x_2)*...*(x-x_n), expand the
- // derivatives like automatic differentiation
- // does.
+ // to compute the value and all derivatives of a polynomial of the
+ // form (x-x_1)*(x-x_2)*...*(x-x_n), expand the derivatives like
+ // automatic differentiation does.
const unsigned int n_supp = lagrange_support_points.size();
switch (values_size)
{
{
const number v = x-lagrange_support_points[i];
- // multiply by (x-x_i) and compute action on
- // all derivatives, too (inspired from
- // automatic differentiation: implement the
- // product rule for the old value and the new
- // variable 'v', i.e., expand value v and
- // derivative one). since we reuse a value
- // from the next lower derivative from the
- // steps before, need to start from the
- // highest derivative
+ // multiply by (x-x_i) and compute action on all derivatives,
+ // too (inspired from automatic differentiation: implement the
+ // product rule for the old value and the new variable 'v',
+ // i.e., expand value v and derivative one). since we reuse a
+ // value from the next lower derivative from the steps before,
+ // need to start from the highest derivative
for (unsigned int k=values_size-1; k>0; --k)
values[k] = (values[k] * v + values[k-1]);
values[0] *= v;
}
- // finally, multiply by the weight in the
- // Lagrange denominator. Could be done instead
- // of setting values[0] = 1 above, but that
- // gives different accumulation of round-off
- // errors (multiplication is not associative)
- // compared to when we computed the weight,
- // and hence a basis function might not be
- // exactly one at the center point, which is
- // nice to have. We also multiply derivatives
- // by k! to transform the product p_n =
- // p^(n)(x)/k! into the actual form of the
- // derivative
+ // finally, multiply by the weight in the Lagrange
+ // denominator. Could be done instead of setting values[0] = 1
+ // above, but that gives different accumulation of round-off
+ // errors (multiplication is not associative) compared to when we
+ // computed the weight, and hence a basis function might not be
+ // exactly one at the center point, which is nice to have. We also
+ // multiply derivatives by k! to transform the product p_n =
+ // p^(n)(x)/k! into the actual form of the derivative
{
number k_faculty = 1;
for (unsigned int k=0; k<values_size; ++k)
}
break;
- // manually implement size 1 (values only),
- // size 2 (value + first derivative), and size
- // 3 (up to second derivative) since they
- // might be called often. then, we can unroll
- // the loop.
+ // manually implement size 1 (values only), size 2 (value + first
+ // derivative), and size 3 (up to second derivative) since they
+ // might be called often. then, we can unroll the loop.
case 1:
values[0] = 1;
for (unsigned int i=0; i<n_supp; ++i)
Assert (coefficients.size() > 0, ExcEmptyObject());
- // if we only need the value, then
- // call the other function since
- // that is significantly faster
- // (there is no need to allocate
- // and free memory, which is really
- // expensive compared to all the
- // other operations!)
+ // if we only need the value, then call the other function since that is
+ // significantly faster (there is no need to allocate and free memory,
+ // which is really expensive compared to all the other operations!)
if (values_size == 1)
{
values[0] = value(x);
return;
};
- // if there are derivatives needed,
- // then do it properly by the
- // full Horner scheme
+ // if there are derivatives needed, then do it properly by the full Horner
+ // scheme
const unsigned int m=coefficients.size();
std::vector<number> a(coefficients);
unsigned int j_faculty=1;
- // loop over all requested
- // derivatives. note that
- // derivatives @p{j>m} are
- // necessarily zero, as they
- // differentiate the polynomial
- // more often than the highest
- // power is
+ // loop over all requested derivatives. note that derivatives @p{j>m} are
+ // necessarily zero, as they differentiate the polynomial more often than
+ // the highest power is
const unsigned int min_valuessize_m=std::min(values_size, m);
for (unsigned int j=0; j<min_valuessize_m; ++j)
{
void
Polynomial<number>::transform_into_standard_form ()
{
- // should only be called when the product form
- // is active
+ // should only be called when the product form is active
Assert (in_lagrange_product_form == true, ExcInternalError());
Assert (coefficients.size() == 0, ExcInternalError());
- // compute coefficients by expanding the
- // product (x-x_i) term by term
+ // compute coefficients by expanding the product (x-x_i) term by term
coefficients.resize (lagrange_support_points.size()+1);
if (lagrange_support_points.size() == 0)
coefficients[0] = 1.;
{
Assert (coefficients.size() == 0, ExcInternalError());
- // For polynomial order up to 3, we have
- // precomputed weights. Use these weights
- // instead of the product form
+ // For polynomial order up to 3, we have precomputed weights. Use these
+ // weights instead of the product form
if (n <= 3)
{
this->in_lagrange_product_form = false;
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2010, 2011, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
+
+#include <deal.II/base/polynomials_piecewise.h>
+
+
+DEAL_II_NAMESPACE_OPEN
+
+
+
+namespace Polynomials
+{
+
+ template <typename number>
+ PiecewisePolynomial<number>::PiecewisePolynomial (const Polynomial<number> &coefficients_on_interval,
+ const unsigned int n_intervals,
+ const unsigned int interval,
+ const bool spans_next_interval)
+ :
+ polynomial (coefficients_on_interval),
+ n_intervals (n_intervals),
+ interval (interval),
+ spans_two_intervals (spans_next_interval)
+ {
+ Assert (n_intervals > 0, ExcMessage ("No intervals given"));
+ AssertIndexRange (interval, n_intervals);
+ }
+
+
+
+ template <typename number>
+ void
+ PiecewisePolynomial<number>::value (const number x,
+ std::vector<number> &values) const
+ {
+ Assert (values.size() > 0, ExcZero());
+ const unsigned int values_size=values.size();
+
+ // shift polynomial if necessary
+ number y = x;
+ double derivative_change_sign = 1.;
+ if (n_intervals > 0)
+ {
+ const number step = 1./n_intervals;
+ // polynomial spans over two intervals
+ if (spans_two_intervals)
+ {
+ const double offset = step * interval;
+ if (x<offset || x>offset+step+step)
+ {
+ for (unsigned int k=0; k<values.size(); ++k)
+ values[k] = 0;
+ return;
+ }
+ else if (x<offset+step)
+ y = x-offset;
+ else
+ {
+ derivative_change_sign = -1.;
+ y = offset+step+step-x;
+ }
+ }
+ else
+ {
+ const double offset = step * interval;
+ if (x<offset || x>offset+step)
+ {
+ for (unsigned int k=0; k<values.size(); ++k)
+ values[k] = 0;
+ return;
+ }
+ else
+ y = x-offset;
+ }
+
+ // on subinterval boundaries, cannot evaluate derivatives properly, so
+ // set them to zero
+ if ((std::abs(y)<1e-14 && (interval > 0 ||
+ derivative_change_sign == -1.))
+ ||
+ (std::abs(y-step)<1e-14 &&
+ (interval < n_intervals-1 || derivative_change_sign == -1.)))
+ {
+ values[0] = value(x);
+ for (unsigned int d=1; d<values_size; ++d)
+ values[d] = 0;
+ return;
+ }
+ }
+
+ polynomial.value(y, values);
+
+ // change sign if necessary
+ for (unsigned int j=1; j<values_size; j+=2)
+ values[j] *= derivative_change_sign;
+ }
+
+
+
+ std::vector<PiecewisePolynomial<double> >
+ generate_complete_Lagrange_basis_on_subdivisions (const unsigned int n_subdivisions,
+ const unsigned int base_degree)
+ {
+ std::vector<Polynomial<double> > p_base =
+ LagrangeEquidistant::generate_complete_basis(base_degree);
+ for (unsigned int i=0; i<p_base.size(); ++i)
+ p_base[i].scale(n_subdivisions);
+
+ std::vector<PiecewisePolynomial<double> > p;
+ p.reserve (n_subdivisions * base_degree + 1);
+
+ p.push_back (PiecewisePolynomial<double> (p_base[0], n_subdivisions, 0,
+ false));
+ for (unsigned int s=0; s<n_subdivisions; ++s)
+ for (unsigned int i=0; i<base_degree; ++i)
+ p.push_back (PiecewisePolynomial<double> (p_base[i+1], n_subdivisions,
+ s,
+ i==(base_degree-1) &&
+ s<n_subdivisions-1));
+ return p;
+ }
+
+}
+
+// ------------------ explicit instantiations --------------- //
+
+namespace Polynomials
+{
+ template class PiecewisePolynomial<float>;
+ template class PiecewisePolynomial<double>;
+ template class PiecewisePolynomial<long double>;
+}
+
+DEAL_II_NAMESPACE_CLOSE
#include <deal.II/base/tensor_product_polynomials.h>
+#include <deal.II/base/polynomials_piecewise.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/table.h>
/* ------------------- TensorProductPolynomials -------------- */
-template <>
-inline
-void
-TensorProductPolynomials<1>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[1]) const
-{
- Assert (i<polynomials.size(), ExcInternalError());
- indices[0] = index_map[i];
-}
+namespace internal
+{
+ namespace
+ {
+ template <int dim>
+ inline
+ void compute_tensor_index(const unsigned int,
+ const unsigned int,
+ const unsigned int,
+ unsigned int (&)[dim])
+ {
+ Assert(false, ExcNotImplemented());
+ }
+ inline
+ void compute_tensor_index(const unsigned int n,
+ const unsigned int ,
+ const unsigned int ,
+ unsigned int (&indices)[1])
+ {
+ indices[0] = n;
+ }
-template <>
-inline
-void
-TensorProductPolynomials<2>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[2]) const
-{
- const unsigned int n_pols = polynomials.size();
- Assert (i<n_pols*n_pols, ExcInternalError());
- const unsigned int n=index_map[i];
+ inline
+ void compute_tensor_index(const unsigned int n,
+ const unsigned int n_pols_0,
+ const unsigned int ,
+ unsigned int (&indices)[2])
+ {
+ indices[0] = n % n_pols_0;
+ indices[1] = n / n_pols_0;
+ }
- indices[0] = n % n_pols;
- indices[1] = n / n_pols;
+ inline
+ void compute_tensor_index(const unsigned int n,
+ const unsigned int n_pols_0,
+ const unsigned int n_pols_1,
+ unsigned int (&indices)[3])
+ {
+ indices[0] = n % n_pols_0;
+ indices[1] = (n/n_pols_0) % n_pols_1;
+ indices[2] = n / (n_pols_0*n_pols_1);
+ }
+ }
}
-template <>
+template <int dim, typename POLY>
inline
void
-TensorProductPolynomials<3>::
+TensorProductPolynomials<dim,POLY>::
compute_index (const unsigned int i,
- unsigned int (&indices)[3]) const
+ unsigned int (&indices)[(dim > 0 ? dim : 1)]) const
{
- const unsigned int n_pols = polynomials.size();
- Assert (i<n_pols*n_pols*n_pols, ExcInternalError());
- const unsigned int n=index_map[i];
-
- indices[0] = n % n_pols;
- indices[1] = (n/n_pols) % n_pols;
- indices[2] = n / (n_pols*n_pols);
+ Assert (i<Utilities::fixed_power<dim>(polynomials.size()),ExcInternalError());
+ internal::compute_tensor_index(index_map[i], polynomials.size(),
+ polynomials.size(), indices);
}
-template <int dim>
+
+template <int dim, typename POLY>
void
-TensorProductPolynomials<dim>::output_indices(std::ostream &out) const
+TensorProductPolynomials<dim,POLY>::output_indices(std::ostream &out) const
{
unsigned int ix[dim];
for (unsigned int i=0; i<n_tensor_pols; ++i)
-template <int dim>
+template <int dim, typename POLY>
void
-TensorProductPolynomials<dim>::set_numbering(
+TensorProductPolynomials<dim,POLY>::set_numbering(
const std::vector<unsigned int> &renumber)
{
Assert(renumber.size()==index_map.size(),
-template <int dim>
+template <int dim, typename POLY>
double
-TensorProductPolynomials<dim>::compute_value (const unsigned int i,
- const Point<dim> &p) const
+TensorProductPolynomials<dim,POLY>::compute_value (const unsigned int i,
+ const Point<dim> &p) const
{
+ Assert(dim>0, ExcNotImplemented());
+
unsigned int indices[dim];
compute_index (i, indices);
-template <>
-double
-TensorProductPolynomials<0>::compute_value (const unsigned int ,
- const Point<0> &) const
-{
- Assert (false, ExcNotImplemented());
- return 0.;
-}
-
-
-
-template <int dim>
+template <int dim, typename POLY>
Tensor<1,dim>
-TensorProductPolynomials<dim>::compute_grad (const unsigned int i,
- const Point<dim> &p) const
+TensorProductPolynomials<dim,POLY>::compute_grad (const unsigned int i,
+ const Point<dim> &p) const
{
unsigned int indices[dim];
compute_index (i, indices);
}
-template <int dim>
+
+template <int dim, typename POLY>
Tensor<2,dim>
-TensorProductPolynomials<dim>::compute_grad_grad (const unsigned int i,
- const Point<dim> &p) const
+TensorProductPolynomials<dim,POLY>::compute_grad_grad (const unsigned int i,
+ const Point<dim> &p) const
{
unsigned int indices[dim];
compute_index (i, indices);
-template <int dim>
+template <int dim, typename POLY>
void
-TensorProductPolynomials<dim>::
+TensorProductPolynomials<dim,POLY>::
compute (const Point<dim> &p,
std::vector<double> &values,
std::vector<Tensor<1,dim> > &grads,
-template <int dim>
-unsigned int
-TensorProductPolynomials<dim>::n() const
-{
- return n_tensor_pols;
-}
-
-
-
-template <>
-unsigned int
-TensorProductPolynomials<0>::n() const
-{
- return numbers::invalid_unsigned_int;
-}
-
-
-
/* ------------------- AnisotropicPolynomials -------------- */
-template <>
-void
-AnisotropicPolynomials<1>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[1]) const
-{
- Assert (i<polynomials[0].size(), ExcInternalError());
- indices[0] = i;
-}
-
-
-
-template <>
-void
-AnisotropicPolynomials<2>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[2]) const
-{
- Assert (i < polynomials[0].size()*polynomials[1].size(),
- ExcInternalError());
-
- indices[0] = i % polynomials[0].size();
- indices[1] = i / polynomials[0].size();
-}
-
-
-
-template <>
+template <int dim>
void
-AnisotropicPolynomials<3>::
+AnisotropicPolynomials<dim>::
compute_index (const unsigned int i,
- unsigned int (&indices)[3]) const
+ unsigned int (&indices)[dim]) const
{
- Assert (i < polynomials[0].size()*polynomials[1].size()*polynomials[2].size(),
- ExcInternalError());
+ unsigned int n_poly = 1;
+ for (unsigned int d=0; d<dim; ++d)
+ n_poly *= polynomials[d].size();
+ Assert (i < n_poly, ExcInternalError());
- indices[0] = i % polynomials[0].size();
- indices[1] = (i/polynomials[0].size()) % polynomials[1].size();
- indices[2] = i / (polynomials[0].size()*polynomials[1].size());
+ internal::compute_tensor_index(i, polynomials[0].size(),
+ polynomials[1].size(), indices);
}
/* ------------------- explicit instantiations -------------- */
-template class TensorProductPolynomials<1>;
-template class TensorProductPolynomials<2>;
-template class TensorProductPolynomials<3>;
+template class TensorProductPolynomials<1,Polynomials::Polynomial<double> >;
+template class TensorProductPolynomials<2,Polynomials::Polynomial<double> >;
+template class TensorProductPolynomials<3,Polynomials::Polynomial<double> >;
+
+template class TensorProductPolynomials<1,Polynomials::PiecewisePolynomial<double> >;
+template class TensorProductPolynomials<2,Polynomials::PiecewisePolynomial<double> >;
+template class TensorProductPolynomials<3,Polynomials::PiecewisePolynomial<double> >;
template class AnisotropicPolynomials<1>;
template class AnisotropicPolynomials<2>;
/* ------------------- TensorProductPolynomialsConst -------------- */
-template <>
-inline
-void
-TensorProductPolynomialsConst<1>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[1]) const
-{
- Assert (i<polynomials.size(), ExcInternalError());
- indices[0] = index_map[i];
-}
-
-template <>
-inline
-void
-TensorProductPolynomialsConst<2>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[2]) const
-{
- const unsigned int n_pols = polynomials.size();
- Assert (i<n_pols*n_pols, ExcInternalError());
- const unsigned int n=index_map[i];
-
- indices[0] = n % n_pols;
- indices[1] = n / n_pols;
-}
-
-template <>
-inline
-void
-TensorProductPolynomialsConst<3>::
-compute_index (const unsigned int i,
- unsigned int (&indices)[3]) const
-{
- const unsigned int n_pols = polynomials.size();
- Assert (i<n_pols*n_pols*n_pols, ExcInternalError());
- const unsigned int n=index_map[i];
-
- indices[0] = n % n_pols;
- indices[1] = (n/n_pols) % n_pols;
- indices[2] = n / (n_pols*n_pols);
-}
-
-
-template <int dim>
-void
-TensorProductPolynomialsConst<dim>::output_indices(std::ostream &out) const
-{
- unsigned int ix[dim];
- //ouput without constant locally constant function
- for (unsigned int i=0; i<n_tensor_pols-1; ++i)
- {
- compute_index(i,ix);
- out << i << "\t";
- for (unsigned int d=0; d<dim; ++d)
- out << ix[d] << " ";
- out << std::endl;
- }
-}
-
-
-
-template <int dim>
-void
-TensorProductPolynomialsConst<dim>::set_numbering(
- const std::vector<unsigned int> &renumber)
-{
- Assert(renumber.size()==index_map.size(),
- ExcDimensionMismatch(renumber.size(), index_map.size()));
-
- index_map=renumber;
- for (unsigned int i=0; i<index_map.size(); ++i)
- index_map_inverse[index_map[i]]=i;
-}
template <int dim>
double
TensorProductPolynomialsConst<dim>::compute_value (const unsigned int i,
const Point<dim> &p) const
{
- const unsigned int max_indices = Utilities::fixed_power<dim>(polynomials.size());
-
+ const unsigned int max_indices = this->n_tensor_pols;
Assert (i<=max_indices, ExcInternalError());
// treat the regular basis functions
if (i<max_indices)
- {
- unsigned int indices[dim];
- compute_index (i, indices);
-
- double value=1.;
- for (unsigned int d=0; d<dim; ++d)
- value *= polynomials[indices[d]].value(p(d));
-
- return value;
- }
+ return this->TensorProductPolynomials<dim>::compute_value(i,p);
else
// this is for the constant function
return 1.;
}
+
+
template <>
double
TensorProductPolynomialsConst<0>::compute_value (const unsigned int ,
TensorProductPolynomialsConst<dim>::compute_grad (const unsigned int i,
const Point<dim> &p) const
{
- const unsigned int max_indices = Utilities::fixed_power<dim>(polynomials.size());
-
+ const unsigned int max_indices = this->n_tensor_pols;
Assert (i<=max_indices, ExcInternalError());
// treat the regular basis functions
if (i<max_indices)
- {
- Tensor<1,dim> grad;
-
- unsigned int indices[dim];
- compute_index (i, indices);
-
- // compute values and
- // uni-directional derivatives at
- // the given point in each
- // co-ordinate direction
- double v [dim][2];
- {
- std::vector<double> tmp (2);
- for (unsigned int d=0; d<dim; ++d)
- {
- polynomials[indices[d]].value (p(d), tmp);
- v[d][0] = tmp[0];
- v[d][1] = tmp[1];
- }
- }
-
- for (unsigned int d=0; d<dim; ++d)
- {
- grad[d] = 1.;
- for (unsigned int x=0; x<dim; ++x)
- grad[d] *= v[x][d==x];
- }
-
- return grad; //return 0 for q0 node
- }
+ return this->TensorProductPolynomials<dim>::compute_grad(i,p);
else
// this is for the constant function
return Tensor<1,dim>();
TensorProductPolynomialsConst<dim>::compute_grad_grad (const unsigned int i,
const Point<dim> &p) const
{
- const unsigned int max_indices = Utilities::fixed_power<dim>(polynomials.size());
-
+ const unsigned int max_indices = this->n_tensor_pols;
Assert (i<=max_indices, ExcInternalError());
// treat the regular basis functions
if (i<max_indices)
- {
- Tensor<2,dim> grad_grad;
-
- unsigned int indices[dim];
- compute_index (i, indices);
-
- double v [dim][3];
- {
- std::vector<double> tmp (3);
- for (unsigned int d=0; d<dim; ++d)
- {
- polynomials[indices[d]].value (p(d), tmp);
- v[d][0] = tmp[0];
- v[d][1] = tmp[1];
- v[d][2] = tmp[2];
- }
- }
- for (unsigned int d1=0; d1<dim; ++d1)
- for (unsigned int d2=0; d2<dim; ++d2)
- {
- grad_grad[d1][d2] = 1.;
- for (unsigned int x=0; x<dim; ++x)
- {
- unsigned int derivative=0;
- if (d1==x || d2==x)
- {
- if (d1==d2)
- derivative=2;
- else
- derivative=1;
- }
- grad_grad[d1][d2] *= v[x][derivative];
- }
- }
- return grad_grad; //return 0 for q0 node
- }
+ return this->TensorProductPolynomials<dim>::compute_grad_grad(i,p);
else
// this is for the constant function
return Tensor<2,dim>();
std::vector<Tensor<1,dim> > &grads,
std::vector<Tensor<2,dim> > &grad_grads) const
{
- Assert (values.size()==n_tensor_pols || values.size()==0,
- ExcDimensionMismatch2(values.size(), n_tensor_pols, 0));
- Assert (grads.size()==n_tensor_pols || grads.size()==0,
- ExcDimensionMismatch2(grads.size(), n_tensor_pols, 0));
- Assert (grad_grads.size()==n_tensor_pols|| grad_grads.size()==0,
- ExcDimensionMismatch2(grad_grads.size(), n_tensor_pols, 0));
-
- const bool update_values = (values.size() == n_tensor_pols),
- update_grads = (grads.size()==n_tensor_pols),
- update_grad_grads = (grad_grads.size()==n_tensor_pols);
-
- // check how many
- // values/derivatives we have to
- // compute
- unsigned int n_values_and_derivatives = 0;
- if (update_values)
- n_values_and_derivatives = 1;
- if (update_grads)
- n_values_and_derivatives = 2;
- if (update_grad_grads)
- n_values_and_derivatives = 3;
-
-
- // compute the values (and derivatives, if
- // necessary) of all polynomials at this
- // evaluation point. to avoid many
- // reallocation, use one std::vector for
- // polynomial evaluation and store the
- // result as Tensor<1,3> (that has enough
- // fields for any evaluation of values and
- // derivatives)
- Table<2,Tensor<1,3> > v(dim, polynomials.size()+1);
- {
- std::vector<double> tmp (n_values_and_derivatives);
- for (unsigned int d=0; d<dim; ++d)
- for (unsigned int i=0; i<polynomials.size(); ++i)
- {
- polynomials[i].value(p(d), tmp);
- for (unsigned int e=0; e<n_values_and_derivatives; ++e)
- v(d,i)[e] = tmp[e];
- };
- }
-
- for (unsigned int i=0; i<n_tensor_pols-1; ++i)
+ Assert (values.size()==this->n_tensor_pols+1 || values.size()==0,
+ ExcDimensionMismatch2(values.size(), this->n_tensor_pols+1, 0));
+ Assert (grads.size()==this->n_tensor_pols+1 || grads.size()==0,
+ ExcDimensionMismatch2(grads.size(), this->n_tensor_pols+1, 0));
+ Assert (grad_grads.size()==this->n_tensor_pols+1 || grad_grads.size()==0,
+ ExcDimensionMismatch2(grad_grads.size(), this->n_tensor_pols+1, 0));
+
+ // remove slot for const value, go into the base class compute method and
+ // finally append the const value again
+ bool do_values = false, do_grads = false, do_grad_grads = false;
+ if (values.empty() == false)
{
- // first get the
- // one-dimensional indices of
- // this particular tensor
- // product polynomial
- unsigned int indices[dim];
- compute_index (i, indices);
-
- if (update_values)
- {
- values[i] = 1;
- for (unsigned int x=0; x<dim; ++x)
- values[i] *= v(x,indices[x])[0];
- }
-
- if (update_grads)
- for (unsigned int d=0; d<dim; ++d)
- {
- grads[i][d] = 1.;
- for (unsigned int x=0; x<dim; ++x)
- grads[i][d] *= v(x,indices[x])[d==x];
- }
-
- if (update_grad_grads)
- for (unsigned int d1=0; d1<dim; ++d1)
- for (unsigned int d2=0; d2<dim; ++d2)
- {
- grad_grads[i][d1][d2] = 1.;
- for (unsigned int x=0; x<dim; ++x)
- {
- unsigned int derivative=0;
- if (d1==x || d2==x)
- {
- if (d1==d2)
- derivative=2;
- else
- derivative=1;
- }
- grad_grads[i][d1][d2]
- *= v(x,indices[x])[derivative];
- }
- }
+ values.pop_back();
+ do_values = true;
+ }
+ if (grads.empty() == false)
+ {
+ grads.pop_back();
+ do_grads = true;
+ }
+ if (grad_grads.empty() == false)
+ {
+ grad_grads.pop_back();
+ do_grad_grads = true;
}
- //for dgq node: values =1, grads=0, grads_grads=0
- if (update_values)
- values[n_tensor_pols-1]=1;
-}
-
+ this->TensorProductPolynomials<dim>::compute(p, values, grads, grad_grads);
-template <int dim>
-unsigned int
-TensorProductPolynomialsConst<dim>::n() const
-{
- return n_tensor_pols;
+ //for dgq node: values =1, grads=0, grads_grads=0
+ if (do_values)
+ values.push_back(1.);
+ if (do_grads)
+ grads.push_back(Tensor<1,dim>());
+ if (do_grad_grads)
+ grad_grads.push_back(Tensor<2,dim>());
}
-
-template <>
-unsigned int
-TensorProductPolynomialsConst<0>::n() const
-{
- return numbers::invalid_unsigned_int;
-}
-
/* ------------------- explicit instantiations -------------- */
template class TensorProductPolynomialsConst<1>;
template class TensorProductPolynomialsConst<2>;
fe_poly.cc
fe_poly_tensor.cc
fe_q.cc
+ fe_q_base.cc
fe_q_dg0.cc
fe_q_hierarchical.cc
+ fe_q_iso_q1.cc
fe_raviart_thomas.cc
fe_raviart_thomas_nodal.cc
fe_system.cc
fe_nothing.inst.in
fe_poly.inst.in
fe_poly_tensor.inst.in
+ fe_q_base.inst.in
fe_q_dg0.inst.in
fe_q_hierarchical.inst.in
fe_q.inst.in
+ fe_q_iso_q1.inst.in
fe_raviart_thomas.inst.in
fe_raviart_thomas_nodal.inst.in
fe_system.inst.in
#include <deal.II/base/tensor_product_polynomials.h>
#include <deal.II/base/tensor_product_polynomials_const.h>
#include <deal.II/base/polynomials_p.h>
+#include <deal.II/base/polynomials_piecewise.h>
#include <deal.II/fe/fe_poly.h>
#include <deal.II/fe/fe_values.h>
for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
{
-#if deal_II_dimension <= deal_II_space_dimension
+#if deal_II_dimension <= deal_II_space_dimension
template class FE_Poly<TensorProductPolynomials<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
template class FE_Poly<TensorProductPolynomialsConst<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
+ template class FE_Poly<TensorProductPolynomials<deal_II_dimension,Polynomials::PiecewisePolynomial<double> >, deal_II_dimension, deal_II_space_dimension>;
template class FE_Poly<PolynomialSpace<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
template class FE_Poly<PolynomialsP<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
#endif
// $Id$
// Version: $Name$
//
-// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 by the deal.II authors
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
//
//---------------------------------------------------------------------------
-#include <deal.II/base/quadrature.h>
-#include <deal.II/base/qprojector.h>
-#include <deal.II/base/template_constraints.h>
-#include <deal.II/fe/fe_q.h>
-#include <deal.II/fe/fe_nothing.h>
-#include <deal.II/fe/fe_tools.h>
#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/fe/fe_q.h>
#include <vector>
#include <sstream>
DEAL_II_NAMESPACE_OPEN
-// namespace for some functions that are used in this file. they are
-// specific to numbering conventions used for the FE_Q element, and
-// are thus not very interesting to the outside world. we'd like to
-// simply put them into an anonymous namespace, but that triggers an
-// odd error with icc which can't compile this small snippet if the
-// function is static:
-// --------------------
-// template <int> struct int2type {};
-//
-// namespace {
-// static void SYMBOL (const int2type<1> & ) {}
-// }
-//
-// template <int dim, int spacedim> void g() {
-// SYMBOL(int2type<dim>());
-// }
-//
-// template void g<1>();
-// --------------------
-// the function needs to be static because of another icc bug, though.
-// work around this by packing everything into a namespace of its own
-// and have the anonymous namespace inside
-//
-// this is now intel icc issue 216082
-namespace FE_Q_Helper
-{
- namespace
- {
- // given a permutation array,
- // compute and return the inverse
- // permutation
- inline
- std::vector<unsigned int>
- invert_numbering (const std::vector<unsigned int> &in)
- {
- std::vector<unsigned int> out (in.size());
- for (unsigned int i=0; i<in.size(); ++i)
- out[in[i]]=i;
- return out;
- }
-
-
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons. this clears a
- // dim-array
- template <int dim>
- inline
- void
- zero_indices (unsigned int indices[dim])
- {
- for (unsigned int d=0; d<dim; ++d)
- indices[d] = 0;
- }
-
-
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons. this increments tensor
- // product indices
- template <int dim>
- inline
- void
- increment_indices (unsigned int indices[dim],
- const unsigned int dofs1d)
- {
- ++indices[0];
- for (int d=0; d<dim-1; ++d)
- if (indices[d]==dofs1d)
- {
- indices[d] = 0;
- indices[d+1]++;
- }
- }
-
-
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons, and we need to again
- // access 1D polynomials. This function
- // creates them from dim-dimensional support
- // points.
- template <int dim>
- inline
- std::vector<Polynomials::Polynomial<double> >
- generate_poly_space1d (const std::vector<Point<dim> > &unit_support_points,
- const std::vector<unsigned int> &index_map_inverse,
- const unsigned int dofs1d)
- {
- AssertDimension (Utilities::fixed_power<dim> (dofs1d),
- unit_support_points.size());
- std::vector<Point<1> > points1d (dofs1d);
- for (unsigned int i=0; i<dofs1d; ++i)
- {
- const unsigned int j = index_map_inverse[i];
- points1d[i] = Point<1>(unit_support_points[j](0));
- for (unsigned int d=1; d<dim; ++d)
- Assert (unit_support_points[j][d] == 0.,
- ExcInternalError());
- }
- return Polynomials::generate_complete_Lagrange_basis (points1d);
- }
- }
-}
-
-
-
-/**
- * A class with the same purpose as the similarly named class of the
- * Triangulation class. See there for more information.
- */
-template <int xdim, int xspacedim>
-struct FE_Q<xdim,xspacedim>::Implementation
-{
- /**
- * Initialize the hanging node
- * constraints matrices. Called from the
- * constructor in case the finite element
- * is based on quadrature points.
- */
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &,
- FE_Q<1,spacedim> &)
- {
- // no constraints in 1d
- }
-
-
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &points,
- FE_Q<2,spacedim> &fe)
- {
- const unsigned int dim = 2;
-
- // restricted to each face, the
- // traces of the shape functions is
- // an element of P_{k} (in 2d), or
- // Q_{k} (in 3d), where k is the
- // degree of the element. from
- // this, we interpolate between
- // mother and cell face.
-
- // the interpolation process works
- // as follows: on each subface,
- // we want that finite element
- // solutions from both sides
- // coincide. i.e. if a and b are
- // expansion coefficients for the
- // shape functions from both sides,
- // we seek a relation between a and
- // b such that
- // sum_j a_j phi^c_j(x)
- // == sum_j b_j phi_j(x)
- // for all points x on the
- // interface. here, phi^c_j are the
- // shape functions on the small
- // cell on one side of the face,
- // and phi_j those on the big cell
- // on the other side. To get this
- // relation, it suffices to look at
- // a sufficient number of points
- // for which this has to hold. if
- // there are n functions, then we
- // need n evaluation points, and we
- // choose them equidistantly.
- //
- // we obtain the matrix system
- // A a == B b
- // where
- // A_ij = phi^c_j(x_i)
- // B_ij = phi_j(x_i)
- // and the relation we are looking
- // for is
- // a = A^-1 B b
- //
- // for the special case of Lagrange
- // interpolation polynomials, A_ij
- // reduces to delta_ij, and
- // a_i = B_ij b_j
- // Hence,
- // interface_constraints(i,j)=B_ij.
- //
- // for the general case, where we
- // don't have Lagrange
- // interpolation polynomials, this
- // is a little more
- // complicated. Then we would
- // evaluate at a number of points
- // and invert the interpolation
- // matrix A.
- //
- // Note, that we build up these
- // matrices for all subfaces at
- // once, rather than considering
- // them separately. the reason is
- // that we finally will want to
- // have them in this order anyway,
- // as this is the format we need
- // inside deal.II
-
- // In the following the points x_i
- // are constructed in following
- // order (n=degree-1)
- // *----------*---------*
- // 1..n 0 n+1..2n
- // i.e. first the midpoint of the
- // line, then the support points on
- // subface 0 and on subface 1
- std::vector<Point<dim-1> > constraint_points;
- // Add midpoint
- constraint_points.push_back (Point<dim-1> (0.5));
-
- if (fe.degree>1)
- {
- const unsigned int n=fe.degree-1;
- const double step=1./fe.degree;
- // subface 0
- for (unsigned int i=1; i<=n; ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),0));
- // subface 1
- for (unsigned int i=1; i<=n; ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),1));
- }
-
- // Now construct relation between
- // destination (child) and source (mother)
- // dofs.
- const std::vector<Polynomials::Polynomial<double> > polynomials=
- Polynomials::generate_complete_Lagrange_basis(points.get_points());
-
- fe.interface_constraints
- .TableBase<2,double>::reinit (fe.interface_constraints_size());
-
- for (unsigned int i=0; i<constraint_points.size(); ++i)
- for (unsigned int j=0; j<fe.degree+1; ++j)
- {
- fe.interface_constraints(i,j) =
- polynomials[fe.face_index_map[j]].value (constraint_points[i](0));
-
- // if the value is small up
- // to round-off, then
- // simply set it to zero to
- // avoid unwanted fill-in
- // of the constraint
- // matrices (which would
- // then increase the number
- // of other DoFs a
- // constrained DoF would
- // couple to)
- if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
- fe.interface_constraints(i,j) = 0;
- }
- }
-
-
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &points,
- FE_Q<3,spacedim> &fe)
- {
- const unsigned int dim = 3;
-
- // For a detailed documentation of
- // the interpolation see the
- // FE_Q<2>::initialize_constraints
- // function.
-
- // In the following the points x_i
- // are constructed in the order as
- // described in the documentation
- // of the FiniteElement class
- // (fe_base.h), i.e.
- // *--15--4--16--*
- // | | |
- // 10 19 6 20 12
- // | | |
- // 1--7---0--8---2
- // | | |
- // 9 17 5 18 11
- // | | |
- // *--13--3--14--*
- std::vector<Point<dim-1> > constraint_points;
-
- // Add midpoint
- constraint_points.push_back (Point<dim-1> (0.5, 0.5));
-
- // Add midpoints of lines of
- // "mother-face"
- constraint_points.push_back (Point<dim-1> (0, 0.5));
- constraint_points.push_back (Point<dim-1> (1, 0.5));
- constraint_points.push_back (Point<dim-1> (0.5, 0));
- constraint_points.push_back (Point<dim-1> (0.5, 1));
-
- if (fe.degree>1)
- {
- const unsigned int n=fe.degree-1;
- const double step=1./fe.degree;
- std::vector<Point<dim-2> > line_support_points(n);
- for (unsigned int i=0; i<n; ++i)
- line_support_points[i](0)=(i+1)*step;
- Quadrature<dim-2> qline(line_support_points);
-
- // auxiliary points in 2d
- std::vector<Point<dim-1> > p_line(n);
-
- // Add nodes of lines interior
- // in the "mother-face"
-
- // line 5: use line 9
- QProjector<dim-1>::project_to_subface(qline, 0, 0, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
- // line 6: use line 10
- QProjector<dim-1>::project_to_subface(qline, 0, 1, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
- // line 7: use line 13
- QProjector<dim-1>::project_to_subface(qline, 2, 0, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
- // line 8: use line 14
- QProjector<dim-1>::project_to_subface(qline, 2, 1, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
-
- // DoFs on bordering lines
- // lines 9-16
- for (unsigned int face=0; face<GeometryInfo<dim-1>::faces_per_cell; ++face)
- for (unsigned int subface=0;
- subface<GeometryInfo<dim-1>::max_children_per_face; ++subface)
- {
- QProjector<dim-1>::project_to_subface(qline, face, subface, p_line);
- constraint_points.insert(constraint_points.end(),
- p_line.begin(), p_line.end());
- }
-
- // Create constraints for
- // interior nodes
- std::vector<Point<dim-1> > inner_points(n*n);
- for (unsigned int i=0, iy=1; iy<=n; ++iy)
- for (unsigned int ix=1; ix<=n; ++ix)
- inner_points[i++] = Point<dim-1> (ix*step, iy*step);
-
- // at the moment do this for
- // isotropic face refinement only
- for (unsigned int child=0;
- child<GeometryInfo<dim-1>::max_children_per_cell; ++child)
- for (unsigned int i=0; i<inner_points.size(); ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(inner_points[i], child));
- }
-
- // Now construct relation between
- // destination (child) and source (mother)
- // dofs.
- const unsigned int pnts=(fe.degree+1)*(fe.degree+1);
- const std::vector<Polynomials::Polynomial<double> > polynomial_basis=
- Polynomials::generate_complete_Lagrange_basis(points.get_points());
-
- const TensorProductPolynomials<dim-1> face_polynomials(polynomial_basis);
-
- fe.interface_constraints
- .TableBase<2,double>::reinit (fe.interface_constraints_size());
-
- for (unsigned int i=0; i<constraint_points.size(); ++i)
- {
- const double interval = (double) (fe.degree * 2);
- bool mirror[dim - 1];
- Point<dim-1> constraint_point;
-
- // Eliminate FP errors in constraint
- // points. Due to their origin, they
- // must all be fractions of the unit
- // interval. If we have polynomial
- // degree 4, the refined element has 8
- // intervals. Hence the coordinates
- // must be 0, 0.125, 0.25, 0.375 etc.
- // Now the coordinates of the
- // constraint points will be multiplied
- // by the inverse of the interval size
- // (in the example by 8). After that
- // the coordinates must be integral
- // numbers. Hence a normal truncation
- // is performed and the coordinates
- // will be scaled back. The equal
- // treatment of all coordinates should
- // eliminate any FP errors.
- for (unsigned int k=0; k<dim-1; ++k)
- {
- const int coord_int =
- static_cast<int> (constraint_points[i](k) * interval + 0.25);
- constraint_point(k) = 1.*coord_int / interval;
-
- // The following lines of code
- // should eliminate the problems
- // with the Constraint-Matrix,
- // which appeared for P>=4. The
- // Constraint-Matrix class
- // complained about different
- // constraints for the same entry
- // of the Constraint-Matrix.
- // Actually this difference could
- // be attributed to FP errors, as
- // it was in the range of
- // 1.0e-16. These errors originate
- // in the loss of symmetry in the
- // FP approximation of the
- // shape-functions. Considering a
- // 3rd order shape function in 1D,
- // we have N0(x)=N3(1-x) and
- // N1(x)=N2(1-x). For higher order
- // polynomials the FP
- // approximations of the shape
- // functions do not satisfy these
- // equations any more! Thus in the
- // following code everything is
- // computed in the interval x \in
- // [0..0.5], which is sufficient to
- // express all values that could
- // come out from a computation of
- // any shape function in the full
- // interval [0..1]. If x > 0.5 the
- // computation is done for 1-x with
- // the shape function N_{p-n}
- // instead of N_n. Hence symmetry
- // is preserved and everything
- // works fine...
- //
- // For a different explanation of
- // the problem, see the discussion
- // in the FiniteElement class
- // for constraint matrices in 3d.
- mirror[k] = (constraint_point(k) > 0.5);
- if (mirror[k])
- constraint_point(k) = 1.0 - constraint_point(k);
- }
-
- for (unsigned int j=0; j<pnts; ++j)
- {
- unsigned int indices[2]
- = { fe.face_index_map[j] % (fe.degree + 1),
- fe.face_index_map[j] / (fe.degree + 1)
- };
-
- for (unsigned int k = 0; k<2; ++k)
- if (mirror[k])
- indices[k] = fe.degree - indices[k];
-
- const unsigned int
- new_index = indices[1] * (fe.degree + 1) + indices[0];
-
- fe.interface_constraints(i,j) =
- face_polynomials.compute_value (new_index, constraint_point);
-
- // if the value is small up
- // to round-off, then
- // simply set it to zero to
- // avoid unwanted fill-in
- // of the constraint
- // matrices (which would
- // then increase the number
- // of other DoFs a
- // constrained DoF would
- // couple to)
- if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
- fe.interface_constraints(i,j) = 0;
- }
- }
- }
-};
-
-
-
-
template <int dim, int spacedim>
FE_Q<dim,spacedim>::FE_Q (const unsigned int degree)
:
- FE_Poly<TensorProductPolynomials<dim>, dim, spacedim> (
+ FE_Q_Base<TensorProductPolynomials<dim>, dim, spacedim> (
TensorProductPolynomials<dim>(Polynomials::LagrangeEquidistant::generate_complete_basis(degree)),
- FiniteElementData<dim>(get_dpo_vector(degree),
+ FiniteElementData<dim>(this->get_dpo_vector(degree),
1, degree,
FiniteElementData<dim>::H1),
- std::vector<bool> (1, false),
- std::vector<ComponentMask>(1, std::vector<bool>(1,true))),
- face_index_map(FE_Q_Helper::invert_numbering(face_lexicographic_to_hierarchic_numbering (degree)))
+ std::vector<bool> (1, false))
{
Assert (degree > 0,
ExcMessage ("This element can only be used for polynomial degrees "
"greater than zero"));
+ std::vector<Point<1> > support_points_1d(degree+1);
+ for (unsigned int i=0; i<=degree; ++i)
+ support_points_1d[i][0] = static_cast<double>(i)/degree;
- std::vector<unsigned int> renumber (this->dofs_per_cell);
- FETools::hierarchic_to_lexicographic_numbering (*this, renumber);
- this->poly_space.set_numbering(renumber);
-
- // finally fill in support points
- // on cell and face
- initialize_unit_support_points ();
- initialize_unit_face_support_points ();
-
- // reinit constraints
- initialize_constraints ();
-
- // Reinit the vectors of restriction and
- // prolongation matrices to the right sizes
- // and compute the matrices
- this->reinit_restriction_and_prolongation_matrices();
- initialize_embedding();
- initialize_restriction();
-
- initialize_quad_dof_index_permutation();
+ this->initialize(support_points_1d);
}
template <int dim, int spacedim>
FE_Q<dim,spacedim>::FE_Q (const Quadrature<1> &points)
:
- FE_Poly<TensorProductPolynomials<dim>, dim, spacedim> (
+ FE_Q_Base<TensorProductPolynomials<dim>, dim, spacedim> (
TensorProductPolynomials<dim>(Polynomials::generate_complete_Lagrange_basis(points.get_points())),
- FiniteElementData<dim>(get_dpo_vector(points.size()-1),
+ FiniteElementData<dim>(this->get_dpo_vector(points.size()-1),
1, points.size()-1,
FiniteElementData<dim>::H1),
- std::vector<bool> (1, false),
- std::vector<ComponentMask>(1, std::vector<bool>(1,true))),
- face_index_map(FE_Q_Helper::invert_numbering(face_lexicographic_to_hierarchic_numbering (points.size()-1)))
+ std::vector<bool> (1, false))
{
- const int degree = points.size()-1;
-
+ const unsigned int degree = points.size()-1;
Assert (degree > 0,
ExcMessage ("This element can only be used for polynomial degrees "
"at least zero"));
- Assert (points.point(0)(0) == 0,
- ExcMessage ("The first support point has to be zero."));
- Assert (points.point(degree)(0) == 1,
- ExcMessage ("The last support point has to be one."));
-
- std::vector<unsigned int> renumber (this->dofs_per_cell);
- FETools::hierarchic_to_lexicographic_numbering (*this, renumber);
- this->poly_space.set_numbering(renumber);
-
- // finally fill in support points
- // on cell and face
- initialize_unit_support_points (points);
- initialize_unit_face_support_points (points);
- // reinit constraints
- Implementation::initialize_constraints (points, *this);
-
- // Reinit the vectors of restriction and
- // prolongation matrices to the right sizes
- // and compute the matrices
- this->reinit_restriction_and_prolongation_matrices();
- initialize_embedding();
- initialize_restriction();
-
- initialize_quad_dof_index_permutation();
+ this->initialize(points.get_points());
}
std::string
FE_Q<dim,spacedim>::get_name () const
{
- // note that the
- // FETools::get_fe_from_name
- // function depends on the
- // particular format of the string
- // this function returns, so they
- // have to be kept in synch
+ // note that the FETools::get_fe_from_name function depends on the
+ // particular format of the string this function returns, so they have to be
+ // kept in synch
std::ostringstream namebuf;
bool type = true;
const std::vector<Point<dim> > &unit_support_points = this->unit_support_points;
unsigned int index = 0;
- // Decode the support points
- // in one coordinate direction.
+ // Decode the support points in one coordinate direction.
for (unsigned int j=0; j<dofs_per_cell; j++)
{
if ((dim>1) ? (unit_support_points[j](1)==0 &&
Assert (index == n_points,
ExcMessage ("Could not decode support points in one coordinate direction."));
- // Check whether the support
- // points are equidistant.
+ // Check whether the support points are equidistant.
for (unsigned int j=0; j<n_points; j++)
if (std::fabs(points[j] - (double)j/this->degree) > 1e-15)
{
else
{
- // Check whether the support
- // points come from QGaussLobatto.
+ // Check whether the support points come from QGaussLobatto.
const QGaussLobatto<1> points_gl(n_points);
type = true;
for (unsigned int j=0; j<n_points; j++)
}
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::
-get_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
- FullMatrix<double> &interpolation_matrix) const
-{
- // this is only implemented, if the
- // source FE is also a
- // Q element
- typedef FE_Q<dim,spacedim> FEQ;
- typedef FiniteElement<dim,spacedim> FEL;
-
- AssertThrow ((x_source_fe.get_name().find ("FE_Q<") == 0)
- ||
- (dynamic_cast<const FEQ *>(&x_source_fe) != 0),
- typename FEL::
- ExcInterpolationNotImplemented());
-
- Assert (interpolation_matrix.m() == this->dofs_per_cell,
- ExcDimensionMismatch (interpolation_matrix.m(),
- this->dofs_per_cell));
- Assert (interpolation_matrix.n() == x_source_fe.dofs_per_cell,
- ExcDimensionMismatch (interpolation_matrix.m(),
- x_source_fe.dofs_per_cell));
-
- // ok, source is a Q element, so
- // we will be able to do the work
- const FE_Q<dim,spacedim> &source_fe
- = dynamic_cast<const FE_Q<dim,spacedim>&>(x_source_fe);
-
- // compute the interpolation
- // matrices in much the same way as
- // we do for the embedding matrices
- // from mother to child.
- FullMatrix<double> cell_interpolation (this->dofs_per_cell,
- this->dofs_per_cell);
- FullMatrix<double> source_interpolation (this->dofs_per_cell,
- source_fe.dofs_per_cell);
- FullMatrix<double> tmp (this->dofs_per_cell,
- source_fe.dofs_per_cell);
- for (unsigned int j=0; j<this->dofs_per_cell; ++j)
- {
- // read in a point on this
- // cell and evaluate the
- // shape functions there
- const Point<dim> p = this->unit_support_points[j];
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- cell_interpolation(j,i) = this->poly_space.compute_value (i, p);
-
- for (unsigned int i=0; i<source_fe.dofs_per_cell; ++i)
- source_interpolation(j,i) = source_fe.poly_space.compute_value (i, p);
- }
-
- // then compute the
- // interpolation matrix
- // for this coordinate
- // direction
- cell_interpolation.gauss_jordan ();
- cell_interpolation.mmult (interpolation_matrix,
- source_interpolation);
-
- const double eps = 2e-13*this->degree*dim;
-
- // cut off very small values
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
- if (std::fabs(interpolation_matrix(i,j)) < eps)
- interpolation_matrix(i,j) = 0.;
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- {
- double sum = 0.;
- for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
- sum += interpolation_matrix(i,j);
-
- Assert (std::fabs(sum-1) < eps, ExcInternalError());
- }
-}
-
-
-
-template <>
-void
-FE_Q<1>::
-get_face_interpolation_matrix (const FiniteElement<1,1> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- Assert (false, ExcImpossibleInDim(1));
-}
-
-
-
-template <>
-void
-FE_Q<1>::
-get_subface_interpolation_matrix (const FiniteElement<1,1> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- Assert (false, ExcImpossibleInDim(1));
-}
-
-
-template <>
-void
-FE_Q<1,2>::
-get_face_interpolation_matrix (const FiniteElement<1,2> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,2> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q<1,2>::
-get_subface_interpolation_matrix (const FiniteElement<1,2> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,2> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q<1,3>::
-get_face_interpolation_matrix (const FiniteElement<1,3> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,3> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q<1,3>::
-get_subface_interpolation_matrix (const FiniteElement<1,3> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,3> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::
-get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
- FullMatrix<double> &interpolation_matrix) const
-{
- // this is only implemented, if the
- // source FE is also a
- // Q element
- typedef FE_Q<dim,spacedim> FEQ;
- typedef FiniteElement<dim,spacedim> FEL;
- AssertThrow ((x_source_fe.get_name().find ("FE_Q<") == 0)
- ||
- (dynamic_cast<const FEQ *>(&x_source_fe) != 0),
- typename FEL::
- ExcInterpolationNotImplemented());
-
- Assert (interpolation_matrix.n() == this->dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.n(),
- this->dofs_per_face));
- Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.m(),
- x_source_fe.dofs_per_face));
-
- // ok, source is a Q element, so
- // we will be able to do the work
- const FE_Q<dim,spacedim> &source_fe
- = dynamic_cast<const FE_Q<dim,spacedim>&>(x_source_fe);
-
- // Make sure, that the element,
- // for which the DoFs should be
- // constrained is the one with
- // the higher polynomial degree.
- // Actually the procedure will work
- // also if this assertion is not
- // satisfied. But the matrices
- // produced in that case might
- // lead to problems in the
- // hp procedures, which use this
- // method.
- Assert (this->dofs_per_face <= source_fe.dofs_per_face,
- typename FEL::
- ExcInterpolationNotImplemented ());
-
- // generate a quadrature
- // with the unit support points.
- // This is later based as a
- // basis for the QProjector,
- // which returns the support
- // points on the face.
- Quadrature<dim-1> quad_face_support (source_fe.get_unit_face_support_points ());
-
- // Rule of thumb for FP accuracy,
- // that can be expected for a
- // given polynomial degree.
- // This value is used to cut
- // off values close to zero.
- const double eps = 2e-13*this->degree*(dim-1);
-
- // compute the interpolation
- // matrix by simply taking the
- // value at the support points.
-//TODO: Verify that all faces are the same with respect to
-// these support points. Furthermore, check if something has to
-// be done for the face orientation flag in 3D.
- const Quadrature<dim> face_quadrature
- = QProjector<dim>::project_to_face (quad_face_support, 0);
- for (unsigned int i=0; i<source_fe.dofs_per_face; ++i)
- {
- const Point<dim> &p = face_quadrature.point (i);
-
- for (unsigned int j=0; j<this->dofs_per_face; ++j)
- {
- double matrix_entry = this->shape_value (this->face_to_cell_index(j, 0), p);
-
- // Correct the interpolated
- // value. I.e. if it is close
- // to 1 or 0, make it exactly
- // 1 or 0. Unfortunately, this
- // is required to avoid problems
- // with higher order elements.
- if (std::fabs (matrix_entry - 1.0) < eps)
- matrix_entry = 1.0;
- if (std::fabs (matrix_entry) < eps)
- matrix_entry = 0.0;
-
- interpolation_matrix(i,j) = matrix_entry;
- }
- }
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int j=0; j<source_fe.dofs_per_face; ++j)
- {
- double sum = 0.;
-
- for (unsigned int i=0; i<this->dofs_per_face; ++i)
- sum += interpolation_matrix(j,i);
-
- Assert (std::fabs(sum-1) < 2e-13*this->degree*(dim-1),
- ExcInternalError());
- }
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::
-get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
- const unsigned int subface,
- FullMatrix<double> &interpolation_matrix) const
-{
- Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.m(),
- x_source_fe.dofs_per_face));
-
- // see if source is a Q element
- if (const FE_Q<dim,spacedim> *source_fe
- = dynamic_cast<const FE_Q<dim,spacedim> *>(&x_source_fe))
- {
- // have this test in here since
- // a table of size 2x0 reports
- // its size as 0x0
- Assert (interpolation_matrix.n() == this->dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.n(),
- this->dofs_per_face));
-
- // Make sure, that the element,
- // for which the DoFs should be
- // constrained is the one with
- // the higher polynomial degree.
- // Actually the procedure will work
- // also if this assertion is not
- // satisfied. But the matrices
- // produced in that case might
- // lead to problems in the
- // hp procedures, which use this
- // method.
- Assert (this->dofs_per_face <= source_fe->dofs_per_face,
- (typename FiniteElement<dim,spacedim>::
- ExcInterpolationNotImplemented ()));
-
- // generate a point on this
- // cell and evaluate the
- // shape functions there
- const Quadrature<dim-1>
- quad_face_support (source_fe->get_unit_face_support_points ());
-
- // Rule of thumb for FP accuracy,
- // that can be expected for a
- // given polynomial degree.
- // This value is used to cut
- // off values close to zero.
- double eps = 2e-13*this->degree*(dim-1);
-
- // compute the interpolation
- // matrix by simply taking the
- // value at the support points.
-//TODO: Verify that all faces are the same with respect to
-// these support points. Furthermore, check if something has to
-// be done for the face orientation flag in 3D.
- const Quadrature<dim> subface_quadrature
- = QProjector<dim>::project_to_subface (quad_face_support, 0, subface);
- for (unsigned int i=0; i<source_fe->dofs_per_face; ++i)
- {
- const Point<dim> &p = subface_quadrature.point (i);
-
- for (unsigned int j=0; j<this->dofs_per_face; ++j)
- {
- double matrix_entry = this->shape_value (this->face_to_cell_index(j, 0), p);
-
- // Correct the interpolated
- // value. I.e. if it is close
- // to 1 or 0, make it exactly
- // 1 or 0. Unfortunately, this
- // is required to avoid problems
- // with higher order elements.
- if (std::fabs (matrix_entry - 1.0) < eps)
- matrix_entry = 1.0;
- if (std::fabs (matrix_entry) < eps)
- matrix_entry = 0.0;
-
- interpolation_matrix(i,j) = matrix_entry;
- }
- }
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int j=0; j<source_fe->dofs_per_face; ++j)
- {
- double sum = 0.;
-
- for (unsigned int i=0; i<this->dofs_per_face; ++i)
- sum += interpolation_matrix(j,i);
-
- Assert (std::fabs(sum-1) < 2e-13*this->degree*dim,
- ExcInternalError());
- }
- }
- else if (dynamic_cast<const FE_Nothing<dim> *>(&x_source_fe) != 0)
- {
- // nothing to do here, the
- // FE_Nothing has no degrees of
- // freedom anyway
- }
- else
- AssertThrow (false,
- (typename FiniteElement<dim,spacedim>::
- ExcInterpolationNotImplemented()));
-}
-
-
-
-template <int dim, int spacedim>
-bool
-FE_Q<dim,spacedim>::hp_constraints_are_implemented () const
-{
- return true;
-}
-
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q<dim,spacedim>::
-hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
-{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Qs or if the other one is an
- // FE_Nothing. in the first case,
- // there should be exactly one
- // single DoF of each FE at a
- // vertex, and they should have
- // identical value
- if (dynamic_cast<const FE_Q<dim,spacedim>*>(&fe_other) != 0)
- {
- return
- std::vector<std::pair<unsigned int, unsigned int> >
- (1, std::make_pair (0U, 0U));
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
- else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
-}
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q<dim,spacedim>::
-hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
-{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Qs or if the other one is an
- // FE_Nothing
- if (const FE_Q<dim,spacedim> *fe_q_other = dynamic_cast<const FE_Q<dim,spacedim>*>(&fe_other))
- {
- // dofs are located along lines, so two
- // dofs are identical if they are
- // located at identical positions. if
- // we had only equidistant points, we
- // could simple check for similarity
- // like (i+1)*q == (j+1)*p, but we
- // might have other support points
- // (e.g. Gauss-Lobatto
- // points). Therefore, read the points
- // in unit_support_points for the first
- // coordinate direction. We take the
- // lexicographic ordering of the points
- // in the first direction (i.e.,
- // x-direction), which we access
- // between index 1 and p-1 (index 0 and
- // p are vertex dofs).
- const unsigned int p = this->degree;
- const unsigned int q = fe_q_other->degree;
-
- std::vector<std::pair<unsigned int, unsigned int> > identities;
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
- const std::vector<unsigned int> &index_map_inverse_other=
- fe_q_other->poly_space.get_numbering_inverse();
-
- for (unsigned int i=0; i<p-1; ++i)
- for (unsigned int j=0; j<q-1; ++j)
- if (std::fabs(this->unit_support_points[index_map_inverse[i+1]][0]-
- fe_q_other->unit_support_points[index_map_inverse_other[j+1]][0])
- < 1e-14)
- identities.push_back (std::make_pair(i,j));
-
- return identities;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
- else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
-}
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q<dim,spacedim>::
-hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
-{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Qs or if the other one is an
- // FE_Nothing
- if (const FE_Q<dim,spacedim> *fe_q_other = dynamic_cast<const FE_Q<dim,spacedim>*>(&fe_other))
- {
- // this works exactly like the line
- // case above, except that now we have
- // to have two indices i1, i2 and j1,
- // j2 to characterize the dofs on the
- // face of each of the finite
- // elements. since they are ordered
- // lexicographically along the first
- // line and we have a tensor product,
- // the rest is rather straightforward
- const unsigned int p = this->degree;
- const unsigned int q = fe_q_other->degree;
-
- std::vector<std::pair<unsigned int, unsigned int> > identities;
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
- const std::vector<unsigned int> &index_map_inverse_other=
- fe_q_other->poly_space.get_numbering_inverse();
-
- for (unsigned int i1=0; i1<p-1; ++i1)
- for (unsigned int i2=0; i2<p-1; ++i2)
- for (unsigned int j1=0; j1<q-1; ++j1)
- for (unsigned int j2=0; j2<q-1; ++j2)
- if ((std::fabs(this->unit_support_points[index_map_inverse[i1+1]][0]-
- fe_q_other->unit_support_points[index_map_inverse_other[j1+1]][0])
- < 1e-14)
- &&
- (std::fabs(this->unit_support_points[index_map_inverse[i2+1]][0]-
- fe_q_other->unit_support_points[index_map_inverse_other[j2+1]][0])
- < 1e-14))
- identities.push_back (std::make_pair(i1*(p-1)+i2,
- j1*(q-1)+j2));
-
- return identities;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
- else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
-}
-
-
-
-template <int dim, int spacedim>
-FiniteElementDomination::Domination
-FE_Q<dim,spacedim>::
-compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const
-{
- if (const FE_Q<dim,spacedim> *fe_q_other
- = dynamic_cast<const FE_Q<dim,spacedim>*>(&fe_other))
- {
- if (this->degree < fe_q_other->degree)
- return FiniteElementDomination::this_element_dominates;
- else if (this->degree == fe_q_other->degree)
- return FiniteElementDomination::either_element_can_dominate;
- else
- return FiniteElementDomination::other_element_dominates;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom and it is
- // typically used in a context
- // where we don't require any
- // continuity along the
- // interface
- return FiniteElementDomination::no_requirements;
- }
-
- Assert (false, ExcNotImplemented());
- return FiniteElementDomination::neither_element_dominates;
-}
-
-
-//---------------------------------------------------------------------------
-// Auxiliary functions
-//---------------------------------------------------------------------------
-
-
-
-template <int dim, int spacedim>
-void FE_Q<dim,spacedim>::initialize_unit_support_points ()
-{
- // number of points: (degree+1)^dim
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<dim; ++i)
- n *= this->degree+1;
-
- this->unit_support_points.resize(n);
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
-
- const double step = 1./this->degree;
- Point<dim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((dim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((dim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = ix * step;
- if (dim>1)
- p(1) = iy * step;
- if (dim>2)
- p(2) = iz * step;
-
- this->unit_support_points[index_map_inverse[k++]] = p;
- }
-}
-
-
-
-template <int dim, int spacedim>
-void FE_Q<dim,spacedim>::initialize_unit_support_points (const Quadrature<1> &points)
-{
- // number of points: (degree+1)^dim
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<dim; ++i)
- n *= this->degree+1;
-
- this->unit_support_points.resize(n);
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
-
- Quadrature<dim> support_quadrature(points);
-
- Point<dim> p;
-
- for (unsigned int k=0; k<n ; k++)
- {
- this->unit_support_points[index_map_inverse[k]] = support_quadrature.point(k);
- }
-}
-
-
-
-template <>
-void FE_Q<1>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q<1>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q<1,2>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q<1,2>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q<1,3>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q<1,3>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <int dim, int spacedim>
-void FE_Q<dim,spacedim>::initialize_unit_face_support_points ()
-{
- const unsigned int codim = dim-1;
- this->unit_face_support_points.resize(Utilities::fixed_power<codim>(this->degree+1));
-
- const std::vector<unsigned int> &face_index_map_inverse=
- FE_Q_Helper::invert_numbering(face_index_map);
-
- const double step = 1./this->degree;
- Point<codim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((codim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((codim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = ix * step;
- if (codim>1)
- p(1) = iy * step;
- if (codim>2)
- p(2) = iz * step;
-
- this->unit_face_support_points[face_index_map_inverse[k++]] = p;
- }
-}
-
-
-
-template <int dim, int spacedim>
-void FE_Q<dim,spacedim>::initialize_unit_face_support_points (const Quadrature<1> &points)
-{
- const unsigned int codim = dim-1;
- this->unit_face_support_points.resize(Utilities::fixed_power<codim>(this->degree+1));
-
- const std::vector< Point<1> > &edge = points.get_points();
-
- const std::vector<unsigned int> &face_index_map_inverse=
- FE_Q_Helper::invert_numbering(face_index_map);
-
- Point<codim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((codim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((codim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = edge[ix](0);
- if (codim>1)
- p(1) = edge[iy](0);
- if (codim>2)
- p(2) = edge[iz](0);
-
- this->unit_face_support_points[face_index_map_inverse[k++]] = p;
- }
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::initialize_quad_dof_index_permutation ()
-{
- // general template for 1D and 2D, do nothing
-}
-
-
-
-template <>
-void
-FE_Q<3>::initialize_quad_dof_index_permutation ()
-{
- Assert (adjust_quad_dof_index_for_face_orientation_table.n_elements()==8*this->dofs_per_quad,
- ExcInternalError());
-
- const unsigned int n=this->degree-1;
- Assert(n*n==this->dofs_per_quad, ExcInternalError());
-
- // alias for the table to fill
- Table<2,int> &data=this->adjust_quad_dof_index_for_face_orientation_table;
-
- // the dofs on a face are connected to a n x
- // n matrix. for example, for degree==4 we
- // have the following dofs on a quad
-
- // ___________
- // | |
- // | 6 7 8 |
- // | |
- // | 3 4 5 |
- // | |
- // | 0 1 2 |
- // |___________|
- //
- // we have dof_no=i+n*j with index i in
- // x-direction and index j in y-direction
- // running from 0 to n-1. to extract i and j
- // we can use i=dof_no%n and j=dof_no/n. i
- // and j can then be used to construct the
- // rotated and mirrored numbers.
-
-
- for (unsigned int local=0; local<this->dofs_per_quad; ++local)
- // face support points are in lexicographic
- // ordering with x running fastest. invert
- // that (y running fastest)
- {
- unsigned int i=local%n,
- j=local/n;
-
- // face_orientation=false, face_flip=false, face_rotation=false
- data(local,0)=j + i *n - local;
- // face_orientation=false, face_flip=false, face_rotation=true
- data(local,1)=i + (n-1-j)*n - local;
- // face_orientation=false, face_flip=true, face_rotation=false
- data(local,2)=(n-1-j) + (n-1-i)*n - local;
- // face_orientation=false, face_flip=true, face_rotation=true
- data(local,3)=(n-1-i) + j *n - local;
- // face_orientation=true, face_flip=false, face_rotation=false
- data(local,4)=0;
- // face_orientation=true, face_flip=false, face_rotation=true
- data(local,5)=j + (n-1-i)*n - local;
- // face_orientation=true, face_flip=true, face_rotation=false
- data(local,6)=(n-1-i) + (n-1-j)*n - local;
- // face_orientation=true, face_flip=true, face_rotation=true
- data(local,7)=(n-1-j) + i *n - local;
- }
-
- // aditionally initialize reordering of line
- // dofs
- for (unsigned int i=0; i<this->dofs_per_line; ++i)
- this->adjust_line_dof_index_for_line_orientation_table[i]=this->dofs_per_line-1-i - i;
-}
-
-
-
-
-template <int dim, int spacedim>
-std::vector<unsigned int>
-FE_Q<dim,spacedim>::get_dpo_vector(const unsigned int deg)
-{
- std::vector<unsigned int> dpo(dim+1, 1U);
- for (unsigned int i=1; i<dpo.size(); ++i)
- dpo[i]=dpo[i-1]*(deg-1);
- return dpo;
-}
-
-
-template <int dim, int spacedim>
-std::vector<unsigned int>
-FE_Q<dim,spacedim>::face_lexicographic_to_hierarchic_numbering (const unsigned int degree)
-{
- const FiniteElementData<dim-1> face_data(FE_Q<dim-1>::get_dpo_vector(degree),1,degree);
- std::vector<unsigned int> face_renumber (face_data.dofs_per_cell);
- FETools::lexicographic_to_hierarchic_numbering (face_data, face_renumber);
- return face_renumber;
-}
-
-
-
-template <>
-std::vector<unsigned int>
-FE_Q<1>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-template <>
-std::vector<unsigned int>
-FE_Q<1,2>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-
-template <>
-std::vector<unsigned int>
-FE_Q<1,3>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::initialize_constraints ()
-{
- QTrapez<1> trapez;
- QIterated<1> points (trapez,this->degree);
- Implementation::initialize_constraints (points, *this);
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::initialize_embedding ()
-{
- // compute the interpolation matrices in
- // much the same way as we do for the
- // constraints. it's actually simpler
- // here, since we don't have this weird
- // renumbering stuff going on. The trick
- // is again that we the interpolation
- // matrix is formed by a permutation of
- // the indices of the cell matrix. The
- // value eps is used a threshold to
- // decide when certain evaluations of the
- // Lagrange polynomials are zero or one.
- const double eps = 1e-15*this->degree*dim;
-
-#ifdef DEBUG
- // in DEBUG mode, check that the evaluation of
- // support points in the current numbering
- // gives the identity operation
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- {
- Assert (std::fabs (1.-this->poly_space.compute_value
- (i, this->unit_support_points[i])) < eps,
- ExcInternalError());
- for (unsigned int j=0; j<this->dofs_per_cell; ++j)
- if (j!=i)
- Assert (std::fabs (this->poly_space.compute_value
- (i, this->unit_support_points[j])) < eps,
- ExcInternalError());
- }
-#endif
-
- // to efficiently evaluate the polynomial at
- // the subcell, make use of the tensor product
- // structure of this element and only evaluate
- // 1D information from the polynomial. This
- // makes the cost of this function almost
- // negligible also for high order elements
- const unsigned int dofs1d = this->degree+1;
- std::vector<Table<2,double> >
- subcell_evaluations (dim, Table<2,double>(dofs1d, dofs1d));
- const std::vector<unsigned int> &index_map_inverse =
- this->poly_space.get_numbering_inverse();
-
- // recreate 1D polynomials
- std::vector<Polynomials::Polynomial<double> > poly_space1d =
- FE_Q_Helper::generate_poly_space1d (this->unit_support_points,
- index_map_inverse, dofs1d);
-
- // helper value: step size how to walk through
- // diagonal and how many points we have left
- // apart from the first dimension
- unsigned int step_size_diag = 0;
- {
- unsigned int factor = 1;
- for (unsigned int d=0; d<dim; ++d)
- {
- step_size_diag += factor;
- factor *= dofs1d;
- }
- }
-
- // next evaluate the functions
- // for the different refinement
- // cases.
- for (unsigned int ref=0; ref<RefinementCase<dim>::isotropic_refinement; ++ref)
- for (unsigned int child=0; child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref+1)); ++child)
- {
- // go through the points in diagonal to
- // capture variation in all directions
- // simultaneously
- for (unsigned int j=0; j<dofs1d; ++j)
- {
- const unsigned int diag_comp = index_map_inverse[j*step_size_diag];
- const Point<dim> p_subcell = this->unit_support_points[diag_comp];
- const Point<dim> p_cell =
- GeometryInfo<dim>::child_to_cell_coordinates (p_subcell, child,
- RefinementCase<dim>(ref+1));
- for (unsigned int i=0; i<dofs1d; ++i)
- for (unsigned int d=0; d<dim; ++d)
- {
- const double cell_value = poly_space1d[i].value (p_cell[d]);
-
- // cut off values that are too
- // small. note that we have here Lagrange
- // interpolation functions, so they
- // should be zero at almost all points,
- // and one at the others, at least on the
- // subcells. so set them to their exact
- // values
- //
- // the actual cut-off value is somewhat
- // fuzzy, but it works for
- // 2e-13*degree*dim (see above), which
- // is kind of reasonable given that we
- // compute the values of the polynomials
- // via an degree-step recursion and then
- // multiply the 1d-values. this gives us
- // a linear growth in degree*dim, times a
- // small constant.
- //
- // the embedding matrix is given by
- // applying the inverse of the subcell
- // matrix on the cell_interpolation
- // matrix. since the subcell matrix is
- // actually only a permutation vector,
- // all we need to do is to switch the
- // rows we write the data into. moreover,
- // cut off very small values here
- if (std::fabs(cell_value) < eps)
- subcell_evaluations[d](j,i) = 0;
- else
- subcell_evaluations[d](j,i) = cell_value;
- }
- }
-
- // now expand from 1D info. block innermost
- // dimension (x_0) in order to avoid difficult
- // checks at innermost loop
- unsigned int j_indices[dim];
- FE_Q_Helper::zero_indices<dim> (j_indices);
- for (unsigned int j=0; j<this->dofs_per_cell; j+=dofs1d)
- {
- unsigned int i_indices[dim];
- FE_Q_Helper::zero_indices<dim> (i_indices);
- for (unsigned int i=0; i<this->dofs_per_cell; i+=dofs1d)
- {
- double val_extra_dim = 1.;
- for (unsigned int d=1; d<dim; ++d)
- val_extra_dim *= subcell_evaluations[d](j_indices[d-1],
- i_indices[d-1]);
-
- // innermost sum where we actually
- // compute. the same as
- // this->prolongation[ref][child](j,i) =
- // this->poly_space.compute_value (i, p_cell);
- for (unsigned int jj=0; jj<dofs1d; ++jj)
- {
- const unsigned int j_ind = index_map_inverse[j+jj];
- for (unsigned int ii=0; ii<dofs1d; ++ii)
- this->prolongation[ref][child](j_ind,index_map_inverse[i+ii])
- = val_extra_dim * subcell_evaluations[0](jj,ii);
- }
-
- // update indices that denote the tensor
- // product position. a bit fuzzy and therefore
- // not done for innermost x_0 direction
- FE_Q_Helper::increment_indices<dim> (i_indices, dofs1d);
- }
- Assert (i_indices[dim-1] == 1, ExcInternalError());
- FE_Q_Helper::increment_indices<dim> (j_indices, dofs1d);
- }
-
- // and make sure that the row sum is
- // 1. this must be so since for this
- // element, the shape functions add up to
- // one
-#ifdef DEBUG
- for (unsigned int row=0; row<this->dofs_per_cell; ++row)
- {
- double sum = 0;
- for (unsigned int col=0; col<this->dofs_per_cell; ++col)
- sum += this->prolongation[ref][child](row,col);
- Assert (std::fabs(sum-1.) < eps, ExcInternalError());
- }
-#endif
- }
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q<dim,spacedim>::initialize_restriction ()
-{
- // for these Lagrange interpolation
- // polynomials, construction of the
- // restriction matrices is relatively
- // simple. the reason is that the
- // interpolation points on the mother cell
- // are (except for the case with arbitrary
- // nonequidistant nodes) always also
- // interpolation points for some shape
- // function on one or the other child,
- // because we have chosen equidistant
- // Lagrange interpolation points for the
- // polynomials
- //
- // so the only thing we have to
- // find out is: for each shape
- // function on the mother cell,
- // which is the child cell
- // (possibly more than one) on
- // which it is located, and which
- // is the corresponding shape
- // function there. rather than
- // doing it for the shape functions
- // on the mother cell, we take the
- // interpolation points there
- //
- // note that the interpolation
- // point of a shape function can be
- // on the boundary between
- // subcells. in that case,
- // restriction from children to
- // mother may produce two or more
- // entries for a dof on the mother
- // cell. however, this doesn't
- // hurt: since the element is
- // continuous, the contribution
- // from each child should yield the
- // same result, and since the
- // element is non-additive we just
- // overwrite one value (compute one
- // one child) by the same value
- // (compute on a later child), so
- // we don't have to care about this
-
- const double eps = 1e-15*this->degree*dim;
- const std::vector<unsigned int> &index_map_inverse =
- this->poly_space.get_numbering_inverse();
-
- // recreate 1D polynomials for faster
- // evaluation of polynomial
- const unsigned int dofs1d = this->degree+1;
- std::vector<Polynomials::Polynomial<double> > poly_space1d =
- FE_Q_Helper::generate_poly_space1d (this->unit_support_points,
- index_map_inverse, dofs1d);
- std::vector<Tensor<1,dim> > evaluations1d (dofs1d);
-
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- {
- unsigned int mother_dof = index_map_inverse[i];
- const Point<dim> p_cell = this->unit_support_points[mother_dof];
-
- // then find the children on
- // which the interpolation
- // point is located
- for (unsigned int ref=RefinementCase<dim>::cut_x; ref<=RefinementCase<dim>::isotropic_refinement; ++ref)
- for (unsigned int child=0; child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref)); ++child)
- {
- // check whether this
- // interpolation point is
- // inside this child cell
- const Point<dim> p_subcell
- = GeometryInfo<dim>::cell_to_child_coordinates (p_cell, child, RefinementCase<dim>(ref));
- if (GeometryInfo<dim>::is_inside_unit_cell (p_subcell))
- {
- // same logic as in initialize_embedding to
- // evaluate the polynomial faster than from
- // the tensor product: since we evaluate all
- // polynomials, it is much faster to just
- // compute the 1D values for all polynomials
- // before and then get the dim-data.
- for (unsigned int j=0; j<dofs1d; ++j)
- for (unsigned int d=0; d<dim; ++d)
- evaluations1d[j][d] = poly_space1d[j].value (p_subcell[d]);
- unsigned int j_indices[dim];
- FE_Q_Helper::zero_indices<dim> (j_indices);
- double sum_check = 0;
- for (unsigned int j = 0; j<this->dofs_per_cell; j += dofs1d)
- {
- double val_extra_dim = 1.;
- for (unsigned int d=1; d<dim; ++d)
- val_extra_dim *= evaluations1d[j_indices[d-1]][d];
- for (unsigned int jj=0; jj<dofs1d; ++jj)
- {
-
- // find the child shape
- // function(s) corresponding
- // to this point. Usually
- // this is just one function;
- // however, when we use FE_Q
- // on arbitrary nodes a
- // parent support point might
- // not be a child support
- // point, and then we will
- // get more than one nonzero
- // value per row. Still, the
- // values should sum up to 1
- const double val
- = val_extra_dim * evaluations1d[jj][0];
- const unsigned int child_dof =
- index_map_inverse[j+jj];
- if (std::fabs (val-1.) < eps)
- this->restriction[ref-1][child](mother_dof,child_dof)=1.;
- else if (std::fabs(val) > eps)
- this->restriction[ref-1][child](mother_dof,child_dof)=val;
- sum_check += val;
- }
- FE_Q_Helper::increment_indices<dim> (j_indices, dofs1d);
- }
- Assert (std::fabs(sum_check-1) < eps,
- ExcInternalError());
- }
- }
- }
-}
-
-
-
-//---------------------------------------------------------------------------
-// Data field initialization
-//---------------------------------------------------------------------------
-
-
-template <>
-bool
-FE_Q<1>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-
-template <>
-bool
-FE_Q<1,2>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-template <>
-bool
-FE_Q<1,3>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-
-template <int dim, int spacedim>
-bool
-FE_Q<dim,spacedim>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<dim>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<dim>::faces_per_cell));
-
-
- // first, special-case interior
- // shape functions, since they
- // have no support no-where on
- // the boundary
- if (((dim==2) && (shape_index>=this->first_quad_index))
- ||
- ((dim==3) && (shape_index>=this->first_hex_index)))
- return false;
-
- // let's see whether this is a
- // vertex
- if (shape_index < this->first_line_index)
- {
- // for Q elements, there is
- // one dof per vertex, so
- // shape_index==vertex_number. check
- // whether this vertex is
- // on the given face. thus,
- // for each face, give a
- // list of vertices
- const unsigned int vertex_no = shape_index;
- Assert (vertex_no < GeometryInfo<dim>::vertices_per_cell,
- ExcInternalError());
-
- for (unsigned int v=0; v<GeometryInfo<dim>::vertices_per_face; ++v)
- if (GeometryInfo<dim>::face_to_cell_vertices(face_index, v) == vertex_no)
- return true;
-
- return false;
- }
- else if (shape_index < this->first_quad_index)
- // ok, dof is on a line
- {
- const unsigned int line_index
- = (shape_index - this->first_line_index) / this->dofs_per_line;
- Assert (line_index < GeometryInfo<dim>::lines_per_cell,
- ExcInternalError());
-
- // in 2d, the line is the
- // face, so get the line
- // index
- if (dim == 2)
- return (line_index == face_index);
- else if (dim == 3)
- {
- // see whether the
- // given line is on the
- // given face.
- for (unsigned int l=0; l<GeometryInfo<dim>::lines_per_face; ++l)
- if (GeometryInfo<3>::face_to_cell_lines(face_index, l) == line_index)
- return true;
-
- return false;
- }
- else
- Assert (false, ExcNotImplemented());
- }
- else if (shape_index < this->first_hex_index)
- // dof is on a quad
- {
- const unsigned int quad_index
- = (shape_index - this->first_quad_index) / this->dofs_per_quad;
- Assert (static_cast<signed int>(quad_index) <
- static_cast<signed int>(GeometryInfo<dim>::quads_per_cell),
- ExcInternalError());
-
- // in 2d, cell bubble are
- // zero on all faces. but
- // we have treated this
- // case above already
- Assert (dim != 2, ExcInternalError());
-
- // in 3d,
- // quad_index=face_index
- if (dim == 3)
- return (quad_index == face_index);
- else
- Assert (false, ExcNotImplemented());
- }
- else
- // dof on hex
- {
- // can only happen in 3d,
- // but this case has
- // already been covered
- // above
- Assert (false, ExcNotImplemented());
- return false;
- }
-
- // we should not have gotten here
- Assert (false, ExcInternalError());
- return false;
-}
-
-
-
-template <int dim, int spacedim>
-std::size_t
-FE_Q<dim,spacedim>::memory_consumption () const
-{
- return (FiniteElement<dim,spacedim>::memory_consumption() +
- MemoryConsumption::memory_consumption (face_index_map));
-}
-
-
-
// explicit instantiations
#include "fe_q.inst"
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
+#include <deal.II/base/quadrature.h>
+#include <deal.II/base/qprojector.h>
+#include <deal.II/base/template_constraints.h>
+#include <deal.II/base/tensor_product_polynomials.h>
+#include <deal.II/base/tensor_product_polynomials_const.h>
+#include <deal.II/base/polynomials_piecewise.h>
+#include <deal.II/fe/fe_q_base.h>
+#include <deal.II/fe/fe_nothing.h>
+#include <deal.II/fe/fe_tools.h>
+#include <deal.II/base/quadrature_lib.h>
+
+#include <vector>
+#include <sstream>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+namespace FE_Q_Helper
+{
+ namespace
+ {
+ // get the renumbering for faces
+ template <int dim>
+ inline
+ std::vector<unsigned int>
+ face_lexicographic_to_hierarchic_numbering (const unsigned int degree)
+ {
+ std::vector<unsigned int> dpo(dim, 1U);
+ for (unsigned int i=1; i<dpo.size(); ++i)
+ dpo[i]=dpo[i-1]*(degree-1);
+ const dealii::FiniteElementData<dim-1> face_data(dpo,1,degree);
+ std::vector<unsigned int> face_renumber (face_data.dofs_per_cell);
+ FETools::lexicographic_to_hierarchic_numbering (face_data, face_renumber);
+ return face_renumber;
+ }
+
+
+
+ // in initialize_embedding() and initialize_restriction(), want to undo
+ // tensorization on inner loops for performance reasons. this clears a
+ // dim-array
+ template <int dim>
+ inline
+ void
+ zero_indices (unsigned int indices[dim])
+ {
+ for (unsigned int d=0; d<dim; ++d)
+ indices[d] = 0;
+ }
+
+
+
+ // in initialize_embedding() and initialize_restriction(), want to undo
+ // tensorization on inner loops for performance reasons. this increments
+ // tensor product indices
+ template <int dim>
+ inline
+ void
+ increment_indices (unsigned int indices[dim],
+ const unsigned int dofs1d)
+ {
+ ++indices[0];
+ for (int d=0; d<dim-1; ++d)
+ if (indices[d]==dofs1d)
+ {
+ indices[d] = 0;
+ indices[d+1]++;
+ }
+ }
+ }
+}
+
+
+
+/**
+ * A class with the same purpose as the similarly named class of the
+ * Triangulation class. See there for more information.
+ */
+template <class POLY, int xdim, int xspacedim>
+struct FE_Q_Base<POLY,xdim,xspacedim>::Implementation
+{
+ /**
+ * Initialize the hanging node constraints matrices. Called from the
+ * constructor in case the finite element is based on quadrature points.
+ */
+ template <int spacedim>
+ static
+ void initialize_constraints (const std::vector<Point<1> > &,
+ FE_Q_Base<POLY,1,spacedim> &)
+ {
+ // no constraints in 1d
+ }
+
+
+ template <int spacedim>
+ static
+ void initialize_constraints (const std::vector<Point<1> > &points,
+ FE_Q_Base<POLY,2,spacedim> &fe)
+ {
+ const unsigned int dim = 2;
+
+ // restricted to each face, the traces of the shape functions is an
+ // element of P_{k} (in 2d), or Q_{k} (in 3d), where k is the degree of
+ // the element. from this, we interpolate between mother and cell face.
+
+ // the interpolation process works as follows: on each subface, we want
+ // that finite element solutions from both sides coincide. i.e. if a and b
+ // are expansion coefficients for the shape functions from both sides, we
+ // seek a relation between a and b such that
+ // sum_j a_j phi^c_j(x) == sum_j b_j phi_j(x)
+ // for all points x on the interface. here, phi^c_j are the shape
+ // functions on the small cell on one side of the face, and phi_j those on
+ // the big cell on the other side. To get this relation, it suffices to
+ // look at a sufficient number of points for which this has to hold. if
+ // there are n functions, then we need n evaluation points, and we choose
+ // them equidistantly.
+ //
+ // we obtain the matrix system
+ // A a == B b
+ // where
+ // A_ij = phi^c_j(x_i)
+ // B_ij = phi_j(x_i)
+ // and the relation we are looking for is
+ // a = A^-1 B b
+ //
+ // for the special case of Lagrange interpolation polynomials, A_ij
+ // reduces to delta_ij, and
+ // a_i = B_ij b_j
+ // Hence, interface_constraints(i,j)=B_ij.
+ //
+ // for the general case, where we don't have Lagrange interpolation
+ // polynomials, this is a little more complicated. Then we would evaluate
+ // at a number of points and invert the interpolation matrix A.
+ //
+ // Note, that we build up these matrices for all subfaces at once, rather
+ // than considering them separately. the reason is that we finally will
+ // want to have them in this order anyway, as this is the format we need
+ // inside deal.II
+
+ // In the following the points x_i are constructed in following order
+ // (n=degree-1)
+ // *----------*---------*
+ // 1..n 0 n+1..2n
+ // i.e. first the midpoint of the line, then the support points on subface
+ // 0 and on subface 1
+ std::vector<Point<dim-1> > constraint_points;
+ // Add midpoint
+ constraint_points.push_back (Point<dim-1> (0.5));
+
+ if (fe.degree>1)
+ {
+ const unsigned int n=fe.degree-1;
+ const double step=1./fe.degree;
+ // subface 0
+ for (unsigned int i=1; i<=n; ++i)
+ constraint_points.push_back (
+ GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),0));
+ // subface 1
+ for (unsigned int i=1; i<=n; ++i)
+ constraint_points.push_back (
+ GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),1));
+ }
+
+ // Now construct relation between destination (child) and source (mother)
+ // dofs.
+
+ fe.interface_constraints
+ .TableBase<2,double>::reinit (fe.interface_constraints_size());
+
+ // use that the element evaluates to 1 at index 0 and along the line at
+ // zero
+ const std::vector<unsigned int> &index_map_inverse =
+ fe.poly_space.get_numbering_inverse();
+ const std::vector<unsigned int> face_index_map =
+ FE_Q_Helper::face_lexicographic_to_hierarchic_numbering<2>(fe.degree);
+ Assert(fe.poly_space.compute_value(index_map_inverse[0],Point<2>()) == 1.,
+ ExcInternalError());
+
+ for (unsigned int i=0; i<constraint_points.size(); ++i)
+ for (unsigned int j=0; j<fe.degree+1; ++j)
+ {
+ Point<2> p;
+ p[0] = constraint_points[i](0);
+ fe.interface_constraints(i,face_index_map[j]) =
+ fe.poly_space.compute_value(index_map_inverse[j], p);
+
+ // if the value is small up to round-off, then simply set it to zero
+ // to avoid unwanted fill-in of the constraint matrices (which would
+ // then increase the number of other DoFs a constrained DoF would
+ // couple to)
+ if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
+ fe.interface_constraints(i,j) = 0;
+ }
+ }
+
+
+ template <int spacedim>
+ static
+ void initialize_constraints (const std::vector<Point<1> > &points,
+ FE_Q_Base<POLY,3,spacedim> &fe)
+ {
+ const unsigned int dim = 3;
+
+ // For a detailed documentation of the interpolation see the
+ // FE_Q_Base<2>::initialize_constraints function.
+
+ // In the following the points x_i are constructed in the order as
+ // described in the documentation of the FiniteElement class (fe_base.h),
+ // i.e.
+ // *--15--4--16--*
+ // | | |
+ // 10 19 6 20 12
+ // | | |
+ // 1--7---0--8---2
+ // | | |
+ // 9 17 5 18 11
+ // | | |
+ // *--13--3--14--*
+ std::vector<Point<dim-1> > constraint_points;
+
+ // Add midpoint
+ constraint_points.push_back (Point<dim-1> (0.5, 0.5));
+
+ // Add midpoints of lines of "mother-face"
+ constraint_points.push_back (Point<dim-1> (0, 0.5));
+ constraint_points.push_back (Point<dim-1> (1, 0.5));
+ constraint_points.push_back (Point<dim-1> (0.5, 0));
+ constraint_points.push_back (Point<dim-1> (0.5, 1));
+
+ if (fe.degree>1)
+ {
+ const unsigned int n=fe.degree-1;
+ const double step=1./fe.degree;
+ std::vector<Point<dim-2> > line_support_points(n);
+ for (unsigned int i=0; i<n; ++i)
+ line_support_points[i](0)=(i+1)*step;
+ Quadrature<dim-2> qline(line_support_points);
+
+ // auxiliary points in 2d
+ std::vector<Point<dim-1> > p_line(n);
+
+ // Add nodes of lines interior in the "mother-face"
+
+ // line 5: use line 9
+ QProjector<dim-1>::project_to_subface(qline, 0, 0, p_line);
+ for (unsigned int i=0; i<n; ++i)
+ constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
+ // line 6: use line 10
+ QProjector<dim-1>::project_to_subface(qline, 0, 1, p_line);
+ for (unsigned int i=0; i<n; ++i)
+ constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
+ // line 7: use line 13
+ QProjector<dim-1>::project_to_subface(qline, 2, 0, p_line);
+ for (unsigned int i=0; i<n; ++i)
+ constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
+ // line 8: use line 14
+ QProjector<dim-1>::project_to_subface(qline, 2, 1, p_line);
+ for (unsigned int i=0; i<n; ++i)
+ constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
+
+ // DoFs on bordering lines lines 9-16
+ for (unsigned int face=0; face<GeometryInfo<dim-1>::faces_per_cell; ++face)
+ for (unsigned int subface=0;
+ subface<GeometryInfo<dim-1>::max_children_per_face; ++subface)
+ {
+ QProjector<dim-1>::project_to_subface(qline, face, subface, p_line);
+ constraint_points.insert(constraint_points.end(),
+ p_line.begin(), p_line.end());
+ }
+
+ // Create constraints for interior nodes
+ std::vector<Point<dim-1> > inner_points(n*n);
+ for (unsigned int i=0, iy=1; iy<=n; ++iy)
+ for (unsigned int ix=1; ix<=n; ++ix)
+ inner_points[i++] = Point<dim-1> (ix*step, iy*step);
+
+ // at the moment do this for isotropic face refinement only
+ for (unsigned int child=0;
+ child<GeometryInfo<dim-1>::max_children_per_cell; ++child)
+ for (unsigned int i=0; i<inner_points.size(); ++i)
+ constraint_points.push_back (
+ GeometryInfo<dim-1>::child_to_cell_coordinates(inner_points[i], child));
+ }
+
+ // Now construct relation between destination (child) and source (mother)
+ // dofs.
+ const unsigned int pnts=(fe.degree+1)*(fe.degree+1);
+
+ // use that the element evaluates to 1 at index 0 and along the line at
+ // zero
+ const std::vector<unsigned int> &index_map_inverse =
+ fe.poly_space.get_numbering_inverse();
+ const std::vector<unsigned int> face_index_map =
+ FE_Q_Helper::face_lexicographic_to_hierarchic_numbering<3>(fe.degree);
+ Assert(fe.poly_space.compute_value(index_map_inverse[0],Point<3>()) == 1.,
+ ExcInternalError());
+
+ fe.interface_constraints
+ .TableBase<2,double>::reinit (fe.interface_constraints_size());
+
+ for (unsigned int i=0; i<constraint_points.size(); ++i)
+ {
+ const double interval = (double) (fe.degree * 2);
+ bool mirror[dim - 1];
+ Point<dim> constraint_point;
+
+ // Eliminate FP errors in constraint points. Due to their origin, they
+ // must all be fractions of the unit interval. If we have polynomial
+ // degree 4, the refined element has 8 intervals. Hence the
+ // coordinates must be 0, 0.125, 0.25, 0.375 etc. Now the coordinates
+ // of the constraint points will be multiplied by the inverse of the
+ // interval size (in the example by 8). After that the coordinates
+ // must be integral numbers. Hence a normal truncation is performed
+ // and the coordinates will be scaled back. The equal treatment of all
+ // coordinates should eliminate any FP errors.
+ for (unsigned int k=0; k<dim-1; ++k)
+ {
+ const int coord_int =
+ static_cast<int> (constraint_points[i](k) * interval + 0.25);
+ constraint_point(k) = 1.*coord_int / interval;
+
+ // The following lines of code should eliminate the problems with
+ // the Constraint-Matrix, which appeared for P>=4. The
+ // ConstraintMatrix class complained about different constraints
+ // for the same entry of the Constraint-Matrix. Actually this
+ // difference could be attributed to FP errors, as it was in the
+ // range of 1.0e-16. These errors originate in the loss of
+ // symmetry in the FP approximation of the shape-functions.
+ // Considering a 3rd order shape function in 1D, we have
+ // N0(x)=N3(1-x) and N1(x)=N2(1-x). For higher order polynomials
+ // the FP approximations of the shape functions do not satisfy
+ // these equations any more! Thus in the following code
+ // everything is computed in the interval x \in [0..0.5], which is
+ // sufficient to express all values that could come out from a
+ // computation of any shape function in the full interval
+ // [0..1]. If x > 0.5 the computation is done for 1-x with the
+ // shape function N_{p-n} instead of N_n. Hence symmetry is
+ // preserved and everything works fine...
+ //
+ // For a different explanation of the problem, see the discussion
+ // in the FiniteElement class for constraint matrices in 3d.
+ mirror[k] = (constraint_point(k) > 0.5);
+ if (mirror[k])
+ constraint_point(k) = 1.0 - constraint_point(k);
+ }
+
+ for (unsigned int j=0; j<pnts; ++j)
+ {
+ unsigned int indices[2] = { j % (fe.degree+1), j / (fe.degree+1) };
+
+ for (unsigned int k = 0; k<2; ++k)
+ if (mirror[k])
+ indices[k] = fe.degree - indices[k];
+
+ const unsigned int
+ new_index = indices[1] * (fe.degree + 1) + indices[0];
+
+ fe.interface_constraints(i,face_index_map[j]) =
+ fe.poly_space.compute_value (index_map_inverse[new_index],
+ constraint_point);
+
+ // if the value is small up to round-off, then simply set it to
+ // zero to avoid unwanted fill-in of the constraint matrices
+ // (which would then increase the number of other DoFs a
+ // constrained DoF would couple to)
+ if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
+ fe.interface_constraints(i,j) = 0;
+ }
+ }
+ }
+};
+
+
+
+template <class POLY, int dim, int spacedim>
+FE_Q_Base<POLY,dim,spacedim>::FE_Q_Base (const POLY &poly_space,
+ const FiniteElementData<dim> &fe_data,
+ const std::vector<bool> &restriction_is_additive_flags)
+ :
+ FE_Poly<POLY,dim,spacedim>(poly_space, fe_data, restriction_is_additive_flags,
+ std::vector<ComponentMask>(1, std::vector<bool>(1,true)))
+{}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::initialize (const std::vector<Point<1> > &points)
+{
+ Assert (points[0][0] == 0,
+ ExcMessage ("The first support point has to be zero."));
+ Assert (points.back()[0] == 1,
+ ExcMessage ("The last support point has to be one."));
+
+ // distinguish q/q_dg0 case: need to be flexible enough to allow more
+ // degrees of freedom than there are FE_Q degrees of freedom for derived
+ // class FE_Q_DG0 that otherwise shares 95% of the code.
+ const unsigned int q_dofs_per_cell = Utilities::fixed_power<dim>(this->degree+1);
+ Assert(q_dofs_per_cell == this->dofs_per_cell ||
+ q_dofs_per_cell+1 == this->dofs_per_cell, ExcInternalError());
+
+ {
+ std::vector<unsigned int> renumber(q_dofs_per_cell);
+ const FiniteElementData<dim> fe(get_dpo_vector(this->degree),1,
+ this->degree);
+ FETools::hierarchic_to_lexicographic_numbering (fe, renumber);
+ if (this->dofs_per_cell > q_dofs_per_cell)
+ renumber.push_back(q_dofs_per_cell);
+ this->poly_space.set_numbering(renumber);
+ }
+
+ // finally fill in support points on cell and face
+ initialize_unit_support_points (points);
+ initialize_unit_face_support_points (points);
+
+ // reinit constraints
+ initialize_constraints (points);
+
+ // Reinit the vectors of restriction and prolongation matrices to the right
+ // sizes and compute the matrices
+ this->reinit_restriction_and_prolongation_matrices();
+ initialize_embedding();
+ initialize_restriction();
+
+ this->initialize_quad_dof_index_permutation();
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::
+get_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
+ FullMatrix<double> &interpolation_matrix) const
+{
+ // this is only implemented, if the source FE is also a Q element
+ typedef FE_Q_Base<POLY,dim,spacedim> FEQ;
+ typedef FiniteElement<dim,spacedim> FEL;
+
+ AssertThrow (dynamic_cast<const FEQ *>(&x_source_fe) != 0,
+ typename FEL::ExcInterpolationNotImplemented());
+
+ Assert (interpolation_matrix.m() == this->dofs_per_cell,
+ ExcDimensionMismatch (interpolation_matrix.m(),
+ this->dofs_per_cell));
+ Assert (interpolation_matrix.n() == x_source_fe.dofs_per_cell,
+ ExcDimensionMismatch (interpolation_matrix.m(),
+ x_source_fe.dofs_per_cell));
+
+ // ok, source is a Q element, so we will be able to do the work
+ const FE_Q_Base<POLY,dim,spacedim> &source_fe
+ = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>&>(x_source_fe);
+
+ // compute the interpolation matrices in much the same way as we do for the
+ // embedding matrices from mother to child.
+ FullMatrix<double> cell_interpolation (this->dofs_per_cell,
+ this->dofs_per_cell);
+ FullMatrix<double> source_interpolation (this->dofs_per_cell,
+ source_fe.dofs_per_cell);
+
+ // only evaluate Q dofs
+ const unsigned int q_dofs_per_cell = Utilities::fixed_power<dim>(this->degree+1);
+ const unsigned int source_q_dofs_per_cell = Utilities::fixed_power<dim>(source_fe.degree+1);
+ for (unsigned int j=0; j<q_dofs_per_cell; ++j)
+ {
+ // read in a point on this cell and evaluate the shape functions there
+ const Point<dim> p = this->unit_support_points[j];
+ for (unsigned int i=0; i<q_dofs_per_cell; ++i)
+ cell_interpolation(j,i) = this->poly_space.compute_value (i, p);
+
+ for (unsigned int i=0; i<source_q_dofs_per_cell; ++i)
+ source_interpolation(j,i) = source_fe.poly_space.compute_value (i, p);
+ }
+
+ // for FE_Q_DG0, add one last row of identity
+ if (q_dofs_per_cell < this->dofs_per_cell)
+ {
+ AssertDimension(source_q_dofs_per_cell+1, source_fe.dofs_per_cell);
+ cell_interpolation(q_dofs_per_cell, q_dofs_per_cell) = 1.;
+ source_interpolation(q_dofs_per_cell, source_q_dofs_per_cell) = 1.;
+ }
+
+ // then compute the interpolation matrix for this coordinate direction
+ cell_interpolation.gauss_jordan ();
+ cell_interpolation.mmult (interpolation_matrix, source_interpolation);
+
+ const double eps = 2e-13*this->degree*dim;
+
+ // cut off very small values
+ for (unsigned int i=0; i<this->dofs_per_cell; ++i)
+ for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
+ if (std::fabs(interpolation_matrix(i,j)) < eps)
+ interpolation_matrix(i,j) = 0.;
+
+ // make sure that the row sum of each of the matrices is 1 at this
+ // point. this must be so since the shape functions sum up to 1
+ for (unsigned int i=0; i<this->dofs_per_cell; ++i)
+ {
+ double sum = 0.;
+ for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
+ sum += interpolation_matrix(i,j);
+
+ Assert (std::fabs(sum-1) < eps, ExcInternalError());
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::
+get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
+ FullMatrix<double> &interpolation_matrix) const
+{
+ Assert (dim > 1, ExcImpossibleInDim(1));
+
+ // this is only implemented, if the source FE is also a Q element
+ typedef FE_Q_Base<POLY,dim,spacedim> FEQ;
+ typedef FiniteElement<dim,spacedim> FEL;
+ AssertThrow (dynamic_cast<const FEQ *>(&x_source_fe) != 0,
+ typename FEL::ExcInterpolationNotImplemented());
+
+ Assert (interpolation_matrix.n() == this->dofs_per_face,
+ ExcDimensionMismatch (interpolation_matrix.n(),
+ this->dofs_per_face));
+ Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
+ ExcDimensionMismatch (interpolation_matrix.m(),
+ x_source_fe.dofs_per_face));
+
+ // ok, source is a Q element, so we will be able to do the work
+ const FE_Q_Base<POLY,dim,spacedim> &source_fe
+ = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>&>(x_source_fe);
+
+ // Make sure that the element for which the DoFs should be constrained is
+ // the one with the higher polynomial degree. Actually the procedure will
+ // work also if this assertion is not satisfied. But the matrices produced
+ // in that case might lead to problems in the hp procedures, which use this
+ // method.
+ Assert (this->dofs_per_face <= source_fe.dofs_per_face,
+ typename FEL::
+ ExcInterpolationNotImplemented ());
+
+ // generate a quadrature with the unit support points. This is later based
+ // as a basis for the QProjector, which returns the support points on the
+ // face.
+ Quadrature<dim-1> quad_face_support (source_fe.get_unit_face_support_points ());
+
+ // Rule of thumb for FP accuracy, that can be expected for a given
+ // polynomial degree. This value is used to cut off values close to zero.
+ const double eps = 2e-13*this->degree*(dim-1);
+
+ // compute the interpolation matrix by simply taking the value at the
+ // support points.
+//TODO: Verify that all faces are the same with respect to
+// these support points. Furthermore, check if something has to
+// be done for the face orientation flag in 3D.
+ const Quadrature<dim> face_quadrature
+ = QProjector<dim>::project_to_face (quad_face_support, 0);
+ for (unsigned int i=0; i<source_fe.dofs_per_face; ++i)
+ {
+ const Point<dim> &p = face_quadrature.point (i);
+
+ for (unsigned int j=0; j<this->dofs_per_face; ++j)
+ {
+ double matrix_entry = this->poly_space.compute_value (this->face_to_cell_index(j, 0), p);
+
+ // Correct the interpolated value. I.e. if it is close to 1 or 0,
+ // make it exactly 1 or 0. Unfortunately, this is required to avoid
+ // problems with higher order elements.
+ if (std::fabs (matrix_entry - 1.0) < eps)
+ matrix_entry = 1.0;
+ if (std::fabs (matrix_entry) < eps)
+ matrix_entry = 0.0;
+
+ interpolation_matrix(i,j) = matrix_entry;
+ }
+ }
+
+ // make sure that the row sum of each of the matrices is 1 at this
+ // point. this must be so since the shape functions sum up to 1
+ for (unsigned int j=0; j<source_fe.dofs_per_face; ++j)
+ {
+ double sum = 0.;
+
+ for (unsigned int i=0; i<this->dofs_per_face; ++i)
+ sum += interpolation_matrix(j,i);
+
+ Assert (std::fabs(sum-1) < 2e-13*this->degree*(dim-1),
+ ExcInternalError());
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::
+get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
+ const unsigned int subface,
+ FullMatrix<double> &interpolation_matrix) const
+{
+ Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
+ ExcDimensionMismatch (interpolation_matrix.m(),
+ x_source_fe.dofs_per_face));
+
+ // see if source is a Q element
+ if (const FE_Q_Base<POLY,dim,spacedim> *source_fe
+ = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim> *>(&x_source_fe))
+ {
+ // have this test in here since a table of size 2x0 reports its size as
+ // 0x0
+ Assert (interpolation_matrix.n() == this->dofs_per_face,
+ ExcDimensionMismatch (interpolation_matrix.n(),
+ this->dofs_per_face));
+
+ // Make sure that the element for which the DoFs should be constrained
+ // is the one with the higher polynomial degree. Actually the procedure
+ // will work also if this assertion is not satisfied. But the matrices
+ // produced in that case might lead to problems in the hp procedures,
+ // which use this method.
+ Assert (this->dofs_per_face <= source_fe->dofs_per_face,
+ (typename FiniteElement<dim,spacedim>::
+ ExcInterpolationNotImplemented ()));
+
+ // generate a point on this cell and evaluate the shape functions there
+ const Quadrature<dim-1>
+ quad_face_support (source_fe->get_unit_face_support_points ());
+
+ // Rule of thumb for FP accuracy, that can be expected for a given
+ // polynomial degree. This value is used to cut off values close to
+ // zero.
+ double eps = 2e-13*this->degree*(dim-1);
+
+ // compute the interpolation matrix by simply taking the value at the
+ // support points.
+//TODO: Verify that all faces are the same with respect to
+// these support points. Furthermore, check if something has to
+// be done for the face orientation flag in 3D.
+ const Quadrature<dim> subface_quadrature
+ = QProjector<dim>::project_to_subface (quad_face_support, 0, subface);
+ for (unsigned int i=0; i<source_fe->dofs_per_face; ++i)
+ {
+ const Point<dim> &p = subface_quadrature.point (i);
+
+ for (unsigned int j=0; j<this->dofs_per_face; ++j)
+ {
+ double matrix_entry = this->shape_value (this->face_to_cell_index(j, 0), p);
+
+ // Correct the interpolated value. I.e. if it is close to 1 or
+ // 0, make it exactly 1 or 0. Unfortunately, this is required to
+ // avoid problems with higher order elements.
+ if (std::fabs (matrix_entry - 1.0) < eps)
+ matrix_entry = 1.0;
+ if (std::fabs (matrix_entry) < eps)
+ matrix_entry = 0.0;
+
+ interpolation_matrix(i,j) = matrix_entry;
+ }
+ }
+
+ // make sure that the row sum of each of the matrices is 1 at this
+ // point. this must be so since the shape functions sum up to 1
+ for (unsigned int j=0; j<source_fe->dofs_per_face; ++j)
+ {
+ double sum = 0.;
+
+ for (unsigned int i=0; i<this->dofs_per_face; ++i)
+ sum += interpolation_matrix(j,i);
+
+ Assert (std::fabs(sum-1) < 2e-13*this->degree*dim,
+ ExcInternalError());
+ }
+ }
+ else if (dynamic_cast<const FE_Nothing<dim> *>(&x_source_fe) != 0)
+ {
+ // nothing to do here, the FE_Nothing has no degrees of freedom anyway
+ }
+ else
+ AssertThrow (false,(typename FiniteElement<dim,spacedim>::
+ ExcInterpolationNotImplemented()));
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+bool
+FE_Q_Base<POLY,dim,spacedim>::hp_constraints_are_implemented () const
+{
+ return true;
+}
+
+
+
+
+template <class POLY, int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int> >
+FE_Q_Base<POLY,dim,spacedim>::
+hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
+{
+ // we can presently only compute these identities if both FEs are FE_Qs or
+ // if the other one is an FE_Nothing. in the first case, there should be
+ // exactly one single DoF of each FE at a vertex, and they should have
+ // identical value
+ if (dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>*>(&fe_other) != 0)
+ {
+ return
+ std::vector<std::pair<unsigned int, unsigned int> >
+ (1, std::make_pair (0U, 0U));
+ }
+ else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
+ {
+ // the FE_Nothing has no degrees of freedom, so there are no
+ // equivalencies to be recorded
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+ else
+ {
+ Assert (false, ExcNotImplemented());
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int> >
+FE_Q_Base<POLY,dim,spacedim>::
+hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
+{
+ // we can presently only compute these identities if both FEs are FE_Qs or
+ // if the other one is an FE_Nothing
+ if (const FE_Q_Base<POLY,dim,spacedim> *fe_q_other = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>*>(&fe_other))
+ {
+ // dofs are located along lines, so two dofs are identical if they are
+ // located at identical positions. if we had only equidistant points, we
+ // could simple check for similarity like (i+1)*q == (j+1)*p, but we
+ // might have other support points (e.g. Gauss-Lobatto
+ // points). Therefore, read the points in unit_support_points for the
+ // first coordinate direction. We take the lexicographic ordering of the
+ // points in the first direction (i.e., x-direction), which we access
+ // between index 1 and p-1 (index 0 and p are vertex dofs).
+ const unsigned int p = this->degree;
+ const unsigned int q = fe_q_other->degree;
+
+ std::vector<std::pair<unsigned int, unsigned int> > identities;
+
+ const std::vector<unsigned int> &index_map_inverse=
+ this->poly_space.get_numbering_inverse();
+ const std::vector<unsigned int> &index_map_inverse_other=
+ fe_q_other->poly_space.get_numbering_inverse();
+
+ for (unsigned int i=0; i<p-1; ++i)
+ for (unsigned int j=0; j<q-1; ++j)
+ if (std::fabs(this->unit_support_points[index_map_inverse[i+1]][0]-
+ fe_q_other->unit_support_points[index_map_inverse_other[j+1]][0])
+ < 1e-14)
+ identities.push_back (std::make_pair(i,j));
+
+ return identities;
+ }
+ else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
+ {
+ // the FE_Nothing has no degrees of freedom, so there are no
+ // equivalencies to be recorded
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+ else
+ {
+ Assert (false, ExcNotImplemented());
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+std::vector<std::pair<unsigned int, unsigned int> >
+FE_Q_Base<POLY,dim,spacedim>::
+hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
+{
+ // we can presently only compute these identities if both FEs are FE_Qs or
+ // if the other one is an FE_Nothing
+ if (const FE_Q_Base<POLY,dim,spacedim> *fe_q_other = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>*>(&fe_other))
+ {
+ // this works exactly like the line case above, except that now we have
+ // to have two indices i1, i2 and j1, j2 to characterize the dofs on the
+ // face of each of the finite elements. since they are ordered
+ // lexicographically along the first line and we have a tensor product,
+ // the rest is rather straightforward
+ const unsigned int p = this->degree;
+ const unsigned int q = fe_q_other->degree;
+
+ std::vector<std::pair<unsigned int, unsigned int> > identities;
+
+ const std::vector<unsigned int> &index_map_inverse=
+ this->poly_space.get_numbering_inverse();
+ const std::vector<unsigned int> &index_map_inverse_other=
+ fe_q_other->poly_space.get_numbering_inverse();
+
+ for (unsigned int i1=0; i1<p-1; ++i1)
+ for (unsigned int i2=0; i2<p-1; ++i2)
+ for (unsigned int j1=0; j1<q-1; ++j1)
+ for (unsigned int j2=0; j2<q-1; ++j2)
+ if ((std::fabs(this->unit_support_points[index_map_inverse[i1+1]][0]-
+ fe_q_other->unit_support_points[index_map_inverse_other[j1+1]][0])
+ < 1e-14)
+ &&
+ (std::fabs(this->unit_support_points[index_map_inverse[i2+1]][0]-
+ fe_q_other->unit_support_points[index_map_inverse_other[j2+1]][0])
+ < 1e-14))
+ identities.push_back (std::make_pair(i1*(p-1)+i2,
+ j1*(q-1)+j2));
+
+ return identities;
+ }
+ else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
+ {
+ // the FE_Nothing has no degrees of freedom, so there are no
+ // equivalencies to be recorded
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+ else
+ {
+ Assert (false, ExcNotImplemented());
+ return std::vector<std::pair<unsigned int, unsigned int> > ();
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+FiniteElementDomination::Domination
+FE_Q_Base<POLY,dim,spacedim>::
+compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const
+{
+ if (const FE_Q_Base<POLY,dim,spacedim> *fe_q_other
+ = dynamic_cast<const FE_Q_Base<POLY,dim,spacedim>*>(&fe_other))
+ {
+ if (this->degree < fe_q_other->degree)
+ return FiniteElementDomination::this_element_dominates;
+ else if (this->degree == fe_q_other->degree)
+ return FiniteElementDomination::either_element_can_dominate;
+ else
+ return FiniteElementDomination::other_element_dominates;
+ }
+ else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
+ {
+ // the FE_Nothing has no degrees of freedom and it is typically used in
+ // a context where we don't require any continuity along the interface
+ return FiniteElementDomination::no_requirements;
+ }
+
+ Assert (false, ExcNotImplemented());
+ return FiniteElementDomination::neither_element_dominates;
+}
+
+
+//---------------------------------------------------------------------------
+// Auxiliary functions
+//---------------------------------------------------------------------------
+
+
+
+template <class POLY, int dim, int spacedim>
+void FE_Q_Base<POLY,dim,spacedim>::initialize_unit_support_points (const std::vector<Point<1> > &points)
+{
+ const std::vector<unsigned int> &index_map_inverse=
+ this->poly_space.get_numbering_inverse();
+
+ Quadrature<1> support_1d(points);
+ Quadrature<dim> support_quadrature(support_1d);
+ this->unit_support_points.resize(support_quadrature.size());
+
+ for (unsigned int k=0; k<support_quadrature.size(); k++)
+ this->unit_support_points[index_map_inverse[k]] = support_quadrature.point(k);
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void FE_Q_Base<POLY,dim,spacedim>::initialize_unit_face_support_points (const std::vector<Point<1> > &points)
+{
+ // no faces in 1d, so nothing to do
+ if (dim == 1)
+ return;
+
+ const unsigned int codim = dim-1;
+ this->unit_face_support_points.resize(Utilities::fixed_power<codim>(this->degree+1));
+
+ // find renumbering of faces and assign from values of quadrature
+ std::vector<unsigned int> face_index_map =
+ FE_Q_Helper::face_lexicographic_to_hierarchic_numbering<dim>(this->degree);
+ Quadrature<1> support_1d(points);
+ Quadrature<codim> support_quadrature(support_1d);
+ this->unit_face_support_points.resize(support_quadrature.size());
+
+ for (unsigned int k=0; k<support_quadrature.size(); k++)
+ this->unit_face_support_points[face_index_map[k]] = support_quadrature.point(k);
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::initialize_quad_dof_index_permutation ()
+{
+ // for 1D and 2D, do nothing
+ if (dim < 3)
+ return;
+
+ Assert (this->adjust_quad_dof_index_for_face_orientation_table.n_elements()==8*this->dofs_per_quad,
+ ExcInternalError());
+
+ const unsigned int n=this->degree-1;
+ Assert(n*n==this->dofs_per_quad, ExcInternalError());
+
+ // alias for the table to fill
+ Table<2,int> &data=this->adjust_quad_dof_index_for_face_orientation_table;
+
+ // the dofs on a face are connected to a n x n matrix. for example, for
+ // degree==4 we have the following dofs on a quad
+
+ // ___________
+ // | |
+ // | 6 7 8 |
+ // | |
+ // | 3 4 5 |
+ // | |
+ // | 0 1 2 |
+ // |___________|
+ //
+ // we have dof_no=i+n*j with index i in x-direction and index j in
+ // y-direction running from 0 to n-1. to extract i and j we can use
+ // i=dof_no%n and j=dof_no/n. i and j can then be used to construct the
+ // rotated and mirrored numbers.
+
+
+ for (unsigned int local=0; local<this->dofs_per_quad; ++local)
+ // face support points are in lexicographic ordering with x running
+ // fastest. invert that (y running fastest)
+ {
+ unsigned int i=local%n,
+ j=local/n;
+
+ // face_orientation=false, face_flip=false, face_rotation=false
+ data(local,0)=j + i *n - local;
+ // face_orientation=false, face_flip=false, face_rotation=true
+ data(local,1)=i + (n-1-j)*n - local;
+ // face_orientation=false, face_flip=true, face_rotation=false
+ data(local,2)=(n-1-j) + (n-1-i)*n - local;
+ // face_orientation=false, face_flip=true, face_rotation=true
+ data(local,3)=(n-1-i) + j *n - local;
+ // face_orientation=true, face_flip=false, face_rotation=false
+ data(local,4)=0;
+ // face_orientation=true, face_flip=false, face_rotation=true
+ data(local,5)=j + (n-1-i)*n - local;
+ // face_orientation=true, face_flip=true, face_rotation=false
+ data(local,6)=(n-1-i) + (n-1-j)*n - local;
+ // face_orientation=true, face_flip=true, face_rotation=true
+ data(local,7)=(n-1-j) + i *n - local;
+ }
+
+ // aditionally initialize reordering of line dofs
+ for (unsigned int i=0; i<this->dofs_per_line; ++i)
+ this->adjust_line_dof_index_for_line_orientation_table[i]=this->dofs_per_line-1-i - i;
+}
+
+
+
+
+template <class POLY, int dim, int spacedim>
+std::vector<unsigned int>
+FE_Q_Base<POLY,dim,spacedim>::get_dpo_vector(const unsigned int deg)
+{
+ std::vector<unsigned int> dpo(dim+1, 1U);
+ for (unsigned int i=1; i<dpo.size(); ++i)
+ dpo[i]=dpo[i-1]*(deg-1);
+ return dpo;
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::initialize_constraints (const std::vector<Point<1> > &points)
+{
+ Implementation::initialize_constraints (points, *this);
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::initialize_embedding ()
+{
+ // distinguish q/q_dg0 case: only treat Q dofs first
+ const unsigned int q_dofs_per_cell = Utilities::fixed_power<dim>(this->degree+1);
+
+ // compute the interpolation matrices in much the same way as we do for the
+ // constraints. it's actually simpler here, since we don't have this weird
+ // renumbering stuff going on. The trick is again that we the interpolation
+ // matrix is formed by a permutation of the indices of the cell matrix. The
+ // value eps is used a threshold to decide when certain evaluations of the
+ // Lagrange polynomials are zero or one.
+ const double eps = 1e-15*this->degree*dim;
+
+#ifdef DEBUG
+ // in DEBUG mode, check that the evaluation of support points in the current
+ // numbering gives the identity operation
+ for (unsigned int i=0; i<q_dofs_per_cell; ++i)
+ {
+ Assert (std::fabs (1.-this->poly_space.compute_value
+ (i, this->unit_support_points[i])) < eps,
+ ExcInternalError());
+ for (unsigned int j=0; j<q_dofs_per_cell; ++j)
+ if (j!=i)
+ Assert (std::fabs (this->poly_space.compute_value
+ (i, this->unit_support_points[j])) < eps,
+ ExcInternalError());
+ }
+#endif
+
+ // to efficiently evaluate the polynomial at the subcell, make use of the
+ // tensor product structure of this element and only evaluate 1D information
+ // from the polynomial. This makes the cost of this function almost
+ // negligible also for high order elements
+ const unsigned int dofs1d = this->degree+1;
+ std::vector<Table<2,double> >
+ subcell_evaluations (dim, Table<2,double>(dofs1d, dofs1d));
+ const std::vector<unsigned int> &index_map_inverse =
+ this->poly_space.get_numbering_inverse();
+
+ // helper value: step size how to walk through diagonal and how many points
+ // we have left apart from the first dimension
+ unsigned int step_size_diag = 0;
+ {
+ unsigned int factor = 1;
+ for (unsigned int d=0; d<dim; ++d)
+ {
+ step_size_diag += factor;
+ factor *= dofs1d;
+ }
+ }
+
+ // next evaluate the functions for the different refinement cases.
+ for (unsigned int ref=0; ref<RefinementCase<dim>::isotropic_refinement; ++ref)
+ for (unsigned int child=0; child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref+1)); ++child)
+ {
+ // go through the points in diagonal to capture variation in all
+ // directions simultaneously
+ for (unsigned int j=0; j<dofs1d; ++j)
+ {
+ const unsigned int diag_comp = index_map_inverse[j*step_size_diag];
+ const Point<dim> p_subcell = this->unit_support_points[diag_comp];
+ const Point<dim> p_cell =
+ GeometryInfo<dim>::child_to_cell_coordinates (p_subcell, child,
+ RefinementCase<dim>(ref+1));
+ for (unsigned int i=0; i<dofs1d; ++i)
+ for (unsigned int d=0; d<dim; ++d)
+ {
+ // evaluate along line where only x is different from zero
+ Point<dim> point;
+ point[0] = p_cell[d];
+ const double cell_value =
+ this->poly_space.compute_value(index_map_inverse[i], point);
+
+ // cut off values that are too small. note that we have here
+ // Lagrange interpolation functions, so they should be zero
+ // at almost all points, and one at the others, at least on
+ // the subcells. so set them to their exact values
+ //
+ // the actual cut-off value is somewhat fuzzy, but it works
+ // for 2e-13*degree*dim (see above), which is kind of
+ // reasonable given that we compute the values of the
+ // polynomials via an degree-step recursion and then
+ // multiply the 1d-values. this gives us a linear growth in
+ // degree*dim, times a small constant.
+ //
+ // the embedding matrix is given by applying the inverse of
+ // the subcell matrix on the cell_interpolation
+ // matrix. since the subcell matrix is actually only a
+ // permutation vector, all we need to do is to switch the
+ // rows we write the data into. moreover, cut off very small
+ // values here
+ if (std::fabs(cell_value) < eps)
+ subcell_evaluations[d](j,i) = 0;
+ else
+ subcell_evaluations[d](j,i) = cell_value;
+ }
+ }
+
+ // now expand from 1D info. block innermost dimension (x_0) in order
+ // to avoid difficult checks at innermost loop
+ unsigned int j_indices[dim];
+ FE_Q_Helper::zero_indices<dim> (j_indices);
+ for (unsigned int j=0; j<q_dofs_per_cell; j+=dofs1d)
+ {
+ unsigned int i_indices[dim];
+ FE_Q_Helper::zero_indices<dim> (i_indices);
+ for (unsigned int i=0; i<q_dofs_per_cell; i+=dofs1d)
+ {
+ double val_extra_dim = 1.;
+ for (unsigned int d=1; d<dim; ++d)
+ val_extra_dim *= subcell_evaluations[d](j_indices[d-1],
+ i_indices[d-1]);
+
+ // innermost sum where we actually compute. the same as
+ // this->prolongation[ref][child](j,i) =
+ // this->poly_space.compute_value (i, p_cell);
+ for (unsigned int jj=0; jj<dofs1d; ++jj)
+ {
+ const unsigned int j_ind = index_map_inverse[j+jj];
+ for (unsigned int ii=0; ii<dofs1d; ++ii)
+ this->prolongation[ref][child](j_ind,index_map_inverse[i+ii])
+ = val_extra_dim * subcell_evaluations[0](jj,ii);
+ }
+
+ // update indices that denote the tensor product position. a
+ // bit fuzzy and therefore not done for innermost x_0
+ // direction
+ FE_Q_Helper::increment_indices<dim> (i_indices, dofs1d);
+ }
+ Assert (i_indices[dim-1] == 1, ExcInternalError());
+ FE_Q_Helper::increment_indices<dim> (j_indices, dofs1d);
+ }
+
+ // the discontinuous node is simply mapped on the discontinuous node
+ // on the child element
+ if (q_dofs_per_cell < this->dofs_per_cell)
+ this->prolongation[ref][child](q_dofs_per_cell,q_dofs_per_cell) = 1.;
+
+ // and make sure that the row sum is 1. this must be so since for this
+ // element, the shape functions add up to one
+#ifdef DEBUG
+ for (unsigned int row=0; row<this->dofs_per_cell; ++row)
+ {
+ double sum = 0;
+ for (unsigned int col=0; col<this->dofs_per_cell; ++col)
+ sum += this->prolongation[ref][child](row,col);
+ Assert (std::fabs(sum-1.) < eps, ExcInternalError());
+ }
+#endif
+ }
+}
+
+
+
+template <class POLY, int dim, int spacedim>
+void
+FE_Q_Base<POLY,dim,spacedim>::initialize_restriction ()
+{
+ // distinguish q/q_dg0 case
+ const unsigned int q_dofs_per_cell = Utilities::fixed_power<dim>(this->degree+1);
+
+ // for these Lagrange interpolation polynomials, construction of the
+ // restriction matrices is relatively simple. the reason is that the
+ // interpolation points on the mother cell are (except for the case with
+ // arbitrary nonequidistant nodes) always also interpolation points for some
+ // shape function on one or the other child, because we have chosen
+ // equidistant Lagrange interpolation points for the polynomials
+ //
+ // so the only thing we have to find out is: for each shape function on the
+ // mother cell, which is the child cell (possibly more than one) on which it
+ // is located, and which is the corresponding shape function there. rather
+ // than doing it for the shape functions on the mother cell, we take the
+ // interpolation points there
+ //
+ // note that the interpolation point of a shape function can be on the
+ // boundary between subcells. in that case, restriction from children to
+ // mother may produce two or more entries for a dof on the mother
+ // cell. however, this doesn't hurt: since the element is continuous, the
+ // contribution from each child should yield the same result, and since the
+ // element is non-additive we just overwrite one value (compute on one
+ // child) by the same value (compute on a later child), so we don't have to
+ // care about this
+
+ const double eps = 1e-15*this->degree*dim;
+ const std::vector<unsigned int> &index_map_inverse =
+ this->poly_space.get_numbering_inverse();
+
+ // recreate 1D polynomials for faster evaluation of polynomial
+ const unsigned int dofs1d = this->degree+1;
+ std::vector<Tensor<1,dim> > evaluations1d (dofs1d);
+
+ for (unsigned int i=0; i<q_dofs_per_cell; ++i)
+ {
+ unsigned int mother_dof = index_map_inverse[i];
+ const Point<dim> p_cell = this->unit_support_points[mother_dof];
+
+ // then find the children on which the interpolation point is located
+ for (unsigned int ref=RefinementCase<dim>::cut_x; ref<=RefinementCase<dim>::isotropic_refinement; ++ref)
+ for (unsigned int child=0; child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref)); ++child)
+ {
+ // check whether this interpolation point is inside this child
+ // cell
+ const Point<dim> p_subcell
+ = GeometryInfo<dim>::cell_to_child_coordinates (p_cell, child, RefinementCase<dim>(ref));
+ if (GeometryInfo<dim>::is_inside_unit_cell (p_subcell))
+ {
+ // same logic as in initialize_embedding to evaluate the
+ // polynomial faster than from the tensor product: since we
+ // evaluate all polynomials, it is much faster to just compute
+ // the 1D values for all polynomials before and then get the
+ // dim-data.
+ for (unsigned int j=0; j<dofs1d; ++j)
+ for (unsigned int d=0; d<dim; ++d)
+ {
+ Point<dim> point;
+ point[0] = p_subcell[d];
+ evaluations1d[j][d] =
+ this->poly_space.compute_value(index_map_inverse[j], point);
+ }
+ unsigned int j_indices[dim];
+ FE_Q_Helper::zero_indices<dim> (j_indices);
+ double sum_check = 0;
+ for (unsigned int j = 0; j<q_dofs_per_cell; j += dofs1d)
+ {
+ double val_extra_dim = 1.;
+ for (unsigned int d=1; d<dim; ++d)
+ val_extra_dim *= evaluations1d[j_indices[d-1]][d];
+ for (unsigned int jj=0; jj<dofs1d; ++jj)
+ {
+
+ // find the child shape function(s) corresponding to
+ // this point. Usually this is just one function;
+ // however, when we use FE_Q on arbitrary nodes a
+ // parent support point might not be a child support
+ // point, and then we will get more than one nonzero
+ // value per row. Still, the values should sum up to 1
+ const double val
+ = val_extra_dim * evaluations1d[jj][0];
+ const unsigned int child_dof =
+ index_map_inverse[j+jj];
+ if (std::fabs (val-1.) < eps)
+ this->restriction[ref-1][child](mother_dof,child_dof)=1.;
+ else if (std::fabs(val) > eps)
+ this->restriction[ref-1][child](mother_dof,child_dof)=val;
+ sum_check += val;
+ }
+ FE_Q_Helper::increment_indices<dim> (j_indices, dofs1d);
+ }
+ Assert (std::fabs(sum_check-1) < eps,
+ ExcInternalError());
+ }
+
+ // part for FE_Q_DG0
+ if (q_dofs_per_cell < this->dofs_per_cell)
+ this->restriction[ref-1][child]
+ (this->dofs_per_cell-1,this->dofs_per_cell-1) =
+ 1./(double)GeometryInfo<dim>::n_children
+ (RefinementCase<dim>(ref));
+ }
+ }
+}
+
+
+
+//---------------------------------------------------------------------------
+// Data field initialization
+//---------------------------------------------------------------------------
+
+
+template <class POLY, int dim, int spacedim>
+bool
+FE_Q_Base<POLY,dim,spacedim>::has_support_on_face (const unsigned int shape_index,
+ const unsigned int face_index) const
+{
+ Assert (shape_index < this->dofs_per_cell,
+ ExcIndexRange (shape_index, 0, this->dofs_per_cell));
+ Assert (face_index < GeometryInfo<dim>::faces_per_cell,
+ ExcIndexRange (face_index, 0, GeometryInfo<dim>::faces_per_cell));
+
+ // in 1d, things are simple. since there is only one degree of freedom per
+ // vertex in this class, the first is on vertex 0 (==face 0 in some sense),
+ // the second on face 1:
+ if (dim == 1)
+ return (((shape_index == 0) && (face_index == 0)) ||
+ ((shape_index == 1) && (face_index == 1)));
+
+ // first, special-case interior shape functions, since they have no support
+ // no-where on the boundary
+ if (((dim==2) && (shape_index>=this->first_quad_index))
+ ||
+ ((dim==3) && (shape_index>=this->first_hex_index)))
+ return false;
+
+ // let's see whether this is a vertex
+ if (shape_index < this->first_line_index)
+ {
+ // for Q elements, there is one dof per vertex, so
+ // shape_index==vertex_number. check whether this vertex is on the given
+ // face. thus, for each face, give a list of vertices
+ const unsigned int vertex_no = shape_index;
+ Assert (vertex_no < GeometryInfo<dim>::vertices_per_cell,
+ ExcInternalError());
+
+ for (unsigned int v=0; v<GeometryInfo<dim>::vertices_per_face; ++v)
+ if (GeometryInfo<dim>::face_to_cell_vertices(face_index, v) == vertex_no)
+ return true;
+
+ return false;
+ }
+ else if (shape_index < this->first_quad_index)
+ // ok, dof is on a line
+ {
+ const unsigned int line_index
+ = (shape_index - this->first_line_index) / this->dofs_per_line;
+ Assert (line_index < GeometryInfo<dim>::lines_per_cell,
+ ExcInternalError());
+
+ // in 2d, the line is the face, so get the line index
+ if (dim == 2)
+ return (line_index == face_index);
+ else if (dim == 3)
+ {
+ // silence compiler warning
+ const unsigned int lines_per_face =
+ dim == 3 ? GeometryInfo<dim>::lines_per_face : 1;
+ // see whether the given line is on the given face.
+ for (unsigned int l=0; l<lines_per_face; ++l)
+ if (GeometryInfo<3>::face_to_cell_lines(face_index, l) == line_index)
+ return true;
+
+ return false;
+ }
+ else
+ Assert (false, ExcNotImplemented());
+ }
+ else if (shape_index < this->first_hex_index)
+ // dof is on a quad
+ {
+ const unsigned int quad_index
+ = (shape_index - this->first_quad_index) / this->dofs_per_quad;
+ Assert (static_cast<signed int>(quad_index) <
+ static_cast<signed int>(GeometryInfo<dim>::quads_per_cell),
+ ExcInternalError());
+
+ // in 2d, cell bubble are zero on all faces. but we have treated this
+ // case above already
+ Assert (dim != 2, ExcInternalError());
+
+ // in 3d, quad_index=face_index
+ if (dim == 3)
+ return (quad_index == face_index);
+ else
+ Assert (false, ExcNotImplemented());
+ }
+ else
+ // dof on hex
+ {
+ // can only happen in 3d, but this case has already been covered above
+ Assert (false, ExcNotImplemented());
+ return false;
+ }
+
+ // we should not have gotten here
+ Assert (false, ExcInternalError());
+ return false;
+}
+
+
+
+// explicit instantiations
+#include "fe_q_base.inst"
+
+DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class FE_Q_Base<TensorProductPolynomials<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
+ template class FE_Q_Base<TensorProductPolynomialsConst<deal_II_dimension>, deal_II_dimension, deal_II_space_dimension>;
+ template class FE_Q_Base<TensorProductPolynomials<deal_II_dimension,Polynomials::PiecewisePolynomial<double> >, deal_II_dimension, deal_II_space_dimension>;
+#endif
+ }
+
+
+//---------------------------------------------------------------------------
+// $Id: fe_q.cc 28798 2013-03-07 17:00:02Z maier $
+// Version: $Name$
+//
+// Copyright (C) 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
#include <deal.II/base/quadrature.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/fe/fe_q_dg0.h>
#include <deal.II/fe/fe_nothing.h>
-#include <deal.II/fe/fe_tools.h>
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/dofs/dof_accessor.h>
DEAL_II_NAMESPACE_OPEN
-// namespace for some functions that are used in this file. they are
-// specific to numbering conventions used for the FE_Q element, and
-// are thus not very interesting to the outside world. we'd like to
-// simply put them into an anonymous namespace, but that triggers an
-// odd error with icc which can't compile this small snippet if the
-// function is static:
-// --------------------
-// template <int> struct int2type {};
-//
-// namespace {
-// static void SYMBOL (const int2type<1> & ) {}
-// }
-//
-// template <int dim, int spacedim> void g() {
-// SYMBOL(int2type<dim>());
-// }
-//
-// template void g<1>();
-// --------------------
-// the function needs to be static because of another icc bug, though.
-// work around this by packing everything into a namespace of its own
-// and have the anonymous namespace inside
-//
-// this is now intel icc issue 216082
-namespace FE_Q_DG0_Helper
-{
- namespace
- {
- // given a permutation array,
- // compute and return the inverse
- // permutation
-#ifdef DEAL_II_ANON_NAMESPACE_BUG
- static
-#endif
- inline
- std::vector<unsigned int>
- invert_numbering (const std::vector<unsigned int> &in)
- {
- std::vector<unsigned int> out (in.size());
- for (unsigned int i=0; i<in.size(); ++i)
- out[in[i]]=i;
- return out;
- }
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons. this clears a
- // dim-array
- template <int dim>
- inline
- void
- zero_indices (unsigned int indices[dim])
- {
- for (unsigned int d=0; d<dim; ++d)
- indices[d] = 0;
- }
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons. this increments tensor
- // product indices
- template <int dim>
- inline
- void
- increment_indices (unsigned int indices[dim],
- const unsigned int dofs1d)
- {
- ++indices[0];
- for (int d=0; d<dim-1; ++d)
- if (indices[d]==dofs1d)
- {
- indices[d] = 0;
- indices[d+1]++;
- }
- }
-
- // in initialize_embedding() and
- // initialize_restriction(), want to undo
- // tensorization on inner loops for
- // performance reasons, and we need to again
- // access 1D polynomials. This function
- // creates them from dim-dimensional support
- // points.
- template <int dim>
- inline
- std::vector<Polynomials::Polynomial<double> >
- generate_poly_space1d (const std::vector<Point<dim> > &unit_support_points,
- const std::vector<unsigned int> &index_map_inverse,
- const unsigned int dofs1d)
- {
- AssertDimension (Utilities::fixed_power<dim> (dofs1d)+1,
- unit_support_points.size());
- std::vector<Point<1> > points1d (dofs1d);
- for (unsigned int i=0; i<dofs1d; ++i)
- {
- const unsigned int j = index_map_inverse[i];
- points1d[i] = Point<1>(unit_support_points[j](0));
- for (unsigned int d=1; d<dim; ++d)
- Assert (unit_support_points[j][d] == 0.,
- ExcInternalError());
- }
- return Polynomials::generate_complete_Lagrange_basis (points1d);
- }
- }
-}
-
-
-
-/**
- * A class with the same purpose as the similarly named class of the
- * Triangulation class. See there for more information.
- */
-template <int xdim, int xspacedim>
-struct FE_Q_DG0<xdim,xspacedim>::Implementation
-{
- /**
- * Initialize the hanging node
- * constraints matrices. Called from the
- * constructor in case the finite element
- * is based on quadrature points.
- */
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &,
- FE_Q_DG0<1,spacedim> &)
- {
- // no constraints in 1d
- }
-
-
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &points,
- FE_Q_DG0<2,spacedim> &fe)
- {
- const unsigned int dim = 2;
-
- // restricted to each face, the
- // traces of the shape functions is
- // an element of P_{k} (in 2d), or
- // Q_{k} (in 3d), where k is the
- // degree of the element. from
- // this, we interpolate between
- // mother and cell face.
-
- // the interpolation process works
- // as follows: on each subface,
- // we want that finite element
- // solutions from both sides
- // coincide. i.e. if a and b are
- // expansion coefficients for the
- // shape functions from both sides,
- // we seek a relation between a and
- // b such that
- // sum_j a_j phi^c_j(x)
- // == sum_j b_j phi_j(x)
- // for all points x on the
- // interface. here, phi^c_j are the
- // shape functions on the small
- // cell on one side of the face,
- // and phi_j those on the big cell
- // on the other side. To get this
- // relation, it suffices to look at
- // a sufficient number of points
- // for which this has to hold. if
- // there are n functions, then we
- // need n evaluation points, and we
- // choose them equidistantly.
- //
- // we obtain the matrix system
- // A a == B b
- // where
- // A_ij = phi^c_j(x_i)
- // B_ij = phi_j(x_i)
- // and the relation we are looking
- // for is
- // a = A^-1 B b
- //
- // for the special case of Lagrange
- // interpolation polynomials, A_ij
- // reduces to delta_ij, and
- // a_i = B_ij b_j
- // Hence,
- // interface_constraints(i,j)=B_ij.
- //
- // for the general case, where we
- // don't have Lagrange
- // interpolation polynomials, this
- // is a little more
- // complicated. Then we would
- // evaluate at a number of points
- // and invert the interpolation
- // matrix A.
- //
- // Note, that we build up these
- // matrices for all subfaces at
- // once, rather than considering
- // them separately. the reason is
- // that we finally will want to
- // have them in this order anyway,
- // as this is the format we need
- // inside deal.II
-
- // In the following the points x_i
- // are constructed in following
- // order (n=degree-1)
- // *----------*---------*
- // 1..n 0 n+1..2n
- // i.e. first the midpoint of the
- // line, then the support points on
- // subface 0 and on subface 1
- std::vector<Point<dim-1> > constraint_points;
- // Add midpoint
- constraint_points.push_back (Point<dim-1> (0.5));
-
- if (fe.degree>1)
- {
- const unsigned int n=fe.degree-1;
- const double step=1./fe.degree;
- // subface 0
- for (unsigned int i=1; i<=n; ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),0));
- // subface 1
- for (unsigned int i=1; i<=n; ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(Point<dim-1>(i*step),1));
- }
-
- // Now construct relation between
- // destination (child) and source (mother)
- // dofs.
- const std::vector<Polynomials::Polynomial<double> > polynomials=
- Polynomials::generate_complete_Lagrange_basis(points.get_points());
-
- fe.interface_constraints
- .TableBase<2,double>::reinit (fe.interface_constraints_size());
-
- for (unsigned int i=0; i<constraint_points.size(); ++i)
- for (unsigned int j=0; j<fe.degree+1; ++j)
- {
- fe.interface_constraints(i,j) =
- polynomials[fe.face_index_map[j]].value (constraint_points[i](0));
-
- // if the value is small up
- // to round-off, then
- // simply set it to zero to
- // avoid unwanted fill-in
- // of the constraint
- // matrices (which would
- // then increase the number
- // of other DoFs a
- // constrained DoF would
- // couple to)
- if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
- fe.interface_constraints(i,j) = 0;
- }
- }
-
-
- template <int spacedim>
- static
- void initialize_constraints (const Quadrature<1> &points,
- FE_Q_DG0<3,spacedim> &fe)
- {
- const unsigned int dim = 3;
-
- // For a detailed documentation of
- // the interpolation see the
- // FE_Q<2>::initialize_constraints
- // function.
-
- // In the following the points x_i
- // are constructed in the order as
- // described in the documentation
- // of the FiniteElement class
- // (fe_base.h), i.e.
- // *--15--4--16--*
- // | | |
- // 10 19 6 20 12
- // | | |
- // 1--7---0--8---2
- // | | |
- // 9 17 5 18 11
- // | | |
- // *--13--3--14--*
- std::vector<Point<dim-1> > constraint_points;
-
- // Add midpoint
- constraint_points.push_back (Point<dim-1> (0.5, 0.5));
-
- // Add midpoints of lines of
- // "mother-face"
- constraint_points.push_back (Point<dim-1> (0, 0.5));
- constraint_points.push_back (Point<dim-1> (1, 0.5));
- constraint_points.push_back (Point<dim-1> (0.5, 0));
- constraint_points.push_back (Point<dim-1> (0.5, 1));
-
- if (fe.degree>1)
- {
- const unsigned int n=fe.degree-1;
- const double step=1./fe.degree;
- std::vector<Point<dim-2> > line_support_points(n);
- for (unsigned int i=0; i<n; ++i)
- line_support_points[i](0)=(i+1)*step;
- Quadrature<dim-2> qline(line_support_points);
-
- // auxiliary points in 2d
- std::vector<Point<dim-1> > p_line(n);
-
- // Add nodes of lines interior
- // in the "mother-face"
-
- // line 5: use line 9
- QProjector<dim-1>::project_to_subface(qline, 0, 0, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
- // line 6: use line 10
- QProjector<dim-1>::project_to_subface(qline, 0, 1, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0.5, 0));
- // line 7: use line 13
- QProjector<dim-1>::project_to_subface(qline, 2, 0, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
- // line 8: use line 14
- QProjector<dim-1>::project_to_subface(qline, 2, 1, p_line);
- for (unsigned int i=0; i<n; ++i)
- constraint_points.push_back (p_line[i] + Point<dim-1> (0, 0.5));
-
- // DoFs on bordering lines
- // lines 9-16
- for (unsigned int face=0; face<GeometryInfo<dim-1>::faces_per_cell; ++face)
- for (unsigned int subface=0;
- subface<GeometryInfo<dim-1>::max_children_per_face; ++subface)
- {
- QProjector<dim-1>::project_to_subface(qline, face, subface, p_line);
- constraint_points.insert(constraint_points.end(),
- p_line.begin(), p_line.end());
- }
-
- // Create constraints for
- // interior nodes
- std::vector<Point<dim-1> > inner_points(n*n);
- for (unsigned int i=0, iy=1; iy<=n; ++iy)
- for (unsigned int ix=1; ix<=n; ++ix)
- inner_points[i++] = Point<dim-1> (ix*step, iy*step);
-
- // at the moment do this for
- // isotropic face refinement only
- for (unsigned int child=0;
- child<GeometryInfo<dim-1>::max_children_per_cell; ++child)
- for (unsigned int i=0; i<inner_points.size(); ++i)
- constraint_points.push_back (
- GeometryInfo<dim-1>::child_to_cell_coordinates(inner_points[i], child));
- }
-
- // Now construct relation between
- // destination (child) and source (mother)
- // dofs.
- const unsigned int pnts=(fe.degree+1)*(fe.degree+1);
- const std::vector<Polynomials::Polynomial<double> > polynomial_basis=
- Polynomials::generate_complete_Lagrange_basis(points.get_points());
-
- const TensorProductPolynomialsConst<dim-1> face_polynomials(polynomial_basis);
-
- fe.interface_constraints
- .TableBase<2,double>::reinit (fe.interface_constraints_size());
-
- for (unsigned int i=0; i<constraint_points.size(); ++i)
- {
- const double interval = (double) (fe.degree * 2);
- bool mirror[dim - 1];
- Point<dim-1> constraint_point;
-
- // Eliminate FP errors in constraint
- // points. Due to their origin, they
- // must all be fractions of the unit
- // interval. If we have polynomial
- // degree 4, the refined element has 8
- // intervals. Hence the coordinates
- // must be 0, 0.125, 0.25, 0.375 etc.
- // Now the coordinates of the
- // constraint points will be multiplied
- // by the inverse of the interval size
- // (in the example by 8). After that
- // the coordinates must be integral
- // numbers. Hence a normal truncation
- // is performed and the coordinates
- // will be scaled back. The equal
- // treatment of all coordinates should
- // eliminate any FP errors.
- for (unsigned int k=0; k<dim-1; ++k)
- {
- const int coord_int =
- static_cast<int> (constraint_points[i](k) * interval + 0.25);
- constraint_point(k) = 1.*coord_int / interval;
-
- // The following lines of code
- // should eliminate the problems
- // with the Constraint-Matrix,
- // which appeared for P>=4. The
- // Constraint-Matrix class
- // complained about different
- // constraints for the same entry
- // of the Constraint-Matrix.
- // Actually this difference could
- // be attributed to FP errors, as
- // it was in the range of
- // 1.0e-16. These errors originate
- // in the loss of symmetry in the
- // FP approximation of the
- // shape-functions. Considering a
- // 3rd order shape function in 1D,
- // we have N0(x)=N3(1-x) and
- // N1(x)=N2(1-x). For higher order
- // polynomials the FP
- // approximations of the shape
- // functions do not satisfy these
- // equations any more! Thus in the
- // following code everything is
- // computed in the interval x \in
- // [0..0.5], which is sufficient to
- // express all values that could
- // come out from a computation of
- // any shape function in the full
- // interval [0..1]. If x > 0.5 the
- // computation is done for 1-x with
- // the shape function N_{p-n}
- // instead of N_n. Hence symmetry
- // is preserved and everything
- // works fine...
- //
- // For a different explanation of
- // the problem, see the discussion
- // in the FiniteElement class
- // for constraint matrices in 3d.
- mirror[k] = (constraint_point(k) > 0.5);
- if (mirror[k])
- constraint_point(k) = 1.0 - constraint_point(k);
- }
-
- for (unsigned int j=0; j<pnts; ++j)
- {
- unsigned int indices[2]
- = { fe.face_index_map[j] % (fe.degree + 1),
- fe.face_index_map[j] / (fe.degree + 1)
- };
-
- for (unsigned int k = 0; k<2; ++k)
- if (mirror[k])
- indices[k] = fe.degree - indices[k];
-
- const unsigned int
- new_index = indices[1] * (fe.degree + 1) + indices[0];
-
- fe.interface_constraints(i,j) =
- face_polynomials.compute_value (new_index, constraint_point);
-
- // if the value is small up
- // to round-off, then
- // simply set it to zero to
- // avoid unwanted fill-in
- // of the constraint
- // matrices (which would
- // then increase the number
- // of other DoFs a
- // constrained DoF would
- // couple to)
- if (std::fabs(fe.interface_constraints(i,j)) < 1e-14)
- fe.interface_constraints(i,j) = 0;
- }
- }
- }
-};
-
-
template <int dim, int spacedim>
FE_Q_DG0<dim,spacedim>::FE_Q_DG0 (const unsigned int degree)
:
- FE_Poly<TensorProductPolynomialsConst<dim>, dim, spacedim> (
+ FE_Q_Base<TensorProductPolynomialsConst<dim>, dim, spacedim> (
TensorProductPolynomialsConst<dim>(Polynomials::LagrangeEquidistant::generate_complete_basis(degree)),
FiniteElementData<dim>(get_dpo_vector(degree),
1, degree,
FiniteElementData<dim>::L2),
- get_riaf_vector(degree),
- std::vector<ComponentMask>(1ll , std::vector<bool>(1,true))),
- face_index_map(FE_Q_DG0_Helper::invert_numbering(
- face_lexicographic_to_hierarchic_numbering (degree)))
+ get_riaf_vector(degree))
{
Assert (degree > 0,
ExcMessage ("This element can only be used for polynomial degrees "
"greater than zero"));
- std::vector<unsigned int> renumber (this->dofs_per_cell-1);
- const FiniteElementData<dim> fe(get_dpo_vector_q(degree),1,degree);
- FETools::hierarchic_to_lexicographic_numbering (fe, renumber);
- renumber.push_back(this->dofs_per_cell-1);
- this->poly_space.set_numbering(renumber);
-
- // finally fill in support points
- // on cell and face
- initialize_unit_support_points ();
- initialize_unit_face_support_points ();
+ std::vector<Point<1> > support_points_1d(degree+1);
+ for (unsigned int i=0; i<=degree; ++i)
+ support_points_1d[i][0] = static_cast<double>(i)/degree;
- // reinit constraints
- initialize_constraints ();
+ this->initialize(support_points_1d);
- // Reinit the vectors of restriction and prolongation matrices to the
- // right sizes and compute the matrices
- this->reinit_restriction_and_prolongation_matrices();
- initialize_embedding();
- initialize_restriction();
-
- initialize_quad_dof_index_permutation();
+ // adjust unit support point for discontinuous node
+ Point<dim> point;
+ for (unsigned int d=0; d<dim; ++d)
+ point[d] = 0.5;
+ this->unit_support_points.push_back(point);
+ AssertDimension(this->dofs_per_cell, this->unit_support_points.size());
}
template <int dim, int spacedim>
FE_Q_DG0<dim,spacedim>::FE_Q_DG0 (const Quadrature<1> &points)
:
- FE_Poly<TensorProductPolynomialsConst<dim>, dim, spacedim> (
+ FE_Q_Base<TensorProductPolynomialsConst<dim>, dim, spacedim> (
TensorProductPolynomialsConst<dim>(Polynomials::generate_complete_Lagrange_basis(points.get_points())),
FiniteElementData<dim>(get_dpo_vector(points.size()-1),
1, points.size()-1,
FiniteElementData<dim>::L2),
- get_riaf_vector(points.size()-1),
- std::vector<ComponentMask>(1, std::vector<bool>(1,true))),
-
- face_index_map(FE_Q_DG0_Helper::invert_numbering(
- face_lexicographic_to_hierarchic_numbering (points.size()-1)))
+ get_riaf_vector(points.size()-1))
{
const int degree = points.size()-1;
Assert (degree > 0,
ExcMessage ("This element can only be used for polynomial degrees "
"at least zero"));
- Assert (points.point(0)(0) == 0,
- ExcMessage ("The first support point has to be zero."));
- Assert (points.point(degree)(0) == 1,
- ExcMessage ("The last support point has to be one."));
-
- std::vector<unsigned int> renumber (this->dofs_per_cell-1);
- const FiniteElementData<dim> fe(get_dpo_vector_q(degree),1,degree);
- FETools::hierarchic_to_lexicographic_numbering (fe, renumber);
- renumber.push_back(this->dofs_per_cell-1);
- this->poly_space.set_numbering(renumber);
-
- // finally fill in support points
- // on cell and face
- initialize_unit_support_points (points);
- initialize_unit_face_support_points (points);
- // reinit constraints
- Implementation::initialize_constraints (points, *this);
+ this->initialize(points.get_points());
- // Reinit the vectors of restriction and prolongation matrices to the
- // right sizes and compute the matrices
- this->reinit_restriction_and_prolongation_matrices();
- initialize_embedding();
- initialize_restriction();
-
- initialize_quad_dof_index_permutation();
+ // adjust unit support point for discontinuous node
+ Point<dim> point;
+ for (unsigned int d=0; d<dim; ++d)
+ point[d] = 0.5;
+ this->unit_support_points.push_back(point);
+ AssertDimension(this->dofs_per_cell, this->unit_support_points.size());
}
std::string
FE_Q_DG0<dim,spacedim>::get_name () const
{
- // note that the
- // FETools::get_fe_from_name
- // function depends on the
- // particular format of the string
- // this function returns, so they
- // have to be kept in synch
+ // note that the FETools::get_fe_from_name function depends on the
+ // particular format of the string this function returns, so they have to be
+ // kept in synch
std::ostringstream namebuf;
bool type = true;
const std::vector<Point<dim> > &unit_support_points = this->unit_support_points;
unsigned int index = 0;
- // Decode the support points
- // in one coordinate direction.
+ // Decode the support points in one coordinate direction.
for (unsigned int j=0; j<dofs_per_cell; j++)
{
if ((dim>1) ? (unit_support_points[j](1)==0 &&
index++;
}
}
- //Do not consider the discontinuous node for dimension 1
+ // Do not consider the discontinuous node for dimension 1
Assert (index == n_points || (dim==1 && index == n_points+1),
ExcMessage ("Could not decode support points in one coordinate direction."));
- // Check whether the support
- // points are equidistant.
+ // Check whether the support points are equidistant.
for (unsigned int j=0; j<n_points; j++)
if (std::fabs(points[j] - (double)j/this->degree) > 1e-15)
{
else
{
- // Check whether the support
- // points come from QGaussLobatto.
+ // Check whether the support points come from QGaussLobatto.
const QGaussLobatto<1> points_gl(n_points);
type = true;
for (unsigned int j=0; j<n_points; j++)
return new FE_Q_DG0<dim,spacedim>(*this);
}
+
+
template <int dim, int spacedim>
void
FE_Q_DG0<dim,spacedim>::interpolate(std::vector<double> &local_dofs,
ExcDimensionMismatch(this->n_components(), 1));
std::copy(values.begin(), values.end(), local_dofs.begin());
- //We don't need the discontinuous function for local interpolation
- local_dofs[local_dofs.size()-1]=0.;
+
+ // We don't need the discontinuous function for local interpolation
+ local_dofs[local_dofs.size()-1] = 0.;
}
+
+
template <int dim, int spacedim>
void
FE_Q_DG0<dim,spacedim>::interpolate(std::vector<double> &local_dofs,
= this->system_to_component_index(i);
local_dofs[i] = values[i](offset+index.first);
}
- //We don't need the discontinuous function for local interpolation
- local_dofs[local_dofs.size()-1]=0.;
-}
+ // We don't need the discontinuous function for local interpolation
+ local_dofs[local_dofs.size()-1] = 0.;
+}
= this->system_to_component_index(i);
local_dofs[i] = values[index.first][i];
}
- //We don't need the discontinuous function for local interpolation
- local_dofs[local_dofs.size()-1]=0.;
+
+ // We don't need the discontinuous function for local interpolation
+ local_dofs[local_dofs.size()-1] = 0.;
}
+
+
template <int dim, int spacedim>
void
FE_Q_DG0<dim,spacedim>::
get_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
FullMatrix<double> &interpolation_matrix) const
{
- // this is only implemented, if the
- // source FE is also a
- // Q_DG0 element
+ // this is only implemented, if the source FE is also a Q_DG0 element
typedef FE_Q_DG0<dim,spacedim> FEQDG0;
typedef FiniteElement<dim,spacedim> FEL;
ExcDimensionMismatch (interpolation_matrix.m(),
x_source_fe.dofs_per_cell));
- // ok, source is a Q_DG0 element, so
- // we will be able to do the work
- const FE_Q_DG0<dim,spacedim> &source_fe
- = dynamic_cast<const FE_Q_DG0<dim,spacedim>&>(x_source_fe);
-
- // compute the interpolation
- // matrices in much the same way as
- // we do for the embedding matrices
- // from mother to child.
- FullMatrix<double> cell_interpolation (this->dofs_per_cell,
- this->dofs_per_cell);
- FullMatrix<double> source_interpolation (this->dofs_per_cell,
- source_fe.dofs_per_cell);
- FullMatrix<double> tmp (this->dofs_per_cell,
- source_fe.dofs_per_cell);
- for (unsigned int j=0; j<this->dofs_per_cell-1; ++j)
- {
- // read in a point on this
- // cell and evaluate the
- // shape functions there
- const Point<dim> p = this->unit_support_points[j];
- for (unsigned int i=0; i<this->dofs_per_cell-1; ++i)
- cell_interpolation(j,i) = this->poly_space.compute_value (i, p);
-
- for (unsigned int i=0; i<source_fe.dofs_per_cell-1; ++i)
- source_interpolation(j,i) = source_fe.poly_space.compute_value (i,p);
-
- }
- //the discontinuous node is not transformed
- cell_interpolation(this->dofs_per_cell-1,
- this->dofs_per_cell-1)=1.;
- source_interpolation(this->dofs_per_cell-1,
- source_fe.dofs_per_cell-1)=1.;
-
- // then compute the
- // interpolation matrix
- // for this coordinate
- // direction
- cell_interpolation.gauss_jordan ();
- cell_interpolation.mmult (interpolation_matrix,
- source_interpolation);
-
- const double eps = 2e-13*this->degree*dim;
-
- // cut off very small values
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
- if (std::fabs(interpolation_matrix(i,j)) < eps)
- interpolation_matrix(i,j) = 0.;
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int i=0; i<this->dofs_per_cell; ++i)
- {
- double sum = 0.;
- for (unsigned int j=0; j<source_fe.dofs_per_cell; ++j)
- sum += interpolation_matrix(i,j);
-
- Assert (std::fabs(sum-1) < eps, ExcInternalError());
- }
-}
-
-
-
-template <>
-void
-FE_Q_DG0<1>::
-get_face_interpolation_matrix (const FiniteElement<1,1> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- Assert (false, ExcImpossibleInDim(1));
-}
-
-
-
-template <>
-void
-FE_Q_DG0<1>::
-get_subface_interpolation_matrix (const FiniteElement<1,1> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- Assert (false, ExcImpossibleInDim(1));
-}
-
-
-template <>
-void
-FE_Q_DG0<1,2>::
-get_face_interpolation_matrix (const FiniteElement<1,2> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,2> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q_DG0<1,2>::
-get_subface_interpolation_matrix (const FiniteElement<1,2> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,2> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q_DG0<1,3>::
-get_face_interpolation_matrix (const FiniteElement<1,3> &/*x_source_fe*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,3> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
-}
-
-
-template <>
-void
-FE_Q_DG0<1,3>::
-get_subface_interpolation_matrix (const FiniteElement<1,3> &/*x_source_fe*/,
- const unsigned int /*subface*/,
- FullMatrix<double> &/*interpolation_matrix*/) const
-{
- typedef FiniteElement<1,3> FEL;
- Assert (false,
- FEL::
- ExcInterpolationNotImplemented ());
+ this->FE_Q_Base<TensorProductPolynomialsConst<dim>,dim,spacedim>::
+ get_interpolation_matrix(x_source_fe, interpolation_matrix);
}
template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::
-get_face_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
- FullMatrix<double> &interpolation_matrix) const
+std::vector<bool>
+FE_Q_DG0<dim,spacedim>::get_riaf_vector(const unsigned int deg)
{
- // this is only implemented, if the
- // source FE is also a
- // Q_DG0 element
- typedef FE_Q_DG0<dim,spacedim> FEQDG0;
- typedef FiniteElement<dim,spacedim> FEL;
- AssertThrow ((x_source_fe.get_name().find ("FE_Q_DG0<") == 0)
- ||
- (dynamic_cast<const FEQDG0 *>(&x_source_fe) != 0),
- typename FEL::
- ExcInterpolationNotImplemented());
-
- Assert (interpolation_matrix.n() == this->dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.n(),
- this->dofs_per_face));
- Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.m(),
- x_source_fe.dofs_per_face));
-
- // ok, source is a Q_DG0 element, so
- // we will be able to do the work
- const FE_Q_DG0<dim,spacedim> &source_fe
- = dynamic_cast<const FE_Q_DG0<dim,spacedim>&>(x_source_fe);
-
- // Make sure, that the element,
- // for which the DoFs should be
- // constrained is the one with
- // the higher polynomial degree.
- // Actually the procedure will work
- // also if this assertion is not
- // satisfied. But the matrices
- // produced in that case might
- // lead to problems in the
- // hp procedures, which use this
- // method.
- Assert (this->dofs_per_face <= source_fe.dofs_per_face,
- typename FEL::
- ExcInterpolationNotImplemented ());
-
- // generate a quadrature
- // with the unit support points.
- // This is later based as a
- // basis for the QProjector,
- // which returns the support
- // points on the face.
- Quadrature<dim-1> quad_face_support (source_fe.get_unit_face_support_points ());
-
- // Rule of thumb for FP accuracy,
- // that can be expected for a
- // given polynomial degree.
- // This value is used to cut
- // off values close to zero.
- const double eps = 2e-13*this->degree*(dim-1);
-
- // compute the interpolation
- // matrix by simply taking the
- // value at the support points.
-//TODO: Verify that all faces are the same with respect to
-// these support points. Furthermore, check if something has to
-// be done for the face orientation flag in 3D.
- const Quadrature<dim> face_quadrature
- = QProjector<dim>::project_to_face (quad_face_support, 0);
- for (unsigned int i=0; i<source_fe.dofs_per_face; ++i)
- {
- const Point<dim> &p = face_quadrature.point (i);
-
- for (unsigned int j=0; j<this->dofs_per_face; ++j)
- {
- double matrix_entry = this->shape_value (this->face_to_cell_index(j, 0), p);
-
- // Correct the interpolated
- // value. I.e. if it is close
- // to 1 or 0, make it exactly
- // 1 or 0. Unfortunately, this
- // is required to avoid problems
- // with higher order elements.
- if (std::fabs (matrix_entry - 1.0) < eps)
- matrix_entry = 1.0;
- if (std::fabs (matrix_entry) < eps)
- matrix_entry = 0.0;
-
- interpolation_matrix(i,j) = matrix_entry;
- }
- }
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int j=0; j<source_fe.dofs_per_face; ++j)
- {
- double sum = 0.;
-
- for (unsigned int i=0; i<this->dofs_per_face; ++i)
- sum += interpolation_matrix(j,i);
-
- Assert (std::fabs(sum-1) < 2e-13*this->degree*(dim-1),
- ExcInternalError());
- }
+ std::vector<bool> riaf(Utilities::fixed_power<dim>(deg+1)+1,false);
+ riaf[riaf.size()-1]=true;
+ return riaf;
}
template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::
-get_subface_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
- const unsigned int subface,
- FullMatrix<double> &interpolation_matrix) const
+std::vector<unsigned int>
+FE_Q_DG0<dim,spacedim>::get_dpo_vector(const unsigned int deg)
{
- Assert (interpolation_matrix.m() == x_source_fe.dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.m(),
- x_source_fe.dofs_per_face));
-
- // see if source is a Q_DG0 element
- if (const FE_Q_DG0<dim,spacedim> *source_fe
- = dynamic_cast<const FE_Q_DG0<dim,spacedim> *>(&x_source_fe))
- {
- // have this test in here since
- // a table of size 2x0 reports
- // its size as 0x0
- Assert (interpolation_matrix.n() == this->dofs_per_face,
- ExcDimensionMismatch (interpolation_matrix.n(),
- this->dofs_per_face));
-
- // Make sure, that the element,
- // for which the DoFs should be
- // constrained is the one with
- // the higher polynomial degree.
- // Actually the procedure will work
- // also if this assertion is not
- // satisfied. But the matrices
- // produced in that case might
- // lead to problems in the
- // hp procedures, which use this
- // method.
- Assert (this->dofs_per_face <= source_fe->dofs_per_face,
- (typename FiniteElement<dim,spacedim>::
- ExcInterpolationNotImplemented ()));
-
- // generate a point on this
- // cell and evaluate the
- // shape functions there
- const Quadrature<dim-1>
- quad_face_support (source_fe->get_unit_face_support_points ());
-
- // Rule of thumb for FP accuracy,
- // that can be expected for a
- // given polynomial degree.
- // This value is used to cut
- // off values close to zero.
- double eps = 2e-13*this->degree*(dim-1);
-
- // compute the interpolation
- // matrix by simply taking the
- // value at the support points.
-//TODO: Verify that all faces are the same with respect to
-// these support points. Furthermore, check if something has to
-// be done for the face orientation flag in 3D.
- const Quadrature<dim> subface_quadrature
- = QProjector<dim>::project_to_subface (quad_face_support, 0, subface);
- for (unsigned int i=0; i<source_fe->dofs_per_face; ++i)
- {
- const Point<dim> &p = subface_quadrature.point (i);
-
- for (unsigned int j=0; j<this->dofs_per_face; ++j)
- {
- double matrix_entry = this->shape_value (this->face_to_cell_index(j, 0), p);
-
- // Correct the interpolated
- // value. I.e. if it is close
- // to 1 or 0, make it exactly
- // 1 or 0. Unfortunately, this
- // is required to avoid problems
- // with higher order elements.
- if (std::fabs (matrix_entry - 1.0) < eps)
- matrix_entry = 1.0;
- if (std::fabs (matrix_entry) < eps)
- matrix_entry = 0.0;
-
- interpolation_matrix(i,j) = matrix_entry;
- }
- }
-
- // make sure that the row sum of
- // each of the matrices is 1 at
- // this point. this must be so
- // since the shape functions sum up
- // to 1
- for (unsigned int j=0; j<source_fe->dofs_per_face; ++j)
- {
- double sum = 0.;
-
- for (unsigned int i=0; i<this->dofs_per_face; ++i)
- sum += interpolation_matrix(j,i);
+ std::vector<unsigned int> dpo(dim+1, 1U);
+ for (unsigned int i=1; i<dpo.size(); ++i)
+ dpo[i]=dpo[i-1]*(deg-1);
- Assert (std::fabs(sum-1) < 2e-13*this->degree*dim,
- ExcInternalError());
- }
- }
- else if (dynamic_cast<const FE_Nothing<dim> *>(&x_source_fe) != 0)
- {
- // nothing to do here, the
- // FE_Nothing has no degrees of
- // freedom anyway
- }
- else
- AssertThrow (false,
- (typename FiniteElement<dim,spacedim>::
- ExcInterpolationNotImplemented()));
+ dpo[dim]++;//we need an additional DG0-node for a dim-dimensional object
+ return dpo;
}
template <int dim, int spacedim>
bool
-FE_Q_DG0<dim,spacedim>::hp_constraints_are_implemented () const
-{
- return true;
-}
-
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q_DG0<dim,spacedim>::
-hp_vertex_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
-{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Q_DG0s or if the other one is an
- // FE_Nothing. in the first case,
- // there should be exactly one
- // single DoF of each FE at a
- // vertex, and they should have
- // identical value
- if (dynamic_cast<const FE_Q_DG0<dim,spacedim>*>(&fe_other) != 0)
- {
- return
- std::vector<std::pair<unsigned int, unsigned int> >
- (1, std::make_pair (0U, 0U));
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
- else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
-}
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q_DG0<dim,spacedim>::
-hp_line_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
-{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Q_DG0s or if the other one is an
- // FE_Nothing
- if (const FE_Q_DG0<dim,spacedim> *fe_q_dg0_other = dynamic_cast<const
- FE_Q_DG0<dim,spacedim> *>(&fe_other))
- {
- // dofs are located along lines, so two
- // dofs are identical if they are
- // located at identical positions. if
- // we had only equidistant points, we
- // could simple check for similarity
- // like (i+1)*q == (j+1)*p, but we
- // might have other support points
- // (e.g. Gauss-Lobatto
- // points). Therefore, read the points
- // in unit_support_points for the first
- // coordinate direction. We take the
- // lexicographic ordering of the points
- // in the first direction (i.e.,
- // x-direction), which we access
- // between index 1 and p-1 (index 0 and
- // p are vertex dofs).
- const unsigned int p = this->degree;
- const unsigned int q = fe_q_dg0_other->degree;
-
- std::vector<std::pair<unsigned int, unsigned int> > identities;
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
- const std::vector<unsigned int> &index_map_inverse_other=
- fe_q_dg0_other->poly_space.get_numbering_inverse();
-
- for (unsigned int i=0; i<p-1; ++i)
- for (unsigned int j=0; j<q-1; ++j)
- if (std::fabs(this->unit_support_points[index_map_inverse[i+1]][0]-
-
- fe_q_dg0_other->unit_support_points[index_map_inverse_other[j+1]][0])
- < 1e-14)
- identities.push_back (std::make_pair(i,j));
-
- return identities;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
- else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
-}
-
-
-
-template <int dim, int spacedim>
-std::vector<std::pair<unsigned int, unsigned int> >
-FE_Q_DG0<dim,spacedim>::
-hp_quad_dof_identities (const FiniteElement<dim,spacedim> &fe_other) const
+FE_Q_DG0<dim,spacedim>::has_support_on_face (const unsigned int shape_index,
+ const unsigned int face_index) const
{
- // we can presently only compute
- // these identities if both FEs are
- // FE_Q_DG0s or if the other one is an
- // FE_Nothing
- if (const FE_Q_DG0<dim,spacedim> *fe_q_dg0_other = dynamic_cast<const
- FE_Q_DG0<dim,spacedim> *>(&fe_other))
- {
- // this works exactly like the line
- // case above, except that now we have
- // to have two indices i1, i2 and j1,
- // j2 to characterize the dofs on the
- // face of each of the finite
- // elements. since they are ordered
- // lexicographically along the first
- // line and we have a tensor product,
- // the rest is rather straightforward
- const unsigned int p = this->degree;
- const unsigned int q = fe_q_dg0_other->degree;
-
- std::vector<std::pair<unsigned int, unsigned int> > identities;
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
- const std::vector<unsigned int> &index_map_inverse_other=
- fe_q_dg0_other->poly_space.get_numbering_inverse();
-
- for (unsigned int i1=0; i1<p-1; ++i1)
- for (unsigned int i2=0; i2<p-1; ++i2)
- for (unsigned int j1=0; j1<q-1; ++j1)
- for (unsigned int j2=0; j2<q-1; ++j2)
- if ((std::fabs(this->unit_support_points[index_map_inverse[i1+1]][0]-
-
- fe_q_dg0_other->unit_support_points[index_map_inverse_other[j1+1]][0])
- < 1e-14)
- &&
- (std::fabs(this->unit_support_points[index_map_inverse[i2+1]][0]-
-
- fe_q_dg0_other->unit_support_points[index_map_inverse_other[j2+1]][0])
- < 1e-14))
- identities.push_back (std::make_pair(i1*(p-1)+i2,
- j1*(q-1)+j2));
-
- return identities;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom, so there
- // are no equivalencies to be
- // recorded
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
+ // discontinuous function has support on all faces
+ if (shape_index == this->dofs_per_cell-1)
+ return true;
else
- {
- Assert (false, ExcNotImplemented());
- return std::vector<std::pair<unsigned int, unsigned int> > ();
- }
+ return FE_Q_Base<TensorProductPolynomialsConst<dim>,dim,spacedim>::has_support_on_face(shape_index, face_index);
}
-
-template <int dim, int spacedim>
-FiniteElementDomination::Domination
-FE_Q_DG0<dim,spacedim>::
-compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const
-{
- if (const FE_Q_DG0<dim,spacedim> *fe_q_dg0_other
- = dynamic_cast<const FE_Q_DG0<dim,spacedim>*>(&fe_other))
- {
- if (this->degree < fe_q_dg0_other->degree)
- return FiniteElementDomination::this_element_dominates;
- else if (this->degree == fe_q_dg0_other->degree)
- return FiniteElementDomination::either_element_can_dominate;
- else
- return FiniteElementDomination::other_element_dominates;
- }
- else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
- {
- // the FE_Nothing has no
- // degrees of freedom and it is
- // typically used in a context
- // where we don't require any
- // continuity along the
- // interface
- return FiniteElementDomination::no_requirements;
- }
-
- Assert (false, ExcNotImplemented());
- return FiniteElementDomination::neither_element_dominates;
-}
-
-
-//---------------------------------------------------------------------------
-// Auxiliary functions
-//---------------------------------------------------------------------------
-
-
-
-template <int dim, int spacedim>
-void FE_Q_DG0<dim,spacedim>::initialize_unit_support_points ()
-{
- // number of points: (degree+1)^dim+1
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<dim; ++i)
- n *= this->degree+1;
- n++;
-
- this->unit_support_points.resize(n);
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
-
- const double step = 1./this->degree;
- Point<dim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((dim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((dim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = ix * step;
- if (dim>1)
- p(1) = iy * step;
- if (dim>2)
- p(2) = iz * step;
-
- this->unit_support_points[index_map_inverse[k++]] = p;
- }
- //dg0 support point in the center of the cell
- p(0) = .5;
- if (dim>1)
- {
- p(1) = .5;
- if (dim>2)
- p(2) = .5;
- }
-
- this->unit_support_points[index_map_inverse[k]] = p;
-}
-
-
-
-template <int dim, int spacedim>
-void FE_Q_DG0<dim,spacedim>::initialize_unit_support_points (const Quadrature<1> &points)
-{
- // number of points: (degree+1)^dim+1
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<dim; ++i)
- n *= this->degree+1;
- n++;
-
- this->unit_support_points.resize(n);
-
- const std::vector<unsigned int> &index_map_inverse=
- this->poly_space.get_numbering_inverse();
-
- Quadrature<dim> support_quadrature(points);
-
- Point<dim> p;
-
- for (unsigned int k=0; k<n ; k++)
- {
- this->unit_support_points[index_map_inverse[k]] = support_quadrature.point(k);
- }
-}
-
-
-
-template <>
-void FE_Q_DG0<1>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q_DG0<1>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q_DG0<1,2>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q_DG0<1,2>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q_DG0<1,3>::initialize_unit_face_support_points ()
-{
- // no faces in 1d, so nothing to do
-}
-
-template <>
-void FE_Q_DG0<1,3>::initialize_unit_face_support_points (const Quadrature<1> &/*points*/)
-{
- // no faces in 1d, so nothing to do
-}
-
-template <int dim, int spacedim>
-void FE_Q_DG0<dim,spacedim>::initialize_unit_face_support_points ()
-{
- const unsigned int codim = dim-1;
-
- // number of points: (degree+1)^codim
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<codim; ++i)
- n *= this->degree+1;
-
- this->unit_face_support_points.resize(n);
-
- const std::vector<unsigned int> &face_index_map_inverse=
- FE_Q_DG0_Helper::invert_numbering(face_index_map);
-
- const double step = 1./this->degree;
- Point<codim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((codim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((codim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = ix * step;
- if (codim>1)
- p(1) = iy * step;
- if (codim>2)
- p(2) = iz * step;
-
- this->unit_face_support_points[face_index_map_inverse[k++]] = p;
- }
-}
-
-
-
-template <int dim, int spacedim>
-void FE_Q_DG0<dim,spacedim>::initialize_unit_face_support_points (const Quadrature<1> &points)
-{
- const unsigned int codim = dim-1;
-
- // number of points: (degree+1)^codim
- unsigned int n = this->degree+1;
- for (unsigned int i=1; i<codim; ++i)
- n *= this->degree+1;
-
- this->unit_face_support_points.resize(n);
-
- const std::vector< Point<1> > edge = points.get_points();
-
- const std::vector<unsigned int> &face_index_map_inverse=
- FE_Q_DG0_Helper::invert_numbering(face_index_map);
-
- Point<codim> p;
-
- unsigned int k=0;
- for (unsigned int iz=0; iz <= ((codim>2) ? this->degree : 0) ; ++iz)
- for (unsigned int iy=0; iy <= ((codim>1) ? this->degree : 0) ; ++iy)
- for (unsigned int ix=0; ix<=this->degree; ++ix)
- {
- p(0) = edge[ix](0);
- if (codim>1)
- p(1) = edge[iy](0);
- if (codim>2)
- p(2) = edge[iz](0);
-
- this->unit_face_support_points[face_index_map_inverse[k++]] = p;
- }
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::initialize_quad_dof_index_permutation ()
-{
- // general template for 1D and 2D, do nothing
-}
-
-
-
-template <>
-void
-FE_Q_DG0<3>::initialize_quad_dof_index_permutation ()
-{
- Assert (adjust_quad_dof_index_for_face_orientation_table.n_elements()==8*this->dofs_per_quad,
- ExcInternalError());
-
- const unsigned int n=this->degree-1;
- Assert(n*n==this->dofs_per_quad, ExcInternalError());
-
- // alias for the table to fill
- Table<2,int> &data=this->adjust_quad_dof_index_for_face_orientation_table;
-
- // the dofs on a face are connected to a n x
- // n matrix. for example, for degree==4 we
- // have the following dofs on a quad
-
- // ___________
- // | |
- // | 6 7 8 |
- // | |
- // | 3 4 5 |
- // | |
- // | 0 1 2 |
- // |___________|
- //
- // we have dof_no=i+n*j with index i in
- // x-direction and index j in y-direction
- // running from 0 to n-1. to extract i and j
- // we can use i=dof_no%n and j=dof_no/n. i
- // and j can then be used to construct the
- // rotated and mirrored numbers.
-
-
- for (unsigned int local=0; local<this->dofs_per_quad; ++local)
- // face support points are in lexicographic
- // ordering with x running fastest. invert
- // that (y running fastest)
- {
- unsigned int i=local%n,
- j=local/n;
-
- // face_orientation=false, face_flip=false, face_rotation=false
- data(local,0)=j + i *n - local;
- // face_orientation=false, face_flip=false, face_rotation=true
- data(local,1)=i + (n-1-j)*n - local;
- // face_orientation=false, face_flip=true, face_rotation=false
- data(local,2)=(n-1-j) + (n-1-i)*n - local;
- // face_orientation=false, face_flip=true, face_rotation=true
- data(local,3)=(n-1-i) + j *n - local;
- // face_orientation=true, face_flip=false, face_rotation=false
- data(local,4)=0;
- // face_orientation=true, face_flip=false, face_rotation=true
- data(local,5)=j + (n-1-i)*n - local;
- // face_orientation=true, face_flip=true, face_rotation=false
- data(local,6)=(n-1-i) + (n-1-j)*n - local;
- // face_orientation=true, face_flip=true, face_rotation=true
- data(local,7)=(n-1-j) + i *n - local;
- }
-
- // aditionally initialize reordering of line
- // dofs
- for (unsigned int i=0; i<this->dofs_per_line; ++i)
- this->adjust_line_dof_index_for_line_orientation_table[i]=this->dofs_per_line-1-i - i;
-}
-
-template <int dim, int spacedim>
-std::vector<bool>
-FE_Q_DG0<dim,spacedim>::get_riaf_vector(const unsigned int deg)
-{
- std::vector<bool> riaf
- (FiniteElementData<dim> (get_dpo_vector(deg),1,deg).dofs_per_cell,
- false);
- riaf[riaf.size()-1]=true;
- return riaf;
-}
-
-
-template <int dim, int spacedim>
-std::vector<unsigned int>
-FE_Q_DG0<dim,spacedim>::get_dpo_vector(const unsigned int deg)
-{
- std::vector<unsigned int> dpo(dim+1, 1U);
- for (unsigned int i=1; i<dpo.size(); ++i)
- dpo[i]=dpo[i-1]*(deg-1);
-
- dpo[dim]++;//we need an additional DG0-node for a dim-dimensional object
- return dpo;
-}
-
-template <int dim, int spacedim>
-std::vector<unsigned int>
-FE_Q_DG0<dim,spacedim>::get_dpo_vector_q(const unsigned int deg)
-{
- std::vector<unsigned int> dpo(dim+1, 1U);
- for (unsigned int i=1; i<dpo.size(); ++i)
- dpo[i]=dpo[i-1]*(deg-1);
-
- return dpo;
-}
-
-
-template <int dim, int spacedim>
-std::vector<unsigned int>
-FE_Q_DG0<dim,spacedim>::face_lexicographic_to_hierarchic_numbering (const unsigned int degree)
-{
- const FiniteElementData<dim-1> face_data(FE_Q_DG0<dim-1>::get_dpo_vector_q(degree),1,degree);
- std::vector<unsigned int> face_renumber (face_data.dofs_per_cell);
- FETools::lexicographic_to_hierarchic_numbering (face_data, face_renumber);
- return face_renumber;
-}
-
-
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1,2>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-
-template <>
-std::vector<unsigned int>
-FE_Q_DG0<1,3>::face_lexicographic_to_hierarchic_numbering (const unsigned int)
-{
- return std::vector<unsigned int>();
-}
-
-
-template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::initialize_constraints ()
-{
- QTrapez<1> trapez;
- QIterated<1> points (trapez,this->degree);
- Implementation::initialize_constraints (points, *this);
-}
-
-
-template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::initialize_embedding ()
-{
- // compute the interpolation matrices in much the same way as we do for
- // the constraints. it's actually simpler here, since we don't have this
- // weird renumbering stuff going on. The trick is again that the
- // interpolation matrix is formed by a permutation of the indices of the
- // cell matrix. The value eps is used a threshold to decide when certain
- // evaluations of the Lagrange polynomials are zero or one.
- const double eps = 1e-15*this->degree*dim;
-
-#ifdef DEBUG
- // in DEBUG mode, check that the evaluation of support points (except for
- // the discontinuous node) in the current numbering gives the identity
- // operation
- for (unsigned int i=0; i<this->dofs_per_cell-1; ++i)
- {
- Assert (std::fabs (1.-this->poly_space.compute_value
- (i, this->unit_support_points[i])) < eps,
- ExcInternalError());
- for (unsigned int j=0; j<this->dofs_per_cell-1; ++j)
- if (j!=i)
- Assert (std::fabs (this->poly_space.compute_value
- (i, this->unit_support_points[j])) < eps,
- ExcInternalError());
- }
-#endif
-
- // to efficiently evaluate the polynomial at the subcell, make use of the
- // tensor product structure of this element and only evaluate 1D
- // information from the polynomial. This makes the cost of this function
- // almost negligible also for high order elements
- const unsigned int dofs1d = this->degree+1;
- std::vector<Table<2,double> > subcell_evaluations
- (dim, Table<2,double>(dofs1d, dofs1d));
- const std::vector<unsigned int> &index_map_inverse =
- this->poly_space.get_numbering_inverse();
-
- // recreate 1D polynomials, the last entry is not used
- std::vector<Polynomials::Polynomial<double> > poly_space1d =
- FE_Q_DG0_Helper::generate_poly_space1d (this->unit_support_points,
- index_map_inverse, dofs1d);
-
- // helper value: step size how to walk through diagonal and how many
- // points we have left apart from the first dimension
- unsigned int step_size_diag = 0;
- {
- unsigned int factor = 1;
- for (unsigned int d=0; d<dim; ++d)
- {
- step_size_diag += factor;
- factor *= dofs1d;
- }
- }
-
- // next evaluate the functions for the different refinement cases.
- for (unsigned int ref=0; ref<RefinementCase<dim>::isotropic_refinement; ++ref)
- for (unsigned int child=0;
- child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref+1));
- ++child)
- {
- // go through the points in diagonal to capture variation in all
- // directions simultaneously
- for (unsigned int j=0; j<dofs1d; ++j)
- {
- const unsigned int diag_comp = index_map_inverse[j*step_size_diag];
- const Point<dim> p_subcell = this->unit_support_points[diag_comp];
- const Point<dim> p_cell =
- GeometryInfo<dim>::child_to_cell_coordinates
- (p_subcell, child, RefinementCase<dim>(ref+1));
- for (unsigned int i=0; i<dofs1d; ++i)
- for (unsigned int d=0; d<dim; ++d)
- {
- const double cell_value = poly_space1d[i].value (p_cell[d]);
-
- // cut off values that are too small. note that we have here
- // Lagrange interpolation functions, so they should be zero at
- // almost all points, and one at the others, at least on the
- // subcells. so set them to their exact values.
- //
- // the actual cut-off value is somewhat fuzzy, but it works for
- // 2e-13*degree*dim (see above), which is kind of reasonable
- // given that we compute the values of the polynomials via an
- // degree-step recursion and then multiply the 1d-values. this
- // gives us a linear growth in degree*dim, times a small
- // constant.
- //
- // the embedding matrix is given by applying the inverse of the
- // subcell matrix on the cell_interpolation matrix. since the
- // subcell matrix is actually only a permutation vector, all we
- // need to do is to switch the rows we write the data into.
- // moreover, cut off very small values here
- if (std::fabs(cell_value) < eps)
- subcell_evaluations[d](j,i) = 0;
- else
- subcell_evaluations[d](j,i) = cell_value;
- }
- }
-
- // now expand from 1D info. block innermost dimension (x_0) in order
- // to avoid difficult checks at innermost loop
- unsigned int j_indices[dim];
- FE_Q_DG0_Helper::zero_indices<dim> (j_indices);
- for (unsigned int j=0; j<this->dofs_per_cell-1; j+=dofs1d)
- {
- unsigned int i_indices[dim];
- FE_Q_DG0_Helper::zero_indices<dim> (i_indices);
- for (unsigned int i=0; i<this->dofs_per_cell-1; i+=dofs1d)
- {
- double val_extra_dim = 1.;
- for (unsigned int d=1; d<dim; ++d)
- val_extra_dim *= subcell_evaluations[d](j_indices[d-1],
- i_indices[d-1]);
-
- // innermost sum where we actually compute. the same as
- // this->prolongation[ref][child](j,i) =
- // this->poly_space.compute_value (i, p_cell);
- for (unsigned int jj=0; jj<dofs1d; ++jj)
- {
- const unsigned int j_ind = index_map_inverse[j+jj];
- for (unsigned int ii=0; ii<dofs1d; ++ii)
-
- this->prolongation[ref][child](j_ind,index_map_inverse[i+ii])
- = val_extra_dim * subcell_evaluations[0](jj,ii);
- }
-
- // update indices that denote the tensor product position. a bit
- // fuzzy and therefore not done for innermost x_0 direction
- FE_Q_DG0_Helper::increment_indices<dim> (i_indices, dofs1d);
- }
- Assert (i_indices[dim-1] == 1, ExcInternalError());
- FE_Q_DG0_Helper::increment_indices<dim> (j_indices, dofs1d);
- }
-
- // the discontinuous node is simply mapped on the discontinuous node
- // on the child cell
- for (unsigned int j=0; j<this->dofs_per_cell-1; j++)
- {
- this->prolongation[ref][child](j,this->dofs_per_cell-1) = 0.;
- this->prolongation[ref][child](this->dofs_per_cell-1,j) = 0.;
- }
- this->prolongation[ref][child](this->dofs_per_cell-1,
- this->dofs_per_cell-1) = 1.;
-
- // make sure that the row sum is one. this must be so since for this
- // element, the continuous shape functions add up to one and the
- // discontinuous node is mapped to the discontinuous node on the
- // child cell
-#ifdef DEBUG
- for (unsigned int row=0; row<this->dofs_per_cell; ++row)
- {
- double sum = 0;
- for (unsigned int col=0; col<this->dofs_per_cell; ++col)
- sum += this->prolongation[ref][child](row,col);
- Assert (std::fabs(sum-1.) < eps, ExcInternalError());
- }
-#endif
- }
-}
-
-
-
-template <int dim, int spacedim>
-void
-FE_Q_DG0<dim,spacedim>::initialize_restriction ()
-{
- // for these Lagrange interpolation polynomials, construction of the
- // restriction matrices is relatively simple. the reason is that the
- // interpolation points on the mother cell are (except for the case with
- // arbitrary nonequidistant nodes and for the discontinuous node) always also
- // interpolation points for some shape function on one or the other child,
- // because we have chosen equidistant Lagrange interpolation points for the
- // polynomials
- //
- // so the only thing we have to find out is: for each shape function on
- // the mother cell, which is the child cell (possibly more than one) on
- // which it is located, and which is the corresponding shape function there.
- // rather than doing it for the shape functions on the mother cell, we take
- // the interpolation points there
- //
- // note that the interpolation point of a shape function can be on the
- // boundary between subcells. in that case, restriction from children to
- // mother may produce two or more entries for a dof on the mother cell.
- // however, this doesn't hurt: since the element is continuous (apart from
- // the discontinuous node), the contribution from each child should yield
- // the same result, and since the element (apart from the discontinuous
- // node) is non-additive we just overwrite one value (compute one one child)
- // by the same value (compute on a later child), so we don't have to care
- // about this.
- // for the discontinuous node we just take the mean of all the child cells'
- // contributions.
-
- const double eps = 1e-15*this->degree*dim;
- const std::vector<unsigned int> &index_map_inverse =
- this->poly_space.get_numbering_inverse();
-
- // recreate 1D polynomials for faster evaluation of polynomial, the last
- // entry is not used
- const unsigned int dofs1d = this->degree+1;
- std::vector<Polynomials::Polynomial<double> > poly_space1d =
- FE_Q_DG0_Helper::generate_poly_space1d (this->unit_support_points,
- index_map_inverse, dofs1d);
- std::vector<Tensor<1,dim> > evaluations1d (dofs1d);
-
- for (unsigned int i=0; i<this->dofs_per_cell-1; ++i)
- {
- unsigned int mother_dof = index_map_inverse[i];
- const Point<dim> p_cell = this->unit_support_points[mother_dof];
-
- // then find the children on which the interpolation point is located
- for (unsigned int ref=RefinementCase<dim>::cut_x;
- ref<=RefinementCase<dim>::isotropic_refinement; ++ref)
- for (unsigned int child=0;
- child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref));
- ++child)
- {
- // check whether this interpolation point is inside this child
- // cell
- const Point<dim> p_subcell =
- GeometryInfo<dim>::cell_to_child_coordinates
- (p_cell, child, RefinementCase<dim>(ref));
- if (GeometryInfo<dim>::is_inside_unit_cell (p_subcell))
- {
- // same logic as in initialize_embedding to evaluate the
- // polynomial faster than from the tensor product: since we
- // evaluate all polynomials, it is much faster to just
- // compute the 1D values for all polynomials before and then
- // get the dim-data.
- for (unsigned int j=0; j<dofs1d; ++j)
- for (unsigned int d=0; d<dim; ++d)
- evaluations1d[j][d] = poly_space1d[j].value(p_subcell[d]);
- unsigned int j_indices[dim];
- FE_Q_DG0_Helper::zero_indices<dim> (j_indices);
- double sum_check = 0;
- for (unsigned int j = 0; j<this->dofs_per_cell-1; j += dofs1d)
- {
- double val_extra_dim = 1.;
- for (unsigned int d=1; d<dim; ++d)
- val_extra_dim *= evaluations1d[j_indices[d-1]][d];
- for (unsigned int jj=0; jj<dofs1d; ++jj)
- {
-
- // find the child shape function(s) corresponding
- // to this point. Usually this is just one function;
- // however, when we use FE_Q_DG0 on arbitrary nodes a
- // parent support point might not be a child support
- // point, and then we will get more than one nonzero
- // value per row. Still, the values should sum up to 1
- const double val = val_extra_dim * evaluations1d[jj][0];
- const unsigned int child_dof = index_map_inverse[j+jj];
- if (std::fabs (val-1.) < eps)
- this->restriction[ref-1][child]
- (mother_dof,child_dof)=1.;
- else if (std::fabs(val) > eps)
- this->restriction[ref-1][child]
- (mother_dof,child_dof)=val;
- sum_check += val;
- }
- FE_Q_DG0_Helper::increment_indices<dim> (j_indices, dofs1d);
- }
- Assert (std::fabs(sum_check-1) < eps,
- ExcInternalError());
- }
- }
- }
- // for the discontinuous node we take the mean of all the child cells'
- // contributions. so there is also a diagonal entry equal to the inverse of
- // n_children.
- for (unsigned int ref=RefinementCase<dim>::cut_x;
- ref<=RefinementCase<dim>::isotropic_refinement; ++ref)
- for (unsigned int child=0;
- child<GeometryInfo<dim>::n_children(RefinementCase<dim>(ref));
- ++child)
- {
- for (unsigned int i=0; i<this->dofs_per_cell-1; ++i)
- {
- this->restriction[ref-1][child](i,this->dofs_per_cell-1)=0.;
- this->restriction[ref-1][child](this->dofs_per_cell-1,i)=0.;
- }
- this->restriction[ref-1][child]
- (this->dofs_per_cell-1,this->dofs_per_cell-1) =
- 1./(double)GeometryInfo<dim>::n_children
- (RefinementCase<dim>(ref));
- }
-
-}
-
-
-
-//---------------------------------------------------------------------------
-// Data field initialization
-//---------------------------------------------------------------------------
-
-
-template <>
-bool
-FE_Q_DG0<1>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-
-template <>
-bool
-FE_Q_DG0<1,2>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-template <>
-bool
-FE_Q_DG0<1,3>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<1>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<1>::faces_per_cell));
-
-
- // in 1d, things are simple. since
- // there is only one degree of
- // freedom per vertex in this
- // class, the first is on vertex 0
- // (==face 0 in some sense), the
- // second on face 1:
- return (((shape_index == 0) && (face_index == 0)) ||
- ((shape_index == 1) && (face_index == 1)));
-}
-
-
-template <int dim, int spacedim>
-bool
-FE_Q_DG0<dim,spacedim>::has_support_on_face (const unsigned int shape_index,
- const unsigned int face_index) const
-{
- Assert (shape_index < this->dofs_per_cell,
- ExcIndexRange (shape_index, 0, this->dofs_per_cell));
- Assert (face_index < GeometryInfo<dim>::faces_per_cell,
- ExcIndexRange (face_index, 0, GeometryInfo<dim>::faces_per_cell));
-
-
- // first, special-case interior
- // shape functions, since they
- // have no support no-where on
- // the boundary
- if (((dim==2) && (shape_index>=this->first_quad_index))
- ||
- ((dim==3) && (shape_index>=this->first_hex_index)))
- return false;
-
- // let's see whether this is a
- // vertex
- if (shape_index < this->first_line_index)
- {
- // for Q_DG0 elements, there is
- // one dof per vertex, so
- // shape_index==vertex_number. check
- // whether this vertex is
- // on the given face. thus,
- // for each face, give a
- // list of vertices
- const unsigned int vertex_no = shape_index;
- Assert (vertex_no < GeometryInfo<dim>::vertices_per_cell,
- ExcInternalError());
-
- for (unsigned int v=0; v<GeometryInfo<dim>::vertices_per_face; ++v)
- if (GeometryInfo<dim>::face_to_cell_vertices(face_index, v) == vertex_no)
- return true;
-
- return false;
- }
- else if (shape_index < this->first_quad_index)
- // ok, dof is on a line
- {
- const unsigned int line_index
- = (shape_index - this->first_line_index) / this->dofs_per_line;
- Assert (line_index < GeometryInfo<dim>::lines_per_cell,
- ExcInternalError());
-
- // in 2d, the line is the
- // face, so get the line
- // index
- if (dim == 2)
- return (line_index == face_index);
- else if (dim == 3)
- {
- // see whether the
- // given line is on the
- // given face.
- for (unsigned int l=0; l<GeometryInfo<dim>::lines_per_face; ++l)
- if (GeometryInfo<3>::face_to_cell_lines(face_index, l) == line_index)
- return true;
-
- return false;
- }
- else
- Assert (false, ExcNotImplemented());
- }
- else if (shape_index < this->first_hex_index)
- // dof is on a quad
- {
- const unsigned int quad_index
- = (shape_index - this->first_quad_index) / this->dofs_per_quad;
- Assert (static_cast<signed int>(quad_index) <
- static_cast<signed int>(GeometryInfo<dim>::quads_per_cell),
- ExcInternalError());
-
- // in 2d, cell bubble are
- // zero on all faces. but
- // we have treated this
- // case above already
- Assert (dim != 2, ExcInternalError());
-
- // in 3d,
- // quad_index=face_index
- if (dim == 3)
- return (quad_index == face_index);
- else
- Assert (false, ExcNotImplemented());
- }
- else
- // dof on hex
- {
- // can only happen in 3d,
- // but this case has
- // already been covered
- // above
- Assert (false, ExcNotImplemented());
- return false;
- }
-
- // we should not have gotten here
- Assert (false, ExcInternalError());
- return false;
-}
-
-
-
-template <int dim, int spacedim>
-std::size_t
-FE_Q_DG0<dim,spacedim>::memory_consumption () const
-{
- return (FiniteElement<dim,spacedim>::memory_consumption() +
- MemoryConsumption::memory_consumption (face_index_map));
-}
-
-
-
// explicit instantiations
#include "fe_q_dg0.inst"
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/fe/fe_q_iso_q1.h>
+#include <deal.II/fe/fe_nothing.h>
+
+#include <vector>
+#include <sstream>
+
+DEAL_II_NAMESPACE_OPEN
+
+
+
+
+
+template <int dim, int spacedim>
+FE_Q_iso_Q1<dim,spacedim>::FE_Q_iso_Q1 (const unsigned int subdivisions)
+ :
+ FE_Q_Base<TensorProductPolynomials<dim,Polynomials::PiecewisePolynomial<double>>, dim, spacedim> (
+ TensorProductPolynomials<dim,Polynomials::PiecewisePolynomial<double> >
+ (Polynomials::generate_complete_Lagrange_basis_on_subdivisions(subdivisions, 1)),
+ FiniteElementData<dim>(this->get_dpo_vector(subdivisions),
+ 1, subdivisions,
+ FiniteElementData<dim>::H1),
+ std::vector<bool> (1, false))
+{
+ Assert (subdivisions > 0,
+ ExcMessage ("This element can only be used with a positive number of "
+ "subelements"));
+
+ QTrapez<1> trapez;
+ QIterated<1> points (trapez, subdivisions);
+
+ this->initialize(points.get_points());
+}
+
+
+
+template <int dim, int spacedim>
+std::string
+FE_Q_iso_Q1<dim,spacedim>::get_name () const
+{
+ // note that the FETools::get_fe_from_name function depends on the
+ // particular format of the string this function returns, so they have to be
+ // kept in synch
+
+ std::ostringstream namebuf;
+ namebuf << "FE_Q_iso_Q1<" << dim << ">(" << this->degree << ")";
+ return namebuf.str();
+}
+
+
+
+template <int dim, int spacedim>
+FiniteElement<dim,spacedim> *
+FE_Q_iso_Q1<dim,spacedim>::clone() const
+{
+ return new FE_Q_iso_Q1<dim,spacedim>(*this);
+}
+
+
+
+template <int dim, int spacedim>
+FiniteElementDomination::Domination
+FE_Q_iso_Q1<dim,spacedim>::
+compare_for_face_domination (const FiniteElement<dim,spacedim> &fe_other) const
+{
+ if (const FE_Q_iso_Q1<dim,spacedim> *fe_q_iso_q1_other
+ = dynamic_cast<const FE_Q_iso_Q1<dim,spacedim>*>(&fe_other))
+ {
+ // different behavior as in FE_Q: as FE_Q_iso_Q1(2) is not a subspace of
+ // FE_Q_iso_Q1(3), need that the element degrees are multiples of each
+ // other
+ if (this->degree < fe_q_iso_q1_other->degree &&
+ fe_q_iso_q1_other->degree % this->degree == 0)
+ return FiniteElementDomination::this_element_dominates;
+ else if (this->degree == fe_q_iso_q1_other->degree)
+ return FiniteElementDomination::either_element_can_dominate;
+ else if (this->degree > fe_q_iso_q1_other->degree &&
+ this->degree % fe_q_iso_q1_other->degree == 0)
+ return FiniteElementDomination::other_element_dominates;
+ else
+ return FiniteElementDomination::neither_element_dominates;
+ }
+ else if (dynamic_cast<const FE_Nothing<dim>*>(&fe_other) != 0)
+ {
+ // the FE_Nothing has no degrees of freedom and it is typically used in
+ // a context where we don't require any continuity along the interface
+ return FiniteElementDomination::no_requirements;
+ }
+
+ Assert (false, ExcNotImplemented());
+ return FiniteElementDomination::neither_element_dominates;
+}
+
+
+// explicit instantiations
+#include "fe_q_iso_q1.inst"
+
+DEAL_II_NAMESPACE_CLOSE
--- /dev/null
+//---------------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------------------------------------------------------
+
+
+for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS)
+ {
+#if deal_II_dimension <= deal_II_space_dimension
+ template class FE_Q_iso_Q1<deal_II_dimension, deal_II_space_dimension>;
+#endif
+ }
+
+
#include <deal.II/grid/grid_generator.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_q_dg0.h>
+#include <deal.II/fe/fe_q_iso_q1.h>
#include <deal.II/fe/fe_q_hierarchical.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_dgp.h>
--- /dev/null
+//----------------------------------------------------------------------
+// $Id: fe_prolongation_dgq.cc 26378 2012-09-14 12:58:04Z bangerth $
+// Version: $Name$
+//
+// Copyright (C) 2007, 2008, 2010, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+//
+// Compute support points
+
+#include "../tests.h"
+#include "fe_prolongation_common.h"
+
+
+
+int
+main()
+{
+ initlog(__FILE__);
+ deallog.threshold_double(1.e-10);
+
+ CHECK_ALL(Q_iso_Q1,1,2);
+ CHECK_ALL(Q_iso_Q1,2,2);
+ CHECK_ALL(Q_iso_Q1,3,2);
+ CHECK_ALL(Q_iso_Q1,4,2);
+
+ CHECK_ALL(Q_iso_Q1,1,3);
+ CHECK_ALL(Q_iso_Q1,2,3);
+}
--- /dev/null
+
+DEAL::Q_iso_Q11<2> constraint
+DEAL::0.5000000 0.5000000
+DEAL::Q_iso_Q11<2> prolongation 0
+DEAL::1.00000 ~ ~ ~
+DEAL::0.500000 0.500000 ~ ~
+DEAL::0.500000 ~ 0.500000 ~
+DEAL::0.250000 0.250000 0.250000 0.250000
+DEAL::Q_iso_Q11<2> prolongation 1
+DEAL::0.500000 0.500000 ~ ~
+DEAL::~ 1.00000 ~ ~
+DEAL::0.250000 0.250000 0.250000 0.250000
+DEAL::~ 0.500000 ~ 0.500000
+DEAL::Q_iso_Q11<2> prolongation 2
+DEAL::0.500000 ~ 0.500000 ~
+DEAL::0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ 1.00000 ~
+DEAL::~ ~ 0.500000 0.500000
+DEAL::Q_iso_Q11<2> prolongation 3
+DEAL::0.250000 0.250000 0.250000 0.250000
+DEAL::~ 0.500000 ~ 0.500000
+DEAL::~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ 1.00000
+DEAL::Q_iso_Q12<2> constraint
+DEAL::~ ~ 1.000000
+DEAL::0.5000000 ~ 0.5000000
+DEAL::~ 0.5000000 0.5000000
+DEAL::Q_iso_Q12<2> prolongation 0
+DEAL::1.00000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::0.250000 ~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::Q_iso_Q12<2> prolongation 1
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ 0.250000 ~ ~ ~ 0.250000 0.250000 ~ 0.250000
+DEAL::Q_iso_Q12<2> prolongation 2
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ 0.500000 ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ 0.250000 ~ 0.250000 ~ ~ 0.250000 0.250000
+DEAL::Q_iso_Q12<2> prolongation 3
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ 0.500000 ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000 0.250000
+DEAL::Q_iso_Q13<2> constraint
+DEAL::~ ~ 0.5000000 0.5000000
+DEAL::0.5000000 ~ 0.5000000 ~
+DEAL::~ ~ 1.000000 ~
+DEAL::~ ~ ~ 1.000000
+DEAL::~ 0.5000000 ~ 0.5000000
+DEAL::Q_iso_Q13<2> prolongation 0
+DEAL::1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~
+DEAL::0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~
+DEAL::0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::Q_iso_Q13<2> prolongation 1
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~
+DEAL::~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::Q_iso_Q13<2> prolongation 2
+DEAL::~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ 0.500000 ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~
+DEAL::Q_iso_Q13<2> prolongation 3
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000
+DEAL::Q_iso_Q14<2> constraint
+DEAL::~ ~ ~ 1.000000 ~
+DEAL::0.5000000 ~ 0.5000000 ~ ~
+DEAL::~ ~ 1.000000 ~ ~
+DEAL::~ ~ 0.5000000 0.5000000 ~
+DEAL::~ ~ ~ 0.5000000 0.5000000
+DEAL::~ ~ ~ ~ 1.000000
+DEAL::~ 0.5000000 ~ ~ 0.5000000
+DEAL::Q_iso_Q14<2> prolongation 0
+DEAL::1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000 0.250000 ~ ~ ~ ~
+DEAL::Q_iso_Q14<2> prolongation 1
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000 0.250000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~
+DEAL::Q_iso_Q14<2> prolongation 2
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~
+DEAL::~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~
+DEAL::Q_iso_Q14<2> prolongation 3
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000
+DEAL::Q_iso_Q11<3> constraint
+DEAL::0.2500000 0.2500000 0.2500000 0.2500000
+DEAL::0.5000000 ~ 0.5000000 ~
+DEAL::~ 0.5000000 ~ 0.5000000
+DEAL::0.5000000 0.5000000 ~ ~
+DEAL::~ ~ 0.5000000 0.5000000
+DEAL::Q_iso_Q11<3> prolongation 0
+DEAL::1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::0.500000 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::0.500000 ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::0.250000 0.250000 0.250000 0.250000 ~ ~ ~ ~
+DEAL::0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000 ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::Q_iso_Q11<3> prolongation 1
+DEAL::0.500000 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::0.250000 0.250000 0.250000 0.250000 ~ ~ ~ ~
+DEAL::~ 0.500000 ~ 0.500000 ~ ~ ~ ~
+DEAL::0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::~ 0.500000 ~ ~ ~ 0.500000 ~ ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ 0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::Q_iso_Q11<3> prolongation 2
+DEAL::0.500000 ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::0.250000 0.250000 0.250000 0.250000 ~ ~ ~ ~
+DEAL::~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000 ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ ~ 0.500000 ~ ~ ~ 0.500000 ~
+DEAL::~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::Q_iso_Q11<3> prolongation 3
+DEAL::0.250000 0.250000 0.250000 0.250000 ~ ~ ~ ~
+DEAL::~ 0.500000 ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ 0.500000 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ 0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::Q_iso_Q11<3> prolongation 4
+DEAL::0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000 ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ 0.500000 0.500000 ~ ~
+DEAL::~ ~ ~ ~ 0.500000 ~ 0.500000 ~
+DEAL::~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::Q_iso_Q11<3> prolongation 5
+DEAL::0.250000 0.250000 ~ ~ 0.250000 0.250000 ~ ~
+DEAL::~ 0.500000 ~ ~ ~ 0.500000 ~ ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ 0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::~ ~ ~ ~ 0.500000 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::Q_iso_Q11<3> prolongation 6
+DEAL::0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000 ~
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ ~ 0.500000 ~ ~ ~ 0.500000 ~
+DEAL::~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ 0.500000 ~ 0.500000 ~
+DEAL::~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::Q_iso_Q11<3> prolongation 7
+DEAL::0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000 0.125000
+DEAL::~ 0.250000 ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::~ ~ 0.250000 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ 0.250000 0.250000 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::Q_iso_Q12<3> constraint
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.000000
+DEAL::~ ~ ~ ~ 1.000000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 1.000000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 1.000000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.000000 ~
+DEAL::~ ~ ~ ~ ~ ~ 0.5000000 ~ 0.5000000
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.5000000 0.5000000
+DEAL::~ ~ ~ ~ 0.5000000 ~ ~ ~ 0.5000000
+DEAL::~ ~ ~ ~ ~ 0.5000000 ~ ~ 0.5000000
+DEAL::0.5000000 ~ ~ ~ 0.5000000 ~ ~ ~ ~
+DEAL::~ ~ 0.5000000 ~ 0.5000000 ~ ~ ~ ~
+DEAL::~ 0.5000000 ~ ~ ~ 0.5000000 ~ ~ ~
+DEAL::~ ~ ~ 0.5000000 ~ 0.5000000 ~ ~ ~
+DEAL::0.5000000 ~ ~ ~ ~ ~ 0.5000000 ~ ~
+DEAL::~ 0.5000000 ~ ~ ~ ~ 0.5000000 ~ ~
+DEAL::~ ~ 0.5000000 ~ ~ ~ ~ 0.5000000 ~
+DEAL::~ ~ ~ 0.5000000 ~ ~ ~ 0.5000000 ~
+DEAL::0.2500000 ~ ~ ~ 0.2500000 ~ 0.2500000 ~ 0.2500000
+DEAL::~ 0.2500000 ~ ~ ~ 0.2500000 0.2500000 ~ 0.2500000
+DEAL::~ ~ 0.2500000 ~ 0.2500000 ~ ~ 0.2500000 0.2500000
+DEAL::~ ~ ~ 0.2500000 ~ 0.2500000 ~ 0.2500000 0.2500000
+DEAL::Q_iso_Q12<3> prolongation 0
+DEAL::1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000
+DEAL::0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ 0.250000
+DEAL::0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ 0.250000
+DEAL::0.125000 ~ ~ ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ ~ ~ ~ ~ 0.125000 ~ ~ ~ 0.125000 ~ 0.125000 ~ 0.125000 ~ 0.125000
+DEAL::Q_iso_Q12<3> prolongation 1
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000
+DEAL::~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~
+DEAL::~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ 0.250000
+DEAL::~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 0.250000 ~ ~ ~ 0.250000
+DEAL::~ 0.125000 ~ ~ ~ ~ ~ ~ ~ 0.125000 0.125000 ~ ~ ~ ~ ~ ~ 0.125000 ~ ~ ~ 0.125000 0.125000 ~ 0.125000 ~ 0.125000
+DEAL::Q_iso_Q12<3> prolongation 2
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ 0.250000
+DEAL::~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~
+DEAL::~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ 0.250000 ~ ~ 0.250000
+DEAL::~ ~ 0.125000 ~ ~ ~ ~ ~ 0.125000 ~ ~ 0.125000 ~ ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ ~ 0.125000 0.125000 ~ 0.125000
+DEAL::Q_iso_Q12<3> prolongation 3
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ 0.250000
+DEAL::~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ 0.250000
+DEAL::~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~
+DEAL::~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000 ~ ~ 0.250000
+DEAL::~ ~ ~ 0.125000 ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ ~ ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ 0.125000 0.125000 ~ 0.125000
+DEAL::Q_iso_Q12<3> prolongation 4
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ 0.250000
+DEAL::~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~
+DEAL::~ ~ ~ ~ 0.125000 ~ ~ ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ 0.125000 ~ ~ ~ 0.125000 ~ 0.125000 ~ ~ 0.125000 0.125000
+DEAL::Q_iso_Q12<3> prolongation 5
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 0.250000 ~ ~ ~ 0.250000
+DEAL::~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ 0.125000 ~ ~ ~ ~ ~ ~ ~ 0.125000 0.125000 ~ ~ 0.125000 ~ ~ ~ 0.125000 0.125000 ~ ~ 0.125000 0.125000
+DEAL::Q_iso_Q12<3> prolongation 6
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ 0.250000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ 0.250000 ~ ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ 0.125000 ~ ~ ~ ~ ~ 0.125000 ~ ~ 0.125000 ~ ~ 0.125000 ~ 0.125000 ~ ~ 0.125000 ~ 0.125000 0.125000
+DEAL::Q_iso_Q12<3> prolongation 7
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 1.00000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 0.500000
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.500000 ~ ~ ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~ 0.250000 ~ ~ ~
+DEAL::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ 0.250000 ~ ~ 0.250000
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.250000 ~ ~ ~ ~ ~ 0.250000 ~ 0.250000 ~ ~ ~ ~ ~ ~ ~ ~ ~ 0.250000 ~
+DEAL::~ ~ ~ ~ ~ ~ ~ 0.125000 ~ ~ ~ ~ ~ 0.125000 ~ 0.125000 ~ ~ ~ 0.125000 ~ 0.125000 ~ 0.125000 ~ 0.125000 0.125000
#include <deal.II/grid/grid_generator.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_q_dg0.h>
+#include <deal.II/fe/fe_q_iso_q1.h>
#include <deal.II/fe/fe_q_hierarchical.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_dgp.h>
--- /dev/null
+//----------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2007, 2008, 2010, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+//
+// Compute support points
+
+#include "../tests.h"
+#include "fe_restriction_common.h"
+
+
+
+int
+main()
+{
+ initlog(__FILE__);
+ deallog.threshold_double(1.e-10);
+
+ CHECK_ALL(Q_iso_Q1,1,2);
+ CHECK_ALL(Q_iso_Q1,2,2);
+ CHECK_ALL(Q_iso_Q1,3,2);
+
+ CHECK_ALL(Q_iso_Q1,1,3);
+ CHECK_ALL(Q_iso_Q1,2,3);
+}
--- /dev/null
+
+DEAL::Q_iso_Q11<2> constraint
+DEAL::0.5000000 0.5000000
+DEAL::Q_iso_Q11<2> restriction 0
+DEAL::1.00000 0 0 0
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::Q_iso_Q11<2> restriction 1
+DEAL::0 0 0 0
+DEAL::0 1.00000 0 0
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::Q_iso_Q11<2> restriction 2
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::0 0 1.00000 0
+DEAL::0 0 0 0
+DEAL::Q_iso_Q11<2> restriction 3
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::0 0 0 0
+DEAL::0 0 0 1.00000
+DEAL::Q_iso_Q12<2> constraint
+DEAL::0 0 1.000000
+DEAL::0.5000000 0 0.5000000
+DEAL::0 0.5000000 0.5000000
+DEAL::Q_iso_Q12<2> restriction 0
+DEAL::1.00000 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0 0
+DEAL::Q_iso_Q12<2> restriction 1
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0 0
+DEAL::1.00000 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0
+DEAL::Q_iso_Q12<2> restriction 2
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::1.00000 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0 0
+DEAL::Q_iso_Q12<2> restriction 3
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0
+DEAL::1.00000 0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q13<2> constraint
+DEAL::0 0 0.5000000 0.5000000
+DEAL::0.5000000 0 0.5000000 0
+DEAL::0 0 1.000000 0
+DEAL::0 0 0 1.000000
+DEAL::0 0.5000000 0 0.5000000
+DEAL::Q_iso_Q13<2> restriction 0
+DEAL::1.00000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 1.00000 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 1.00000 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00000
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q13<2> restriction 1
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 1.00000 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 1.00000 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.00000 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q13<2> restriction 2
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 1.00000 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 1.00000 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 1.00000 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q13<2> restriction 3
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 1.00000 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 1.00000 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 1.00000 0 0 0
+DEAL::Q_iso_Q11<3> constraint
+DEAL::0.2500000 0.2500000 0.2500000 0.2500000
+DEAL::0.5000000 0 0.5000000 0
+DEAL::0 0.5000000 0 0.5000000
+DEAL::0.5000000 0.5000000 0 0
+DEAL::0 0 0.5000000 0.5000000
+DEAL::Q_iso_Q11<3> restriction 0
+DEAL::1.00000 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 1
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 1.00000 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 2
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 3
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 1.00000 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 4
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 1.00000 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 5
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 1.00000 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 6
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 1.00000 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::Q_iso_Q11<3> restriction 7
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 1.00000
+DEAL::Q_iso_Q12<3> constraint
+DEAL::0 0 0 0 0 0 0 0 1.000000
+DEAL::0 0 0 0 1.000000 0 0 0 0
+DEAL::0 0 0 0 0 1.000000 0 0 0
+DEAL::0 0 0 0 0 0 1.000000 0 0
+DEAL::0 0 0 0 0 0 0 1.000000 0
+DEAL::0 0 0 0 0 0 0.5000000 0 0.5000000
+DEAL::0 0 0 0 0 0 0 0.5000000 0.5000000
+DEAL::0 0 0 0 0.5000000 0 0 0 0.5000000
+DEAL::0 0 0 0 0 0.5000000 0 0 0.5000000
+DEAL::0.5000000 0 0 0 0.5000000 0 0 0 0
+DEAL::0 0 0.5000000 0 0.5000000 0 0 0 0
+DEAL::0 0.5000000 0 0 0 0.5000000 0 0 0
+DEAL::0 0 0 0.5000000 0 0.5000000 0 0 0
+DEAL::0.5000000 0 0 0 0 0 0.5000000 0 0
+DEAL::0 0.5000000 0 0 0 0 0.5000000 0 0
+DEAL::0 0 0.5000000 0 0 0 0 0.5000000 0
+DEAL::0 0 0 0.5000000 0 0 0 0.5000000 0
+DEAL::0.2500000 0 0 0 0.2500000 0 0.2500000 0 0.2500000
+DEAL::0 0.2500000 0 0 0 0.2500000 0.2500000 0 0.2500000
+DEAL::0 0 0.2500000 0 0.2500000 0 0 0.2500000 0.2500000
+DEAL::0 0 0 0.2500000 0 0.2500000 0 0.2500000 0.2500000
+DEAL::Q_iso_Q12<3> restriction 0
+DEAL::1.00000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 1.00000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+DEAL::0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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#include <deal.II/fe/fe_q_hierarchical.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_q_dg0.h>
+#include <deal.II/fe/fe_q_iso_q1.h>
#include <deal.II/fe/fe_dgp.h>
#include <deal.II/fe/fe_nedelec.h>
#include <deal.II/fe/fe_bdm.h>
--- /dev/null
+//----------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2007, 2008, 2010, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+//
+// Compute support points
+
+#include "../tests.h"
+#include "fe_support_points_common.h"
+
+
+
+int
+main()
+{
+ initlog(__FILE__);
+ deallog.threshold_double(1.e-10);
+
+ CHECK_ALL(Q_iso_Q1,1,2);
+ CHECK_ALL(Q_iso_Q1,2,2);
+ CHECK_ALL(Q_iso_Q1,3,2);
+
+ CHECK_ALL(Q_iso_Q1,1,3);
+ CHECK_ALL(Q_iso_Q1,2,3);
+}
--- /dev/null
+
+DEAL::Q_iso_Q11<2> cell support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::Q_iso_Q11<2> face 0 support points
+DEAL::0.00 0.00
+DEAL::0.00 1.00
+DEAL::Q_iso_Q11<2> face 1 support points
+DEAL::1.00 0.00
+DEAL::1.00 1.00
+DEAL::Q_iso_Q11<2> face 2 support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::Q_iso_Q11<2> face 3 support points
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::Q_iso_Q12<2> cell support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::0.00 0.500
+DEAL::1.00 0.500
+DEAL::0.500 0.00
+DEAL::0.500 1.00
+DEAL::0.500 0.500
+DEAL::Q_iso_Q12<2> face 0 support points
+DEAL::0.00 0.00
+DEAL::0.00 1.00
+DEAL::0.00 0.500
+DEAL::Q_iso_Q12<2> face 1 support points
+DEAL::1.00 0.00
+DEAL::1.00 1.00
+DEAL::1.00 0.500
+DEAL::Q_iso_Q12<2> face 2 support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::0.500 0.00
+DEAL::Q_iso_Q12<2> face 3 support points
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::0.500 1.00
+DEAL::Q_iso_Q13<2> cell support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::0.00 0.333
+DEAL::0.00 0.667
+DEAL::1.00 0.333
+DEAL::1.00 0.667
+DEAL::0.333 0.00
+DEAL::0.667 0.00
+DEAL::0.333 1.00
+DEAL::0.667 1.00
+DEAL::0.333 0.333
+DEAL::0.667 0.333
+DEAL::0.333 0.667
+DEAL::0.667 0.667
+DEAL::Q_iso_Q13<2> face 0 support points
+DEAL::0.00 0.00
+DEAL::0.00 1.00
+DEAL::0.00 0.333
+DEAL::0.00 0.667
+DEAL::Q_iso_Q13<2> face 1 support points
+DEAL::1.00 0.00
+DEAL::1.00 1.00
+DEAL::1.00 0.333
+DEAL::1.00 0.667
+DEAL::Q_iso_Q13<2> face 2 support points
+DEAL::0.00 0.00
+DEAL::1.00 0.00
+DEAL::0.333 0.00
+DEAL::0.667 0.00
+DEAL::Q_iso_Q13<2> face 3 support points
+DEAL::0.00 1.00
+DEAL::1.00 1.00
+DEAL::0.333 1.00
+DEAL::0.667 1.00
+DEAL::Q_iso_Q11<3> cell support points
+DEAL::0.00 0.00 0.00
+DEAL::1.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::Q_iso_Q11<3> face 0 support points
+DEAL::0.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::Q_iso_Q11<3> face 1 support points
+DEAL::1.00 0.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::1.00 0.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::Q_iso_Q11<3> face 2 support points
+DEAL::0.00 0.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 0.00
+DEAL::1.00 0.00 1.00
+DEAL::Q_iso_Q11<3> face 3 support points
+DEAL::0.00 1.00 0.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 0.00
+DEAL::1.00 1.00 1.00
+DEAL::Q_iso_Q11<3> face 4 support points
+DEAL::0.00 0.00 0.00
+DEAL::1.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::Q_iso_Q11<3> face 5 support points
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::Q_iso_Q12<3> cell support points
+DEAL::0.00 0.00 0.00
+DEAL::1.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::0.00 0.500 0.00
+DEAL::1.00 0.500 0.00
+DEAL::0.500 0.00 0.00
+DEAL::0.500 1.00 0.00
+DEAL::0.00 0.500 1.00
+DEAL::1.00 0.500 1.00
+DEAL::0.500 0.00 1.00
+DEAL::0.500 1.00 1.00
+DEAL::0.00 0.00 0.500
+DEAL::1.00 0.00 0.500
+DEAL::0.00 1.00 0.500
+DEAL::1.00 1.00 0.500
+DEAL::0.00 0.500 0.500
+DEAL::1.00 0.500 0.500
+DEAL::0.500 0.00 0.500
+DEAL::0.500 1.00 0.500
+DEAL::0.500 0.500 0.00
+DEAL::0.500 0.500 1.00
+DEAL::0.500 0.500 0.500
+DEAL::Q_iso_Q12<3> face 0 support points
+DEAL::0.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::0.00 0.00 0.500
+DEAL::0.00 1.00 0.500
+DEAL::0.00 0.500 0.00
+DEAL::0.00 0.500 1.00
+DEAL::0.00 0.500 0.500
+DEAL::Q_iso_Q12<3> face 1 support points
+DEAL::1.00 0.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::1.00 0.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::1.00 0.00 0.500
+DEAL::1.00 1.00 0.500
+DEAL::1.00 0.500 0.00
+DEAL::1.00 0.500 1.00
+DEAL::1.00 0.500 0.500
+DEAL::Q_iso_Q12<3> face 2 support points
+DEAL::0.00 0.00 0.00
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 0.00
+DEAL::1.00 0.00 1.00
+DEAL::0.500 0.00 0.00
+DEAL::0.500 0.00 1.00
+DEAL::0.00 0.00 0.500
+DEAL::1.00 0.00 0.500
+DEAL::0.500 0.00 0.500
+DEAL::Q_iso_Q12<3> face 3 support points
+DEAL::0.00 1.00 0.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 0.00
+DEAL::1.00 1.00 1.00
+DEAL::0.500 1.00 0.00
+DEAL::0.500 1.00 1.00
+DEAL::0.00 1.00 0.500
+DEAL::1.00 1.00 0.500
+DEAL::0.500 1.00 0.500
+DEAL::Q_iso_Q12<3> face 4 support points
+DEAL::0.00 0.00 0.00
+DEAL::1.00 0.00 0.00
+DEAL::0.00 1.00 0.00
+DEAL::1.00 1.00 0.00
+DEAL::0.00 0.500 0.00
+DEAL::1.00 0.500 0.00
+DEAL::0.500 0.00 0.00
+DEAL::0.500 1.00 0.00
+DEAL::0.500 0.500 0.00
+DEAL::Q_iso_Q12<3> face 5 support points
+DEAL::0.00 0.00 1.00
+DEAL::1.00 0.00 1.00
+DEAL::0.00 1.00 1.00
+DEAL::1.00 1.00 1.00
+DEAL::0.00 0.500 1.00
+DEAL::1.00 0.500 1.00
+DEAL::0.500 0.00 1.00
+DEAL::0.500 1.00 1.00
+DEAL::0.500 0.500 1.00
--- /dev/null
+//---------------------------- injection_q.cc ---------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2006, 2007, 2012 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------- injection_q.cc ---------------------------
+
+
+char logname[] = "injection_q_iso_q1/output";
+
+
+#include "injection_common.h"
+#include <deal.II/fe/fe_q_iso_q1.h>
+
+template <int dim>
+void test ()
+{
+ deallog << std::setprecision (4);
+ for (unsigned int i=1; i<4; ++i)
+ for (unsigned int j=i; j<4; ++j)
+ if (j%i == 0)
+ do_check (FE_Q_iso_Q1<dim>(i), FE_Q_iso_Q1<dim>(j));
+}
--- /dev/null
+
+DEAL::0 0 0.7500
+DEAL::0 1 0.2500
+DEAL::1 0 0.5000
+DEAL::1 1 0.5000
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::0 0 0.7500
+DEAL::0 1 0.2500
+DEAL::1 0 0.5000
+DEAL::1 1 0.5000
+DEAL::2 0 0.6250
+DEAL::2 1 0.3750
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::2 0 0.3750
+DEAL::2 1 0.6250
+DEAL::0 0 0.7500
+DEAL::0 1 0.2500
+DEAL::1 0 0.5000
+DEAL::1 1 0.5000
+DEAL::2 0 0.6667
+DEAL::2 1 0.3333
+DEAL::3 0 0.5833
+DEAL::3 1 0.4167
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::2 0 0.4167
+DEAL::2 1 0.5833
+DEAL::3 0 0.3333
+DEAL::3 1 0.6667
+DEAL::0 0 0.5000
+DEAL::0 1 0
+DEAL::0 2 0.5000
+DEAL::1 0 0
+DEAL::1 1 0
+DEAL::1 2 1.000
+DEAL::2 0 0.2500
+DEAL::2 1 0
+DEAL::2 2 0.7500
+DEAL::0 0 0
+DEAL::0 1 0
+DEAL::0 2 1.000
+DEAL::1 0 0
+DEAL::1 1 0.5000
+DEAL::1 2 0.5000
+DEAL::2 0 0
+DEAL::2 1 0.2500
+DEAL::2 2 0.7500
+DEAL::0 0 0.2500
+DEAL::0 1 0
+DEAL::0 2 0.7500
+DEAL::0 3 0
+DEAL::1 0 0
+DEAL::1 1 0
+DEAL::1 2 0.5000
+DEAL::1 3 0.5000
+DEAL::2 0 0
+DEAL::2 1 0
+DEAL::2 2 1.000
+DEAL::2 3 0
+DEAL::3 0 0
+DEAL::3 1 0
+DEAL::3 2 0.7500
+DEAL::3 3 0.2500
+DEAL::0 0 0
+DEAL::0 1 0
+DEAL::0 2 0.5000
+DEAL::0 3 0.5000
+DEAL::1 0 0
+DEAL::1 1 0.2500
+DEAL::1 2 0
+DEAL::1 3 0.7500
+DEAL::2 0 0
+DEAL::2 1 0
+DEAL::2 2 0.2500
+DEAL::2 3 0.7500
+DEAL::3 0 0
+DEAL::3 1 0
+DEAL::3 2 0
+DEAL::3 3 1.000
+DEAL::0 0 0.5625
+DEAL::0 1 0.1875
+DEAL::0 2 0.1875
+DEAL::0 3 0.06250
+DEAL::1 0 0.3750
+DEAL::1 1 0.3750
+DEAL::1 2 0.1250
+DEAL::1 3 0.1250
+DEAL::2 0 0.3750
+DEAL::2 1 0.1250
+DEAL::2 2 0.3750
+DEAL::2 3 0.1250
+DEAL::3 0 0.2500
+DEAL::3 1 0.2500
+DEAL::3 2 0.2500
+DEAL::3 3 0.2500
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::0 2 0
+DEAL::0 3 0
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::1 2 0
+DEAL::1 3 0
+DEAL::2 0 0.3750
+DEAL::2 1 0.3750
+DEAL::2 2 0.1250
+DEAL::2 3 0.1250
+DEAL::3 0 0.1875
+DEAL::3 1 0.5625
+DEAL::3 2 0.06250
+DEAL::3 3 0.1875
+DEAL::0 0 0.3750
+DEAL::0 1 0.1250
+DEAL::0 2 0.3750
+DEAL::0 3 0.1250
+DEAL::1 0 0.2500
+DEAL::1 1 0.2500
+DEAL::1 2 0.2500
+DEAL::1 3 0.2500
+DEAL::2 0 0.1875
+DEAL::2 1 0.06250
+DEAL::2 2 0.5625
+DEAL::2 3 0.1875
+DEAL::3 0 0.1250
+DEAL::3 1 0.1250
+DEAL::3 2 0.3750
+DEAL::3 3 0.3750
+DEAL::0 0 0.1250
+DEAL::0 1 0.1250
+DEAL::0 2 0.3750
+DEAL::0 3 0.3750
+DEAL::1 0 0.06250
+DEAL::1 1 0.1875
+DEAL::1 2 0.1875
+DEAL::1 3 0.5625
+DEAL::2 0 0
+DEAL::2 1 0
+DEAL::2 2 0.5000
+DEAL::2 3 0.5000
+DEAL::3 0 0
+DEAL::3 1 0
+DEAL::3 2 0.2500
+DEAL::3 3 0.7500
+DEAL::0 0 0.5625
+DEAL::0 1 0.1875
+DEAL::0 2 0.1875
+DEAL::0 3 0.06250
+DEAL::1 0 0.3750
+DEAL::1 1 0.3750
+DEAL::1 2 0.1250
+DEAL::1 3 0.1250
+DEAL::2 0 0.3750
+DEAL::2 1 0.1250
+DEAL::2 2 0.3750
+DEAL::2 3 0.1250
+DEAL::3 0 0.2500
+DEAL::3 1 0.2500
+DEAL::3 2 0.2500
+DEAL::3 3 0.2500
+DEAL::4 0 0.4688
+DEAL::4 1 0.1562
+DEAL::4 2 0.2812
+DEAL::4 3 0.09375
+DEAL::5 0 0.3125
+DEAL::5 1 0.3125
+DEAL::5 2 0.1875
+DEAL::5 3 0.1875
+DEAL::6 0 0.4688
+DEAL::6 1 0.2812
+DEAL::6 2 0.1562
+DEAL::6 3 0.09375
+DEAL::7 0 0.3125
+DEAL::7 1 0.1875
+DEAL::7 2 0.3125
+DEAL::7 3 0.1875
+DEAL::8 0 0.3906
+DEAL::8 1 0.2344
+DEAL::8 2 0.2344
+DEAL::8 3 0.1406
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::0 2 0
+DEAL::0 3 0
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::1 2 0
+DEAL::1 3 0
+DEAL::2 0 0.3750
+DEAL::2 1 0.3750
+DEAL::2 2 0.1250
+DEAL::2 3 0.1250
+DEAL::3 0 0.1875
+DEAL::3 1 0.5625
+DEAL::3 2 0.06250
+DEAL::3 3 0.1875
+DEAL::4 0 0.4375
+DEAL::4 1 0.4375
+DEAL::4 2 0.06250
+DEAL::4 3 0.06250
+DEAL::5 0 0.2188
+DEAL::5 1 0.6562
+DEAL::5 2 0.03125
+DEAL::5 3 0.09375
+DEAL::6 0 0.3750
+DEAL::6 1 0.6250
+DEAL::6 2 0
+DEAL::6 3 0
+DEAL::7 0 0.2812
+DEAL::7 1 0.4688
+DEAL::7 2 0.09375
+DEAL::7 3 0.1562
+DEAL::8 0 0.3281
+DEAL::8 1 0.5469
+DEAL::8 2 0.04688
+DEAL::8 3 0.07812
+DEAL::0 0 0.3750
+DEAL::0 1 0.1250
+DEAL::0 2 0.3750
+DEAL::0 3 0.1250
+DEAL::1 0 0.2500
+DEAL::1 1 0.2500
+DEAL::1 2 0.2500
+DEAL::1 3 0.2500
+DEAL::2 0 0.1875
+DEAL::2 1 0.06250
+DEAL::2 2 0.5625
+DEAL::2 3 0.1875
+DEAL::3 0 0.1250
+DEAL::3 1 0.1250
+DEAL::3 2 0.3750
+DEAL::3 3 0.3750
+DEAL::4 0 0.2812
+DEAL::4 1 0.09375
+DEAL::4 2 0.4688
+DEAL::4 3 0.1562
+DEAL::5 0 0.1875
+DEAL::5 1 0.1875
+DEAL::5 2 0.3125
+DEAL::5 3 0.3125
+DEAL::6 0 0.3125
+DEAL::6 1 0.1875
+DEAL::6 2 0.3125
+DEAL::6 3 0.1875
+DEAL::7 0 0.1562
+DEAL::7 1 0.09375
+DEAL::7 2 0.4688
+DEAL::7 3 0.2812
+DEAL::8 0 0.2344
+DEAL::8 1 0.1406
+DEAL::8 2 0.3906
+DEAL::8 3 0.2344
+DEAL::0 0 0.1250
+DEAL::0 1 0.1250
+DEAL::0 2 0.3750
+DEAL::0 3 0.3750
+DEAL::1 0 0.06250
+DEAL::1 1 0.1875
+DEAL::1 2 0.1875
+DEAL::1 3 0.5625
+DEAL::2 0 0
+DEAL::2 1 0
+DEAL::2 2 0.5000
+DEAL::2 3 0.5000
+DEAL::3 0 0
+DEAL::3 1 0
+DEAL::3 2 0.2500
+DEAL::3 3 0.7500
+DEAL::4 0 0.06250
+DEAL::4 1 0.06250
+DEAL::4 2 0.4375
+DEAL::4 3 0.4375
+DEAL::5 0 0.03125
+DEAL::5 1 0.09375
+DEAL::5 2 0.2188
+DEAL::5 3 0.6562
+DEAL::6 0 0.09375
+DEAL::6 1 0.1562
+DEAL::6 2 0.2812
+DEAL::6 3 0.4688
+DEAL::7 0 0
+DEAL::7 1 0
+DEAL::7 2 0.3750
+DEAL::7 3 0.6250
+DEAL::8 0 0.04688
+DEAL::8 1 0.07812
+DEAL::8 2 0.3281
+DEAL::8 3 0.5469
+DEAL::0 0 0.5625
+DEAL::0 1 0.1875
+DEAL::0 2 0.1875
+DEAL::0 3 0.06250
+DEAL::1 0 0.3750
+DEAL::1 1 0.3750
+DEAL::1 2 0.1250
+DEAL::1 3 0.1250
+DEAL::2 0 0.3750
+DEAL::2 1 0.1250
+DEAL::2 2 0.3750
+DEAL::2 3 0.1250
+DEAL::3 0 0.2500
+DEAL::3 1 0.2500
+DEAL::3 2 0.2500
+DEAL::3 3 0.2500
+DEAL::4 0 0.5000
+DEAL::4 1 0.1667
+DEAL::4 2 0.2500
+DEAL::4 3 0.08333
+DEAL::5 0 0.4375
+DEAL::5 1 0.1458
+DEAL::5 2 0.3125
+DEAL::5 3 0.1042
+DEAL::6 0 0.3333
+DEAL::6 1 0.3333
+DEAL::6 2 0.1667
+DEAL::6 3 0.1667
+DEAL::7 0 0.2917
+DEAL::7 1 0.2917
+DEAL::7 2 0.2083
+DEAL::7 3 0.2083
+DEAL::8 0 0.5000
+DEAL::8 1 0.2500
+DEAL::8 2 0.1667
+DEAL::8 3 0.08333
+DEAL::9 0 0.4375
+DEAL::9 1 0.3125
+DEAL::9 2 0.1458
+DEAL::9 3 0.1042
+DEAL::10 0 0.3333
+DEAL::10 1 0.1667
+DEAL::10 2 0.3333
+DEAL::10 3 0.1667
+DEAL::11 0 0.2917
+DEAL::11 1 0.2083
+DEAL::11 2 0.2917
+DEAL::11 3 0.2083
+DEAL::12 0 0.4444
+DEAL::12 1 0.2222
+DEAL::12 2 0.2222
+DEAL::12 3 0.1111
+DEAL::13 0 0.3889
+DEAL::13 1 0.2778
+DEAL::13 2 0.1944
+DEAL::13 3 0.1389
+DEAL::14 0 0.3889
+DEAL::14 1 0.1944
+DEAL::14 2 0.2778
+DEAL::14 3 0.1389
+DEAL::15 0 0.3403
+DEAL::15 1 0.2431
+DEAL::15 2 0.2431
+DEAL::15 3 0.1736
+DEAL::0 0 0.5000
+DEAL::0 1 0.5000
+DEAL::0 2 0
+DEAL::0 3 0
+DEAL::1 0 0.2500
+DEAL::1 1 0.7500
+DEAL::1 2 0
+DEAL::1 3 0
+DEAL::2 0 0.3750
+DEAL::2 1 0.3750
+DEAL::2 2 0.1250
+DEAL::2 3 0.1250
+DEAL::3 0 0.1875
+DEAL::3 1 0.5625
+DEAL::3 2 0.06250
+DEAL::3 3 0.1875
+DEAL::4 0 0.4583
+DEAL::4 1 0.4583
+DEAL::4 2 0.04167
+DEAL::4 3 0.04167
+DEAL::5 0 0.4167
+DEAL::5 1 0.4167
+DEAL::5 2 0.08333
+DEAL::5 3 0.08333
+DEAL::6 0 0.2292
+DEAL::6 1 0.6875
+DEAL::6 2 0.02083
+DEAL::6 3 0.06250
+DEAL::7 0 0.2083
+DEAL::7 1 0.6250
+DEAL::7 2 0.04167
+DEAL::7 3 0.1250
+DEAL::8 0 0.4167
+DEAL::8 1 0.5833
+DEAL::8 2 0
+DEAL::8 3 0
+DEAL::9 0 0.3333
+DEAL::9 1 0.6667
+DEAL::9 2 0
+DEAL::9 3 0
+DEAL::10 0 0.3125
+DEAL::10 1 0.4375
+DEAL::10 2 0.1042
+DEAL::10 3 0.1458
+DEAL::11 0 0.2500
+DEAL::11 1 0.5000
+DEAL::11 2 0.08333
+DEAL::11 3 0.1667
+DEAL::12 0 0.3819
+DEAL::12 1 0.5347
+DEAL::12 2 0.03472
+DEAL::12 3 0.04861
+DEAL::13 0 0.3056
+DEAL::13 1 0.6111
+DEAL::13 2 0.02778
+DEAL::13 3 0.05556
+DEAL::14 0 0.3472
+DEAL::14 1 0.4861
+DEAL::14 2 0.06944
+DEAL::14 3 0.09722
+DEAL::15 0 0.2778
+DEAL::15 1 0.5556
+DEAL::15 2 0.05556
+DEAL::15 3 0.1111
+DEAL::0 0 0.3750
+DEAL::0 1 0.1250
+DEAL::0 2 0.3750
+DEAL::0 3 0.1250
+DEAL::1 0 0.2500
+DEAL::1 1 0.2500
+DEAL::1 2 0.2500
+DEAL::1 3 0.2500
+DEAL::2 0 0.1875
+DEAL::2 1 0.06250
+DEAL::2 2 0.5625
+DEAL::2 3 0.1875
+DEAL::3 0 0.1250
+DEAL::3 1 0.1250
+DEAL::3 2 0.3750
+DEAL::3 3 0.3750
+DEAL::4 0 0.3125
+DEAL::4 1 0.1042
+DEAL::4 2 0.4375
+DEAL::4 3 0.1458
+DEAL::5 0 0.2500
+DEAL::5 1 0.08333
+DEAL::5 2 0.5000
+DEAL::5 3 0.1667
+DEAL::6 0 0.2083
+DEAL::6 1 0.2083
+DEAL::6 2 0.2917
+DEAL::6 3 0.2917
+DEAL::7 0 0.1667
+DEAL::7 1 0.1667
+DEAL::7 2 0.3333
+DEAL::7 3 0.3333
+DEAL::8 0 0.3333
+DEAL::8 1 0.1667
+DEAL::8 2 0.3333
+DEAL::8 3 0.1667
+DEAL::9 0 0.2917
+DEAL::9 1 0.2083
+DEAL::9 2 0.2917
+DEAL::9 3 0.2083
+DEAL::10 0 0.1667
+DEAL::10 1 0.08333
+DEAL::10 2 0.5000
+DEAL::10 3 0.2500
+DEAL::11 0 0.1458
+DEAL::11 1 0.1042
+DEAL::11 2 0.4375
+DEAL::11 3 0.3125
+DEAL::12 0 0.2778
+DEAL::12 1 0.1389
+DEAL::12 2 0.3889
+DEAL::12 3 0.1944
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+DEAL::60 60 0
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--- /dev/null
+//----------------------------------------------------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2005 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//----------------------------------------------------------------------
+
+#include "interpolate_common.h"
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/quadrature_lib.h>
+
+#include <deal.II/fe/fe_q_iso_q1.h>
+
+#include <fstream>
+
+// FE_Q_iso_Q1<dim>::interpolate(...)
+
+template <int dim>
+void check(const Function<dim>& f,
+ const unsigned int degree)
+{
+ FE_Q_iso_Q1<dim> fe(degree);
+ deallog << fe.get_name() << ' ';
+
+ std::vector<double> dofs(fe.dofs_per_cell);
+
+ std::vector<std::vector<double> >
+ values(1, std::vector<double>(fe.get_unit_support_points().size()));
+ f.value_list(fe.get_unit_support_points(), values[0]);
+ fe.interpolate(dofs, values[0]);
+ deallog << " value " << difference(fe,dofs,f);
+ fe.interpolate(dofs, values);
+ deallog << " vector " << difference(fe,dofs,f);
+
+ std::vector<Vector<double> >
+ vectors(fe.get_unit_support_points().size(), Vector<double>(1));
+ f.vector_value_list(fe.get_unit_support_points(), vectors);
+ fe.interpolate(dofs, vectors, 0);
+ deallog << " Vector " << difference(fe,dofs,f) << std::endl;
+}
+
+int main()
+{
+ std::ofstream logfile ("interpolate_q_iso_q1/output");
+ deallog.attach(logfile);
+ deallog.depth_console(0);
+ deallog.threshold_double(2.e-15);
+
+ Q1WedgeFunction<1,1> w1;
+ check(w1,1);
+ check(w1,2);
+ check(w1,3);
+ Q1WedgeFunction<2,1> w2;
+ check(w2,1);
+ check(w2,2);
+ check(w2,3);
+ Q1WedgeFunction<3,1> w3;
+ check(w3,1);
+ check(w3,2);
+ check(w3,3);
+}
--- /dev/null
+
+DEAL::FE_Q_iso_Q1<1>(1) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<1>(2) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<1>(3) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<2>(1) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<2>(2) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<2>(3) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<3>(1) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<3>(2) value 0 vector 0 Vector 0
+DEAL::FE_Q_iso_Q1<3>(3) value 0 vector 0 Vector 0
--- /dev/null
+//---------------------------- abfprojection_01.cc ---------------------------
+// abfprojection_01.cc,v 1.3 2003/06/09 16:00:38 wolf Exp
+// Version:
+//
+// Copyright (C) 2003, 2005, 2006 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------- abfprojection_01.cc ---------------------------
+
+/*
+ * Project the function [1,1] onto a deformed grid and see whether the
+ * FESystem elements can represent it exactly. This shouldn't be a surprise,
+ * but it is nice to compare with the RT and ABF elements
+ */
+
+
+
+char logname[] = "q_iso_q1_projection_01/output";
+#include "deformed_projection.h"
+#include <deal.II/fe/fe_q_iso_q1.h>
+
+void test ()
+{
+ FESystem<2> fe (FE_Q_iso_Q1<2>(3), 2);
+ check (fe);
+}
--- /dev/null
+
+DEAL::Dofs/cell 32Dofs/face 8
+DEAL::Dofs total 32
+DEAL::MM created
+DEAL::RHS created
+DEAL:cg::Starting value 0.14
+DEAL:cg::Convergence step 14 value 0
+# This file was generated by the deal.II library.
+
+
+#
+# For a description of the GNUPLOT format see the GNUPLOT manual.
+#
+# <x> <y> <solution_0> <solution_1>
+0.00 0.00 1.00 1.00
+0.25 0.00 1.00 1.00
+0.50 0.00 1.00 1.00
+0.75 0.00 1.00 1.00
+1.00 0.00 1.00 1.00
+
+0.15 0.12 1.00 1.00
+0.36 0.12 1.00 1.00
+0.57 0.12 1.00 1.00
+0.79 0.12 1.00 1.00
+1.00 0.12 1.00 1.00
+
+0.30 0.25 1.00 1.00
+0.47 0.25 1.00 1.00
+0.65 0.25 1.00 1.00
+0.82 0.25 1.00 1.00
+1.00 0.25 1.00 1.00
+
+0.45 0.38 1.00 1.00
+0.59 0.38 1.00 1.00
+0.73 0.38 1.00 1.00
+0.86 0.38 1.00 1.00
+1.00 0.38 1.00 1.00
+
+0.60 0.50 1.00 1.00
+0.70 0.50 1.00 1.00
+0.80 0.50 1.00 1.00
+0.90 0.50 1.00 1.00
+1.00 0.50 1.00 1.00
+
+
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
+DEAL::Dofs total 32
+DEAL::MM created
+DEAL::RHS created
+DEAL:cg::Starting value 0.09
+DEAL:cg::Convergence step 14 value 0
+# This file was generated by the deal.II library.
+
+
+#
+# For a description of the GNUPLOT format see the GNUPLOT manual.
+#
+# <x> <y> <solution_0> <solution_1>
+0.60 0.50 1.00 1.00
+0.70 0.50 1.00 1.00
+0.80 0.50 1.00 1.00
+0.90 0.50 1.00 1.00
+1.00 0.50 1.00 1.00
+
+0.57 0.62 1.00 1.00
+0.68 0.62 1.00 1.00
+0.79 0.62 1.00 1.00
+0.89 0.62 1.00 1.00
+1.00 0.62 1.00 1.00
+
+0.55 0.75 1.00 1.00
+0.66 0.75 1.00 1.00
+0.78 0.75 1.00 1.00
+0.89 0.75 1.00 1.00
+1.00 0.75 1.00 1.00
+
+0.53 0.88 1.00 1.00
+0.64 0.88 1.00 1.00
+0.76 0.88 1.00 1.00
+0.88 0.88 1.00 1.00
+1.00 0.88 1.00 1.00
+
+0.50 1.00 1.00 1.00
+0.62 1.00 1.00 1.00
+0.75 1.00 1.00 1.00
+0.88 1.00 1.00 1.00
+1.00 1.00 1.00 1.00
+
+
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
+DEAL::Dofs total 32
+DEAL::MM created
+DEAL::RHS created
+DEAL:cg::Starting value 0.17
+DEAL:cg::Convergence step 14 value 0
+# This file was generated by the deal.II library.
+
+
+#
+# For a description of the GNUPLOT format see the GNUPLOT manual.
+#
+# <x> <y> <solution_0> <solution_1>
+0.00 0.00 1.00 1.00
+0.15 0.12 1.00 1.00
+0.30 0.25 1.00 1.00
+0.45 0.38 1.00 1.00
+0.60 0.50 1.00 1.00
+
+0.00 0.25 1.00 1.00
+0.14 0.34 1.00 1.00
+0.29 0.44 1.00 1.00
+0.43 0.53 1.00 1.00
+0.57 0.62 1.00 1.00
+
+0.00 0.50 1.00 1.00
+0.14 0.56 1.00 1.00
+0.28 0.62 1.00 1.00
+0.41 0.69 1.00 1.00
+0.55 0.75 1.00 1.00
+
+0.00 0.75 1.00 1.00
+0.13 0.78 1.00 1.00
+0.26 0.81 1.00 1.00
+0.39 0.84 1.00 1.00
+0.53 0.88 1.00 1.00
+
+0.00 1.00 1.00 1.00
+0.12 1.00 1.00 1.00
+0.25 1.00 1.00 1.00
+0.38 1.00 1.00 1.00
+0.50 1.00 1.00 1.00
+
+
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
+DEAL::1.00
--- /dev/null
+#!/usr/bin/gnuplot -persist
+#
+#
+# G N U P L O T
+# Version 4.0 patchlevel 0
+# last modified Thu Apr 15 14:44:22 CEST 2004
+# System: Linux 2.6.11.4-21.11-smp
+#
+# Copyright (C) 1986 - 1993, 1998, 2004, 2006
+# Thomas Williams, Colin Kelley and many others
+#
+# This is gnuplot version 4.0. Please refer to the documentation
+# for command syntax changes. The old syntax will be accepted
+# throughout the 4.0 series, but all save files use the new syntax.
+#
+# Type `help` to access the on-line reference manual.
+# The gnuplot FAQ is available from
+# http://www.gnuplot.info/faq/
+#
+# Send comments and requests for help to
+# <gnuplot-info@lists.sourceforge.net>
+# Send bugs, suggestions and mods to
+# <gnuplot-bugs@lists.sourceforge.net>
+#
+set terminal postscript eps color solid enhanced 20
+# set output
+unset clip points
+set clip one
+unset clip two
+set bar 1.000000
+set border 31 lt -1 lw 1.000
+set xdata
+set ydata
+set zdata
+set x2data
+set y2data
+set timefmt x "%d/%m/%y,%H:%M"
+set timefmt y "%d/%m/%y,%H:%M"
+set timefmt z "%d/%m/%y,%H:%M"
+set timefmt x2 "%d/%m/%y,%H:%M"
+set timefmt y2 "%d/%m/%y,%H:%M"
+set timefmt cb "%d/%m/%y,%H:%M"
+set boxwidth
+set style fill empty border
+set dummy x,y
+set format x "% g"
+set format y "% g"
+set format x2 "% g"
+set format y2 "% g"
+set format z "% g"
+set format cb "% g"
+set angles radians
+unset grid
+set key title ""
+set key right top Right noreverse enhanced box linetype -2 linewidth 1.000 samplen 4 spacing 1 width 0 height 0 autotitles
+unset label
+unset arrow
+unset style line
+unset style arrow
+unset logscale
+set offsets 0, 0, 0, 0
+set pointsize 1
+set encoding default
+unset polar
+unset parametric
+unset decimalsign
+set view 60, 20, 1, 1
+set samples 100, 100
+set isosamples 10, 10
+set surface
+unset contour
+set clabel '%8.3g'
+set mapping cartesian
+set datafile separator whitespace
+set hidden3d offset 1 trianglepattern 3 undefined 1 altdiagonal bentover
+set cntrparam order 4
+set cntrparam linear
+set cntrparam levels auto 5
+set cntrparam points 5
+set size ratio 0 1,1
+set origin 0,0
+set style data lines
+set style function lines
+set xzeroaxis lt -2 lw 1.000
+set yzeroaxis lt -2 lw 1.000
+set x2zeroaxis lt -2 lw 1.000
+set y2zeroaxis lt -2 lw 1.000
+set tics in
+set ticslevel 0.5
+set ticscale 1 0.5
+set mxtics default
+set mytics default
+set mztics default
+set mx2tics default
+set my2tics default
+set mcbtics default
+set xtics border mirror norotate autofreq
+set ytics border mirror norotate autofreq
+set ztics border nomirror norotate autofreq
+set nox2tics
+set noy2tics
+set cbtics border mirror norotate autofreq
+set title "" 0.000000,0.000000 font ""
+set timestamp "" bottom norotate 0.000000,0.000000 ""
+set rrange [ * : * ] noreverse nowriteback # (currently [0.00000:10.0000] )
+set trange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] )
+set urange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] )
+set vrange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] )
+set xlabel "x" 0.000000,0.000000 font ""
+set x2label "" 0.000000,0.000000 font ""
+set xrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set x2range [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set ylabel "y" 0.000000,0.000000 font ""
+set y2label "" 0.000000,0.000000 font ""
+set yrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set y2range [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set zlabel "" 0.000000,0.000000 font ""
+set zrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set cblabel "" 0.000000,0.000000 font ""
+set cbrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] )
+set zero 1e-08
+set lmargin -1
+set bmargin -1
+set rmargin -1
+set tmargin -1
+set locale "C"
+set pm3d scansautomatic flush begin noftriangles nohidden3d implicit corners2color mean
+unset pm3d
+set palette positive nops_allcF maxcolors 0 gamma 1.5 color model RGB
+set palette rgbformulae 7, 5, 15
+set colorbox default
+set colorbox vertical origin 0.9,0.2 size 0.1,0.63 bdefault
+set loadpath
+set fontpath
+set fit noerrorvariables
+set output "RT1_00_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:3 title "v0_x"
+set output "RT1_00_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:4 title "v0_y"
+set output "RT1_01_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:5 title "v1_x"
+set output "RT1_01_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:6 title "v1_y"
+set output "RT1_02_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:7 title "v2_x"
+set output "RT1_02_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:8 title "v2_y"
+set output "RT1_03_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:9 title "v3_x"
+set output "RT1_03_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:10 title "v3_y"
+set output "RT1_04_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:11 title "v4_x"
+set output "RT1_04_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:12 title "v4_y"
+set output "RT1_05_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:13 title "v5_x"
+set output "RT1_05_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:14 title "v5_y"
+set output "RT1_06_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:15 title "v6_x"
+set output "RT1_06_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:16 title "v6_y"
+set output "RT1_07_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:17 title "v7_x"
+set output "RT1_07_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:18 title "v7_y"
+set output "RT1_08_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:19 title "v8_x"
+set output "RT1_08_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:20 title "v8_y"
+set output "RT1_09_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:21 title "v9_x"
+set output "RT1_09_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:22 title "v9_y"
+set output "RT1_10_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:23 title "v10_x"
+set output "RT1_10_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:24 title "v10_y"
+set output "RT1_11_x.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:25 title "v11_x"
+set output "RT1_11_y.eps"
+splot "< perl -ne \'print if s/DEAL:RT<2>.1.::value//;\' output" using 1:2:26 title "v11_y"
+# EOF
--- /dev/null
+// shapes.cc,v 1.18 2003/04/09 15:49:55 wolf Exp
+// (c) Guido Kanschat
+//
+// Show the shape functions implemented.
+
+#include "../tests.h"
+#include "shapes.h"
+#include <deal.II/fe/fe_q_iso_q1.h>
+#include <deal.II/fe/mapping_q1.h>
+#include <fstream>
+#include <string>
+
+#define PRECISION 2
+
+
+template<int dim>
+void plot_FE_Q_shape_functions()
+{
+ MappingQ1<dim> m;
+
+ FE_Q_iso_Q1<dim> q1(1);
+ plot_shape_functions(m, q1, "Q1");
+ plot_face_shape_functions(m, q1, "Q1");
+ test_compute_functions(m, q1, "Q1");
+
+ FE_Q_iso_Q1<dim> q2(2);
+ plot_shape_functions(m, q2, "Q2_iso_Q1");
+ plot_face_shape_functions(m, q2, "Q2_iso_Q1");
+ test_compute_functions(m, q2, "Q2_iso_Q1");
+
+ // skip the following tests to
+ // reduce run-time
+ if (dim < 3)
+ {
+ FE_Q_iso_Q1<dim> q3(3);
+ plot_shape_functions(m, q3, "Q3_iso_Q1");
+ plot_face_shape_functions(m, q3, "Q3_iso_Q1");
+ test_compute_functions(m, q3, "Q3_iso_Q1");
+
+ FE_Q_iso_Q1<dim> q4(4);
+ plot_shape_functions(m, q4, "Q4_iso_Q1");
+ plot_face_shape_functions(m, q4, "Q4_iso_Q1");
+ test_compute_functions(m, q4, "Q4_iso_Q1");
+ };
+
+}
+
+
+int
+main()
+{
+ std::ofstream logfile ("shapes_q_iso_q1/output");
+ deallog << std::setprecision(PRECISION) << std::fixed;
+ deallog.attach(logfile);
+ deallog.depth_console(0);
+ deallog.threshold_double(1.e-10);
+
+ plot_FE_Q_shape_functions<1>();
+ plot_FE_Q_shape_functions<2>();
+ plot_FE_Q_shape_functions<3>();
+
+ return 0;
+}
+
+
+
--- /dev/null
+
+DEAL:Cell1d-Q1::0.00 2.00 1.00
+DEAL:Cell1d-Q1::0.25 1.75 1.25
+DEAL:Cell1d-Q1::0.50 1.50 1.50
+DEAL:Cell1d-Q1::0.75 1.25 1.75
+DEAL:Cell1d-Q1::1.00 1.00 2.00
+DEAL:Cell1d-Q1::
+DEAL:Cell1d-Q1::
+DEAL:Cell1d-Q2_iso_Q1::0.00 2.00 1.00 1.00
+DEAL:Cell1d-Q2_iso_Q1::0.25 1.50 1.00 1.50
+DEAL:Cell1d-Q2_iso_Q1::0.50 1.00 1.00 2.00
+DEAL:Cell1d-Q2_iso_Q1::0.75 1.00 1.50 1.50
+DEAL:Cell1d-Q2_iso_Q1::1.00 1.00 2.00 1.00
+DEAL:Cell1d-Q2_iso_Q1::
+DEAL:Cell1d-Q2_iso_Q1::
+DEAL:Cell1d-Q3_iso_Q1::0.00 2.00 1.00 1.00 1.00
+DEAL:Cell1d-Q3_iso_Q1::0.25 1.25 1.00 1.75 1.00
+DEAL:Cell1d-Q3_iso_Q1::0.50 1.00 1.00 1.50 1.50
+DEAL:Cell1d-Q3_iso_Q1::0.75 1.00 1.25 1.00 1.75
+DEAL:Cell1d-Q3_iso_Q1::1.00 1.00 2.00 1.00 1.00
+DEAL:Cell1d-Q3_iso_Q1::
+DEAL:Cell1d-Q3_iso_Q1::
+DEAL:Cell1d-Q4_iso_Q1::0.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Cell1d-Q4_iso_Q1::0.25 1.00 1.00 2.00 1.00 1.00
+DEAL:Cell1d-Q4_iso_Q1::0.50 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell1d-Q4_iso_Q1::0.75 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell1d-Q4_iso_Q1::1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Cell1d-Q4_iso_Q1::
+DEAL:Cell1d-Q4_iso_Q1::
+DEAL:Cell2d-Q1::0.00 0.00 2.00 1.00 1.00 1.00
+DEAL:Cell2d-Q1::0.25 0.00 1.75 1.25 1.00 1.00
+DEAL:Cell2d-Q1::0.50 0.00 1.50 1.50 1.00 1.00
+DEAL:Cell2d-Q1::0.75 0.00 1.25 1.75 1.00 1.00
+DEAL:Cell2d-Q1::1.00 0.00 1.00 2.00 1.00 1.00
+DEAL:Cell2d-Q1::
+DEAL:Cell2d-Q1::0.00 0.25 1.75 1.00 1.25 1.00
+DEAL:Cell2d-Q1::0.25 0.25 1.56 1.19 1.19 1.06
+DEAL:Cell2d-Q1::0.50 0.25 1.38 1.38 1.12 1.12
+DEAL:Cell2d-Q1::0.75 0.25 1.19 1.56 1.06 1.19
+DEAL:Cell2d-Q1::1.00 0.25 1.00 1.75 1.00 1.25
+DEAL:Cell2d-Q1::
+DEAL:Cell2d-Q1::0.00 0.50 1.50 1.00 1.50 1.00
+DEAL:Cell2d-Q1::0.25 0.50 1.38 1.12 1.38 1.12
+DEAL:Cell2d-Q1::0.50 0.50 1.25 1.25 1.25 1.25
+DEAL:Cell2d-Q1::0.75 0.50 1.12 1.38 1.12 1.38
+DEAL:Cell2d-Q1::1.00 0.50 1.00 1.50 1.00 1.50
+DEAL:Cell2d-Q1::
+DEAL:Cell2d-Q1::0.00 0.75 1.25 1.00 1.75 1.00
+DEAL:Cell2d-Q1::0.25 0.75 1.19 1.06 1.56 1.19
+DEAL:Cell2d-Q1::0.50 0.75 1.12 1.12 1.38 1.38
+DEAL:Cell2d-Q1::0.75 0.75 1.06 1.19 1.19 1.56
+DEAL:Cell2d-Q1::1.00 0.75 1.00 1.25 1.00 1.75
+DEAL:Cell2d-Q1::
+DEAL:Cell2d-Q1::0.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell2d-Q1::0.25 1.00 1.00 1.00 1.75 1.25
+DEAL:Cell2d-Q1::0.50 1.00 1.00 1.00 1.50 1.50
+DEAL:Cell2d-Q1::0.75 1.00 1.00 1.00 1.25 1.75
+DEAL:Cell2d-Q1::1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell2d-Q1::
+DEAL:Cell2d-Q1::
+DEAL:Face2d-Q1::0.00 0.00 2.00 1.00 1.00 1.00
+DEAL:Face2d-Q1::0.00 0.12 1.75 1.00 1.25 1.00
+DEAL:Face2d-Q1::0.00 0.25 1.50 1.00 1.50 1.00
+DEAL:Face2d-Q1::0.00 0.38 1.25 1.00 1.75 1.00
+DEAL:Face2d-Q1::0.00 0.50 1.00 1.00 2.00 1.00
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::0.50 0.00 1.00 2.00 1.00 1.00
+DEAL:Face2d-Q1::0.50 0.06 1.00 1.88 1.00 1.12
+DEAL:Face2d-Q1::0.50 0.12 1.00 1.75 1.00 1.25
+DEAL:Face2d-Q1::0.50 0.19 1.00 1.62 1.00 1.38
+DEAL:Face2d-Q1::0.50 0.25 1.00 1.50 1.00 1.50
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::0.50 0.25 1.00 1.50 1.00 1.50
+DEAL:Face2d-Q1::0.50 0.31 1.00 1.38 1.00 1.62
+DEAL:Face2d-Q1::0.50 0.38 1.00 1.25 1.00 1.75
+DEAL:Face2d-Q1::0.50 0.44 1.00 1.12 1.00 1.88
+DEAL:Face2d-Q1::0.50 0.50 1.00 1.00 1.00 2.00
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::0.00 0.00 2.00 1.00 1.00 1.00
+DEAL:Face2d-Q1::0.12 0.00 1.75 1.25 1.00 1.00
+DEAL:Face2d-Q1::0.25 0.00 1.50 1.50 1.00 1.00
+DEAL:Face2d-Q1::0.38 0.00 1.25 1.75 1.00 1.00
+DEAL:Face2d-Q1::0.50 0.00 1.00 2.00 1.00 1.00
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::0.00 0.50 1.00 1.00 2.00 1.00
+DEAL:Face2d-Q1::0.12 0.50 1.00 1.00 1.75 1.25
+DEAL:Face2d-Q1::0.25 0.50 1.00 1.00 1.50 1.50
+DEAL:Face2d-Q1::0.38 0.50 1.00 1.00 1.25 1.75
+DEAL:Face2d-Q1::0.50 0.50 1.00 1.00 1.00 2.00
+DEAL:Face2d-Q1::
+DEAL:Face2d-Q1::
+DEAL:Cell2d-Q2_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.25 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.75 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::1.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Cell2d-Q2_iso_Q1::0.00 0.25 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.25 0.25 1.25 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.25
+DEAL:Cell2d-Q2_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50
+DEAL:Cell2d-Q2_iso_Q1::0.75 0.25 1.00 1.25 1.00 1.00 1.00 1.25 1.25 1.00 1.25
+DEAL:Cell2d-Q2_iso_Q1::1.00 0.25 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Cell2d-Q2_iso_Q1::0.00 0.50 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50
+DEAL:Cell2d-Q2_iso_Q1::0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell2d-Q2_iso_Q1::0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.50
+DEAL:Cell2d-Q2_iso_Q1::1.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Cell2d-Q2_iso_Q1::0.00 0.75 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.25 0.75 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.25 1.25
+DEAL:Cell2d-Q2_iso_Q1::0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50
+DEAL:Cell2d-Q2_iso_Q1::0.75 0.75 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.25 1.25
+DEAL:Cell2d-Q2_iso_Q1::1.00 0.75 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Cell2d-Q2_iso_Q1::0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.25 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::0.75 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell2d-Q2_iso_Q1::1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Cell2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.00 0.12 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.00 0.25 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.00 0.38 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.06 1.00 1.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.12 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.19 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.31 1.00 1.00 1.00 1.25 1.00 1.75 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.38 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.44 1.00 1.00 1.00 1.75 1.00 1.25 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.12 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.38 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.12 0.50 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Face2d-Q2_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Face2d-Q2_iso_Q1::0.38 0.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00
+DEAL:Face2d-Q2_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Face2d-Q2_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.25 0.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.75 0.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::1.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::0.00 0.25 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.25 0.25 1.06 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.56 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.38 1.38 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.75 0.25 1.00 1.06 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.19 1.00 1.00 1.00 1.56 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::1.00 0.25 1.00 1.25 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::0.00 0.50 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.38 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25
+DEAL:Cell2d-Q3_iso_Q1::0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.38
+DEAL:Cell2d-Q3_iso_Q1::1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::0.00 0.75 1.00 1.00 1.25 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.25 0.75 1.00 1.00 1.06 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.56 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.38 1.38
+DEAL:Cell2d-Q3_iso_Q1::0.75 0.75 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.56
+DEAL:Cell2d-Q3_iso_Q1::1.00 0.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::0.75 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Cell2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.00 0.12 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.00 0.25 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.00 0.38 1.00 1.00 1.25 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.06 1.00 1.62 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.12 1.00 1.25 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.19 1.00 1.00 1.00 1.00 1.00 1.00 1.88 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.88 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.38 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.44 1.00 1.00 1.00 1.62 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.12 0.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.38 0.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.12 0.50 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.38 0.50 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Face2d-Q3_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.75 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::1.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::0.00 0.25 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.75 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::1.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::0.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell2d-Q4_iso_Q1::1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Cell2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.00 0.12 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.00 0.25 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.00 0.38 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.06 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.44 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.12 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.38 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::0.00 0.50 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.12 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::0.50 0.50 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Face2d-Q4_iso_Q1::
+DEAL:Cell3d-Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.00 0.00 1.75 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.00 0.00 1.50 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.00 0.00 1.25 1.75 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.25 0.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.25 0.00 1.56 1.19 1.19 1.06 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.25 0.00 1.38 1.38 1.12 1.12 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.25 0.00 1.19 1.56 1.06 1.19 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.25 0.00 1.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.50 0.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.50 0.00 1.38 1.12 1.38 1.12 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.50 0.00 1.25 1.25 1.25 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.50 0.00 1.12 1.38 1.12 1.38 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.50 0.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.75 0.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.75 0.00 1.19 1.06 1.56 1.19 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.75 0.00 1.12 1.12 1.38 1.38 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.75 0.00 1.06 1.19 1.19 1.56 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.75 0.00 1.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 1.00 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 1.00 0.00 1.00 1.00 1.75 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.50 1.00 0.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.75 1.00 0.00 1.00 1.00 1.25 1.75 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::1.00 1.00 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.00 0.25 1.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.00 0.25 1.56 1.19 1.00 1.00 1.19 1.06 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.00 0.25 1.38 1.38 1.00 1.00 1.12 1.12 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.00 0.25 1.19 1.56 1.00 1.00 1.06 1.19 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.00 0.25 1.00 1.75 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.25 0.25 1.56 1.00 1.19 1.00 1.19 1.00 1.06 1.00
+DEAL:Cell3d-Q1::0.25 0.25 0.25 1.42 1.14 1.14 1.05 1.14 1.05 1.05 1.02
+DEAL:Cell3d-Q1::0.50 0.25 0.25 1.28 1.28 1.09 1.09 1.09 1.09 1.03 1.03
+DEAL:Cell3d-Q1::0.75 0.25 0.25 1.14 1.42 1.05 1.14 1.05 1.14 1.02 1.05
+DEAL:Cell3d-Q1::1.00 0.25 0.25 1.00 1.56 1.00 1.19 1.00 1.19 1.00 1.06
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.50 0.25 1.38 1.00 1.38 1.00 1.12 1.00 1.12 1.00
+DEAL:Cell3d-Q1::0.25 0.50 0.25 1.28 1.09 1.28 1.09 1.09 1.03 1.09 1.03
+DEAL:Cell3d-Q1::0.50 0.50 0.25 1.19 1.19 1.19 1.19 1.06 1.06 1.06 1.06
+DEAL:Cell3d-Q1::0.75 0.50 0.25 1.09 1.28 1.09 1.28 1.03 1.09 1.03 1.09
+DEAL:Cell3d-Q1::1.00 0.50 0.25 1.00 1.38 1.00 1.38 1.00 1.12 1.00 1.12
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.75 0.25 1.19 1.00 1.56 1.00 1.06 1.00 1.19 1.00
+DEAL:Cell3d-Q1::0.25 0.75 0.25 1.14 1.05 1.42 1.14 1.05 1.02 1.14 1.05
+DEAL:Cell3d-Q1::0.50 0.75 0.25 1.09 1.09 1.28 1.28 1.03 1.03 1.09 1.09
+DEAL:Cell3d-Q1::0.75 0.75 0.25 1.05 1.14 1.14 1.42 1.02 1.05 1.05 1.14
+DEAL:Cell3d-Q1::1.00 0.75 0.25 1.00 1.19 1.00 1.56 1.00 1.06 1.00 1.19
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 1.00 0.25 1.00 1.00 1.75 1.00 1.00 1.00 1.25 1.00
+DEAL:Cell3d-Q1::0.25 1.00 0.25 1.00 1.00 1.56 1.19 1.00 1.00 1.19 1.06
+DEAL:Cell3d-Q1::0.50 1.00 0.25 1.00 1.00 1.38 1.38 1.00 1.00 1.12 1.12
+DEAL:Cell3d-Q1::0.75 1.00 0.25 1.00 1.00 1.19 1.56 1.00 1.00 1.06 1.19
+DEAL:Cell3d-Q1::1.00 1.00 0.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.25
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.00 0.50 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.00 0.50 1.38 1.12 1.00 1.00 1.38 1.12 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.00 0.50 1.25 1.25 1.00 1.00 1.25 1.25 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.00 0.50 1.12 1.38 1.00 1.00 1.12 1.38 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.00 0.50 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.25 0.50 1.38 1.00 1.12 1.00 1.38 1.00 1.12 1.00
+DEAL:Cell3d-Q1::0.25 0.25 0.50 1.28 1.09 1.09 1.03 1.28 1.09 1.09 1.03
+DEAL:Cell3d-Q1::0.50 0.25 0.50 1.19 1.19 1.06 1.06 1.19 1.19 1.06 1.06
+DEAL:Cell3d-Q1::0.75 0.25 0.50 1.09 1.28 1.03 1.09 1.09 1.28 1.03 1.09
+DEAL:Cell3d-Q1::1.00 0.25 0.50 1.00 1.38 1.00 1.12 1.00 1.38 1.00 1.12
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.50 0.50 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00
+DEAL:Cell3d-Q1::0.25 0.50 0.50 1.19 1.06 1.19 1.06 1.19 1.06 1.19 1.06
+DEAL:Cell3d-Q1::0.50 0.50 0.50 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12
+DEAL:Cell3d-Q1::0.75 0.50 0.50 1.06 1.19 1.06 1.19 1.06 1.19 1.06 1.19
+DEAL:Cell3d-Q1::1.00 0.50 0.50 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.75 0.50 1.12 1.00 1.38 1.00 1.12 1.00 1.38 1.00
+DEAL:Cell3d-Q1::0.25 0.75 0.50 1.09 1.03 1.28 1.09 1.09 1.03 1.28 1.09
+DEAL:Cell3d-Q1::0.50 0.75 0.50 1.06 1.06 1.19 1.19 1.06 1.06 1.19 1.19
+DEAL:Cell3d-Q1::0.75 0.75 0.50 1.03 1.09 1.09 1.28 1.03 1.09 1.09 1.28
+DEAL:Cell3d-Q1::1.00 0.75 0.50 1.00 1.12 1.00 1.38 1.00 1.12 1.00 1.38
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 1.00 0.50 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell3d-Q1::0.25 1.00 0.50 1.00 1.00 1.38 1.12 1.00 1.00 1.38 1.12
+DEAL:Cell3d-Q1::0.50 1.00 0.50 1.00 1.00 1.25 1.25 1.00 1.00 1.25 1.25
+DEAL:Cell3d-Q1::0.75 1.00 0.50 1.00 1.00 1.12 1.38 1.00 1.00 1.12 1.38
+DEAL:Cell3d-Q1::1.00 1.00 0.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.00 0.75 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.00 0.75 1.19 1.06 1.00 1.00 1.56 1.19 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.00 0.75 1.12 1.12 1.00 1.00 1.38 1.38 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.00 0.75 1.06 1.19 1.00 1.00 1.19 1.56 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.00 0.75 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.25 0.75 1.19 1.00 1.06 1.00 1.56 1.00 1.19 1.00
+DEAL:Cell3d-Q1::0.25 0.25 0.75 1.14 1.05 1.05 1.02 1.42 1.14 1.14 1.05
+DEAL:Cell3d-Q1::0.50 0.25 0.75 1.09 1.09 1.03 1.03 1.28 1.28 1.09 1.09
+DEAL:Cell3d-Q1::0.75 0.25 0.75 1.05 1.14 1.02 1.05 1.14 1.42 1.05 1.14
+DEAL:Cell3d-Q1::1.00 0.25 0.75 1.00 1.19 1.00 1.06 1.00 1.56 1.00 1.19
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.50 0.75 1.12 1.00 1.12 1.00 1.38 1.00 1.38 1.00
+DEAL:Cell3d-Q1::0.25 0.50 0.75 1.09 1.03 1.09 1.03 1.28 1.09 1.28 1.09
+DEAL:Cell3d-Q1::0.50 0.50 0.75 1.06 1.06 1.06 1.06 1.19 1.19 1.19 1.19
+DEAL:Cell3d-Q1::0.75 0.50 0.75 1.03 1.09 1.03 1.09 1.09 1.28 1.09 1.28
+DEAL:Cell3d-Q1::1.00 0.50 0.75 1.00 1.12 1.00 1.12 1.00 1.38 1.00 1.38
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.75 0.75 1.06 1.00 1.19 1.00 1.19 1.00 1.56 1.00
+DEAL:Cell3d-Q1::0.25 0.75 0.75 1.05 1.02 1.14 1.05 1.14 1.05 1.42 1.14
+DEAL:Cell3d-Q1::0.50 0.75 0.75 1.03 1.03 1.09 1.09 1.09 1.09 1.28 1.28
+DEAL:Cell3d-Q1::0.75 0.75 0.75 1.02 1.05 1.05 1.14 1.05 1.14 1.14 1.42
+DEAL:Cell3d-Q1::1.00 0.75 0.75 1.00 1.06 1.00 1.19 1.00 1.19 1.00 1.56
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 1.00 0.75 1.00 1.00 1.25 1.00 1.00 1.00 1.75 1.00
+DEAL:Cell3d-Q1::0.25 1.00 0.75 1.00 1.00 1.19 1.06 1.00 1.00 1.56 1.19
+DEAL:Cell3d-Q1::0.50 1.00 0.75 1.00 1.00 1.12 1.12 1.00 1.00 1.38 1.38
+DEAL:Cell3d-Q1::0.75 1.00 0.75 1.00 1.00 1.06 1.19 1.00 1.00 1.19 1.56
+DEAL:Cell3d-Q1::1.00 1.00 0.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.75
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Cell3d-Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.75 1.25 1.00 1.00
+DEAL:Cell3d-Q1::0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00
+DEAL:Cell3d-Q1::0.75 0.00 1.00 1.00 1.00 1.00 1.00 1.25 1.75 1.00 1.00
+DEAL:Cell3d-Q1::1.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.25 1.00
+DEAL:Cell3d-Q1::0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.56 1.19 1.19 1.06
+DEAL:Cell3d-Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.38 1.38 1.12 1.12
+DEAL:Cell3d-Q1::0.75 0.25 1.00 1.00 1.00 1.00 1.00 1.19 1.56 1.06 1.19
+DEAL:Cell3d-Q1::1.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.25
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00
+DEAL:Cell3d-Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.38 1.12 1.38 1.12
+DEAL:Cell3d-Q1::0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25
+DEAL:Cell3d-Q1::0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.12 1.38 1.12 1.38
+DEAL:Cell3d-Q1::1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.75 1.00
+DEAL:Cell3d-Q1::0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.19 1.06 1.56 1.19
+DEAL:Cell3d-Q1::0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.38 1.38
+DEAL:Cell3d-Q1::0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.06 1.19 1.19 1.56
+DEAL:Cell3d-Q1::1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.75
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell3d-Q1::0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.25
+DEAL:Cell3d-Q1::0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50
+DEAL:Cell3d-Q1::0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.75
+DEAL:Cell3d-Q1::1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell3d-Q1::
+DEAL:Cell3d-Q1::
+DEAL:Face3d-Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.12 0.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.25 0.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.38 0.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.50 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.00 0.12 1.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.12 0.12 1.56 1.00 1.19 1.00 1.19 1.00 1.06 1.00
+DEAL:Face3d-Q1::0.00 0.25 0.12 1.38 1.00 1.38 1.00 1.12 1.00 1.12 1.00
+DEAL:Face3d-Q1::0.00 0.38 0.12 1.19 1.00 1.56 1.00 1.06 1.00 1.19 1.00
+DEAL:Face3d-Q1::0.00 0.50 0.12 1.00 1.00 1.75 1.00 1.00 1.00 1.25 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.00 0.25 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.12 0.25 1.38 1.00 1.12 1.00 1.38 1.00 1.12 1.00
+DEAL:Face3d-Q1::0.00 0.25 0.25 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00
+DEAL:Face3d-Q1::0.00 0.38 0.25 1.12 1.00 1.38 1.00 1.12 1.00 1.38 1.00
+DEAL:Face3d-Q1::0.00 0.50 0.25 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.00 0.38 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.00 0.12 0.38 1.19 1.00 1.06 1.00 1.56 1.00 1.19 1.00
+DEAL:Face3d-Q1::0.00 0.25 0.38 1.12 1.00 1.12 1.00 1.38 1.00 1.38 1.00
+DEAL:Face3d-Q1::0.00 0.38 0.38 1.06 1.00 1.19 1.00 1.19 1.00 1.56 1.00
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+DEAL:Face3d-Q1::
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+DEAL:Face3d-Q1::0.50 0.12 0.00 1.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.25 0.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.12 0.25 0.00 1.38 1.12 1.38 1.12 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.25 0.25 0.00 1.25 1.25 1.25 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.38 0.25 0.00 1.12 1.38 1.12 1.38 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.50 0.25 0.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.38 0.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.12 0.38 0.00 1.19 1.06 1.56 1.19 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.25 0.38 0.00 1.12 1.12 1.38 1.38 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.38 0.38 0.00 1.06 1.19 1.19 1.56 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.50 0.38 0.00 1.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.50 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.12 0.50 0.00 1.00 1.00 1.75 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.25 0.50 0.00 1.00 1.00 1.50 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.38 0.50 0.00 1.00 1.00 1.25 1.75 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.50 0.50 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.00 0.50 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Face3d-Q1::0.12 0.00 0.50 1.00 1.00 1.00 1.00 1.75 1.25 1.00 1.00
+DEAL:Face3d-Q1::0.25 0.00 0.50 1.00 1.00 1.00 1.00 1.50 1.50 1.00 1.00
+DEAL:Face3d-Q1::0.38 0.00 0.50 1.00 1.00 1.00 1.00 1.25 1.75 1.00 1.00
+DEAL:Face3d-Q1::0.50 0.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.12 0.50 1.00 1.00 1.00 1.00 1.75 1.00 1.25 1.00
+DEAL:Face3d-Q1::0.12 0.12 0.50 1.00 1.00 1.00 1.00 1.56 1.19 1.19 1.06
+DEAL:Face3d-Q1::0.25 0.12 0.50 1.00 1.00 1.00 1.00 1.38 1.38 1.12 1.12
+DEAL:Face3d-Q1::0.38 0.13 0.50 1.00 1.00 1.00 1.00 1.19 1.56 1.06 1.19
+DEAL:Face3d-Q1::0.50 0.13 0.50 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.25
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.25 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00
+DEAL:Face3d-Q1::0.12 0.25 0.50 1.00 1.00 1.00 1.00 1.38 1.12 1.38 1.12
+DEAL:Face3d-Q1::0.25 0.25 0.50 1.00 1.00 1.00 1.00 1.25 1.25 1.25 1.25
+DEAL:Face3d-Q1::0.38 0.25 0.50 1.00 1.00 1.00 1.00 1.12 1.38 1.12 1.38
+DEAL:Face3d-Q1::0.50 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.38 0.50 1.00 1.00 1.00 1.00 1.25 1.00 1.75 1.00
+DEAL:Face3d-Q1::0.12 0.38 0.50 1.00 1.00 1.00 1.00 1.19 1.06 1.56 1.19
+DEAL:Face3d-Q1::0.25 0.38 0.50 1.00 1.00 1.00 1.00 1.12 1.12 1.38 1.38
+DEAL:Face3d-Q1::0.38 0.38 0.50 1.00 1.00 1.00 1.00 1.06 1.19 1.19 1.56
+DEAL:Face3d-Q1::0.50 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.75
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::0.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Face3d-Q1::0.12 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.25
+DEAL:Face3d-Q1::0.25 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50
+DEAL:Face3d-Q1::0.38 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.75
+DEAL:Face3d-Q1::0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Face3d-Q1::
+DEAL:Face3d-Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.00 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.00 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.25 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.25 0.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.25 0.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.25 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.75 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.75 0.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.75 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.75 0.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.75 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 1.00 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 1.00 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 1.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 1.00 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 1.00 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.00 0.25 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.00 0.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.00 0.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.00 0.25 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.25 0.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.25 0.25 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.12 1.00 1.12
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.25 0.25 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.12 1.12 1.00 1.12 1.00 1.12
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.25 0.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.75 0.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.75 0.25 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.12 1.12 1.00 1.12
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.75 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.75 0.25 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.12 1.12 1.00 1.12
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.75 0.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 1.00 0.25 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 1.00 0.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 1.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 1.00 0.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 1.00 0.25 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.25 1.00 1.00 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.50
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.00 1.25
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.00 0.75 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.00 0.75 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.25 0.75 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.25 0.75 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.12 1.12
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.12 1.00 1.00 1.00 1.12 1.12 1.00 1.00 1.12 1.12
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.50
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.25
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.12 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.12 1.00 1.12 1.12
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.25
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.12 1.00 1.12 1.12
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Cell3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.12 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.12 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.12 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.12 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.38 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.38 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.50 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.06 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.06 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.06 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.06 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.12 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.12 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.12 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.12 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.19 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.19 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.19 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.19 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.06 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.06 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.06 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.06 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.06 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.12 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.13 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.13 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.13 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.19 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.56 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.19 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.19 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.19 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.31 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.31 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.31 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.31 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.38 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.38 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.13 0.38 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.38 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.44 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.44 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.13 0.44 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.44 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.06 0.50 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.13 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.19 0.50 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.56 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.31 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.38 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.06 1.00 1.19 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.38 1.00 1.00 1.00 1.00 1.00 1.12 1.00 1.12 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.56 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.00 1.00 1.00 1.00 1.19 1.00 1.06 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.44 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.31 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.44 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.75 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.12 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.38 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.50 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.12 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.38 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.12 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.38 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.12 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.38 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.12 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.12 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.13 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.50 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.25 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.13 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.38 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.12 0.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.12 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.12 0.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.12 0.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.38 0.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.38 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.38 0.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.00 0.50 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.00 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.00 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.00 0.50 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.12 0.50 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.12 0.50 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.12 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.13 0.50 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.13 0.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.25 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.25 1.00 1.25 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.25 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.38 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::0.00 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.12 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.25 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.38 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
+DEAL:Face3d-Q2_iso_Q1::
+DEAL:Face3d-Q2_iso_Q1::
--- /dev/null
+//---------------------------- precondition_q_iso_q1.cc ---------------------------
+// $Id$
+// Version: $Name$
+//
+// Copyright (C) 2013 by the deal.II authors
+//
+// This file is subject to QPL and may not be distributed
+// without copyright and license information. Please refer
+// to the file deal.II/doc/license.html for the text and
+// further information on this license.
+//
+//---------------------------- precondition_q_iso_q1.cc ---------------------------
+
+
+// solves a 2D Poisson equation for FE_Q elements using FE_Q_iso_Q1 elements
+// for the AMG preconditioner. The problem is taken from step-4
+
+#include "../tests.h"
+#include <deal.II/lac/trilinos_sparse_matrix.h>
+#include <deal.II/fe/fe_q.h>
+#include <deal.II/fe/fe_q_iso_q1.h>
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/lac/vector.h>
+#include <deal.II/lac/constraint_matrix.h>
+#include <deal.II/lac/compressed_sparsity_pattern.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/dofs/dof_tools.h>
+#include <deal.II/numerics/vector_tools.h>
+#include <deal.II/lac/trilinos_precondition.h>
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/base/function.h>
+#include <deal.II/grid/tria.h>
+
+#include <fstream>
+#include <iomanip>
+
+
+template <int dim>
+class Step4
+{
+public:
+ Step4 ();
+ void run ();
+
+private:
+ void make_grid ();
+ void setup_system();
+ void assemble_system ();
+ void assemble_preconditioner ();
+ void solve ();
+
+ Triangulation<dim> triangulation;
+ FE_Q<dim> fe;
+ DoFHandler<dim> dof_handler;
+
+ FE_Q_iso_Q1<dim> fe_precondition;
+ DoFHandler<dim> dof_handler_precondition;
+
+ ConstraintMatrix constraints;
+
+ TrilinosWrappers::SparseMatrix system_matrix;
+ TrilinosWrappers::SparseMatrix preconditioner_matrix;
+
+ Vector<double> solution;
+ Vector<double> system_rhs;
+};
+
+
+template <int dim>
+class RightHandSide : public Function<dim>
+{
+public:
+ RightHandSide () : Function<dim>() {}
+
+ virtual double value (const Point<dim> &p,
+ const unsigned int component = 0) const;
+};
+
+
+
+template <int dim>
+class BoundaryValues : public Function<dim>
+{
+public:
+ BoundaryValues () : Function<dim>() {}
+
+ virtual double value (const Point<dim> &p,
+ const unsigned int component = 0) const;
+};
+
+
+
+
+template <int dim>
+double RightHandSide<dim>::value (const Point<dim> &p,
+ const unsigned int /*component*/) const
+{
+ double return_value = 0;
+ for (unsigned int i=0; i<dim; ++i)
+ return_value += 4*std::pow(p(i), 4);
+
+ return return_value;
+}
+
+
+
+template <int dim>
+double BoundaryValues<dim>::value (const Point<dim> &p,
+ const unsigned int /*component*/) const
+{
+ return p.square();
+}
+
+
+
+template <int dim>
+Step4<dim>::Step4 ()
+ :
+ fe (3),
+ dof_handler (triangulation),
+ fe_precondition (3),
+ dof_handler_precondition(triangulation)
+{}
+
+
+template <int dim>
+void Step4<dim>::make_grid ()
+{
+ GridGenerator::hyper_cube (triangulation, -1, 1);
+ triangulation.refine_global (4);
+}
+
+
+
+template <int dim>
+void Step4<dim>::setup_system ()
+{
+ dof_handler.distribute_dofs (fe);
+ dof_handler_precondition.distribute_dofs (fe_precondition);
+
+ constraints.clear();
+ std::map<unsigned int,double> boundary_values;
+ VectorTools::interpolate_boundary_values (dof_handler,
+ 0,
+ BoundaryValues<dim>(),
+ constraints);
+ constraints.close();
+
+ CompressedSparsityPattern c_sparsity(dof_handler.n_dofs());
+ DoFTools::make_sparsity_pattern (dof_handler, c_sparsity, constraints, false);
+ system_matrix.reinit (c_sparsity);
+ preconditioner_matrix.reinit(c_sparsity);
+
+ solution.reinit (dof_handler.n_dofs());
+ system_rhs.reinit (dof_handler.n_dofs());
+}
+
+
+template <int dim>
+void Step4<dim>::assemble_system ()
+{
+ QGauss<dim> quadrature_formula(fe.degree+1);
+
+ const RightHandSide<dim> right_hand_side;
+
+ FEValues<dim> fe_values (fe, quadrature_formula,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ const unsigned int dofs_per_cell = fe.dofs_per_cell;
+ const unsigned int n_q_points = quadrature_formula.size();
+
+ FullMatrix<double> cell_matrix (dofs_per_cell, dofs_per_cell);
+ Vector<double> cell_rhs (dofs_per_cell);
+
+ std::vector<unsigned int> local_dof_indices (dofs_per_cell);
+
+ typename DoFHandler<dim>::active_cell_iterator
+ cell = dof_handler.begin_active(),
+ endc = dof_handler.end();
+
+ for (; cell!=endc; ++cell)
+ {
+ fe_values.reinit (cell);
+ cell_matrix = 0;
+ cell_rhs = 0;
+
+ for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
+ for (unsigned int i=0; i<dofs_per_cell; ++i)
+ {
+ for (unsigned int j=0; j<dofs_per_cell; ++j)
+ cell_matrix(i,j) += (fe_values.shape_grad (i, q_point) *
+ fe_values.shape_grad (j, q_point) *
+ fe_values.JxW (q_point));
+
+ cell_rhs(i) += (fe_values.shape_value (i, q_point) *
+ right_hand_side.value (fe_values.quadrature_point (q_point)) *
+ fe_values.JxW (q_point));
+ }
+
+ cell->get_dof_indices (local_dof_indices);
+ constraints.distribute_local_to_global(cell_matrix, cell_rhs,
+ local_dof_indices,
+ system_matrix, system_rhs);
+ }
+ system_matrix.compress(VectorOperation::add);
+}
+
+
+
+
+template <int dim>
+void Step4<dim>::assemble_preconditioner ()
+{
+ QIterated<dim> quadrature_formula(QGauss<1>(2), fe.degree);
+
+ const RightHandSide<dim> right_hand_side;
+
+ FEValues<dim> fe_values (fe_precondition, quadrature_formula,
+ update_values | update_gradients |
+ update_quadrature_points | update_JxW_values);
+
+ const unsigned int dofs_per_cell = fe_precondition.dofs_per_cell;
+ const unsigned int n_q_points = quadrature_formula.size();
+
+ FullMatrix<double> cell_matrix (dofs_per_cell, dofs_per_cell);
+ Vector<double> cell_rhs (dofs_per_cell);
+
+ std::vector<unsigned int> local_dof_indices (dofs_per_cell);
+
+ typename DoFHandler<dim>::active_cell_iterator
+ cell = dof_handler_precondition.begin_active(),
+ endc = dof_handler_precondition.end();
+
+ for (; cell!=endc; ++cell)
+ {
+ fe_values.reinit (cell);
+ cell_matrix = 0;
+ cell_rhs = 0;
+
+ for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
+ for (unsigned int i=0; i<dofs_per_cell; ++i)
+ {
+ for (unsigned int j=0; j<dofs_per_cell; ++j)
+ cell_matrix(i,j) += (fe_values.shape_grad (i, q_point) *
+ fe_values.shape_grad (j, q_point) *
+ fe_values.JxW (q_point));
+ }
+
+ cell->get_dof_indices (local_dof_indices);
+ constraints.distribute_local_to_global(cell_matrix,
+ local_dof_indices,
+ preconditioner_matrix);
+ }
+ preconditioner_matrix.compress(VectorOperation::add);
+}
+
+
+template <int dim>
+void Step4<dim>::solve ()
+{
+
+ // variant 1: solve with AMG
+ deallog.push(Utilities::int_to_string(dof_handler.n_dofs(),5));
+ deallog.push("AMG_Q");
+ {
+ solution = 0;
+ SolverControl solver_control (1000, 1e-12);
+ SolverCG<> solver (solver_control);
+ TrilinosWrappers::PreconditionAMG preconditioner;
+ preconditioner.initialize(system_matrix);
+ solver.solve (system_matrix, solution, system_rhs,
+ preconditioner);
+ }
+ deallog.pop();
+
+ deallog.push("AMG_Q_iso_Q1");
+ {
+ solution = 0;
+ SolverControl solver_control (1000, 1e-12);
+ SolverCG<> solver (solver_control);
+ TrilinosWrappers::PreconditionAMG preconditioner;
+ preconditioner.initialize(preconditioner_matrix);
+ solver.solve (system_matrix, solution, system_rhs,
+ preconditioner);
+ }
+ deallog.pop();
+ deallog << std::endl;
+ deallog.pop();
+}
+
+
+
+template <int dim>
+void Step4<dim>::run()
+{
+ for (unsigned int cycle = 0; cycle < 2; ++cycle)
+ {
+ if (cycle == 0)
+ make_grid();
+ else
+ triangulation.refine_global(1);
+
+ setup_system();
+ assemble_system();
+ assemble_preconditioner();
+ solve();
+ }
+}
+
+
+int main (int argc, char **argv)
+{
+ std::ofstream logfile("precondition_q_iso_q1/output");
+ deallog.attach(logfile);
+ deallog.depth_console(0);
+ deallog.threshold_double(1.e-10);
+
+ Utilities::MPI::MPI_InitFinalize mpi_initialization (argc, argv);
+
+ try
+ {
+ Step4<2> test;
+ test.run();
+ }
+ catch (std::exception &exc)
+ {
+ deallog << std::endl << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ deallog << "Exception on processing: " << std::endl
+ << exc.what() << std::endl
+ << "Aborting!" << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+
+ return 1;
+ }
+ catch (...)
+ {
+ deallog << std::endl << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ deallog << "Unknown exception!" << std::endl
+ << "Aborting!" << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ return 1;
+ };
+}
--- /dev/null
+
+DEAL:02401:AMG_Q:cg::Starting value 31.4643
+DEAL:02401:AMG_Q:cg::Convergence step 45 value 0
+DEAL:02401:AMG_Q_iso_Q1:cg::Starting value 31.4643
+DEAL:02401:AMG_Q_iso_Q1:cg::Convergence step 36 value 0
+DEAL:02401::
+DEAL:09409:AMG_Q:cg::Starting value 44.5271
+DEAL:09409:AMG_Q:cg::Convergence step 51 value 0
+DEAL:09409:AMG_Q_iso_Q1:cg::Starting value 44.5271
+DEAL:09409:AMG_Q_iso_Q1:cg::Convergence step 36 value 0
+DEAL:09409::
projects / dealii-svn.git / commitdiff