From: Marc Fehling Date: Mon, 7 Jan 2019 12:00:03 +0000 (+0100) Subject: SmoothnessEstimator: Enhancements regarding applicability. X-Git-Tag: v9.2.0-rc1~12^2~3 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=372e4f94d6d167c13e0404ea45c95ee4515c3614;p=dealii.git SmoothnessEstimator: Enhancements regarding applicability. --- diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index 3df58194b8..e4d9a420ad 100644 --- a/doc/doxygen/references.bib +++ b/doc/doxygen/references.bib @@ -715,6 +715,77 @@ year = {2008}, Address = {University of Minnesota}, Number = {91-279}, Url = {citeseer.ist.psu.edu/saad93flexible.html} + +@article{ainsworth1998hp, + author = {Ainsworth, Mark and Senior, Bill}, + title = {An adaptive refinement strategy for hp-finite element computations}, + journal = {{Applied Numerical Mathematics}}, + volume = {26}, + number = {1--2}, + pages = {165--178}, + publisher = {Elsevier}, + year = {1998}, + doi = {10.1016/S0168-9274(97)00083-4} +} + +@article{melenk2001hp, + author = {Melenk, Jens Markus and Wohlmuth, Barbara I.}, + title = {{On residual-based a posteriori error estimation in hp-FEM}}, + journal = {{Advances in Computational Mathematics}}, + volume = {15}, + number = {1}, + pages = {311--331}, + publisher = {Springer US}, + year = {2001}, + doi = {10.1023/A:1014268310921} +} + +@article{mavriplis1994hp, + author = {Mavriplis, Catherine}, + title = {Adaptive mesh strategies for the spectral element method}, + journal = {{Computer Methods in Applied Mechanics and Engineering}}, + year = {1994}, + volume = {116}, + number = {1}, + pages = {77--86}, + publisher = {Elsevier}, + doi = {10.1016/S0045-7825(94)80010-3} +} + +@article{houston2005hp, + author = {Houston, Paul and S{\"u}li, Endre}, + title = {A note on the design of hp-adaptive finite element methods for elliptic partial differential equations}, + journal = {{Computer Methods in Applied Mechanics and Engineering}}, + number = {2}, + pages = {229--243}, + publisher = {Elsevier}, + volume = {194}, + year = {2005}, + doi = {10.1016/j.cma.2004.04.009} +} + +@article{eibner2007hp, + author = {Eibner, Tino and Melenk, Jens Markus}, + title = {{An adaptive strategy for hp-FEM based on testing for analyticity}}, + journal = {{Computational Mechanics}}, + year = {2007}, + volume = {39}, + number = {5}, + pages = {575--595}, + publisher = {Springer}, + doi = {10.1007/s00466-006-0107-0} +} + +@article{davydov2017hp, + author = {Davydov, Denis and Gerasimov, Tymofiy and Pelteret, Jean-Paul and Steinmann, Paul}, + title = {{Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics}}, + journal = {{Advanced Modeling and Simulation in Engineering Sciences}}, + year = {2017}, + volume = {4}, + number = {1}, + pages = {7}, + issn = {2213-7467}, + doi = {10.1186/s40323-017-0093-0} } @Article{clevenger_par_gmg, diff --git a/doc/news/changes/incompatibilities/20200122Fehling b/doc/news/changes/incompatibilities/20200122Fehling new file mode 100644 index 0000000000..2905eb5bd0 --- /dev/null +++ b/doc/news/changes/incompatibilities/20200122Fehling @@ -0,0 +1,5 @@ +Changed: FESeries::Fourier and FESeries::Legendre now require to use an +individual number of coefficients per direction for each finite element +registered in the provided FECollection. +
+(Marc Fehling, 2020/01/22) diff --git a/doc/news/changes/major/20191030FehlingDavydov b/doc/news/changes/major/20191030FehlingDavydov new file mode 100644 index 0000000000..192d2cf755 --- /dev/null +++ b/doc/news/changes/major/20191030FehlingDavydov @@ -0,0 +1,4 @@ +New: Namespace SmoothnessEstimator providing smoothness estimation strategies +for hp-adaptive FEM based on Fourier and Legendre series expansions. +
+(Marc Fehling, Denis Davydov, 2019/10/30) diff --git a/doc/news/changes/minor/20181227DenisDavydov b/doc/news/changes/minor/20181227DenisDavydov index b45077c2c9..88a2c69b27 100644 --- a/doc/news/changes/minor/20181227DenisDavydov +++ b/doc/news/changes/minor/20181227DenisDavydov @@ -1,6 +1,7 @@ -New: Add FESeries::Legendre::get_size_in_each_direction() to retrieve +New: Add FESeries::Legendre::get_n_coefficients_per_direction() and +FESeries::Fourier::get_n_coefficients_per_direction() to retrieve the number of coefficients in each direction. Also add an Assert in -FESeries::Legendre::calculate() to check the dimension of the table to store -coefficients. +FESeries::Legendre::calculate() and FESeries::Fourier::calculate() to +check the dimension of the table to store coefficients.
(Denis Davydov, 2018/12/27) diff --git a/doc/news/changes/minor/20191030Fehling b/doc/news/changes/minor/20191030Fehling new file mode 100644 index 0000000000..09d88a1c0c --- /dev/null +++ b/doc/news/changes/minor/20191030Fehling @@ -0,0 +1,4 @@ +New: Member function hp::FECollection::max_degree() that returns the +maximal polynomial degree of all finite elements in the collection. +
+(Marc Fehling, 2019/10/30) diff --git a/doc/news/changes/minor/20200121Fehling-1 b/doc/news/changes/minor/20200121Fehling-1 new file mode 100644 index 0000000000..15f4f7d57d --- /dev/null +++ b/doc/news/changes/minor/20200121Fehling-1 @@ -0,0 +1,4 @@ +New: Both classes FESeries::Fourier and FESeries::Legendre now offer a +default constructor and can be initialized after their instantiation. +
+(Marc Fehling, 2020/01/21) diff --git a/doc/news/changes/minor/20200121Fehling-2 b/doc/news/changes/minor/20200121Fehling-2 new file mode 100644 index 0000000000..7a13044e72 --- /dev/null +++ b/doc/news/changes/minor/20200121Fehling-2 @@ -0,0 +1,4 @@ +New: FESeries::process_coefficients() now allows to ignore coefficients +below an absolute threshold, which has to be provided as a parameter. +
+(Marc Fehling, 2020/01/21) diff --git a/examples/step-27/doc/intro.dox b/examples/step-27/doc/intro.dox index 9e1cca1b68..0597da6789 100644 --- a/examples/step-27/doc/intro.dox +++ b/examples/step-27/doc/intro.dox @@ -367,7 +367,7 @@ and see how fast they decay. If they decay as then consequently the function we had here was in $H^{\mu-d/2}$. -

What has to be done

+

What we have to do

So what do we have to do to estimate the local smoothness of $u({\bf x})$ on a cell $K$? Clearly, the first step is to compute the Fourier coefficients @@ -403,9 +403,9 @@ cell. In other words, we can write it as a matrix-vector product @f] with the matrix @f[ - {\cal F}_{{\bf k},j} - = - \int_{\hat K} e^{i {\bf k}\cdot \hat{\bf x}} \hat \varphi_j(\hat{\bf x}) d\hat{\bf x}. + {\cal F}_{{\bf k},j} + = + \int_{\hat K} e^{i {\bf k}\cdot \hat{\bf x}} \hat \varphi_j(\hat{\bf x}) d\hat{\bf x}. @f] This matrix is easily computed for a given number of shape functions $\varphi_j$ and Fourier modes $N$. Consequently, finding the @@ -509,10 +509,6 @@ $\beta$, the formula above gives us a mean to calculate the value of the exponent $\mu$ that we can then use to determine that $\hat u(\hat{\bf x})$ is in $H^s(\hat K)$ with $s=\mu-\frac d2$. -These steps outlined above are applicable on many different scenarios, which -motivated the introduction of a generic function -SmoothnessEstimator::estimate_by_coefficient_decay() in deal.II, that combines all -the tasks described in this section in one simple function call.

Compensating for anisotropy

diff --git a/examples/step-27/step-27.cc b/examples/step-27/step-27.cc index 39cad4e140..70d163efdb 100644 --- a/examples/step-27/step-27.cc +++ b/examples/step-27/step-27.cc @@ -49,11 +49,11 @@ // These are the new files we need. The first and second provide hp // versions of the DoFHandler and FEValues classes as described in the -// introduction of this program. The last one provides the smoothness estimation -// algorithms on decaying series expansion coefficients. +// introduction of this program. The last one provides Fourier transformation +// class on the unit cell. #include #include -#include +#include // The last set of include files are standard C++ headers. We need support for // complex numbers when we compute the Fourier transform. @@ -99,7 +99,9 @@ namespace Step27 void assemble_system(); void solve(); void create_coarse_grid(); + void estimate_smoothness(Vector &smoothness_indicators); void postprocess(const unsigned int cycle); + std::pair predicate(const TableIndices &indices); Triangulation triangulation; @@ -108,6 +110,11 @@ namespace Step27 hp::QCollection quadrature_collection; hp::QCollection face_quadrature_collection; + hp::QCollection fourier_q_collection; + FESeries::Fourier fourier; + std::vector ln_k; + Table> fourier_coefficients; + AffineConstraints constraints; SparsityPattern sparsity_pattern; @@ -160,6 +167,23 @@ namespace Step27 // face quadrature objects. We start with quadratic elements, and each // quadrature formula is chosen so that it is appropriate for the matching // finite element in the hp::FECollection object. + // + // Finally, we initialize FESeries::Fourier object which will be used to + // calculate coefficient in Fourier series as described in the introduction. + // In addition to the hp::FECollection, we need to provide quadrature rules + // hp::QCollection for integration on the reference cell. + // + // In order to resize fourier_coefficients Table, we use the following + // auxiliary function + template + void resize(Table &coeff, const unsigned int N) + { + TableIndices size; + for (unsigned int d = 0; d < dim; d++) + size[d] = N; + coeff.reinit(size); + } + template LaplaceProblem::LaplaceProblem() : dof_handler(triangulation) @@ -171,6 +195,50 @@ namespace Step27 quadrature_collection.push_back(QGauss(degree + 1)); face_quadrature_collection.push_back(QGauss(degree + 1)); } + + // As described in the introduction, we define the Fourier vectors ${\bf + // k}$ for which we want to compute Fourier coefficients of the solution + // on each cell as follows. In 2d, we will need coefficients corresponding + // to vectors ${\bf k}=(2 \pi i, 2\pi j)^T$ for which $\sqrt{i^2+j^2}\le N$, + // with $i,j$ integers and $N$ being the maximal polynomial degree we use + // for the finite elements in this program. The FESeries::Fourier class' + // constructor first parameter $N$ defines the number of coefficients in 1D + // with the total number of coefficients being $N^{dim}$. Although we will + // not use coefficients corresponding to + // $\sqrt{i^2+j^2}> N$ and $i+j==0$, the overhead of their calculation is + // minimal. The transformation matrices for each FiniteElement will be + // calculated only once the first time they are required in the course of + // hp-adaptive refinement. Because we work on the unit cell, we can do all + // this work without a mapping from reference to real cell and consequently + // can precalculate these matrices. The calculation of expansion + // coefficients for a particular set of local degrees of freedom on a given + // cell then follows as a simple matrix-vector product. + // The 3d case is handled analogously. + const unsigned int N = max_degree; + + // We will need to assemble the matrices that do the Fourier transforms + // for each of the finite elements we deal with, i.e. the matrices ${\cal + // F}_{{\bf k},j}$ defined in the introduction. We have to do that for + // each of the finite elements in use. To that end we need a quadrature + // rule. In this example we use the same quadrature formula for each + // finite element, namely that is obtained by iterating a + // 2-point Gauss formula as many times as the maximal exponent we use for + // the term $e^{i{\bf k}\cdot{\bf x}}$: + QGauss<1> base_quadrature(2); + QIterated quadrature(base_quadrature, N); + for (unsigned int i = 0; i < fe_collection.size(); i++) + fourier_q_collection.push_back(quadrature); + + // Now we are ready to set-up the FESeries::Fourier object + const std::vector n_coefficients_per_direction( + fe_collection.size(), N); + fourier.initialize(n_coefficients_per_direction, + fe_collection, + fourier_q_collection); + + // We need to resize the matrix of fourier coefficients according to the + // number of modes N. + resize(fourier_coefficients, N); } @@ -337,7 +405,8 @@ namespace Step27 // Let us start with computing estimated error and smoothness indicators, // which each are one number for each active cell of our // triangulation. For the error indicator, we use the KellyErrorEstimator - // class as always. + // class as always. Estimating the smoothness is done in the respective + // function of this class; that function is discussed further down below: Vector estimated_error_per_cell(triangulation.n_active_cells()); KellyErrorEstimator::estimate( dof_handler, @@ -346,12 +415,9 @@ namespace Step27 solution, estimated_error_per_cell); - // Estimating the smoothness is performed with the method of decaing - // expansion coefficients as outlined in the introduction. - Vector smoothness_indicators; - SmoothnessEstimator::fourier_coefficient_decay(dof_handler, - solution, - smoothness_indicators); + + Vector smoothness_indicators(triangulation.n_active_cells()); + estimate_smoothness(smoothness_indicators); // Next we want to generate graphical output. In addition to the two // estimated quantities derived above, we would also like to output the @@ -549,6 +615,118 @@ namespace Step27 postprocess(cycle); } } + + + // @sect4{LaplaceProblem::estimate_smoothness} + + // As described in the introduction, we will need to take the maximum + // absolute value of fourier coefficients which correspond to $k$-vector + // $|{\bf k}|= const$. To filter the coefficients Table we + // will use the FESeries::process_coefficients() which requires a predicate + // to be specified. The predicate should operate on TableIndices and return + // a pair of bool and unsigned int. The latter + // is the value of the map from TableIndicies to unsigned int. It is + // used to define subsets of coefficients from which we search for the one + // with highest absolute value, i.e. $l^\infty$-norm. The bool + // parameter defines which indices should be used in processing. In the + // current case we are interested in coefficients which correspond to + // $0 < i*i+j*j < N*N$ and $0 < i*i+j*j+k*k < N*N$ in 2D and 3D, respectively. + template + std::pair + LaplaceProblem::predicate(const TableIndices &ind) + { + unsigned int v = 0; + for (unsigned int i = 0; i < dim; i++) + v += ind[i] * ind[i]; + if (v > 0 && v < max_degree * max_degree) + return std::make_pair(true, v); + else + return std::make_pair(false, v); + } + + // This last function of significance implements the algorithm to estimate + // the smoothness exponent using the algorithms explained in detail in the + // introduction. We will therefore only comment on those points that are of + // implementational importance. + template + void + LaplaceProblem::estimate_smoothness(Vector &smoothness_indicators) + { + // Since most of the hard work is done for us in FESeries::Fourier and + // we set up the object of this class in the constructor, what we are left + // to do here is apply this class to calculate coefficients and then + // perform linear regression to fit their decay slope. + + + // First thing to do is to loop over all cells and do our work there, i.e. + // to locally do the Fourier transform and estimate the decay coefficient. + // We will use the following array as a scratch array in the loop to store + // local DoF values: + Vector local_dof_values; + + // Then here is the loop: + for (const auto &cell : dof_handler.active_cell_iterators()) + { + // Inside the loop, we first need to get the values of the local + // degrees of freedom (which we put into the + // local_dof_values array after setting it to the right + // size) and then need to compute the Fourier transform by multiplying + // this vector with the matrix ${\cal F}$ corresponding to this finite + // element. This is done by calling FESeries::Fourier::calculate(), + // that has to be provided with the local_dof_values, + // cell->active_fe_index() and a Table to store + // coefficients. + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(solution, local_dof_values); + + fourier.calculate(local_dof_values, + cell->active_fe_index(), + fourier_coefficients); + + // The next thing, as explained in the introduction, is that we wanted + // to only fit our exponential decay of Fourier coefficients to the + // largest coefficients for each possible value of $|{\bf k}|$. To + // this end, we use FESeries::process_coefficients() to rework + // coefficients into the desired format. We'll only take those Fourier + // coefficients with the largest magnitude for a given value of $|{\bf + // k}|$ and thereby need to use VectorTools::Linfty_norm: + std::pair, std::vector> res = + FESeries::process_coefficients( + fourier_coefficients, + [this](const TableIndices &indices) { + return this->predicate(indices); + }, + VectorTools::Linfty_norm); + + Assert(res.first.size() == res.second.size(), ExcInternalError()); + + // The first vector in the std::pair will store values of + // the predicate, that is $i*i+j*j= const$ or $i*i+j*j+k*k = const$ in + // 2D or 3D respectively. This vector will be the same for all the cells + // so we can calculate logarithms of the corresponding Fourier vectors + // $|{\bf k}|$ only once in the whole hp-refinement cycle: + if (ln_k.size() == 0) + { + ln_k.resize(res.first.size(), 0); + for (unsigned int f = 0; f < ln_k.size(); f++) + ln_k[f] = + std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f])); + } + + // We have to calculate the logarithms of absolute values of + // coefficients and use it in a linear regression fit to obtain $\mu$. + for (double &residual_element : res.second) + residual_element = std::log(residual_element); + + std::pair fit = + FESeries::linear_regression(ln_k, res.second); + + // The final step is to compute the Sobolev index $s=\mu-\frac d2$ and + // store it in the vector of estimated values for each cell: + smoothness_indicators(cell->active_cell_index()) = + -fit.first - 1. * dim / 2; + } + } } // namespace Step27 diff --git a/include/deal.II/fe/fe_series.h b/include/deal.II/fe/fe_series.h index 229604c51e..dc4fe34234 100644 --- a/include/deal.II/fe/fe_series.h +++ b/include/deal.II/fe/fe_series.h @@ -13,8 +13,8 @@ // // --------------------------------------------------------------------- -#ifndef dealii_fe_series_H -#define dealii_fe_series_H +#ifndef dealii_fe_series_h +#define dealii_fe_series_h @@ -94,16 +94,50 @@ namespace FESeries using CoefficientType = typename std::complex; /** - * A non-default constructor. The @p size_in_each_direction defines the number - * of modes in each direction, @p fe_collection is the hp::FECollection + * Default constructor. + * + * Leaves all members in an empty or uninitialized state. Please call + * initialize() once you have all data structures ready. + */ + Fourier() = default; + + /** + * Constructor that initializes all required data structures. + * + * The @p n_coefficients_per_direction defines the number of coefficients in + * each direction, @p fe_collection is the hp::FECollection for which + * expansion will be used and @p q_collection is the hp::QCollection used to + * integrate the expansion for each FiniteElement in @p fe_collection. + */ + Fourier(const std::vector & n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection); + + /** + * A non-default constructor. The @p n_coefficients_per_direction defines the + * number of modes in each direction, @p fe_collection is the hp::FECollection * for which expansion will be used and @p q_collection is the hp::QCollection * used to integrate the expansion for each FiniteElement * in @p fe_collection. + * + * @deprecated Use a different constructor instead. */ - Fourier(const unsigned int size_in_each_direction, + DEAL_II_DEPRECATED + Fourier(const unsigned int n_coefficients_per_direction, const hp::FECollection &fe_collection, const hp::QCollection & q_collection); + /** + * Initialize and overwrite all mandatory data structures for the + * calculation calculation of transformation matrices. + * + * All previously calculated transformation matrices will be cleared. + */ + void + initialize(const std::vector &n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection); + /** * Calculate @p fourier_coefficients of the cell vector field given by * @p local_dof_values corresponding to FiniteElement with @@ -115,6 +149,13 @@ namespace FESeries const unsigned int cell_active_fe_index, Table & fourier_coefficients); + /** + * Return the number of coefficients in each coordinate direction for the + * finite element associated with @p index in the provided hp::FECollection. + */ + unsigned int + get_n_coefficients_per_direction(const unsigned int index) const; + /** * Calculate all transformation matrices to transfer the finite element * solution to the series expansion representation. @@ -156,6 +197,12 @@ namespace FESeries operator==(const Fourier &fourier) const; private: + /** + * Number of coefficients in each direction for each finite element in the + * registered hp::FECollection. + */ + std::vector n_coefficients_per_direction; + /** * hp::FECollection for which transformation matrices will be calculated. */ @@ -182,6 +229,8 @@ namespace FESeries std::vector unrolled_coefficients; }; + + /** * A class to calculate expansion of a scalar FE field into series of Legendre * functions on a reference element. @@ -231,17 +280,50 @@ namespace FESeries public: using CoefficientType = double; + /** + * Default constructor. + * + * Leaves all members in an empty or uninitialized state. Please call + * initialize() once you have all data structures ready. + */ + Legendre() = default; + + /** + * Constructor that initializes all required data structures. + * + * The @p n_coefficients_per_direction defines the number of coefficients in + * each direction, @p fe_collection is the hp::FECollection for which + * expansion will be used and @p q_collection is the hp::QCollection used to + * integrate the expansion for each FiniteElement in @p fe_collection. + */ + Legendre(const std::vector &n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection); + /** * A non-default constructor. The @p size_in_each_direction defines the number * of coefficients in each direction, @p fe_collection is the hp::FECollection * for which expansion will be used and @p q_collection is the hp::QCollection - * used to integrate the expansion for each FiniteElement - * in @p fe_collection. + * used to integrate the expansion for each FiniteElement in @p fe_collection. + * + * @deprecated Use a different constructor instead. */ - Legendre(const unsigned int size_in_each_direction, + DEAL_II_DEPRECATED + Legendre(const unsigned int n_coefficients_per_direction, const hp::FECollection &fe_collection, const hp::QCollection & q_collection); + /** + * Initialize and overwrite all mandatory data structures for the + * calculation of transformation matrices. + * + * All previously calculated transformation matrices will be cleared. + */ + void + initialize(const std::vector &n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection); + /** * Calculate @p legendre_coefficients of the cell vector field given by * @p local_dof_values corresponding to FiniteElement with @@ -254,10 +336,11 @@ namespace FESeries Table & legendre_coefficients); /** - * Return number of coefficients in each coordinate direction. + * Return the number of coefficients in each coordinate direction for the + * finite element associated with @p index in the provided hp::FECollection. */ unsigned int - get_size_in_each_direction() const; + get_n_coefficients_per_direction(const unsigned int index) const; /** * Calculate all transformation matrices to transfer the finite element @@ -301,9 +384,10 @@ namespace FESeries private: /** - * Number of coefficients in each direction + * Number of coefficients in each direction for each finite element in the + * registered hp::FECollection. */ - const unsigned int N; + std::vector n_coefficients_per_direction; /** * hp::FECollection for which transformation matrices will be calculated. @@ -327,6 +411,7 @@ namespace FESeries }; + /** * Calculate the @p norm of subsets of @p coefficients defined by * @p predicate being constant. Return the pair of vectors of predicate values @@ -337,18 +422,20 @@ namespace FESeries * used in calculation, whereas the latter is the unrolled value of indices * according to which the subsets of coefficients will be formed. * - * @note Only the following values of @p norm are implemented and make sense - * in this case: mean, L1_norm, L2_norm, Linfty_norm. The mean norm can only - * be applied to real valued coefficients. + * Only those coefficients will be considered which are larger than + * @p smallest_abs_coefficient. + * + * @note Only the following values of @p norm_type are implemented and make + * sense in this case: mean, L1_norm, L2_norm, Linfty_norm. The mean norm ca + * only be applied to real valued coefficients. */ template std::pair, std::vector> process_coefficients(const Table &coefficients, const std::function( const TableIndices &)> & predicate, - const VectorTools::NormType norm); - - + const VectorTools::NormType norm_type, + const double smallest_abs_coefficient = 1e-10); /** * Linear regression least-square fit of $y = k \, x + b$. @@ -362,6 +449,8 @@ namespace FESeries /*@}*/ + + #ifndef DOXYGEN // ------------------- inline and template functions ---------------- @@ -391,6 +480,8 @@ namespace internal pred_to_values[pred_value].push_back(coeff_value); } + + template void fill_map( @@ -406,6 +497,8 @@ namespace internal } } + + template void fill_map( @@ -422,6 +515,8 @@ namespace internal } } + + template void fill_map( @@ -440,6 +535,7 @@ namespace internal } + template double complex_mean_value(const Number &value) @@ -447,6 +543,8 @@ namespace internal return value; } + + template double complex_mean_value(const std::complex &value) @@ -461,14 +559,19 @@ namespace internal } // namespace internal + template std::pair, std::vector> FESeries::process_coefficients( const Table &coefficients, const std::function(const TableIndices &)> & predicate, - const VectorTools::NormType norm) + const VectorTools::NormType norm_type, + const double smallest_abs_coefficient) { + Assert(smallest_abs_coefficient >= 0., + ExcMessage("smallest_abs_coefficient should be non-negative.")); + std::vector predicate_values; std::vector norm_values; @@ -483,38 +586,44 @@ FESeries::process_coefficients( // now go through the map and populate the @p norm_values based on @p norm: for (const auto &pred_to_value : pred_to_values) { - predicate_values.push_back(pred_to_value.first); Vector values(pred_to_value.second.cbegin(), pred_to_value.second.cend()); - switch (norm) + double norm_value = 0; + switch (norm_type) { case VectorTools::L2_norm: { - norm_values.push_back(values.l2_norm()); + norm_value = values.l2_norm(); break; } case VectorTools::L1_norm: { - norm_values.push_back(values.l1_norm()); + norm_value = values.l1_norm(); break; } case VectorTools::Linfty_norm: { - norm_values.push_back(values.linfty_norm()); + norm_value = values.linfty_norm(); break; } case VectorTools::mean: { - norm_values.push_back( - internal::FESeriesImplementation::complex_mean_value( - values.mean_value())); + norm_value = internal::FESeriesImplementation::complex_mean_value( + values.mean_value()); break; } default: AssertThrow(false, ExcNotImplemented()); break; } + + // will use all non-zero coefficients + if (std::abs(norm_value) > smallest_abs_coefficient) + { + predicate_values.push_back(pred_to_value.first); + norm_values.push_back(norm_value); + } } return std::make_pair(predicate_values, norm_values); @@ -532,12 +641,11 @@ FESeries::Fourier::save_transformation_matrices( // Store information about those resources which have been used to generate // the transformation matrices. // mode vector - unsigned int size = k_vectors.n_elements(); - ar & size; + ar &n_coefficients_per_direction; // finite element collection - size = fe_collection->size(); - ar &size; + unsigned int size = fe_collection->size(); + ar & size; for (unsigned int i = 0; i < size; ++i) ar &(*fe_collection)[i].get_name(); @@ -563,12 +671,16 @@ FESeries::Fourier::load_transformation_matrices( // Check whether the currently registered resources are compatible with // the transformation matrices to load. // mode vector - unsigned int size; - ar & size; - AssertDimension(size, k_vectors.n_elements()); + std::vector compare_coefficients; + ar & compare_coefficients; + Assert(compare_coefficients == n_coefficients_per_direction, + ExcMessage("A different number of coefficients vector has been used " + "to generate the transformation matrices you are about " + "to load!")); // finite element collection - ar &size; + unsigned int size; + ar & size; AssertDimension(size, fe_collection->size()); std::string name; for (unsigned int i = 0; i < size; ++i) @@ -607,12 +719,11 @@ FESeries::Legendre::save_transformation_matrices( // Store information about those resources which have been used to generate // the transformation matrices. // mode vector - unsigned int size = N; - ar & size; + ar &n_coefficients_per_direction; // finite element collection - size = fe_collection->size(); - ar &size; + unsigned int size = fe_collection->size(); + ar & size; for (unsigned int i = 0; i < size; ++i) ar &(*fe_collection)[i].get_name(); @@ -638,12 +749,16 @@ FESeries::Legendre::load_transformation_matrices( // Check whether the currently registered resources are compatible with // the transformation matrices to load. // mode vector - unsigned int size; - ar & size; - AssertDimension(size, N); + std::vector compare_coefficients; + ar & compare_coefficients; + Assert(compare_coefficients == n_coefficients_per_direction, + ExcMessage("A different number of coefficients vector has been used " + "to generate the transformation matrices you are about " + "to load!")); // finite element collection - ar &size; + unsigned int size; + ar & size; AssertDimension(size, fe_collection->size()); std::string name; for (unsigned int i = 0; i < size; ++i) @@ -675,4 +790,4 @@ FESeries::Legendre::load_transformation_matrices( DEAL_II_NAMESPACE_CLOSE -#endif // dealii_fe_series_H +#endif // dealii_fe_series_h diff --git a/include/deal.II/hp/fe_collection.h b/include/deal.II/hp/fe_collection.h index 746fea8972..d7cdb068d9 100644 --- a/include/deal.II/hp/fe_collection.h +++ b/include/deal.II/hp/fe_collection.h @@ -228,7 +228,8 @@ namespace hp n_blocks() const; /** - * Return the maximal degree over all elements of this collection. + * Return the maximum of values returned by FiniteElement::get_degree() + * over all elements of this collection. */ unsigned int max_degree() const; diff --git a/include/deal.II/numerics/smoothness_estimator.h b/include/deal.II/numerics/smoothness_estimator.h index cdb30ab702..659ad33049 100644 --- a/include/deal.II/numerics/smoothness_estimator.h +++ b/include/deal.II/numerics/smoothness_estimator.h @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 1998 - 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -19,306 +19,476 @@ #include -#include -#include - -#include - -#include - -#include +#include #include +#include #include DEAL_II_NAMESPACE_OPEN +// forward declarations +#ifndef DOXYGEN +template +class Vector; + +namespace FESeries +{ + template + class Fourier; + template + class Legendre; +} // namespace FESeries + +namespace hp +{ + template + class DoFHandler; + template + class FECollection; +} // namespace hp +#endif + + /** * A namespace for various smoothness estimation strategies for hp-adaptive FEM. + * + * Smoothness estimation is one strategy to decide whether a cell with a large + * error estimate should undergo h- or p-refinement. Typical strategies decide + * to increase the polynomial degree on a cell if the solution is particularly + * smooth, whereas one would refine the mesh if the solution on the cell is + * singular, has kinks in some derivative, or is otherwise not particularly + * smooth. All of these strategies rely on a way to identify how "smooth" a + * function is on a given cell. */ namespace SmoothnessEstimator { /** - * Estimate smoothness from decay of Legendre absolute values of coefficients - * on the reference cell. + * Smoothness estimation strategy based on the decay of Legendre expansion + * coefficients. * - * In one dimension, the finite element solution on the reference element with - * polynomial degree $p$ can be written as + * In one dimension, the finite element solution on cell $K$ with polynomial + * degree $p$ can be written as + * @f{eqnarray*} + * u_h(x) &=& \sum_j u_j \varphi_j (x) \\ + * u_{h, k}(x) &=& \sum_{k=0}^{p} a_k \widetilde P_k (x), + * \quad a_k = \sum_j {\cal L}_{k,j} u_j + * @f} + * where $u_j$ are degrees of freedom and $\varphi_j$ are the corresponding + * shape functions. $\{\widetilde P_k(x)\}$ are Legendre polynomials on cell + * $K$. $a_k$ and ${\cal L}_{k,j}$ are coefficients and transformation + * matrices from the Legendre expansion of each shape function. For practical + * reasons, we will perform the calculation of these matrices and coefficients + * only on the reference cell $\hat K$. We only have to calculate the + * transformation matrices once this way. However, results are only applicable + * if mapping from the reference cell to the actual cell is linear. We use + * the class FESeries::Legendre to determine all coefficients $a_k$. + * + * A function is analytic, i.e., representable by a power series, if and only + * if their Legendre expansion coefficients decay as (see @cite eibner2007hp) * @f[ - * u_h(\hat x) = \sum_{j=0}^{p} a_j P_j(\hat x) + * |a_k| \sim c \, \exp(-\sigma k) * @f] - * where $\{P_j(x)\}$ are Legendre polynomials. The decay of the coefficients - * is estimated by performing the linear regression fit of + * We determine their decay rate $\sigma$ by performing the linear regression + * fit of * @f[ - * \ln |a_j| \sim C - \sigma j + * \ln |a_k| \sim C - \sigma k * @f] - * for $j=0,..,p$. The rate of the decay $\sigma$ can be used to estimate the - * smoothness. For example, one strategy to implement hp-refinement - * criteria is to perform p-refinement if $\sigma>1$. - * - * Extension to higher dimension is done by performing the fit in each - * coordinate direction separately and then taking the lowest value of - * $\sigma$. - * - * For each input vector of degrees of freedom defined on a DoFHandler, - * this function returns a vector with as many elements as there are cells - * where each element contains $\exp(-\sigma)$, which is a so-called - * analyticity (see references below). - * - * @param [in] fe_series FESeries::Legendre object to calculate coefficients. - * This object needs to be initialized to have at least $p+1$ coefficients in - * each direction, where $p$ is the maximum polynomial degree to be used. - * @param [in] dof_hander An hp::DoFHandler - * @param [in] all_solutions A vector of pointers to the solution vectors - * @param [out] all_smoothness_indicators A vector of pointers to the smoothness indicators for each @p all_solutions. - * @param [in] coefficients_predicate A predicate to select Legendre - * coefficients $a_j \;\; j=0\dots p$ for linear regression in each coordinate - * direction. The user is responsible for updating the vector of `flags` - * provided to this function. Note that its size is $p+1$, where $p$ is the - * polynomial degree of the FE basis on a given element. Default - * implementation will use all Legendre coefficients in each coordinate - * direction, i.e. set all elements of the vector to `true`. - * @param [in] smallest_abs_coefficient The smallest absolute value of the - * coefficient to be used in linear regression in each coordinate direction. - * Note that Legendre coefficients of some functions may have a repeating - * pattern of zero coefficients (i.e. for functions that are locally symmetric - * or antisymmetric about the midpoint of the element in any coordinate - * direction). Thus this parameters allows to ingore small (in absolute value) - * coefficients within the linear regression fit. In case there are less than - * two non-zero coefficients for a coordinate direction, this direction will - * be skipped. If all coefficients are zero, the returned value for this cell - * will be zero (i.e. corresponding to the $\sigma=\infty$). - * - * For more theoretical details see - * @code{.bib} - * @Article{Mavriplis1994, - * author = {Mavriplis, Catherine}, - * title = {Adaptive mesh strategies for the spectral element method}, - * journal = {{Computer Methods in Applied Mechanics and Engineering}}, - * year = {1994}, - * volume = {116}, - * number = {1}, - * pages = {77--86}, - * publisher = {Elsevier}, - * } - * @article{Houston2005, - * author = {Houston, Paul and S{\"u}li, Endre}, - * title = {A note on the design of hp-adaptive finite element - * methods for elliptic partial differential equations}, - * journal = {{Computer Methods in Applied Mechanics and Engineering}}, - * number = {2}, - * pages = {229--243}, - * publisher = {Elsevier}, - * volume = {194}, - * year = {2005} - * } - * @article{Eibner2007, - * author = {Eibner, Tino and Melenk, Jens Markus}, - * title = {An adaptive strategy for hp-FEM based on testing for - * analyticity}, - * journal = {{Computational Mechanics}}, - * year = {2007}, - * volume = {39}, - * number = {5}, - * pages = {575--595}, - * publisher = {Springer}, - * } - * @endcode - * and for the application within the deal.II: - * @code{.bib} - * @article{Davydov2017, - * author = {Denis Davydov and Tymofiy Gerasimov and Jean-Paul Pelteret and - * Paul Steinmann}, - * title = {Convergence study of the h-adaptive PUM and the hp-adaptive - * FEM applied to eigenvalue problems in quantum mechanics}, - * journal = {{Advanced Modeling and Simulation in Engineering Sciences}}, - * year = {2017}, - * volume = {4}, - * number = {1}, - * pages = {7}, - * issn = {2213-7467}, - * doi = {10.1186/s40323-017-0093-0}, - * } - * @endcode + * for $k=0,\ldots,p$, with $p$ the polynomial degree of the finite element. + * The rate of the decay $\sigma$ can be used to estimate the smoothness. For + * example, one strategy to implement hp-refinement criteria is to perform + * p-refinement if $\sigma>1$ (see @cite mavriplis1994hp). * - * @ingroup numerics - * @author Denis Davydov, 2018 + * @author Denis Davydov, 2018, Marc Fehling, 2018 - 2020 */ - template - void - legendre_coefficient_decay( - FESeries::Legendre & fe_series, - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const std::function &flags)> coefficients_predicate = - [](std::vector &flags) -> void { - std::fill(flags.begin(), flags.end(), true); - }, - const double smallest_abs_coefficient = 1e-10); + namespace Legendre + { + /** + * In this variant of the estimation strategy for higher dimensions, we will + * consider all mode vectors $\bf k$ describing Legendre polynomials + * $\widetilde P_{\bf k}$ and perform one least-squares fit over all + * coefficients at once. If there are multiple coefficients corresponding to + * the same absolute value of modes $\|{\bf k}\|_1$, we take the maximum + * among those. Thus, the least-squares fit is performed on + * @f{eqnarray*} + * \widetilde P_{\bf k}({\bf x}) &=& + * \widetilde P_{k_1} (x_1) \ldots \widetilde P_{k_d} (x_d) \\ + * \ln \left( \max\limits_{\|{\bf k}\|_1} |a_{\bf k}| \right) &\sim& + * C - \sigma \|{\bf k}\|_1 + * @f} + * for ${\bf k}=(k_1,\ldots,k_d)$ and $k_i=0,\ldots,p$, with $p$ the + * polynomial degree of the finite element. + * + * For a finite element approximation @p solution, this function writes the + * decay rate for every cell into the output vector @p smoothness_indicators. + * + * @param [in] fe_legendre FESeries::Legendre object to calculate coefficients. + * This object needs to be initialized to have at least $p+1$ coefficients + * in each direction for every finite element in the collection, where $p$ + * is its polynomial degree. + * @param [in] dof_handler An hp::DoFHandler. + * @param [in] solution A solution vector. + * @param [out] smoothness_indicators A vector for smoothness indicators. + * @param [in] regression_strategy Determines which norm will be used on the + * subset of coefficients $\mathbf{k}$ with the same absolute value + * $\|{\bf k}\|_1$. Default is VectorTools::Linfty_norm for a maximum + * approximation. + * @param [in] smallest_abs_coefficient The smallest absolute value of the + * coefficient to be used in linear regression. Note that Legendre + * coefficients of some functions may have a repeating pattern of zero + * coefficients (i.e. for functions that are locally symmetric or + * antisymmetric about the midpoint of the element in any coordinate + * direction). Thus this parameters allows to ingore small (in absolute + * value) coefficients within the linear regression fit. In case there are + * less than two nonzero coefficients, the returned value for this cell will + * be $\sigma=\infty$. + * @param [in] only_flagged_cells Smoothness indicators are usually used to + * decide whether to perform h- or p-adaptation. So in most cases, we only + * need to calculate those indicators on cells flagged for refinement or + * coarsening. This parameter controls whether this particular subset or all + * cells will be considered. By default, all cells will be considered. When + * only flagged cells are supposed to be considered, smoothness indicators + * will only be set on those vector entries of flagged cells; the others + * will be set to a signaling NaN. + * + * For more theoretical details see @cite mavriplis1994hp + * @cite houston2005hp @cite eibner2007hp. + */ + template + void + coefficient_decay(FESeries::Legendre & fe_legendre, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const VectorTools::NormType regression_strategy = + VectorTools::Linfty_norm, + const double smallest_abs_coefficient = 1e-10, + const bool only_flagged_cells = false); + + /** + * In this variant of the estimation strategy for higher dimensions, we only + * consider modes in each coordinate direction, i.e., only mode vectors + * $\bf k$ with one nonzero entry. We perform the least-squares fit in + * each coordinate direction separately and take the lowest decay rate + * $\sigma$ among them. + * + * For a finite element approximation @p solution, this function writes the + * decay rate for every cell into the output vector @p smoothness_indicators. + * + * @param [in] fe_legendre FESeries::Legendre object to calculate coefficients. + * This object needs to be initialized to have at least $p+1$ coefficients + * in each direction, where $p$ is the maximum polynomial degree to be used. + * @param [in] dof_handler An hp::DoFHandler + * @param [in] solution A solution vector + * @param [out] smoothness_indicators A vector for smoothness indicators + * @param [in] coefficients_predicate A predicate to select Legendre + * coefficients $a_j$, $j=0,\ldots,p$ for linear regression in each + * coordinate direction. The user is responsible for updating the vector of + * `flags` provided to this function. Note that its size is $p+1$, where $p$ + * is the polynomial degree of the FE basis on a given element. The default + * implementation will use all Legendre coefficients in each coordinate + * direction, i.e. set all elements of the vector to `true`. + * @param [in] smallest_abs_coefficient The smallest absolute value of the + * coefficient to be used in linear regression in each coordinate direction. + * Note that Legendre coefficients of some functions may have a repeating + * pattern of zero coefficients (i.e. for functions that are locally + * symmetric or antisymmetric about the midpoint of the element in any + * coordinate direction). Thus this parameters allows to ingore small (in + * absolute value) coefficients within the linear regression fit. In case + * there are less than two nonzero coefficients for a coordinate direction, + * this direction will be skipped. If all coefficients are zero, the + * returned value for this cell will be $\sigma=\infty$. + * @param [in] only_flagged_cells Smoothness indicators are usually used to + * decide whether to perform h- or p-adaptation. So in most cases, we only + * need to calculate those indicators on cells flagged for refinement or + * coarsening. This parameter controls whether this particular subset or all + * cells will be considered. By default, all cells will be considered. When + * only flagged cells are supposed to be considered, smoothness indicators + * will only be set on those vector entries of flagged cells; the others + * will be set to NaN. + * + * For more theoretical details and the application within the deal.II + * library see @cite davydov2017hp. + */ + template + void + coefficient_decay_per_direction( + FESeries::Legendre & fe_legendre, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const ComponentMask &coefficients_predicate = ComponentMask(), + const double smallest_abs_coefficient = 1e-10, + const bool only_flagged_cells = false); + + /** + * Number of modes for the default configuration of the Legendre series + * expansion FESeries::Legendre. + * + * We use as many modes as the highest polynomial of all finite elements + * used plus one, since we start with the first Legendre polynomial which is + * just a constant. + */ + template + std::vector + default_number_of_coefficients_per_direction( + const hp::FECollection &fe_collection); + + /** + * Quadrature collection for the default configuration of the Legendre + * series expansion FESeries::Legendre. + * + * We use Gaussian quadrature designed to yield exact results for the + * highest order Legendre polynomial used. + */ + template + hp::QCollection + default_quadrature_collection( + const hp::FECollection &fe_collection); + } // namespace Legendre - /** - * Same as above, but for a single solution vector. - */ - template - void - legendre_coefficient_decay( - FESeries::Legendre & fe_series, - const hp::DoFHandler & dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const std::function &flags)> coefficients_predicate = - [](std::vector &flags) -> void { - std::fill(flags.begin(), flags.end(), true); - }, - const double smallest_abs_coefficient = 1e-10); - /** - * Same as above, but for a single solution vector and with the default - * FESeries::Legendre. - */ - template - void - legendre_coefficient_decay( - const hp::DoFHandler & dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const std::function &flags)> coefficients_predicate = - [](std::vector &flags) -> void { - std::fill(flags.begin(), flags.end(), true); - }, - const double smallest_abs_coefficient = 1e-10); /** - * Estimate the smoothness of a solution based on the decay of coefficients - * from a series expansion. + * Smoothness estimation strategy based on the decay of Fourier expansion + * coefficients. * - * From the definition, we can write our series expansion $\hat U_{\bf k}$ as - * a matrix product + * From the definition, we can write our Fourier series expansion + * $a_{\bf k}$ of the finite element solution on cell $K$ with polynomial + * degree $p$ as a matrix product + * @f{eqnarray*} + * u_h({\bf x}) &=& \sum_j u_j \varphi_j ({\bf x}) \\ + * u_{h, {\bf k}}({\bf x}) &=& + * \sum_{{\bf k}, \|{\bf k}\|\le p} a_{\bf k} \phi_{\bf k}({\bf x}), + * \quad a_{\bf k} = \sum_j {\cal F}_{{\bf k},j} u_j + * @f} + * with $u_j$ the degrees of freedom and $\varphi_j$ the corresponding shape + * functions. $\{\phi_{\bf k}({\bf x}) = \exp(i \, 2 \pi \, {\bf k} \cdot + * {\bf x}) \}$ are expoential functions on cell $K$. $a_{\bf k}$ and ${\cal + * F}_{{\bf k},j}$ are coefficients and transformation matrices from the + * Fourier expansion of each shape function. For practical reasons, we will + * perform the calculation of these matrices and coefficients only on the + * reference cell $\hat K$. We only have to calculate the transformation + * matrices once this way. However, results are only applicable if mapping + * from the reference cell to the actual cell is linear. We use the class + * FESeries::Fourier to determine all coefficients $a_{\bf k}$. + * + * If the finite element approximation on cell $K$ is part of the Hilbert + * space $H^s(K)$, then the following integral must exit for both finite + * element and spectral representation of our solution + * @f{eqnarray*} + * \| \nabla^s u_h({\bf x}) \|_{L^2(K)}^2 &=& + * \int\limits_K \left| \nabla^s u_h({\bf x}) \right|^2 d{\bf x} < + * \infty \\ + * \| \nabla^s u_{h, {\bf k}}({\bf x}) \|_{L^2(K)}^2 &=& + * \int\limits_K \left| \sum\limits_{\bf k} (-i \, 2 \pi {\bf k})^s \, + * a_{\bf k} \, \phi_{\bf k}({\bf x}) \right|^2 d{\bf x} = + * (2 \pi)^{2s} \sum\limits_{\bf k} \left| a_{\bf k} \right|^2 + * \|{\bf k}\|_2^{2s} < \infty + * @f} + * The sum is finite only if the summands decay at least with order * @f[ - * \hat U_{\bf k} - * = {\cal F}_{{\bf k},j} u_j, + * |a_{\bf k}|^2 \|{\bf k}\|_2^{2s} \|{\bf k}\|_2^{d - 1} = + * {\cal O}\left( \|{\bf k}\|_2^{-1-\epsilon} \right) + * @f] + * for all $\epsilon > 0$. The additional factor stems from the fact that, + * since we sum over all multi-indices ${\bf k}$ that are located on a + * dim-dimensional sphere, we actually have, up to a constant, + * $\|{\bf k}\|_2^{d-1}$ modes located in each increment $\|{\bf k}\|_2 + + * d\|{\bf k}\|_2$ that need to be taken into account. By a comparison of + * exponents, we can rewrite this condition as + * @f[ + * |a_{\bf k}| = {\cal O}\left(\|{\bf k}\|_2^ + * {-\left(s + \frac d2 + \epsilon \right)} \right) * @f] - * with $u_j$ the coefficients and ${\cal F}_{{\bf k},j}$ the transformation - * matrix from the expansion. We use the classes FESeries::Fourier and - * FESeries::Legendre to determine all coefficients $u_j$. * * The next step is that we have to estimate how fast these coefficients - * decay with $|{\bf k}|$. Thus, we perform a least-squares fit + * decay with $\|{\bf k}\|_2$. Thus, we perform a least-squares fit * @f[ - * \min_{\alpha,\mu} - * \frac 12 \sum_{{\bf k}, |{\bf k}|\le N} - * \left( |\hat U_{\bf k}| - \alpha |{\bf k}|^{-\mu}\right)^2 + * \min_{\alpha,\sigma} + * \frac 12 \sum_{{\bf k}, \|{\bf k}\|_2 \le p} + * \left( |a_{\bf k}| - \alpha \|{\bf k}\|_2^{-\sigma}\right)^2 * @f] - * with linear regressions coefficients $\alpha$ and $\mu$. For - * simplification, we apply a logarithm on our minimization problem + * with regression coefficients $\alpha$ and $\sigma$. For simplification, we + * apply a logarithm on our minimization problem * @f[ - * \min_{\beta,\mu} - * Q(\beta,\mu) = - * \frac 12 \sum_{{\bf k}, |{\bf k}|\le N} - * \left( \ln |\hat U_{\bf k}| - \beta + \mu \ln |{\bf k}|\right)^2, + * \min_{\beta,\sigma} + * Q(\beta,\sigma) = + * \frac 12 \sum_{{\bf k}, \|{\bf k}\|_2 \le p} + * \left( \ln |a_{\bf k}| - \beta + \sigma \ln \|{\bf k}\|_2 + * \right)^2, * @f] - * where $\beta=\ln \alpha$. This is now a problem for which the - * optimality conditions $\frac{\partial Q}{\partial\beta}=0, - * \frac{\partial Q}{\partial\mu}=0$, are linear in $\beta,\mu$. We can + * where $\beta=\ln \alpha$. This is now a problem for which the optimality + * conditions $\frac{\partial Q}{\partial\beta}=0, + * \frac{\partial Q}{\partial\sigma}=0$, are linear in $\beta,\sigma$. We can * write these conditions as follows: * @f[ * \left(\begin{array}{cc} - * \sum_{{\bf k}, |{\bf k}|\le N} 1 & - * \sum_{{\bf k}, |{\bf k}|\le N} \ln |{\bf k}| + * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} 1 & + * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} \ln \|{\bf k}\|_2 * \\ - * \sum_{{\bf k}, |{\bf k}|\le N} \ln |{\bf k}| & - * \sum_{{\bf k}, |{\bf k}|\le N} (\ln |{\bf k}|)^2 + * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} \ln \|{\bf k}\|_2 & + * \sum_{{\bf k}, \|{\bf k}\|_2 \le p} (\ln \|{\bf k}\|_2)^2 * \end{array}\right) * \left(\begin{array}{c} - * \beta \\ -\mu + * \beta \\ -\sigma * \end{array}\right) * = * \left(\begin{array}{c} - * \sum_{{\bf k}, |{\bf k}|\le N} \ln |\hat U_{{\bf k}}| + * \sum_{{\bf k}, \|{\bf k}\|_2\le p} \ln |a_{{\bf k}}| * \\ - * \sum_{{\bf k}, |{\bf k}|\le N} \ln |\hat U_{{\bf k}}| \ln |{\bf k}| - * \end{array}\right) + * \sum_{{\bf k}, \|{\bf k}\|_2\le p} \ln |a_{{\bf k}}| \ln \|{\bf + * k}\|_2 \end{array}\right) * @f] - * Solving for $\beta$ and $\mu$ is nothing else but a linear regression fit - * and to do that we will use FESeries::linear_regression(). + * Solving for $\beta$ and $\sigma$ is just a linear regression fit and to do + * that we will use FESeries::linear_regression(). * * While we are not particularly interested in the actual value of - * $\beta$, the formula above gives us a mean to calculate the value of - * the exponent $\mu$ that we can then use to determine that - * $\hat u(\hat{\bf x})$ is in $H^s(\hat K)$ with $s=\mu-\frac d2$. These - * Sobolev indices $s$ will suffice as our smoothness estimators and will be - * calculated on each cell for any provided solution. - * - * @note An extensive demonstration of the use of these functions is provided in step-27. - * - * The @p regression_strategy parameter determines which norm will be used on the subset of - * coeffiecients $\mathbf{k}$ with the same absolute value $|\mathbf{k}|$. - * Default is VectorTools::Linfty_norm for a maximum approximation. + * $\beta$, the formula above gives us a means to calculate the value of + * the exponent $\sigma$ that we can then use to determine that + * $u(\hat{\bf x})$ is in $H^s(K)$ with $s=\sigma-\frac d2$. The + * decay rates $\sigma$ will suffice as our smoothness indicators and + * will be calculated on each cell for any provided solution. * - * Smoothness indicators will be calculated for each solution in @p all_solutions - * and stored in @p all_smoothness_indicators in the same order. - * - * An individual @p fe_series object can be supplied, which has to be constructed with the - * same FECollection object as the @p dof_handler. - * - * @ingroup numerics - * @author Denis Davydov, 2016, Marc Fehling, 2018 + * @note An extensive demonstration of the use of these functions is + * provided in step-27. * + * @author Wolfgang Bangerth 2006 - 2007, Denis Davydov, 2016, + * Marc Fehling, 2018 - 2020 */ - template - void - fourier_coefficient_decay( - FESeries::Fourier & fe_series, - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm); + namespace Fourier + { + /** + * In this variant of the estimation strategy for higher dimensions, we will + * consider all mode vectors $\bf k$ describing Fourier polynomials + * $P_{\bf k}$ and perform one least-squares fit over all coefficients + * at once. If there are multiple coefficients corresponding to the same + * absolute value of modes $\|\bf k\|_2$, we take the maximum among those. + * Thus, the least-squares fit is performed on + * @f[ + * \ln \left( \max\limits_{\|{\bf k}\|_2} |a_{\bf k}| \right) \sim + * C - \sigma \ln \|{\bf k}\|_2 + * @f] + * for ${\bf k}=(k_1,\ldots,k_d)$ and $k_i=0,\ldots,p$, with $p$ the + * polynomial degree of the finite element. We exclude the $\|{\bf k}\|_2=0$ + * modes to avoid the singularity of the logarithm. + * + * The @p regression_strategy parameter determines which norm will be used + * on the subset of coefficients $\bf k$ with the same absolute value + * $\|{\bf k}\|_2$. Default is VectorTools::Linfty_norm for a maximum + * approximation. + * + * For a provided solution vector @p solution defined on a DoFHandler + * @p dof_handler, this function returns a vector @p smoothness_indicators + * with as many elements as there are cells where each element contains the + * estimated regularity $\sigma$. + * + * A series expansion object @p fe_fourier has to be supplied, which needs + * to be constructed with the same FECollection object as the @p dof_handler. + * + * The parameter @p smallest_abs_coefficient allows to ingore small (in + * absolute value) coefficients within the linear regression fit. In case + * there are less than two nonzero coefficients for a coordinate direction, + * this direction will be skipped. If all coefficients are zero, the + * returned value for this cell will be $\sigma=\infty$. + * + * Smoothness indicators are usually used to decide whether to perform h- or + * p-adaptation. So in most cases, we only need to calculate those + * indicators on cells flagged for refinement or coarsening. The parameter + * @p only_flagged_cells controls whether this particular subset or all + * cells will be considered. By default, all cells will be considered. + * When only flagged cells are supposed to be considered, smoothness + * indicators will only be set on those vector entries of flagged cells; + * the others will be set to a signaling NaN. + */ + template + void + coefficient_decay(FESeries::Fourier & fe_fourier, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const VectorTools::NormType regression_strategy = + VectorTools::Linfty_norm, + const double smallest_abs_coefficient = 1e-10, + const bool only_flagged_cells = false); - /** - * Same as the function above, only for one @p solution vector. - */ - template - void - fourier_coefficient_decay( - FESeries::Fourier & fe_series, - const hp::DoFHandler &dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm); + /** + * In this variant of the estimation strategy for higher dimensions, we only + * consider modes in each coordinate direction, i.e., only mode vectors + * $\bf k$ with one nonzero entry. We perform the least-squares fit in + * each coordinate direction separately and take the lowest decay rate + * $\sigma$ among them. + * + * The @p coefficients_predicate parameter selects Fourier coefficients + * $a_j$, $j=0,\ldots,p$ for linear regression in each coordinate + * direction. The user is responsible for updating the vector of `flags` + * provided to this function. Note that its size is $p+1$, where $p$ is the + * polynomial degree of the FE basis on a given element. The default + * implementation will use all Fourier coefficients in each coordinate + * direction, i.e. set all elements of the vector to `true`. + * + * For a provided solution vector @p solution defined on a DoFHandler + * @p dof_handler, this function returns a vector @p smoothness_indicators + * with as many elements as there are cells where each element contains the + * estimated regularity $\sigma$. + * + * A series expansion object @p fe_fourier has to be supplied, which needs + * to be constructed with the same FECollection object as the @p dof_handler. + * + * The parameter @p smallest_abs_coefficient allows to ingore small (in + * absolute value) coefficients within the linear regression fit. In case + * there are less than two nonzero coefficients for a coordinate direction, + * this direction will be skipped. If all coefficients are zero, the + * returned value for this cell will be $\sigma=\infty$. + * + * Smoothness indicators are usually used to decide whether to perform h- or + * p-adaptation. So in most cases, we only need to calculate those + * indicators on cells flagged for refinement or coarsening. The parameter + * @p only_flagged_cells controls whether this particular subset or all + * cells will be considered. By default, all cells will be considered. + * When only flagged cells are supposed to be considered, smoothness + * indicators will only be set on those vector entries of flagged cells; + * the others will be set to a signaling NaN. + */ + template + void + coefficient_decay_per_direction( + FESeries::Fourier & fe_fourier, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const ComponentMask &coefficients_predicate = ComponentMask(), + const double smallest_abs_coefficient = 1e-10, + const bool only_flagged_cells = false); - /** - * Same as the function above, but with a default configuration for the chosen - * series expansion, using 2-point Gaussian quadrature for each finite - * element. - * - * Provide the desired series expansion as a template argument, i.e. - * @code - * SmoothnessEstimator::estimate_by_coefficient_decay>( - * dof_handler, all_solutions, all_smoothness_indicators); - * @endcode - */ - template - void - fourier_coefficient_decay( - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm); + /** + * Number of modes for the default configuration of the Fourier series + * expansion FESeries::Fourier. + * + * We use as many modes as the highest polynomial degree of all finite + * elements used plus one, and at least three modes. + */ + template + std::vector + default_number_of_coefficients_per_direction( + const hp::FECollection &fe_collection); - /** - * Same as the function above, only for one @p solution vector. - */ - template - void - fourier_coefficient_decay( - const hp::DoFHandler &dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const VectorTools::NormType regression_strategy = VectorTools::Linfty_norm); + /** + * Quadrature collection for the default configuration of the Fourier series + * expansion FESeries::Fourier. + * + * We use a 4-point Gaussian quarature iterated in each dimension by the + * maximal wave number, which is the number of modes decresed by one since + * we start with $k = 0$. + */ + template + hp::QCollection + default_quadrature_collection( + const hp::FECollection &fe_collection); + } // namespace Fourier } // namespace SmoothnessEstimator diff --git a/include/deal.II/numerics/smoothness_estimator.templates.h b/include/deal.II/numerics/smoothness_estimator.templates.h deleted file mode 100644 index 48fb8d25cb..0000000000 --- a/include/deal.II/numerics/smoothness_estimator.templates.h +++ /dev/null @@ -1,450 +0,0 @@ -// --------------------------------------------------------------------- -// -// Copyright (C) 1998 - 2018 by the deal.II authors -// -// This file is part of the deal.II library. -// -// The deal.II library is free software; you can use it, redistribute -// it, and/or modify it under the terms of the GNU Lesser General -// Public License as published by the Free Software Foundation; either -// version 2.1 of the License, or (at your option) any later version. -// The full text of the license can be found in the file LICENSE.md at -// the top level directory of deal.II. -// -// --------------------------------------------------------------------- - -#ifndef dealii_smoothness_estimator_templates_h -#define dealii_smoothness_estimator_templates_h - -#include - -#include - -#include - -#include - -#include -#include -#include -#include - - -DEAL_II_NAMESPACE_OPEN - - -namespace SmoothnessEstimator -{ - namespace - { - /** - * Resizes @p coeff to @p N in each dimension. - */ - template - void - resize(Table &coeff, const unsigned int N) - { - TableIndices size; - for (unsigned int d = 0; d < dim; d++) - size[d] = N; - coeff.reinit(size); - } - - - - /** - * we will need to take the maximum - * absolute value of fourier coefficients which correspond to $k$-vector - * $|{\bf k}|= const$. To filter the coefficients Table we - * will use the FESeries::process_coefficients() which requires a predicate - * to be specified. The predicate should operate on TableIndices and return - * a pair of bool and unsigned int. The latter - * is the value of the map from TableIndicies to unsigned int. It is - * used to define subsets of coefficients from which we search for the one - * with highest absolute value, i.e. $l^\infty$-norm. The bool - * parameter defines which indices should be used in processing. In the - * current case we are interested in coefficients which correspond to - * $0 < i*i+j*j < N*N$ and $0 < i*i+j*j+k*k < N*N$ in 2D and 3D, - * respectively. - */ - template - std::pair - predicate(const TableIndices &ind, const unsigned int max_degree) - { - unsigned int v = 0; - for (unsigned int i = 0; i < dim; i++) - v += ind[i] * ind[i]; - if (v > 0 && v < max_degree * max_degree) - return std::make_pair(true, v); - else - return std::make_pair(false, v); - } - } // namespace - - - template - void - legendre_coefficient_decay( - FESeries::Legendre & fe_legendre, - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const std::function &flags)> coefficients_predicate, - const double smallest_abs_coefficient) - { - Assert(smallest_abs_coefficient >= 0., - ExcMessage("smallest_abs_coefficient should be non-negative.")); - - using number = typename VectorType::value_type; - using number_coeff = - typename FESeries::Legendre::CoefficientType; - - AssertDimension(all_solutions.size(), all_smoothness_indicators.size()); - - for (auto &smoothness_indicator : all_smoothness_indicators) - smoothness_indicator->reinit( - dof_handler.get_triangulation().n_active_cells()); - - Table expansion_coefficients; - resize(expansion_coefficients, fe_legendre.get_size_in_each_direction()); - - Vector local_dof_values; - - // auxiliary vector to do linear regression - std::vector x; - std::vector y; - - x.reserve(dof_handler.get_fe_collection().max_degree()); - y.reserve(dof_handler.get_fe_collection().max_degree()); - - // precalculate predicates for each degree: - std::vector> predicates( - dof_handler.get_fe_collection().max_degree()); - for (unsigned int p = 1; p <= dof_handler.get_fe_collection().max_degree(); - ++p) - { - auto &pred = predicates[p - 1]; - // we have p+1 coefficients for degree p - pred.resize(p + 1); - coefficients_predicate(pred); - } - - for (auto &cell : dof_handler.active_cell_iterators()) - if (cell->is_locally_owned()) - { - local_dof_values.reinit(cell->get_fe().dofs_per_cell); - - const unsigned int pe = cell->get_fe().degree; - - Assert(pe > 0, ExcInternalError()); - const auto &pred = predicates[pe - 1]; - - // since we use coefficients with indices [1,pe] in each direction, - // the number of coefficients we need to calculate is at least N=pe+1 - AssertIndexRange(pe, fe_legendre.get_size_in_each_direction()); - - auto solution_it = all_solutions.cbegin(); - auto smoothness_indicators_it = all_smoothness_indicators.begin(); - for (; solution_it != all_solutions.cend(); - ++solution_it, ++smoothness_indicators_it) - { - cell->get_dof_values(*(*solution_it), local_dof_values); - fe_legendre.calculate(local_dof_values, - cell->active_fe_index(), - expansion_coefficients); - - // choose the smallest decay of coefficients in each direction, - // i.e. the maximum decay slope k_v - number_coeff k_v = -std::numeric_limits::max(); - for (unsigned int d = 0; d < dim; d++) - { - x.resize(0); - y.resize(0); - - // will use all non-zero coefficients allowed by the predicate - // function - Assert(pred.size() == pe + 1, ExcInternalError()); - for (unsigned int i = 0; i <= pe; i++) - if (pred[i]) - { - TableIndices ind; - ind[d] = i; - const number_coeff coeff_abs = - std::abs(expansion_coefficients(ind)); - - if (coeff_abs > smallest_abs_coefficient) - { - y.push_back(std::log(coeff_abs)); - x.push_back(i); - } - } - - // in case we don't have enough non-zero coefficient to fit, - // skip this direction - if (x.size() < 2) - continue; - - const std::pair fit = - FESeries::linear_regression(x, y); - - // decay corresponds to negative slope - // take the lesser negative slope along each direction - k_v = std::max(k_v, fit.first); - } - - (*(*smoothness_indicators_it))(cell->active_cell_index()) = - std::exp(k_v); - } - } - } - - - - template - void - legendre_coefficient_decay( - FESeries::Legendre & fe_legendre, - const hp::DoFHandler & dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const std::function &flags)> coefficients_predicate, - const double smallest_abs_coefficient) - { - const std::vector all_solutions(1, &solution); - const std::vector *> all_smoothness_indicators( - 1, &smoothness_indicators); - - legendre_coefficient_decay(fe_legendre, - dof_handler, - all_solutions, - all_smoothness_indicators, - coefficients_predicate, - smallest_abs_coefficient); - } - - - - template - void - legendre_coefficient_decay( - const hp::DoFHandler & dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const std::function &flags)> coefficients_predicate, - const double smallest_abs_coefficient) - { - const unsigned int max_degree = - dof_handler.get_fe_collection().max_degree(); - - // We initialize a FESeries::Legendre expansion object object which will be - // used to calculate the expansion coefficients. In addition to the - // hp::FECollection, we need to provide quadrature rules hp::QCollection for - // integration on the reference cell. - // We will need to assemble the expansion matrices for each of the finite - // elements we deal with, i.e. the matrices F_k,j. We have to do that for - // each of the finite elements in use. To that end we need a quadrature - // rule. As a default, we use the same quadrature formula for each finite - // element, namely one that is obtained by iterating a 2-point Gauss formula - // as many times as the maximal polynomial degree. - QGauss<1> base_quadrature(2); - QIterated quadrature(base_quadrature, max_degree); - - hp::QCollection expansion_q_collection; - for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i) - expansion_q_collection.push_back(quadrature); - - FESeries::Legendre legendre(max_degree + 1, - dof_handler.get_fe_collection(), - expansion_q_collection); - - legendre_coefficient_decay(legendre, - dof_handler, - solution, - smoothness_indicators, - coefficients_predicate, - smallest_abs_coefficient); - } - - - - template - void - fourier_coefficient_decay( - FESeries::Fourier & fe_series, - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const VectorTools::NormType regression_strategy) - { - using number = typename VectorType::value_type; - using number_coeff = - typename FESeries::Fourier::CoefficientType; - - AssertDimension(all_solutions.size(), all_smoothness_indicators.size()); - - for (auto &smoothness_indicator : all_smoothness_indicators) - smoothness_indicator->reinit( - dof_handler.get_triangulation().n_active_cells()); - - const unsigned int max_degree = - dof_handler.get_fe_collection().max_degree(); - - Table expansion_coefficients; - resize(expansion_coefficients, max_degree); - - Vector local_dof_values; - std::vector ln_k; - std::pair, std::vector> res; - for (auto &cell : dof_handler.active_cell_iterators()) - if (cell->is_locally_owned()) - { - local_dof_values.reinit(cell->get_fe().dofs_per_cell); - - auto solution_it = all_solutions.cbegin(); - auto smoothness_indicators_it = all_smoothness_indicators.begin(); - for (; solution_it != all_solutions.cend(); - ++solution_it, ++smoothness_indicators_it) - { - // Inside the loop, we first need to get the values of the local - // degrees of freedom and then need to compute the series - // expansion by multiplying this vector with the matrix ${\cal F}$ - // corresponding to this finite element. - cell->get_dof_values(*(*solution_it), local_dof_values); - - fe_series.calculate(local_dof_values, - cell->active_fe_index(), - expansion_coefficients); - - // We fit our exponential decay of expansion coefficients to the - // provided regression_strategy on each possible value of |k|. To - // this end, we use FESeries::process_coefficients() to rework - // coefficients into the desired format. - res = FESeries::process_coefficients( - expansion_coefficients, - std::bind(&predicate, std::placeholders::_1, max_degree), - regression_strategy); - - Assert(res.first.size() == res.second.size(), ExcInternalError()); - - // Prepare linear equation for the logarithmic least squares fit. - // - // First, calculate ln(|k|). This vector will be the same for all - // the cells so we can calculate ln(|k|) only once. - // - // For Fourier expansion, this translates to - // ln(2*pi*sqrt(predicate)) = ln(2*pi) + 0.5*ln(predicate). Since - // we are just interested in a linear regression later, we omit - // the ln(2*pi) factor. - // For Legendre expansion, this translates to - // 0.5*ln(predicate) as well, without the pi factor. - if (ln_k.empty()) - { - ln_k.resize(res.first.size()); - for (unsigned int f = 0; f < res.first.size(); ++f) - ln_k[f] = 0.5 * std::log((double)res.first[f]); - } - - // Second, calculate ln(U_k). - for (auto &residual_element : res.second) - residual_element = std::log(residual_element); - - // Last, do the linear regression. - const auto fit = FESeries::linear_regression(ln_k, res.second); - - // Compute the Sobolev index s=mu-dim/2 and store it in the vector - // of estimated values for each cell. - (*(*smoothness_indicators_it))(cell->active_cell_index()) = - (float)(-fit.first - .5 * dim); - } - } - } - - - - template - void - fourier_coefficient_decay(FESeries::Fourier & fe_series, - const hp::DoFHandler &dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const VectorTools::NormType regression_strategy) - { - const std::vector all_solutions(1, &solution); - const std::vector *> all_smoothness_indicators( - 1, &smoothness_indicators); - - fourier_coefficient_decay(fe_series, - dof_handler, - all_solutions, - all_smoothness_indicators, - regression_strategy); - } - - - - template - void - fourier_coefficient_decay( - const hp::DoFHandler & dof_handler, - const std::vector &all_solutions, - const std::vector *> & all_smoothness_indicators, - const VectorTools::NormType regression_strategy) - { - const unsigned int max_degree = - dof_handler.get_fe_collection().max_degree(); - - // We initialize a series expansion object object which will be used to - // calculate the expansion coefficients. In addition to the - // hp::FECollection, we need to provide quadrature rules hp::QCollection for - // integration on the reference cell. - // We will need to assemble the expansion matrices for each of the finite - // elements we deal with, i.e. the matrices F_k,j. We have to do that for - // each of the finite elements in use. To that end we need a quadrature - // rule. As a default, we use the same quadrature formula for each finite - // element, namely one that is obtained by iterating a 2-point Gauss formula - // as many times as the maximal exponent we use for the term exp(ikx). - QGauss<1> base_quadrature(2); - QIterated quadrature(base_quadrature, max_degree); - - hp::QCollection expansion_q_collection; - for (unsigned int i = 0; i < dof_handler.get_fe_collection().size(); ++i) - expansion_q_collection.push_back(quadrature); - - // The FESeries::Fourier class' constructor first parameter $N$ defines the - // number of coefficients in 1D with the total number of coefficients being - // $N^{dim}$. - FESeries::Fourier fe_series(max_degree, - dof_handler.get_fe_collection(), - expansion_q_collection); - - fourier_coefficient_decay(fe_series, - dof_handler, - all_solutions, - all_smoothness_indicators, - regression_strategy); - } - - - - template - void - fourier_coefficient_decay(const hp::DoFHandler &dof_handler, - const VectorType & solution, - Vector & smoothness_indicators, - const VectorTools::NormType regression_strategy) - { - const std::vector all_solutions(1, &solution); - const std::vector *> all_smoothness_indicators( - 1, &smoothness_indicators); - - fourier_coefficient_decay(dof_handler, - all_solutions, - all_smoothness_indicators, - regression_strategy); - } -} // namespace SmoothnessEstimator - - -DEAL_II_NAMESPACE_CLOSE - -#endif diff --git a/source/fe/fe_series_fourier.cc b/source/fe/fe_series_fourier.cc index 923f59234e..e810baa426 100644 --- a/source/fe/fe_series_fourier.cc +++ b/source/fe/fe_series_fourier.cc @@ -86,6 +86,7 @@ namespace template void ensure_existence( + const std::vector & n_coefficients_per_direction, const hp::FECollection<1, spacedim> & fe_collection, const hp::QCollection<1> & q_collection, const Table<1, Tensor<1, 1>> & k_vectors, @@ -96,10 +97,10 @@ namespace if (fourier_transform_matrices[fe].m() == 0) { - fourier_transform_matrices[fe].reinit(k_vectors.n_elements(), + fourier_transform_matrices[fe].reinit(n_coefficients_per_direction[fe], fe_collection[fe].dofs_per_cell); - for (unsigned int k = 0; k < k_vectors.size(0); ++k) + for (unsigned int k = 0; k < n_coefficients_per_direction[fe]; ++k) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) fourier_transform_matrices[fe](k, j) = integrate(fe_collection[fe], q_collection[fe], k_vectors(k), j); @@ -109,6 +110,7 @@ namespace template void ensure_existence( + const std::vector & n_coefficients_per_direction, const hp::FECollection<2, spacedim> & fe_collection, const hp::QCollection<2> & q_collection, const Table<2, Tensor<1, 2>> & k_vectors, @@ -119,12 +121,14 @@ namespace if (fourier_transform_matrices[fe].m() == 0) { - fourier_transform_matrices[fe].reinit(k_vectors.n_elements(), - fe_collection[fe].dofs_per_cell); + fourier_transform_matrices[fe].reinit( + Utilities::fixed_power<2>(n_coefficients_per_direction[fe]), + fe_collection[fe].dofs_per_cell); unsigned int k = 0; - for (unsigned int k1 = 0; k1 < k_vectors.size(0); ++k1) - for (unsigned int k2 = 0; k2 < k_vectors.size(1); ++k2, k++) + for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1) + for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; + ++k2, ++k) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) fourier_transform_matrices[fe](k, j) = integrate( fe_collection[fe], q_collection[fe], k_vectors(k1, k2), j); @@ -134,6 +138,7 @@ namespace template void ensure_existence( + const std::vector & n_coefficients_per_direction, const hp::FECollection<3, spacedim> & fe_collection, const hp::QCollection<3> & q_collection, const Table<3, Tensor<1, 3>> & k_vectors, @@ -144,13 +149,15 @@ namespace if (fourier_transform_matrices[fe].m() == 0) { - fourier_transform_matrices[fe].reinit(k_vectors.n_elements(), - fe_collection[fe].dofs_per_cell); + fourier_transform_matrices[fe].reinit( + Utilities::fixed_power<3>(n_coefficients_per_direction[fe]), + fe_collection[fe].dofs_per_cell); unsigned int k = 0; - for (unsigned int k1 = 0; k1 < k_vectors.size(0); ++k1) - for (unsigned int k2 = 0; k2 < k_vectors.size(1); ++k2) - for (unsigned int k3 = 0; k3 < k_vectors.size(2); ++k3, k++) + for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1) + for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; ++k2) + for (unsigned int k3 = 0; k3 < n_coefficients_per_direction[fe]; + ++k3, ++k) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) fourier_transform_matrices[fe](k, j) = integrate(fe_collection[fe], @@ -167,28 +174,54 @@ namespace FESeries { template Fourier::Fourier( - const unsigned int N, + const std::vector & n_coefficients_per_direction, const hp::FECollection &fe_collection, const hp::QCollection & q_collection) - : fe_collection(&fe_collection) + : n_coefficients_per_direction(n_coefficients_per_direction) + , fe_collection(&fe_collection) , q_collection(&q_collection) , fourier_transform_matrices(fe_collection.size()) { - set_k_vectors(k_vectors, N); - unrolled_coefficients.resize(k_vectors.n_elements()); + Assert(n_coefficients_per_direction.size() == fe_collection.size() && + n_coefficients_per_direction.size() == q_collection.size(), + ExcMessage("All parameters are supposed to have the same size.")); + + const unsigned int max_n_coefficients_per_direction = + *std::max_element(n_coefficients_per_direction.cbegin(), + n_coefficients_per_direction.cend()); + set_k_vectors(k_vectors, max_n_coefficients_per_direction); + + // reserve sufficient memory + unrolled_coefficients.reserve(k_vectors.n_elements()); } + template + Fourier::Fourier( + const unsigned int n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection) + : Fourier( + std::vector(fe_collection.size(), + n_coefficients_per_direction), + fe_collection, + q_collection) + {} + + + template inline bool Fourier:: operator==(const Fourier &fourier) const { - return ((*fe_collection == *(fourier.fe_collection)) && - (*q_collection == *(fourier.q_collection)) && - (k_vectors == fourier.k_vectors) && - (fourier_transform_matrices == fourier.fourier_transform_matrices)); + return ( + (n_coefficients_per_direction == fourier.n_coefficients_per_direction) && + (*fe_collection == *(fourier.fe_collection)) && + (*q_collection == *(fourier.q_collection)) && + (k_vectors == fourier.k_vectors) && + (fourier_transform_matrices == fourier.fourier_transform_matrices)); } @@ -200,7 +233,8 @@ namespace FESeries Threads::TaskGroup<> task_group; for (unsigned int fe = 0; fe < fe_collection->size(); ++fe) task_group += Threads::new_task([&, fe]() { - ensure_existence(*fe_collection, + ensure_existence(n_coefficients_per_direction, + *fe_collection, *q_collection, k_vectors, fe, @@ -212,6 +246,51 @@ namespace FESeries + template + void + Fourier::initialize( + const std::vector & n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection) + { + Assert(n_coefficients_per_direction.size() == fe_collection.size() && + n_coefficients_per_direction.size() == q_collection.size(), + ExcMessage("All parameters are supposed to have the same size.")); + + // set members + this->n_coefficients_per_direction = n_coefficients_per_direction; + this->fe_collection = + SmartPointer>(&fe_collection); + this->q_collection = + SmartPointer>(&q_collection); + + // clean up auxiliary members + k_vectors.reset_values(); + const unsigned int max_n_coefficients_per_direction = + *std::max_element(n_coefficients_per_direction.cbegin(), + n_coefficients_per_direction.cend()); + set_k_vectors(k_vectors, max_n_coefficients_per_direction); + + fourier_transform_matrices.clear(); + fourier_transform_matrices.resize(fe_collection.size()); + + unrolled_coefficients.clear(); + // reserve sufficient memory + unrolled_coefficients.reserve(k_vectors.n_elements()); + } + + + + template + unsigned int + Fourier::get_n_coefficients_per_direction( + const unsigned int index) const + { + return n_coefficients_per_direction[index]; + } + + + template template void @@ -220,7 +299,14 @@ namespace FESeries const unsigned int cell_active_fe_index, Table &fourier_coefficients) { - ensure_existence(*fe_collection, + Assert(fe_collection != nullptr && q_collection != nullptr, + ExcMessage("Initialize this FESeries object first!")); + for (unsigned int d = 0; d < dim; ++d) + AssertDimension(fourier_coefficients.size(d), + n_coefficients_per_direction[cell_active_fe_index]); + + ensure_existence(n_coefficients_per_direction, + *fe_collection, *q_collection, k_vectors, cell_active_fe_index, @@ -229,6 +315,8 @@ namespace FESeries const FullMatrix &matrix = fourier_transform_matrices[cell_active_fe_index]; + unrolled_coefficients.resize(Utilities::fixed_power( + n_coefficients_per_direction[cell_active_fe_index])); std::fill(unrolled_coefficients.begin(), unrolled_coefficients.end(), CoefficientType(0.)); diff --git a/source/fe/fe_series_legendre.cc b/source/fe/fe_series_legendre.cc index b23de8ad08..6be7a8a4f0 100644 --- a/source/fe/fe_series_legendre.cc +++ b/source/fe/fe_series_legendre.cc @@ -107,20 +107,21 @@ namespace */ template void - ensure_existence(const hp::FECollection<1, spacedim> &fe_collection, - const hp::QCollection<1> & q_collection, - const unsigned int N, - const unsigned int fe, - std::vector> &legendre_transform_matrices) + ensure_existence( + const std::vector & n_coefficients_per_direction, + const hp::FECollection<1, spacedim> &fe_collection, + const hp::QCollection<1> & q_collection, + const unsigned int fe, + std::vector> & legendre_transform_matrices) { AssertIndexRange(fe, fe_collection.size()); if (legendre_transform_matrices[fe].m() == 0) { - legendre_transform_matrices[fe].reinit(N, + legendre_transform_matrices[fe].reinit(n_coefficients_per_direction[fe], fe_collection[fe].dofs_per_cell); - for (unsigned int k = 0; k < N; ++k) + for (unsigned int k = 0; k < n_coefficients_per_direction[fe]; ++k) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) legendre_transform_matrices[fe](k, j) = integrate( fe_collection[fe], q_collection[fe], TableIndices<1>(k), j); @@ -129,22 +130,25 @@ namespace template void - ensure_existence(const hp::FECollection<2, spacedim> &fe_collection, - const hp::QCollection<2> & q_collection, - const unsigned int N, - const unsigned int fe, - std::vector> &legendre_transform_matrices) + ensure_existence( + const std::vector & n_coefficients_per_direction, + const hp::FECollection<2, spacedim> &fe_collection, + const hp::QCollection<2> & q_collection, + const unsigned int fe, + std::vector> & legendre_transform_matrices) { AssertIndexRange(fe, fe_collection.size()); if (legendre_transform_matrices[fe].m() == 0) { - legendre_transform_matrices[fe].reinit(N * N, - fe_collection[fe].dofs_per_cell); + legendre_transform_matrices[fe].reinit( + Utilities::fixed_power<2>(n_coefficients_per_direction[fe]), + fe_collection[fe].dofs_per_cell); unsigned int k = 0; - for (unsigned int k1 = 0; k1 < N; ++k1) - for (unsigned int k2 = 0; k2 < N; ++k2, k++) + for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1) + for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; + ++k2, k++) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) legendre_transform_matrices[fe](k, j) = integrate(fe_collection[fe], @@ -156,23 +160,26 @@ namespace template void - ensure_existence(const hp::FECollection<3, spacedim> &fe_collection, - const hp::QCollection<3> & q_collection, - const unsigned int N, - const unsigned int fe, - std::vector> &legendre_transform_matrices) + ensure_existence( + const std::vector & n_coefficients_per_direction, + const hp::FECollection<3, spacedim> &fe_collection, + const hp::QCollection<3> & q_collection, + const unsigned int fe, + std::vector> & legendre_transform_matrices) { AssertIndexRange(fe, fe_collection.size()); if (legendre_transform_matrices[fe].m() == 0) { - legendre_transform_matrices[fe].reinit(N * N * N, - fe_collection[fe].dofs_per_cell); + legendre_transform_matrices[fe].reinit( + Utilities::fixed_power<3>(n_coefficients_per_direction[fe]), + fe_collection[fe].dofs_per_cell); unsigned int k = 0; - for (unsigned int k1 = 0; k1 < N; ++k1) - for (unsigned int k2 = 0; k2 < N; ++k2) - for (unsigned int k3 = 0; k3 < N; ++k3, k++) + for (unsigned int k1 = 0; k1 < n_coefficients_per_direction[fe]; ++k1) + for (unsigned int k2 = 0; k2 < n_coefficients_per_direction[fe]; ++k2) + for (unsigned int k3 = 0; k3 < n_coefficients_per_direction[fe]; + ++k3, k++) for (unsigned int j = 0; j < fe_collection[fe].dofs_per_cell; ++j) legendre_transform_matrices[fe](k, j) = integrate(fe_collection[fe], @@ -189,14 +196,38 @@ namespace FESeries { template Legendre::Legendre( - const unsigned int size_in_each_direction, + const std::vector & n_coefficients_per_direction, const hp::FECollection &fe_collection, const hp::QCollection & q_collection) - : N(size_in_each_direction) + : n_coefficients_per_direction(n_coefficients_per_direction) , fe_collection(&fe_collection) , q_collection(&q_collection) , legendre_transform_matrices(fe_collection.size()) - , unrolled_coefficients(Utilities::fixed_power(N), 0.) + { + Assert(n_coefficients_per_direction.size() == fe_collection.size() && + n_coefficients_per_direction.size() == q_collection.size(), + ExcMessage("All parameters are supposed to have the same size.")); + + // reserve sufficient memory + const unsigned int max_n_coefficients_per_direction = + *std::max_element(n_coefficients_per_direction.cbegin(), + n_coefficients_per_direction.cend()); + unrolled_coefficients.reserve( + Utilities::fixed_power(max_n_coefficients_per_direction)); + } + + + + template + Legendre::Legendre( + const unsigned int n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection) + : Legendre( + std::vector(fe_collection.size(), + n_coefficients_per_direction), + fe_collection, + q_collection) {} @@ -207,7 +238,8 @@ namespace FESeries operator==(const Legendre &legendre) const { return ( - (N == legendre.N) && (*fe_collection == *(legendre.fe_collection)) && + (n_coefficients_per_direction == legendre.n_coefficients_per_direction) && + (*fe_collection == *(legendre.fe_collection)) && (*q_collection == *(legendre.q_collection)) && (legendre_transform_matrices == legendre.legendre_transform_matrices)); } @@ -221,8 +253,11 @@ namespace FESeries Threads::TaskGroup<> task_group; for (unsigned int fe = 0; fe < fe_collection->size(); ++fe) task_group += Threads::new_task([&, fe]() { - ensure_existence( - *fe_collection, *q_collection, N, fe, legendre_transform_matrices); + ensure_existence(n_coefficients_per_direction, + *fe_collection, + *q_collection, + fe, + legendre_transform_matrices); }); task_group.join_all(); @@ -230,11 +265,45 @@ namespace FESeries + template + void + Legendre::initialize( + const std::vector & n_coefficients_per_direction, + const hp::FECollection &fe_collection, + const hp::QCollection & q_collection) + { + Assert(n_coefficients_per_direction.size() == fe_collection.size() && + n_coefficients_per_direction.size() == q_collection.size(), + ExcMessage("All parameters are supposed to have the same size.")); + + // set members + this->n_coefficients_per_direction = n_coefficients_per_direction; + this->fe_collection = + SmartPointer>(&fe_collection); + this->q_collection = + SmartPointer>(&q_collection); + + // clean up auxiliary members + legendre_transform_matrices.clear(); + legendre_transform_matrices.resize(fe_collection.size()); + + unrolled_coefficients.clear(); + // reserve sufficient memory + const unsigned int max_n_coefficients_per_direction = + *std::max_element(n_coefficients_per_direction.cbegin(), + n_coefficients_per_direction.cend()); + unrolled_coefficients.reserve( + Utilities::fixed_power(max_n_coefficients_per_direction)); + } + + + template unsigned int - Legendre::get_size_in_each_direction() const + Legendre::get_n_coefficients_per_direction( + const unsigned int index) const { - return N; + return n_coefficients_per_direction[index]; } @@ -247,18 +316,23 @@ namespace FESeries const unsigned int cell_active_fe_index, Table & legendre_coefficients) { + Assert(fe_collection != nullptr && q_collection != nullptr, + ExcMessage("Initialize this FESeries object first!")); for (unsigned int d = 0; d < dim; ++d) - AssertDimension(legendre_coefficients.size(d), N); + AssertDimension(legendre_coefficients.size(d), + n_coefficients_per_direction[cell_active_fe_index]); - ensure_existence(*fe_collection, + ensure_existence(n_coefficients_per_direction, + *fe_collection, *q_collection, - N, cell_active_fe_index, legendre_transform_matrices); const FullMatrix &matrix = legendre_transform_matrices[cell_active_fe_index]; + unrolled_coefficients.resize(Utilities::fixed_power( + n_coefficients_per_direction[cell_active_fe_index])); std::fill(unrolled_coefficients.begin(), unrolled_coefficients.end(), CoefficientType(0.)); diff --git a/source/hp/refinement.cc b/source/hp/refinement.cc index f47d71f4d3..eca416709a 100644 --- a/source/hp/refinement.cc +++ b/source/hp/refinement.cc @@ -137,13 +137,9 @@ namespace hp dealii::GridRefinement::ExcInvalidParameterValue()); // We first have to determine the maximal and minimal values of the - // criteria of all flagged cells. We start with the minimal and maximal - // values of all cells, a range within which the minimal and maximal - // values on cells flagged for refinement must surely lie. - Number max_criterion_refine = - *std::min_element(criteria.begin(), criteria.end()), - min_criterion_refine = - *std::max_element(criteria.begin(), criteria.end()); + // criteria of all flagged cells. + Number max_criterion_refine = std::numeric_limits::lowest(), + min_criterion_refine = std::numeric_limits::max(); Number max_criterion_coarsen = max_criterion_refine, min_criterion_coarsen = min_criterion_refine; diff --git a/source/numerics/CMakeLists.txt b/source/numerics/CMakeLists.txt index 028dab93e7..7ae55dc1ba 100644 --- a/source/numerics/CMakeLists.txt +++ b/source/numerics/CMakeLists.txt @@ -1,6 +1,6 @@ ## --------------------------------------------------------------------- ## -## Copyright (C) 2012 - 2019 by the deal.II authors +## Copyright (C) 2012 - 2020 by the deal.II authors ## ## This file is part of the deal.II library. ## diff --git a/source/numerics/smoothness_estimator.cc b/source/numerics/smoothness_estimator.cc index 8276f8aa85..568765d245 100644 --- a/source/numerics/smoothness_estimator.cc +++ b/source/numerics/smoothness_estimator.cc @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 1998 - 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -13,6 +13,14 @@ // // --------------------------------------------------------------------- +#include +#include + +#include + +#include +#include + #include #include #include @@ -24,10 +32,602 @@ #include #include -#include +#include + +#include +#include +#include +#include + DEAL_II_NAMESPACE_OPEN + +namespace SmoothnessEstimator +{ + namespace + { + /** + * Resizes @p coeff to @p N in each dimension. + */ + template + void + resize(Table &coeff, const unsigned int N) + { + TableIndices size; + for (unsigned int d = 0; d < dim; d++) + size[d] = N; + coeff.reinit(size); + } + } // namespace + + + + namespace Legendre + { + namespace + { + /** + * We will need to take the maximum absolute value of Legendre + * coefficients which correspond to $k$-vector $|{\bf k}|= const$. To + * filter the coefficients Table we will use the + * FESeries::process_coefficients() which requires a predicate to be + * specified. The predicate should operate on TableIndices and return a + * pair of bool and unsigned int. The latter is + * the value of the map from TableIndicies to unsigned int. It is used to + * define subsets of coefficients from which we search for the one with + * highest absolute value, i.e. $l^\infty$-norm. The bool + * parameter defines which indices should be used in processing. In the + * current case we are interested in coefficients which correspond to $0 + * <= i+j < N$ and $0 <= i+j+k < N$ in 2D and 3D, respectively. + */ + template + std::pair + index_sum_less_than_N(const TableIndices &ind, const unsigned int N) + { + unsigned int v = 0; + for (unsigned int i = 0; i < dim; ++i) + v += ind[i]; + + return std::make_pair((v < N), v); + } + } // namespace + + + + template + void + coefficient_decay(FESeries::Legendre & fe_legendre, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const VectorTools::NormType regression_strategy, + const double smallest_abs_coefficient, + const bool only_flagged_cells) + { + using number = typename VectorType::value_type; + using number_coeff = + typename FESeries::Legendre::CoefficientType; + + smoothness_indicators.reinit( + dof_handler.get_triangulation().n_active_cells()); + + unsigned int n_modes; + Table expansion_coefficients; + + Vector local_dof_values; + std::vector converted_indices; + std::pair, std::vector> res; + for (const auto &cell : dof_handler.active_cell_iterators()) + if (cell->is_locally_owned()) + { + if (!only_flagged_cells || cell->refine_flag_set() || + cell->coarsen_flag_set()) + { + n_modes = fe_legendre.get_n_coefficients_per_direction( + cell->active_fe_index()); + resize(expansion_coefficients, n_modes); + + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(solution, local_dof_values); + + fe_legendre.calculate(local_dof_values, + cell->active_fe_index(), + expansion_coefficients); + + // We fit our exponential decay of expansion coefficients to the + // provided regression_strategy on each possible value of |k|. + // To this end, we use FESeries::process_coefficients() to + // rework coefficients into the desired format. + res = FESeries::process_coefficients( + expansion_coefficients, + [n_modes](const TableIndices &indices) { + return index_sum_less_than_N(indices, n_modes); + }, + regression_strategy, + smallest_abs_coefficient); + + Assert(res.first.size() == res.second.size(), + ExcInternalError()); + + // Last, do the linear regression. + float regularity = std::numeric_limits::infinity(); + if (res.first.size() > 1) + { + // Prepare linear equation for the logarithmic least squares + // fit. + converted_indices.assign(res.first.begin(), + res.first.end()); + + for (auto &residual_element : res.second) + residual_element = std::log(residual_element); + + const std::pair fit = + FESeries::linear_regression(converted_indices, + res.second); + regularity = static_cast(-fit.first); + } + + smoothness_indicators(cell->active_cell_index()) = regularity; + } + else + smoothness_indicators(cell->active_cell_index()) = + numbers::signaling_nan(); + } + } + + + + template + void + coefficient_decay_per_direction( + FESeries::Legendre & fe_legendre, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const ComponentMask & coefficients_predicate, + const double smallest_abs_coefficient, + const bool only_flagged_cells) + { + Assert(smallest_abs_coefficient >= 0., + ExcMessage("smallest_abs_coefficient should be non-negative.")); + + using number = typename VectorType::value_type; + using number_coeff = + typename FESeries::Legendre::CoefficientType; + + smoothness_indicators.reinit( + dof_handler.get_triangulation().n_active_cells()); + + unsigned int n_modes; + Table expansion_coefficients; + Vector local_dof_values; + + // auxiliary vector to do linear regression + const unsigned int max_degree = + dof_handler.get_fe_collection().max_degree(); + + std::vector x, y; + x.reserve(max_degree); + y.reserve(max_degree); + + for (const auto &cell : dof_handler.active_cell_iterators()) + if (cell->is_locally_owned()) + { + if (!only_flagged_cells || cell->refine_flag_set() || + cell->coarsen_flag_set()) + { + n_modes = fe_legendre.get_n_coefficients_per_direction( + cell->active_fe_index()); + resize(expansion_coefficients, n_modes); + + const unsigned int pe = cell->get_fe().degree; + Assert(pe > 0, ExcInternalError()); + + // since we use coefficients with indices [1,pe] in each + // direction, the number of coefficients we need to calculate is + // at least N=pe+1 + AssertIndexRange(pe, n_modes); + + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(solution, local_dof_values); + + fe_legendre.calculate(local_dof_values, + cell->active_fe_index(), + expansion_coefficients); + + // choose the smallest decay of coefficients in each direction, + // i.e. the maximum decay slope k_v as in exp(-k_v) + double k_v = std::numeric_limits::infinity(); + for (unsigned int d = 0; d < dim; ++d) + { + x.resize(0); + y.resize(0); + + // will use all non-zero coefficients allowed by the + // predicate function + for (unsigned int i = 0; i <= pe; ++i) + if (coefficients_predicate[i]) + { + TableIndices ind; + ind[d] = i; + const double coeff_abs = + std::abs(expansion_coefficients(ind)); + + if (coeff_abs > smallest_abs_coefficient) + { + x.push_back(i); + y.push_back(std::log(coeff_abs)); + } + } + + // in case we don't have enough non-zero coefficient to fit, + // skip this direction + if (x.size() < 2) + continue; + + const std::pair fit = + FESeries::linear_regression(x, y); + + // decay corresponds to negative slope + // take the lesser negative slope along each direction + k_v = std::min(k_v, -fit.first); + } + + smoothness_indicators(cell->active_cell_index()) = + static_cast(k_v); + } + else + smoothness_indicators(cell->active_cell_index()) = + numbers::signaling_nan(); + } + } + + + + template + std::vector + default_number_of_coefficients_per_direction( + const hp::FECollection &fe_collection) + { + std::vector n_coefficients_per_direction; + for (unsigned int i = 0; i < fe_collection.size(); ++i) + n_coefficients_per_direction.push_back(fe_collection[i].degree + 1); + + return n_coefficients_per_direction; + } + + + + template + hp::QCollection + default_quadrature_collection( + const hp::FECollection &fe_collection) + { + const std::vector n_modes = + default_number_of_coefficients_per_direction(fe_collection); + + // We initialize a FESeries::Legendre expansion object object which will + // be used to calculate the expansion coefficients. In addition to the + // hp::FECollection, we need to provide quadrature rules hp::QCollection + // for integration on the reference cell. + // We will need to assemble the expansion matrices for each of the finite + // elements we deal with, i.e. the matrices F_k,j. We have to do that for + // each of the finite elements in use. To that end we need a quadrature + // rule. As a default, we use the same quadrature formula for each finite + // element, namely a Gauss formula that yields exact results for the + // highest order Legendre polynomial used. + + // We start with the zeroth Legendre polynomial which is just a constant, + // so the highest Legendre polynomial will be of order (n_modes - 1). + hp::QCollection q_collection; + for (unsigned int i = 0; i < fe_collection.size(); ++i) + { + const QGauss quadrature(n_modes[i]); + const QSorted quadrature_sorted(quadrature); + q_collection.push_back(quadrature_sorted); + } + + return q_collection; + } + } // namespace Legendre + + + + namespace Fourier + { + namespace + { + /** + * We will need to take the maximum absolute value of Fourier coefficients + * which correspond to $k$-vector $|{\bf k}|= const$. To filter the + * coefficients Table we will use the FESeries::process_coefficients() + * which requires a predicate to be specified. The predicate should + * operate on TableIndices and return a pair of bool and + * unsigned int. The latter is the value of the map from + * TableIndicies to unsigned int. It is used to define subsets of + * coefficients from which we search for the one with highest absolute + * value, i.e. $l^\infty$-norm. The bool parameter defines + * which indices should be used in processing. In the current case we are + * interested in coefficients which correspond to $0 < i^2+j^2 < N^2$ and + * $0 < i^2+j^2+k^2 < N^2$ in 2D and 3D, respectively. + */ + template + std::pair + index_norm_greater_than_zero_and_less_than_N_squared( + const TableIndices &ind, + const unsigned int N) + { + unsigned int v = 0; + for (unsigned int i = 0; i < dim; ++i) + v += ind[i] * ind[i]; + + return std::make_pair((v > 0 && v < N * N), v); + } + } // namespace + + + + template + void + coefficient_decay(FESeries::Fourier & fe_fourier, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const VectorTools::NormType regression_strategy, + const double smallest_abs_coefficient, + const bool only_flagged_cells) + { + using number = typename VectorType::value_type; + using number_coeff = + typename FESeries::Fourier::CoefficientType; + + smoothness_indicators.reinit( + dof_handler.get_triangulation().n_active_cells()); + + unsigned int n_modes; + Table expansion_coefficients; + + Vector local_dof_values; + std::vector ln_k; + std::pair, std::vector> res; + for (const auto &cell : dof_handler.active_cell_iterators()) + if (cell->is_locally_owned()) + { + if (!only_flagged_cells || cell->refine_flag_set() || + cell->coarsen_flag_set()) + { + n_modes = fe_fourier.get_n_coefficients_per_direction( + cell->active_fe_index()); + resize(expansion_coefficients, n_modes); + + // Inside the loop, we first need to get the values of the local + // degrees of freedom and then need to compute the series + // expansion by multiplying this vector with the matrix ${\cal + // F}$ corresponding to this finite element. + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(solution, local_dof_values); + + fe_fourier.calculate(local_dof_values, + cell->active_fe_index(), + expansion_coefficients); + + // We fit our exponential decay of expansion coefficients to the + // provided regression_strategy on each possible value of |k|. + // To this end, we use FESeries::process_coefficients() to + // rework coefficients into the desired format. + res = FESeries::process_coefficients( + expansion_coefficients, + [n_modes](const TableIndices &indices) { + return index_norm_greater_than_zero_and_less_than_N_squared( + indices, n_modes); + }, + regression_strategy, + smallest_abs_coefficient); + + Assert(res.first.size() == res.second.size(), + ExcInternalError()); + + // Last, do the linear regression. + float regularity = std::numeric_limits::infinity(); + if (res.first.size() > 1) + { + // Prepare linear equation for the logarithmic least squares + // fit. + // + // First, calculate ln(|k|). + // + // For Fourier expansion, this translates to + // ln(2*pi*sqrt(predicate)) = ln(2*pi) + 0.5*ln(predicate). + // Since we are just interested in the slope of a linear + // regression later, we omit the ln(2*pi) factor. + ln_k.resize(res.first.size()); + for (unsigned int f = 0; f < res.first.size(); ++f) + ln_k[f] = + 0.5 * std::log(static_cast(res.first[f])); + + // Second, calculate ln(U_k). + for (auto &residual_element : res.second) + residual_element = std::log(residual_element); + + const std::pair fit = + FESeries::linear_regression(ln_k, res.second); + // Compute regularity s = mu - dim/2 + regularity = static_cast(-fit.first) - + ((dim > 1) ? (.5 * dim) : 0); + } + + // Store result in the vector of estimated values for each cell. + smoothness_indicators(cell->active_cell_index()) = regularity; + } + else + smoothness_indicators(cell->active_cell_index()) = + numbers::signaling_nan(); + } + } + + + + template + void + coefficient_decay_per_direction( + FESeries::Fourier & fe_fourier, + const hp::DoFHandler &dof_handler, + const VectorType & solution, + Vector & smoothness_indicators, + const ComponentMask & coefficients_predicate, + const double smallest_abs_coefficient, + const bool only_flagged_cells) + { + Assert(smallest_abs_coefficient >= 0., + ExcMessage("smallest_abs_coefficient should be non-negative.")); + + using number = typename VectorType::value_type; + using number_coeff = + typename FESeries::Fourier::CoefficientType; + + smoothness_indicators.reinit( + dof_handler.get_triangulation().n_active_cells()); + + unsigned int n_modes; + Table expansion_coefficients; + Vector local_dof_values; + + // auxiliary vector to do linear regression + const unsigned int max_degree = + dof_handler.get_fe_collection().max_degree(); + + std::vector x, y; + x.reserve(max_degree); + y.reserve(max_degree); + + for (const auto &cell : dof_handler.active_cell_iterators()) + if (cell->is_locally_owned()) + { + if (!only_flagged_cells || cell->refine_flag_set() || + cell->coarsen_flag_set()) + { + n_modes = fe_fourier.get_n_coefficients_per_direction( + cell->active_fe_index()); + resize(expansion_coefficients, n_modes); + + const unsigned int pe = cell->get_fe().degree; + Assert(pe > 0, ExcInternalError()); + + // since we use coefficients with indices [1,pe] in each + // direction, the number of coefficients we need to calculate is + // at least N=pe+1 + AssertIndexRange(pe, n_modes); + + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(solution, local_dof_values); + + fe_fourier.calculate(local_dof_values, + cell->active_fe_index(), + expansion_coefficients); + + // choose the smallest decay of coefficients in each direction, + // i.e. the maximum decay slope k_v as in exp(-k_v) + double k_v = std::numeric_limits::infinity(); + for (unsigned int d = 0; d < dim; ++d) + { + x.resize(0); + y.resize(0); + + // will use all non-zero coefficients allowed by the + // predicate function + // + // skip i=0 because of logarithm + for (unsigned int i = 1; i <= pe; ++i) + if (coefficients_predicate[i]) + { + TableIndices ind; + ind[d] = i; + const double coeff_abs = + std::abs(expansion_coefficients(ind)); + + if (coeff_abs > smallest_abs_coefficient) + { + x.push_back(std::log(i)); + y.push_back(std::log(coeff_abs)); + } + } + + // in case we don't have enough non-zero coefficient to fit, + // skip this direction + if (x.size() < 2) + continue; + + const std::pair fit = + FESeries::linear_regression(x, y); + + // decay corresponds to negative slope + // take the lesser negative slope along each direction + k_v = std::min(k_v, -fit.first); + } + + smoothness_indicators(cell->active_cell_index()) = + static_cast(k_v); + } + else + smoothness_indicators(cell->active_cell_index()) = + numbers::signaling_nan(); + } + } + + + + template + std::vector + default_number_of_coefficients_per_direction( + const hp::FECollection &fe_collection) + { + std::vector n_coefficients_per_direction; + for (unsigned int i = 0; i < fe_collection.size(); ++i) + n_coefficients_per_direction.push_back( + std::max(3, fe_collection[i].degree + 1)); + + return n_coefficients_per_direction; + } + + + + template + hp::QCollection + default_quadrature_collection( + const hp::FECollection &fe_collection) + { + const std::vector n_modes = + default_number_of_coefficients_per_direction(fe_collection); + + // We initialize a series expansion object object which will be used to + // calculate the expansion coefficients. In addition to the + // hp::FECollection, we need to provide quadrature rules hp::QCollection + // for integration on the reference cell. + // We will need to assemble the expansion matrices for each of the finite + // elements we deal with, i.e. the matrices F_k,j. We have to do that for + // each of the finite elements in use. To that end we need a quadrature + // rule. As a default, we use the same quadrature formula for each finite + // element, namely one that is obtained by iterating a 4-point Gauss + // formula as many times as the maximal exponent we use for the term + // exp(ikx). Since the first mode corresponds to k = 0, the maximal wave + // number is k = n_modes - 1. + const QGauss<1> base_quadrature(4); + hp::QCollection q_collection; + for (unsigned int i = 0; i < fe_collection.size(); ++i) + { + const QIterated quadrature(base_quadrature, n_modes[i] - 1); + const QSorted quadrature_sorted(quadrature); + q_collection.push_back(quadrature_sorted); + } + + return q_collection; + } + } // namespace Fourier +} // namespace SmoothnessEstimator + + +// explicit instantiations #include "smoothness_estimator.inst" DEAL_II_NAMESPACE_CLOSE diff --git a/source/numerics/smoothness_estimator.inst.in b/source/numerics/smoothness_estimator.inst.in index d6aca7def7..912fa2d70c 100644 --- a/source/numerics/smoothness_estimator.inst.in +++ b/source/numerics/smoothness_estimator.inst.in @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 2010 - 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -17,67 +17,67 @@ for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS; VEC : REAL_VECTOR_TYPES) { -#if deal_II_dimension != 1 && deal_II_dimension <= deal_II_space_dimension - template void SmoothnessEstimator::legendre_coefficient_decay< +#if deal_II_dimension <= deal_II_space_dimension + template void SmoothnessEstimator::Legendre:: + coefficient_decay( + FESeries::Legendre &, + const hp::DoFHandler &, + const VEC &, + Vector &, + const VectorTools::NormType, + const double, + const bool); + template void + SmoothnessEstimator::Legendre::coefficient_decay_per_direction< deal_II_dimension, deal_II_space_dimension, VEC>(FESeries::Legendre &, const hp::DoFHandler &, - const std::vector &, - const std::vector *> &, - const std::function & flags)>, - const double); - - template void SmoothnessEstimator::legendre_coefficient_decay< - deal_II_dimension, - deal_II_space_dimension, - VEC>(FESeries::Legendre &, - const hp::DoFHandler &, - const VEC &, - Vector &, - const std::function & flags)>, - const double); - - template void SmoothnessEstimator::legendre_coefficient_decay< - deal_II_dimension, - deal_II_space_dimension, - VEC>(const hp::DoFHandler &, const VEC &, Vector &, - const std::function & flags)>, - const double); + const ComponentMask &, + const double, + const bool); - template void SmoothnessEstimator::fourier_coefficient_decay< - deal_II_dimension, - deal_II_space_dimension, - VEC>(FESeries::Fourier &, - const hp::DoFHandler &, - const std::vector &, - const std::vector *> &, - const VectorTools::NormType); - template void SmoothnessEstimator::fourier_coefficient_decay< + template void SmoothnessEstimator::Fourier:: + coefficient_decay( + FESeries::Fourier &, + const hp::DoFHandler &, + const VEC &, + Vector &, + const VectorTools::NormType, + const double, + const bool); + template void SmoothnessEstimator::Fourier::coefficient_decay_per_direction< deal_II_dimension, deal_II_space_dimension, VEC>(FESeries::Fourier &, const hp::DoFHandler &, const VEC &, Vector &, - const VectorTools::NormType); + const ComponentMask &, + const double, + const bool); +#endif + } - template void SmoothnessEstimator::fourier_coefficient_decay< - deal_II_dimension, - deal_II_space_dimension, - VEC>(const hp::DoFHandler &, - const std::vector &, - const std::vector *> &, - const VectorTools::NormType); - template void SmoothnessEstimator::fourier_coefficient_decay< - deal_II_dimension, - deal_II_space_dimension, - VEC>(const hp::DoFHandler &, - const VEC &, - Vector &, - const VectorTools::NormType); +for (deal_II_dimension : DIMENSIONS; deal_II_space_dimension : SPACE_DIMENSIONS) + { +#if deal_II_dimension <= deal_II_space_dimension + template std::vector SmoothnessEstimator::Legendre:: + default_number_of_coefficients_per_direction( + const hp::FECollection &); + template hp::QCollection SmoothnessEstimator::Legendre:: + default_quadrature_collection( + const hp::FECollection &); + template std::vector SmoothnessEstimator::Fourier:: + default_number_of_coefficients_per_direction( + const hp::FECollection &); + template hp::QCollection SmoothnessEstimator::Fourier:: + default_quadrature_collection( + const hp::FECollection &); #endif } diff --git a/tests/fe/fe_series_01.cc b/tests/fe/fe_series_01.cc index 9256a9c2d7..b32b753f85 100644 --- a/tests/fe/fe_series_01.cc +++ b/tests/fe/fe_series_01.cc @@ -62,7 +62,11 @@ test_1d() QIterated quadrature(base_quadrature, N); q_collection.push_back(quadrature); - FESeries::Fourier fourier(N, fe_collection, q_collection); + const std::vector n_coefficients_per_direction(1, N); + + FESeries::Fourier fourier(n_coefficients_per_direction, + fe_collection, + q_collection); Vector local_dof_values(2); local_dof_values[0] = 0; diff --git a/tests/fe/fe_series_02.cc b/tests/fe/fe_series_02.cc index 4ee1807b59..3f73031a53 100644 --- a/tests/fe/fe_series_02.cc +++ b/tests/fe/fe_series_02.cc @@ -106,8 +106,12 @@ test(const LegendreFunction &func, const unsigned int poly_degree) VectorTools::interpolate(dof_handler, func, values); - const unsigned int N = poly_degree + 1; - FESeries::Legendre legendre(N, fe_collection, quadrature_formula); + const unsigned int N = poly_degree + 1; + const std::vector n_coefficients_per_direction( + fe_collection.size(), N); + FESeries::Legendre legendre(n_coefficients_per_direction, + fe_collection, + quadrature_formula); const std::vector &coeff_in = func.get_coefficients(); Table<1, double> coeff_out(N); diff --git a/tests/fe/fe_series_03.output b/tests/fe/fe_series_03.output index 0dbfbbeb1f..ff31decf46 100644 --- a/tests/fe/fe_series_03.output +++ b/tests/fe/fe_series_03.output @@ -1,6 +1,5 @@ DEAL::L2_norm -DEAL::0 : 0 DEAL::1 : 4.12311 DEAL::2 : 9.64365 DEAL::3 : 16.4317 @@ -8,7 +7,6 @@ DEAL::4 : 17.8326 DEAL::5 : 17.8045 DEAL::6 : 15.0000 DEAL::L1_norm -DEAL::0 : 0 DEAL::1 : 5.00000 DEAL::2 : 15.0000 DEAL::3 : 30.0000 @@ -16,7 +14,6 @@ DEAL::4 : 30.0000 DEAL::5 : 25.0000 DEAL::6 : 15.0000 DEAL::Linfty_norm -DEAL::0 : 0 DEAL::1 : 4.00000 DEAL::2 : 8.00000 DEAL::3 : 12.0000 @@ -24,7 +21,6 @@ DEAL::4 : 13.0000 DEAL::5 : 14.0000 DEAL::6 : 15.0000 DEAL::mean -DEAL::0 : 0 DEAL::1 : 2.50000 DEAL::2 : 5.00000 DEAL::3 : 7.50000 diff --git a/tests/fe/fe_series_04.cc b/tests/fe/fe_series_04.cc index e75cec9fdc..313e817bfd 100644 --- a/tests/fe/fe_series_04.cc +++ b/tests/fe/fe_series_04.cc @@ -104,8 +104,13 @@ test(const LegendreFunction &func, const unsigned int poly_degree) Vector values(dof_handler.n_dofs()); VectorTools::interpolate(dof_handler, func, values); - const unsigned int N = 4; - FESeries::Legendre legendre(N, fe_collection, quadrature_formula); + + const unsigned int N = 4; + const std::vector n_coefficients_per_direction( + fe_collection.size(), N); + FESeries::Legendre legendre(n_coefficients_per_direction, + fe_collection, + quadrature_formula); Table<1, double> coeff_out(N); Vector local_dof_values; diff --git a/tests/fe/fe_series_05.cc b/tests/fe/fe_series_05.cc index b5c269b08b..20ef90aa27 100644 --- a/tests/fe/fe_series_05.cc +++ b/tests/fe/fe_series_05.cc @@ -171,8 +171,12 @@ test(const LegendreFunction &func, const unsigned int poly_degree) VectorTools::interpolate(dof_handler, func, values); - const unsigned int N = poly_degree + 1; - FESeries::Legendre legendre(N, fe_collection, quadrature_formula); + const unsigned int N = poly_degree + 1; + const std::vector n_coefficients_per_direction( + fe_collection.size(), N); + FESeries::Legendre legendre(n_coefficients_per_direction, + fe_collection, + quadrature_formula); const Table &coeff_in = func.get_coefficients(); Table coeff_out; diff --git a/tests/fe/fe_series_06.cc b/tests/fe/fe_series_06.cc index f88adce0ab..a85d10a630 100644 --- a/tests/fe/fe_series_06.cc +++ b/tests/fe/fe_series_06.cc @@ -43,7 +43,13 @@ test_2d() for (unsigned int i = 0; i < fe_collection.size(); i++) fourier_q_collection.push_back(quadrature); - FESeries::Fourier fourier(N, fe_collection, fourier_q_collection); + const std::vector n_coefficients_per_direction( + fe_collection.size(), N); + + FESeries::Fourier fourier(n_coefficients_per_direction, + fe_collection, + fourier_q_collection); + Table> fourier_coefficients; fourier_coefficients.reinit(N, N); diff --git a/tests/fe/fe_series_07.output b/tests/fe/fe_series_07.output index e7955ac589..8e24248cd1 100644 --- a/tests/fe/fe_series_07.output +++ b/tests/fe/fe_series_07.output @@ -1,21 +1,17 @@ DEAL::L2_norm -DEAL::0 : 0 DEAL::1 : 4.12311 DEAL::2 : 9.64365 DEAL::3 : 16.4317 DEAL::L1_norm -DEAL::0 : 0 DEAL::1 : 5.00000 DEAL::2 : 15.0000 DEAL::3 : 30.0000 DEAL::Linfty_norm -DEAL::0 : 0 DEAL::1 : 4.00000 DEAL::2 : 8.00000 DEAL::3 : 12.0000 DEAL::mean -DEAL::0 : 0 DEAL::1 : 2.50000 DEAL::2 : 5.00000 DEAL::3 : 7.50000 diff --git a/tests/hp/laplace.h b/tests/hp/laplace.h index a87e12949a..bbd65bf0c6 100644 --- a/tests/hp/laplace.h +++ b/tests/hp/laplace.h @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -13,8 +13,11 @@ // // --------------------------------------------------------------------- + + // base header for hp-FEM test on Laplace equation. + #include #include #include @@ -30,6 +33,7 @@ #include #include +#include #include #include #include @@ -42,7 +46,12 @@ #include #include -#include +#include +#include +#include +#include + +#include #include #include #include @@ -68,6 +77,8 @@ #include "../tests.h" + + /** * Basic class for Laplace problem */ @@ -118,13 +129,10 @@ protected: mark_h_cells() = 0; /** - * remove h-refinement flag from some cells and populate @p p_cells with - * iterators to those cells, that shall be p-refined. + * remove h-refinement flag from some cells and flag cells for p-refinement */ - virtual std::pair - substitute_h_for_p( - std::vector::active_cell_iterator> - &p_cells) = 0; + virtual void + substitute_h_for_p() = 0; void refine_grid(const unsigned int cycle); @@ -206,12 +214,16 @@ Laplace::Laplace(const Function &force_function, deallog << std::endl; } + + template Laplace::~Laplace() { dof_handler.clear(); } + + template hp::DoFHandler & Laplace::get_dof_handler() @@ -219,6 +231,8 @@ Laplace::get_dof_handler() return dof_handler; } + + template void Laplace::setup_solve_estimate(Vector &output_estimate) @@ -230,6 +244,8 @@ Laplace::setup_solve_estimate(Vector &output_estimate) output_estimate = estimated_error_per_cell; } + + template void Laplace::setup_system() @@ -285,6 +301,8 @@ Laplace::setup_system() << std::endl; } + + template void Laplace::assemble() @@ -352,6 +370,8 @@ Laplace::assemble() pcout << " done." << std::endl; } + + //#define DIRECT template @@ -397,6 +417,8 @@ Laplace::solve() pcout << " done." << std::endl; } + + template void Laplace::refine_grid(const unsigned int cycle) @@ -407,11 +429,20 @@ Laplace::refine_grid(const unsigned int cycle) mark_h_cells(); // 3.3. Substitute h for p refinement - std::vector::active_cell_iterator> p_cells; - hp_number = substitute_h_for_p(p_cells); + substitute_h_for_p(); + // prepare refinement and store number of flagged cells triangulation.prepare_coarsening_and_refinement(); + hp_number = {0, 0}; + for (const auto &cell : dof_handler.active_cell_iterators()) + { + if (cell->refine_flag_set()) + ++hp_number.first; + if (cell->future_fe_index_set()) + ++hp_number.second; + } + // 3.4. Solution Transfer SolutionTransfer> soltrans(dof_handler); @@ -424,18 +455,7 @@ Laplace::refine_grid(const unsigned int cycle) solution_coarse = solution; soltrans.prepare_for_coarsening_and_refinement(solution_coarse); - // increase fe_index() - for (unsigned int i = 0; i < p_cells.size(); i++) - { - typename hp::DoFHandler::active_cell_iterator cell( - &triangulation, p_cells[i]->level(), p_cells[i]->index(), &dof_handler); - - const unsigned int incremented_index = cell->active_fe_index() + 1; - Assert(incremented_index < fe.size(), ExcInternalError()); - cell->set_active_fe_index(incremented_index); - } - - // 3.5. Refinement + // 3.5. h-refinement and p-refinement triangulation.execute_coarsening_and_refinement(); // FIXME: some hp strategies might need: @@ -448,6 +468,8 @@ Laplace::refine_grid(const unsigned int cycle) soltrans.interpolate(solution_coarse, solution); } + + template void Laplace::calculate_error() @@ -535,6 +557,8 @@ Laplace::calculate_error() Linfty_error = Utilities::MPI::max(Linfty_error, mpi_communicator); } + + template void Laplace::output_results(int cycle) @@ -557,6 +581,8 @@ Laplace::output_results(int cycle) << Linfty_error << sp << total_error << sp << std::endl; } + + template void Laplace::print_errors() @@ -603,6 +629,8 @@ Laplace::print_errors() } } + + template void Laplace::run() diff --git a/tests/hp/laplace_mitchel2014_04_peak.output b/tests/hp/laplace_mitchel2014_04_peak.output deleted file mode 100644 index 7bcfbb54fb..0000000000 --- a/tests/hp/laplace_mitchel2014_04_peak.output +++ /dev/null @@ -1,13 +0,0 @@ - -DEAL:: -DEAL::0 256 0 0 289 0.267081 1.78414 1.23386 1.29971 -DEAL::1 256 0 56 477 0.0236257 0.743531 0.249786 1.18708 -DEAL::2 268 4 48 801 0.00108376 0.293919 0.0466411 0.361176 -DEAL::3 349 22 32 1730 0.000121568 0.0610935 0.00469929 0.0464975 -DEAL::4 421 18 52 2818 3.08015e-05 0.0204013 0.00315297 0.0197872 -DEAL::5 496 11 74 4632 6.14178e-06 0.00604078 0.000714754 0.00530244 -DEAL::6 622 28 72 7303 2.03947e-06 0.00210333 0.000148863 0.00185557 -DEAL::7 712 17 108 10086 5.30920e-07 0.000744017 7.24474e-05 0.000775517 -DEAL::8 811 20 123 13127 2.97635e-07 0.000379131 3.62580e-05 0.000400767 -DEAL::9 901 15 148 17913 1.18544e-07 0.000194730 1.48736e-05 0.000216587 -DEAL::10 1030 20 161 23692 4.78884e-08 0.000106719 9.54505e-06 0.000117773 diff --git a/tests/hp/laplace_mitchel2014_04_peak.cc b/tests/hp/laplace_mitchell2014_04_peak.cc similarity index 78% rename from tests/hp/laplace_mitchel2014_04_peak.cc rename to tests/hp/laplace_mitchell2014_04_peak.cc index 0cc813c60e..26d7c49527 100644 --- a/tests/hp/laplace_mitchel2014_04_peak.cc +++ b/tests/hp/laplace_mitchell2014_04_peak.cc @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -13,8 +13,11 @@ // // --------------------------------------------------------------------- -// test SmoothnessEstimator::legendre_coefficient_decay() on -// problem 4 (peak) in Mitchel 2014. + + +// test SmoothnessEstimator::Legendre::coefficient_decay_per_direction() +// on problem 4 (peak) in Mitchell 2014. + #include "laplace.h" @@ -103,12 +106,14 @@ private: estimate_error(); void mark_h_cells(); - - std::pair - substitute_h_for_p( - std::vector::active_cell_iterator> &p_cells); + void + substitute_h_for_p(); hp::QCollection quadrature_face; + + std::vector n_coefficients_per_direction; + hp::QCollection expansion_q_collection; + FESeries::Legendre legendre; }; template @@ -135,45 +140,42 @@ Problem4::Problem4(const Function &force_function, const QIterated q_iterated(q_trapez, p + 3); Laplace::quadrature_infty.push_back(QSorted(q_iterated)); } + + // after the FECollection has been generated, create a corresponding legendre + // series expansion object + n_coefficients_per_direction = + SmoothnessEstimator::Legendre::default_number_of_coefficients_per_direction( + Laplace::fe); + expansion_q_collection = + SmoothnessEstimator::Legendre::default_quadrature_collection( + Laplace::fe); + legendre.initialize(n_coefficients_per_direction, + Laplace::fe, + expansion_q_collection); } template -std::pair -Problem4::substitute_h_for_p( - std::vector::active_cell_iterator> &p_cells) +void +Problem4::substitute_h_for_p() { Vector smoothness_indicators( Laplace::triangulation.n_active_cells()); - SmoothnessEstimator::legendre_coefficient_decay(Laplace::dof_handler, - Laplace::solution, - smoothness_indicators); - - unsigned int num_p_cells = 0; - unsigned int num_h_cells = 0; - for (auto &cell : Laplace::dof_handler.active_cell_iterators()) - if (cell->refine_flag_set()) - { - typename Triangulation::active_cell_iterator tria_cell( - &(Laplace::triangulation), cell->level(), cell->index()); - - const unsigned int cur_fe_index = cell->active_fe_index(); - const bool p_ref = smoothness_indicators(cell->index()) < exp(-1.); - - if (cur_fe_index < Laplace::fe.size() - 1 && p_ref) - { - ++num_p_cells; - cell->clear_refine_flag(); - p_cells.push_back(tria_cell); - } - else - { - ++num_h_cells; - } - } - - return std::make_pair(num_h_cells, num_p_cells); + SmoothnessEstimator::Legendre::coefficient_decay_per_direction( + legendre, + Laplace::dof_handler, + Laplace::solution, + smoothness_indicators); + + hp::Refinement::p_adaptivity_from_absolute_threshold( + Laplace::dof_handler, + smoothness_indicators, + static_cast(1.), + static_cast(0.), + std::greater(), + std::less()); + hp::Refinement::choose_p_over_h(Laplace::dof_handler); } diff --git a/tests/hp/laplace_mitchell2014_04_peak.with_trilinos=true.output b/tests/hp/laplace_mitchell2014_04_peak.with_trilinos=true.output new file mode 100644 index 0000000000..87817c4fe6 --- /dev/null +++ b/tests/hp/laplace_mitchell2014_04_peak.with_trilinos=true.output @@ -0,0 +1,13 @@ + +DEAL:: +DEAL::0 256 0 0 289 0.267081 1.78414 1.23386 1.29971 +DEAL::1 256 0 56 477 0.0236257 0.743531 0.249786 1.18708 +DEAL::2 268 4 47 790 0.00108376 0.293919 0.0466411 0.361176 +DEAL::3 328 20 33 1498 0.000182183 0.0621768 0.00717343 0.0760236 +DEAL::4 508 60 24 3402 2.40462e-05 0.0134610 0.00111509 0.0224341 +DEAL::5 571 21 80 4536 3.48748e-06 0.00249342 0.000176335 0.00398940 +DEAL::6 595 8 106 5758 7.30475e-07 0.000644499 3.23861e-05 0.000860213 +DEAL::7 673 26 93 7725 3.32389e-07 0.000304984 2.38742e-05 0.000356132 +DEAL::8 721 16 118 9246 5.75345e-08 7.11276e-05 2.90017e-06 0.000111393 +DEAL::9 772 17 127 11000 3.50959e-08 3.79011e-05 2.91724e-06 5.96652e-05 +DEAL::10 796 8 146 13074 9.97255e-09 1.29806e-05 7.46491e-07 1.99550e-05 diff --git a/tests/hp/step-27.cc b/tests/hp/step-27.cc index 857c7163ff..eafadd5c0e 100644 --- a/tests/hp/step-27.cc +++ b/tests/hp/step-27.cc @@ -29,6 +29,7 @@ #include #include +#include #include #include @@ -38,6 +39,8 @@ #include #include +#include +#include #include #include @@ -101,7 +104,9 @@ namespace Step27 Vector solution; Vector system_rhs; - const unsigned int max_degree; + const unsigned int max_degree; + hp::QCollection fourier_q_collection; + FESeries::Fourier fourier; }; @@ -142,6 +147,17 @@ namespace Step27 quadrature_collection.push_back(QGauss(degree + 1)); face_quadrature_collection.push_back(QGauss(degree + 1)); } + + QGauss<1> base_quadrature(2); + QIterated quadrature(base_quadrature, max_degree); + for (unsigned int i = 0; i < fe_collection.size(); ++i) + fourier_q_collection.push_back(quadrature); + + const std::vector n_coefficients_per_direction( + fe_collection.size(), max_degree); + fourier.initialize(n_coefficients_per_direction, + fe_collection, + fourier_q_collection); } @@ -269,10 +285,26 @@ namespace Step27 solution, estimated_error_per_cell); + GridRefinement::refine_and_coarsen_fixed_number(triangulation, + estimated_error_per_cell, + 0.3, + 0.03); + Vector smoothness_indicators; - SmoothnessEstimator::fourier_coefficient_decay(dof_handler, - solution, - smoothness_indicators); + SmoothnessEstimator::Fourier::coefficient_decay( + fourier, + dof_handler, + solution, + smoothness_indicators, + /*regression_strategy=*/VectorTools::Linfty_norm, + /*smallest_abs_coefficient=*/1e-10, + /*only_flagged_cells=*/true); + + hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler, + smoothness_indicators, + 0.5, + 0); + hp::Refinement::choose_p_over_h(dof_handler); // Output to VTK if (false) @@ -302,50 +334,7 @@ namespace Step27 data_out.write_vtk(output); } - { - GridRefinement::refine_and_coarsen_fixed_number(triangulation, - estimated_error_per_cell, - 0.3, - 0.03); - - float max_smoothness = *std::min_element(smoothness_indicators.begin(), - smoothness_indicators.end()), - min_smoothness = *std::max_element(smoothness_indicators.begin(), - smoothness_indicators.end()); - { - typename hp::DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - for (; cell != endc; ++cell) - if (cell->refine_flag_set()) - { - max_smoothness = - std::max(max_smoothness, - smoothness_indicators(cell->active_cell_index())); - min_smoothness = - std::min(min_smoothness, - smoothness_indicators(cell->active_cell_index())); - } - } - const float threshold_smoothness = (max_smoothness + min_smoothness) / 2; - - { - typename hp::DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - for (; cell != endc; ++cell) - if (cell->refine_flag_set() && - (smoothness_indicators(cell->active_cell_index()) > - threshold_smoothness) && - (cell->active_fe_index() + 1 < fe_collection.size())) - { - cell->clear_refine_flag(); - cell->set_active_fe_index(cell->active_fe_index() + 1); - } - } - - triangulation.execute_coarsening_and_refinement(); - } + triangulation.execute_coarsening_and_refinement(); } diff --git a/tests/mpi/petsc_step-27.cc b/tests/mpi/petsc_step-27.cc index 322ba6b338..eb95c3caff 100644 --- a/tests/mpi/petsc_step-27.cc +++ b/tests/mpi/petsc_step-27.cc @@ -47,6 +47,7 @@ namespace LA #include #include +#include #include #include @@ -59,6 +60,7 @@ namespace LA #include #include #include +#include #include #include @@ -105,10 +107,8 @@ namespace Step27 hp::QCollection quadrature_collection; hp::QCollection face_quadrature_collection; - hp::QCollection fourier_q_collection; - std::shared_ptr> fourier; - std::vector ln_k; - Table> fourier_coefficients; + hp::QCollection fourier_q_collection; + FESeries::Fourier fourier; AffineConstraints constraints; @@ -181,18 +181,16 @@ namespace Step27 face_quadrature_collection.push_back(QGauss(degree + 1)); } - const unsigned int N = max_degree; - QGauss<1> base_quadrature(2); - QIterated quadrature(base_quadrature, N); + QIterated quadrature(base_quadrature, max_degree); for (unsigned int i = 0; i < fe_collection.size(); i++) fourier_q_collection.push_back(quadrature); - fourier = std::make_shared>(N, - fe_collection, - fourier_q_collection); - - resize(fourier_coefficients, N); + const std::vector n_coefficients_per_direction( + fe_collection.size(), max_degree); + fourier.initialize(n_coefficients_per_direction, + fe_collection, + fourier_q_collection); } @@ -373,12 +371,19 @@ namespace Step27 solution, estimated_error_per_cell); - Vector smoothness_indicators(triangulation.n_active_cells()); - estimate_smoothness(smoothness_indicators); - parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number( triangulation, estimated_error_per_cell, 0.3, 0.03); + Vector smoothness_indicators(triangulation.n_active_cells()); + SmoothnessEstimator::Fourier::coefficient_decay( + fourier, + dof_handler, + solution, + smoothness_indicators, + /*regression_strategy=*/VectorTools::Linfty_norm, + /*smallest_abs_coefficient=*/1e-10, + /*only_flagged_cells=*/true); + hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler, smoothness_indicators, 0.5, @@ -460,68 +465,6 @@ namespace Step27 postprocess(); } } - - - - template - std::pair - LaplaceProblem::predicate(const TableIndices &ind) - { - unsigned int v = 0; - for (unsigned int i = 0; i < dim; i++) - v += ind[i] * ind[i]; - if (v > 0 && v < max_degree * max_degree) - return std::make_pair(true, v); - else - return std::make_pair(false, v); - } - - - - template - void - LaplaceProblem::estimate_smoothness(Vector &smoothness_indicators) - { - Vector local_dof_values; - - for (const auto &cell : dof_handler.active_cell_iterators()) - if (cell->is_locally_owned()) - { - local_dof_values.reinit(cell->get_fe().dofs_per_cell); - cell->get_dof_values(solution, local_dof_values); - - fourier->calculate(local_dof_values, - cell->active_fe_index(), - fourier_coefficients); - - std::pair, std::vector> res = - FESeries::process_coefficients( - fourier_coefficients, - std::bind(&LaplaceProblem::predicate, - this, - std::placeholders::_1), - VectorTools::Linfty_norm); - - Assert(res.first.size() == res.second.size(), ExcInternalError()); - - if (ln_k.size() == 0) - { - ln_k.resize(res.first.size(), 0); - for (unsigned int f = 0; f < ln_k.size(); f++) - ln_k[f] = - std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f])); - } - - for (double &residual_element : res.second) - residual_element = std::log(residual_element); - - std::pair fit = - FESeries::linear_regression(ln_k, res.second); - - smoothness_indicators(cell->active_cell_index()) = - -fit.first - 1. * dim / 2; - } - } } // namespace Step27 diff --git a/tests/mpi/trilinos_step-27.cc b/tests/mpi/trilinos_step-27.cc index 2dde52358a..692a3c2c61 100644 --- a/tests/mpi/trilinos_step-27.cc +++ b/tests/mpi/trilinos_step-27.cc @@ -47,6 +47,7 @@ namespace LA #include #include +#include #include #include @@ -59,6 +60,7 @@ namespace LA #include #include #include +#include #include #include @@ -105,10 +107,8 @@ namespace Step27 hp::QCollection quadrature_collection; hp::QCollection face_quadrature_collection; - hp::QCollection fourier_q_collection; - std::shared_ptr> fourier; - std::vector ln_k; - Table> fourier_coefficients; + hp::QCollection fourier_q_collection; + FESeries::Fourier fourier; AffineConstraints constraints; @@ -181,18 +181,16 @@ namespace Step27 face_quadrature_collection.push_back(QGauss(degree + 1)); } - const unsigned int N = max_degree; - QGauss<1> base_quadrature(2); - QIterated quadrature(base_quadrature, N); + QIterated quadrature(base_quadrature, max_degree); for (unsigned int i = 0; i < fe_collection.size(); i++) fourier_q_collection.push_back(quadrature); - fourier = std::make_shared>(N, - fe_collection, - fourier_q_collection); - - resize(fourier_coefficients, N); + const std::vector n_coefficients_per_direction( + fe_collection.size(), max_degree); + fourier.initialize(n_coefficients_per_direction, + fe_collection, + fourier_q_collection); } @@ -374,13 +372,19 @@ namespace Step27 solution, estimated_error_per_cell); - - Vector smoothness_indicators(triangulation.n_active_cells()); - estimate_smoothness(smoothness_indicators); - parallel::distributed::GridRefinement::refine_and_coarsen_fixed_number( triangulation, estimated_error_per_cell, 0.3, 0.03); + Vector smoothness_indicators(triangulation.n_active_cells()); + SmoothnessEstimator::Fourier::coefficient_decay( + fourier, + dof_handler, + solution, + smoothness_indicators, + /*regression_strategy=*/VectorTools::Linfty_norm, + /*smallest_abs_coefficient=*/1e-10, + /*only_flagged_cells=*/true); + hp::Refinement::p_adaptivity_from_relative_threshold(dof_handler, smoothness_indicators, 0.5, @@ -462,68 +466,6 @@ namespace Step27 postprocess(); } } - - - - template - std::pair - LaplaceProblem::predicate(const TableIndices &ind) - { - unsigned int v = 0; - for (unsigned int i = 0; i < dim; i++) - v += ind[i] * ind[i]; - if (v > 0 && v < max_degree * max_degree) - return std::make_pair(true, v); - else - return std::make_pair(false, v); - } - - - - template - void - LaplaceProblem::estimate_smoothness(Vector &smoothness_indicators) - { - Vector local_dof_values; - - for (const auto &cell : dof_handler.active_cell_iterators()) - if (cell->is_locally_owned()) - { - local_dof_values.reinit(cell->get_fe().dofs_per_cell); - cell->get_dof_values(solution, local_dof_values); - - fourier->calculate(local_dof_values, - cell->active_fe_index(), - fourier_coefficients); - - std::pair, std::vector> res = - FESeries::process_coefficients( - fourier_coefficients, - std::bind(&LaplaceProblem::predicate, - this, - std::placeholders::_1), - VectorTools::Linfty_norm); - - Assert(res.first.size() == res.second.size(), ExcInternalError()); - - if (ln_k.size() == 0) - { - ln_k.resize(res.first.size(), 0); - for (unsigned int f = 0; f < ln_k.size(); f++) - ln_k[f] = - std::log(2.0 * numbers::PI * std::sqrt(1. * res.first[f])); - } - - for (double &residual_element : res.second) - residual_element = std::log(residual_element); - - std::pair fit = - FESeries::linear_regression(ln_k, res.second); - - smoothness_indicators(cell->active_cell_index()) = - -fit.first - 1. * dim / 2; - } - } } // namespace Step27 diff --git a/tests/numerics/smoothness_estimator_01.cc b/tests/numerics/smoothness_estimator_01.cc index 13b3e41b66..a08429ace5 100644 --- a/tests/numerics/smoothness_estimator_01.cc +++ b/tests/numerics/smoothness_estimator_01.cc @@ -1,6 +1,6 @@ // --------------------------------------------------------------------- // -// Copyright (C) 2018 by the deal.II authors +// Copyright (C) 2018 - 2020 by the deal.II authors // // This file is part of the deal.II library. // @@ -13,10 +13,15 @@ // // --------------------------------------------------------------------- + + // essentially similar to fe/fe_series_05.cc but test smoothness estimation. + + #include #include +#include #include #include @@ -37,8 +42,10 @@ #include "../tests.h" + using namespace dealii; + template class LegendreFunction : public Function { @@ -61,6 +68,8 @@ private: const Table coefficients; }; + + // copy-paste from fe_series.cc template double @@ -78,6 +87,8 @@ Lh(const Point &x_q, const TableIndices &indices) return res; } + + template <> double LegendreFunction<2>::value(const dealii::Point<2> &point, @@ -107,6 +118,8 @@ LegendreFunction<3>::value(const dealii::Point<3> &point, return f; } + + void compare(const Table<2, double> &coeff1, const Table<2, double> &coeff2) { @@ -134,6 +147,8 @@ compare(const Table<3, double> &coeff1, const Table<3, double> &coeff2) << linf << std::endl; } + + void resize(Table<2, double> &coeff, const unsigned int N) { coeff.reinit(N, N); @@ -153,45 +168,48 @@ template void test(const LegendreFunction &func, const unsigned int poly_degree) { - // custom predicate: - // p-ref for linear elements and use j=1,...,pe otherwise. - const auto coefficients_predicate = [](std::vector &flags) -> void { - std::fill(flags.begin(), flags.end(), flags.size() > 2); - flags[0] = false; - }; - const unsigned int max_poly = poly_degree + 3; deallog << "-----------------------------------" << std::endl; deallog << dim << "d, p=" << poly_degree << ", max_p=" << max_poly << std::endl; deallog << "-----------------------------------" << std::endl; - Triangulation triangulation; - hp::DoFHandler dof_handler(triangulation); - hp::FECollection fe_collection; - hp::QCollection quadrature_formula; // add some extra FEs in fe_collection + hp::FECollection fe_collection; for (unsigned int p = 1; p <= max_poly; p++) - { - fe_collection.push_back(FE_Q(p)); - quadrature_formula.push_back(QGauss(p + 1 + 5)); - } + fe_collection.push_back(FE_Q(p)); + + const unsigned int fe_index = poly_degree - 1; + const std::vector n_coefficients_per_direction = + SmoothnessEstimator::Legendre::default_number_of_coefficients_per_direction( + fe_collection); + const unsigned int n_modes = n_coefficients_per_direction[fe_index]; + + // custom predicate: + // p-ref for linear elements and use j=1,...,pe otherwise. + ComponentMask coefficients_predicate(n_modes, true); + coefficients_predicate.set(0, false); + Triangulation triangulation; GridGenerator::hyper_cube(triangulation, 0.0, 1.0); // reference cell - const unsigned int fe_index = poly_degree - 1; + + hp::DoFHandler dof_handler(triangulation); + dof_handler.set_fe(fe_collection); dof_handler.begin_active()->set_active_fe_index(fe_index); dof_handler.distribute_dofs(fe_collection); Vector values(dof_handler.n_dofs()); - VectorTools::interpolate(dof_handler, func, values); - const unsigned int N = poly_degree + 1; - FESeries::Legendre legendre(N, fe_collection, quadrature_formula); + hp::QCollection q_collection = + SmoothnessEstimator::Legendre::default_quadrature_collection(fe_collection); + FESeries::Legendre legendre(n_coefficients_per_direction, + fe_collection, + q_collection); const Table &coeff_in = func.get_coefficients(); Table coeff_out; - resize(coeff_out, N); + resize(coeff_out, n_modes); Vector local_dof_values; @@ -211,14 +229,22 @@ test(const LegendreFunction &func, const unsigned int poly_degree) // finally test smoothness estimator: Vector smoothness(1); - SmoothnessEstimator::legendre_coefficient_decay( - legendre, dof_handler, values, smoothness, coefficients_predicate); + SmoothnessEstimator::Legendre::coefficient_decay_per_direction( + legendre, + dof_handler, + values, + smoothness, + coefficients_predicate, + /*smallest_abs_coefficient=*/1e-10, + /*only_flagged_cells=*/false); deallog << "smoothness:" << std::endl << smoothness[0] << std::endl; dof_handler.clear(); } + + int main() { @@ -239,7 +265,8 @@ main() LegendreFunction function(coeff_in); test(function, p); - deallog << "expected smoothness:" << std::endl << 0. << std::endl; + deallog << "expected smoothness:" << std::endl + << std::numeric_limits::infinity() << std::endl; } // for quadratic we can already assign exponential decay: a_i = C exp ( -k @@ -266,7 +293,7 @@ main() test(function, p); deallog << "expected smoothness:" << std::endl - << exp(-std::min(k1, k2)) << std::endl; + << std::min(k1, k2) << std::endl; } // linear elements in 3D (expect zero output) @@ -283,7 +310,8 @@ main() LegendreFunction function(coeff_in); test(function, p); - deallog << "expected smoothness:" << std::endl << 0. << std::endl; + deallog << "expected smoothness:" << std::endl + << std::numeric_limits::infinity() << std::endl; } // cubic in 3D @@ -311,7 +339,7 @@ main() test(function, p); deallog << "expected smoothness:" << std::endl - << exp(-std::min(k1, std::min(k2, k3))) << std::endl; + << std::min(k1, std::min(k2, k3)) << std::endl; } @@ -342,7 +370,7 @@ main() LegendreFunction function(coeff_in); test(function, p); - deallog << "expected smoothness:" << std::endl << exp(-k1) << std::endl; + deallog << "expected smoothness:" << std::endl << k1 << std::endl; } // cubic in 3D (zero) @@ -355,7 +383,8 @@ main() LegendreFunction function(coeff_in); test(function, p); - deallog << "expected smoothness:" << std::endl << 0. << std::endl; + deallog << "expected smoothness:" << std::endl + << std::numeric_limits::infinity() << std::endl; } dealii::deallog << "Ok" << std::endl; diff --git a/tests/numerics/smoothness_estimator_01.with_gsl=on.output b/tests/numerics/smoothness_estimator_01.with_gsl=on.output index 65d1adcafb..f655fa0cd8 100644 --- a/tests/numerics/smoothness_estimator_01.with_gsl=on.output +++ b/tests/numerics/smoothness_estimator_01.with_gsl=on.output @@ -2,54 +2,54 @@ DEAL::----------------------------------- DEAL::2d, p=1, max_p=4 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: -DEAL::1.77636e-15 +DEAL::1.33227e-15 DEAL::smoothness: -DEAL::0.00000 +DEAL::inf DEAL::expected smoothness: -DEAL::0.00000 +DEAL::inf DEAL::----------------------------------- DEAL::2d, p=2, max_p=5 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: -DEAL::7.21645e-16 +DEAL::7.27596e-12 DEAL::smoothness: -DEAL::0.367879 +DEAL::1.00000 DEAL::expected smoothness: -DEAL::0.367879 +DEAL::1.00000 DEAL::----------------------------------- DEAL::3d, p=1, max_p=4 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: -DEAL::3.55271e-15 +DEAL::1.77636e-15 DEAL::smoothness: -DEAL::0.00000 +DEAL::inf DEAL::expected smoothness: -DEAL::0.00000 +DEAL::inf DEAL::----------------------------------- DEAL::3d, p=3, max_p=6 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: -DEAL::3.19189e-15 +DEAL::1.38017e-11 DEAL::smoothness: -DEAL::0.135335 +DEAL::2.00000 DEAL::expected smoothness: -DEAL::0.135335 +DEAL::2.00000 DEAL::----------------------------------- DEAL::3d, p=4, max_p=7 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: -DEAL::4.31599e-15 +DEAL::1.31877e-11 DEAL::smoothness: -DEAL::0.135335 +DEAL::2.00000 DEAL::expected smoothness: -DEAL::0.135335 +DEAL::2.00000 DEAL::----------------------------------- DEAL::3d, p=3, max_p=6 DEAL::----------------------------------- DEAL::Linf norm in exact and calculate Legendre coefficients: DEAL::0.00000 DEAL::smoothness: -DEAL::0.00000 +DEAL::inf DEAL::expected smoothness: -DEAL::0.00000 +DEAL::inf DEAL::Ok diff --git a/tests/numerics/smoothness_estimator_02.cc b/tests/numerics/smoothness_estimator_02.cc new file mode 100644 index 0000000000..0b59a5299b --- /dev/null +++ b/tests/numerics/smoothness_estimator_02.cc @@ -0,0 +1,329 @@ +// --------------------------------------------------------------------- +// +// Copyright (C) 2019 - 2020 by the deal.II authors +// +// This file is part of the deal.II library. +// +// The deal.II library is free software; you can use it, redistribute +// it, and/or modify it under the terms of the GNU Lesser General +// Public License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. +// The full text of the license can be found in the file LICENSE.md at +// the top level directory of deal.II. +// +// --------------------------------------------------------------------- + + + +// Check Fourier coefficient for simple functions in 1D/2D/3D. +// Further, estimate regularity in 1D/2D/3D. +// Similar to tests fe/fe_series_01.cc and numerics/smoothness_estimator_01.cc. + +// Test functions: 1D: x^p, 2D: (x*y)^p, 3D: (x*y*z)^p +// Below is the MWE in Maxima for x*y*z: +/********************************************************* +integrate2(F,xx,aa,bb,yy,cc,dd):=block(integrate(integrate(F,xx,aa,bb),yy,cc,dd)); +integrate3(F,xx,aa,bb,yy,cc,dd,zz,ee,ff):=block(integrate(integrate(integrate(F,xx,aa,bb),yy,cc,dd),zz,ee,ff)); +a:0; +b:1; +nmax:3; +Phi(xx,nn):=exp(((-2)*%i*%pi*nn*xx)/(b-a)); +Phi2(xx,nn,yy,mm):=Phi(xx,nn)*Phi(yy,mm); +Phi3(xx,nn,yy,mm,zz,ll):=Phi(xx,nn)*Phi(yy,mm)*Phi(zz,ll); +f3:x*y*z; +C3(n,m,l):=integrate3(f3*conjugate(Phi3(x,n,y,m,z,l)),x,a,b,y,a,b,z,a,b)/(b-a)^3; +load(functs); +for i:0 thru nmax do (for j:0 thru nmax do (for k:0 thru nmax do +print([i,j,k],fullratsimp(C3(i,j,k))))); +*********************************************************/ + + +#include +#include +#include + +#include +#include + +#include +#include + +#include +#include + +#include + +#include +#include + +#include +#include + +#include "../tests.h" + + +using namespace dealii; + + +template +class PolyFunction : public Function +{ +public: + PolyFunction(const unsigned int poly_degree) + : Function(1) + , poly_degree(poly_degree) + {} + + virtual double + value(const Point &point, const unsigned int component = 0) const; + +private: + const unsigned int poly_degree; +}; + + + +template +double +PolyFunction::value(const Point &point, const unsigned int) const +{ + double f = 1.; + for (unsigned int d = 0; d < dim; ++d) + f *= std::pow(point[d], poly_degree); + return f; +} + + + +template +void + prepare_symmetric_coefficients(Table<1, CoefficientType> & coeff, + const std::vector &coeff_1d) +{ + Assert(coeff.size(0) == coeff_1d.size(), ExcInternalError()); + + for (unsigned int i = 0; i < coeff.size(0); ++i) + coeff(i) = coeff_1d[i]; +} + +template +void + prepare_symmetric_coefficients(Table<2, CoefficientType> & coeff, + const std::vector &coeff_1d) +{ + for (unsigned int d = 0; d < 2; ++d) + Assert(coeff.size(d) == coeff_1d.size(), ExcInternalError()); + + for (unsigned int i = 0; i < coeff.size(0); ++i) + for (unsigned int j = 0; j < coeff.size(1); ++j) + coeff(i, j) = coeff_1d[i] * coeff_1d[j]; +} + +template +void + prepare_symmetric_coefficients(Table<3, CoefficientType> & coeff, + const std::vector &coeff_1d) +{ + for (unsigned int d = 0; d < 3; ++d) + Assert(coeff.size(d) == coeff_1d.size(), ExcInternalError()); + + for (unsigned int i = 0; i < coeff.size(0); ++i) + for (unsigned int j = 0; j < coeff.size(1); ++j) + for (unsigned int k = 0; k < coeff.size(2); ++k) + coeff(i, j, k) = coeff_1d[i] * coeff_1d[j] * coeff_1d[k]; +} + + + +template +typename CoefficientType::value_type +compare(const Table<1, CoefficientType> &coeff1, + const Table<1, CoefficientType> &coeff2) +{ + Assert(coeff1.size(0) == coeff2.size(0), ExcInternalError()); + + typename CoefficientType::value_type linf = 0.; + for (unsigned int i = 0; i < coeff1.size(0); i++) + linf = std::max(linf, std::abs(coeff1(i) - coeff2(i))); + + return linf; +} + +template +typename CoefficientType::value_type +compare(const Table<2, CoefficientType> &coeff1, + const Table<2, CoefficientType> &coeff2) +{ + for (unsigned int d = 0; d < 2; ++d) + Assert(coeff1.size(d) == coeff2.size(d), ExcInternalError()); + + typename CoefficientType::value_type linf = 0.; + for (unsigned int i = 0; i < coeff1.size(0); i++) + for (unsigned int j = 0; j < coeff1.size(1); j++) + linf = std::max(linf, std::abs(coeff1(i, j) - coeff2(i, j))); + + return linf; +} + +template +typename CoefficientType::value_type +compare(const Table<3, CoefficientType> &coeff1, + const Table<3, CoefficientType> &coeff2) +{ + for (unsigned int d = 0; d < 3; ++d) + Assert(coeff1.size(d) == coeff2.size(d), ExcInternalError()); + + typename CoefficientType::value_type linf = 0.; + for (unsigned int i = 0; i < coeff1.size(0); i++) + for (unsigned int j = 0; j < coeff1.size(1); j++) + for (unsigned int k = 0; k < coeff1.size(2); k++) + linf = std::max(linf, std::abs(coeff1(i, j, k) - coeff2(i, j, k))); + + return linf; +} + + + +template +void +test(const unsigned int poly_degree) +{ + const unsigned int max_poly = 3; + hp::FECollection fe_collection; + for (unsigned int p = 1; p <= max_poly; ++p) + fe_collection.push_back(FE_Q(p)); + + const unsigned int fe_index = poly_degree - 1; + const std::vector n_coefficients_per_direction = + SmoothnessEstimator::Fourier::default_number_of_coefficients_per_direction( + fe_collection); + const unsigned int n_modes = n_coefficients_per_direction[fe_index]; + + Assert((poly_degree >= 1) && (poly_degree <= max_poly), ExcInternalError()); + Assert((n_modes >= 3) && (n_modes <= max_poly + 1), ExcInternalError()); + + deallog << "-----------------------------------" << std::endl; + deallog << dim << "d, p=" << poly_degree << ", max_p=" << max_poly + << ", n_modes=" << n_modes << std::endl; + deallog << "-----------------------------------" << std::endl; + + // --- prepare test function --- + PolyFunction test_function(poly_degree); + + // exact coefficients in 1D case + const double &pi = numbers::PI; + const double pi2 = std::pow(pi, 2); + const double pi3 = std::pow(pi, 3); + + std::vector> exact(n_modes); + switch (poly_degree) + { + case 1: + exact[0] = std::complex(1., 0.) / 2.; + if (n_modes > 1) + exact[1] = std::complex(0., -1.) / (2. * pi); + if (n_modes > 2) + exact[2] = std::complex(0., -1.) / (4. * pi); + if (n_modes > 3) + exact[3] = std::complex(0., -1.) / (6. * pi); + break; + case 2: + exact[0] = std::complex(1., 0.) / 3.; + if (n_modes > 1) + exact[1] = std::complex(1., -pi) / (2. * pi2); + if (n_modes > 2) + exact[2] = std::complex(1., -2. * pi) / (8. * pi2); + if (n_modes > 3) + exact[3] = std::complex(1., -3. * pi) / (18. * pi2); + break; + case 3: + exact[0] = std::complex(1., 0.) / 4.; + if (n_modes > 1) + exact[1] = std::complex(3. * pi, 3. - 2. * pi2) / (4. * pi3); + if (n_modes > 2) + exact[2] = std::complex(6. * pi, 3. - 8. * pi2) / (32. * pi3); + if (n_modes > 3) + exact[3] = std::complex(3. * pi, 1. - 6. * pi2) / (36. * pi3); + break; + default: + Assert(false, ExcNotImplemented()); + break; + } + + // coefficient table for multi-dimensional case + TableIndices size; + for (unsigned int d = 0; d < dim; ++d) + size[d] = n_modes; + Table> coeff_in; + coeff_in.reinit(size); + prepare_symmetric_coefficients(coeff_in, exact); + + + // --- prepare data structures --- + Triangulation tria; + GridGenerator::hyper_cube(tria, 0.0, 1.0); // reference cell + + hp::DoFHandler dof_handler(tria); + dof_handler.begin_active()->set_active_fe_index(fe_index); + dof_handler.distribute_dofs(fe_collection); + + + // --- calculate coefficients from exact solution --- + Vector values(dof_handler.n_dofs()); + VectorTools::interpolate(dof_handler, test_function, values); + + Vector local_dof_values; + auto cell = dof_handler.begin_active(); + local_dof_values.reinit(cell->get_fe().dofs_per_cell); + cell->get_dof_values(values, local_dof_values); + + hp::QCollection q_collection = + SmoothnessEstimator::Fourier::default_quadrature_collection(fe_collection); + FESeries::Fourier fourier(n_coefficients_per_direction, + fe_collection, + q_collection); + + Table> coeff_out; + coeff_out.reinit(size); + fourier.calculate(local_dof_values, cell->active_fe_index(), coeff_out); + + // verify results + const double linf = compare(coeff_in, coeff_out); + deallog << "Linf norm in exact and calculate Fourier coefficients:" + << std::endl + << linf << std::endl; + + // finally test smoothness estimator: + Vector regularity(1); + SmoothnessEstimator::Fourier::coefficient_decay( + fourier, + dof_handler, + values, + regularity, + /*regression_strategy=*/VectorTools::Linfty_norm, + /*smallest_abs_coefficient=*/1e-10, + /*only_flagged_cells=*/false); + + deallog << "estimated regularity:" << std::endl << regularity[0] << std::endl; + + dof_handler.clear(); +} + + + +int +main() +{ + std::ofstream logfile("output"); + dealii::deallog.attach(logfile, /*do not print job id*/ false); + dealii::deallog.depth_console(0); + + for (unsigned int poly_degree = 1; poly_degree <= 3; ++poly_degree) + { + test<1>(poly_degree); + test<2>(poly_degree); + test<3>(poly_degree); + } + + dealii::deallog << "Ok" << std::endl; +} diff --git a/tests/numerics/smoothness_estimator_02.output b/tests/numerics/smoothness_estimator_02.output new file mode 100644 index 0000000000..ff88be2087 --- /dev/null +++ b/tests/numerics/smoothness_estimator_02.output @@ -0,0 +1,64 @@ +DEAL::----------------------------------- +DEAL::1d, p=1, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000831516 +DEAL::estimated regularity: +DEAL::0.988446 +DEAL::----------------------------------- +DEAL::2d, p=1, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000415758 +DEAL::estimated regularity: +DEAL::1.09966 +DEAL::----------------------------------- +DEAL::3d, p=1, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000207879 +DEAL::estimated regularity: +DEAL::0.890459 +DEAL::----------------------------------- +DEAL::1d, p=2, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000637547 +DEAL::estimated regularity: +DEAL::1.04031 +DEAL::----------------------------------- +DEAL::2d, p=2, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000212516 +DEAL::estimated regularity: +DEAL::0.854283 +DEAL::----------------------------------- +DEAL::3d, p=2, max_p=3, n_modes=3 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::7.08386e-05 +DEAL::estimated regularity: +DEAL::0.504717 +DEAL::----------------------------------- +DEAL::1d, p=3, max_p=3, n_modes=4 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::0.000375709 +DEAL::estimated regularity: +DEAL::0.972651 +DEAL::----------------------------------- +DEAL::2d, p=3, max_p=3, n_modes=4 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::9.39272e-05 +DEAL::estimated regularity: +DEAL::0.522683 +DEAL::----------------------------------- +DEAL::3d, p=3, max_p=3, n_modes=4 +DEAL::----------------------------------- +DEAL::Linf norm in exact and calculate Fourier coefficients: +DEAL::2.34818e-05 +DEAL::estimated regularity: +DEAL::0.417735 +DEAL::Ok diff --git a/tests/serialization/fe_series_01.cc b/tests/serialization/fe_series_01.cc index 28276e26c4..4a4bc517d3 100644 --- a/tests/serialization/fe_series_01.cc +++ b/tests/serialization/fe_series_01.cc @@ -32,8 +32,9 @@ void test() { // setup - hp::FECollection hp_fe; - hp::QCollection hp_q; + std::vector n_modes; + hp::FECollection hp_fe; + hp::QCollection hp_q; const unsigned int min_degree = 1, max_degree = 2; const QGauss<1> base_quadrature(4); @@ -41,12 +42,13 @@ test() const QSorted quadrature_sorted(quadrature); for (unsigned int p = min_degree; p <= max_degree; ++p) { + n_modes.push_back(max_degree + 1); hp_fe.push_back(FE_Q(p)); hp_q.push_back(quadrature_sorted); } - FESeries::Fourier fourier_save(max_degree + 1, hp_fe, hp_q); - FESeries::Fourier fourier_load(max_degree + 1, hp_fe, hp_q); + FESeries::Fourier fourier_save(n_modes, hp_fe, hp_q); + FESeries::Fourier fourier_load(n_modes, hp_fe, hp_q); // create transformation matrices fourier_save.precalculate_all_transformation_matrices(); diff --git a/tests/serialization/fe_series_01.output b/tests/serialization/fe_series_01.output index 39398fe863..571787f90c 100644 --- a/tests/serialization/fe_series_01.output +++ b/tests/serialization/fe_series_01.output @@ -1,7 +1,7 @@ -DEAL:1d::3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 8 0 1 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 0 0 2 0 0 0 0 0 6 4.99999999999999944e-01 0.00000000000000000e+00 5.00000000000000000e-01 0.00000000000000000e+00 -1.73472347597680709e-18 1.59153688032438484e-01 0.00000000000000000e+00 -1.59153688032438317e-01 5.33917649563659877e-04 8.02149275159986103e-02 5.33917649563897534e-04 -8.02149275159983605e-02 0 0 2 3 2 9 1.66666666666666657e-01 0.00000000000000000e+00 1.66666666666666713e-01 0.00000000000000000e+00 6.66666666666666630e-01 0.00000000000000000e+00 1.01327477769070529e-01 1.59153688032438401e-01 1.01327477769070570e-01 -1.59153688032438401e-01 -2.02654955538141113e-01 1.38777878078144568e-16 2.48179557444021051e-02 8.02149275159984715e-02 2.48179557444023757e-02 -8.02149275159984576e-02 -4.85680761896770119e-02 2.22044604925031308e-16 2 3 3 +DEAL:1d::2 0 3 3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 8 0 1 3.47159221014868569e-02 1 4.65284077898513171e-01 1 5.34715922101486885e-01 1 9.65284077898513226e-01 1 1.65004739103785936e-01 1 3.34995260896214064e-01 1 6.65004739103785880e-01 1 8.34995260896214120e-01 8 0 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 8.69637112843634624e-02 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 1.63036288715636524e-01 0 0 2 0 0 0 0 0 6 4.99999999999999944e-01 0.00000000000000000e+00 5.00000000000000000e-01 0.00000000000000000e+00 -1.73472347597680709e-18 1.59153688032438484e-01 0.00000000000000000e+00 -1.59153688032438317e-01 5.33917649563659877e-04 8.02149275159986103e-02 5.33917649563897534e-04 -8.02149275159983605e-02 0 0 2 3 2 9 1.66666666666666657e-01 0.00000000000000000e+00 1.66666666666666713e-01 0.00000000000000000e+00 6.66666666666666630e-01 0.00000000000000000e+00 1.01327477769070529e-01 1.59153688032438401e-01 1.01327477769070570e-01 -1.59153688032438401e-01 -2.02654955538141113e-01 1.38777878078144568e-16 2.48179557444021051e-02 8.02149275159984715e-02 2.48179557444023757e-02 -8.02149275159984576e-02 -4.85680761896770119e-02 2.22044604925031308e-16 2 3 3 DEAL:1d::OK -DEAL:2d::9 2 10 FE_Q<2>(1) 10 FE_Q<2>(2) 2 0 0 0 0 0 0 64 0 0 0 0 0 2 0 0 3.47159221014868569e-02 5.34715922101486885e-01 2 9.65284077898513226e-01 9.65284077898513226e-01 2 5.34715922101486885e-01 9.65284077898513226e-01 2 4.65284077898513171e-01 9.65284077898513226e-01 2 3.47159221014868569e-02 9.65284077898513226e-01 2 9.65284077898513226e-01 5.34715922101486885e-01 2 5.34715922101486885e-01 5.34715922101486885e-01 2 4.65284077898513171e-01 5.34715922101486885e-01 2 3.47159221014868569e-02 3.47159221014868569e-02 2 9.65284077898513226e-01 4.65284077898513171e-01 2 5.34715922101486885e-01 4.65284077898513171e-01 2 4.65284077898513171e-01 4.65284077898513171e-01 2 3.47159221014868569e-02 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-3.42422689921490701e-04 -7.00326840789855520e-18 -3.42422689921491406e-04 -6.94101148627536413e-18 2.82593529509100232e-04 1.93375788693044329e-04 5.85420340659764627e-05 1.89215625477152721e-04 5.85420340659924005e-05 -1.89215625477162479e-04 5.85420340659733186e-05 1.89215625477151745e-04 5.85420340659912621e-05 -1.89215625477163021e-04 5.85420340659595492e-05 1.89215625477162018e-04 5.85420340660127022e-05 -1.89215625477167141e-04 -1.14565196267454395e-04 8.86335276006899875e-18 2 27 27 DEAL:3d::OK diff --git a/tests/serialization/fe_series_02.cc b/tests/serialization/fe_series_02.cc index d946db6a81..ef24513e2d 100644 --- a/tests/serialization/fe_series_02.cc +++ b/tests/serialization/fe_series_02.cc @@ -32,20 +32,22 @@ void test() { // setup - hp::FECollection hp_fe; - hp::QCollection hp_q; + std::vector n_modes; + hp::FECollection hp_fe; + hp::QCollection hp_q; const unsigned int min_degree = 1, max_degree = 2; const QGauss quadrature(max_degree + 1); const QSorted quadrature_sorted(quadrature); for (unsigned int p = min_degree; p <= max_degree; ++p) { + n_modes.push_back(max_degree + 1); hp_fe.push_back(FE_Q(p)); hp_q.push_back(quadrature_sorted); } - FESeries::Legendre legendre_save(max_degree + 1, hp_fe, hp_q); - FESeries::Legendre legendre_load(max_degree + 1, hp_fe, hp_q); + FESeries::Legendre legendre_save(n_modes, hp_fe, hp_q); + FESeries::Legendre legendre_load(n_modes, hp_fe, hp_q); // create transformation matrices legendre_save.precalculate_all_transformation_matrices(); diff --git a/tests/serialization/fe_series_02.with_gsl=on.output b/tests/serialization/fe_series_02.with_gsl=on.output index 80688c3a60..edc0ab7e3a 100644 --- a/tests/serialization/fe_series_02.with_gsl=on.output +++ b/tests/serialization/fe_series_02.with_gsl=on.output @@ -1,7 +1,7 @@ -DEAL:1d::3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 3 0 1 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 0 0 2 0 0 0 0 0 6 3.53553390593273786e-01 3.53553390593273786e-01 -3.53553390593273786e-01 3.53553390593273786e-01 1.38777878078144568e-16 1.38777878078144568e-16 0 0 2 3 2 9 1.17851130197757933e-01 1.17851130197757933e-01 4.71404520791031678e-01 -3.53553390593273842e-01 3.53553390593273842e-01 -2.08166817117216851e-17 2.35702260395515950e-01 2.35702260395515950e-01 -4.71404520791031567e-01 2 3 3 +DEAL:1d::2 0 3 3 2 10 FE_Q<1>(1) 10 FE_Q<1>(2) 2 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 3 0 1 1.12701665379258312e-01 1 8.87298334620741702e-01 1 5.00000000000000000e-01 3 0 2.77777777777777790e-01 2.77777777777777790e-01 4.44444444444444420e-01 0 0 2 0 0 0 0 0 6 3.53553390593273786e-01 3.53553390593273786e-01 -3.53553390593273786e-01 3.53553390593273786e-01 1.38777878078144568e-16 1.38777878078144568e-16 0 0 2 3 2 9 1.17851130197757933e-01 1.17851130197757933e-01 4.71404520791031678e-01 -3.53553390593273842e-01 3.53553390593273842e-01 -2.08166817117216851e-17 2.35702260395515950e-01 2.35702260395515950e-01 -4.71404520791031567e-01 2 3 3 DEAL:1d::OK -DEAL:2d::3 2 10 FE_Q<2>(1) 10 FE_Q<2>(2) 2 0 0 0 0 0 0 9 0 0 0 0 0 2 0 0 1.12701665379258312e-01 1.12701665379258312e-01 2 8.87298334620741702e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 8.87298334620741702e-01 2 8.87298334620741702e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 1.12701665379258312e-01 2 1.12701665379258312e-01 5.00000000000000000e-01 2 8.87298334620741702e-01 5.00000000000000000e-01 2 5.00000000000000000e-01 8.87298334620741702e-01 2 5.00000000000000000e-01 5.00000000000000000e-01 9 0 7.71604938271605034e-02 7.71604938271605034e-02 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