From 4ea1726aae1b2d8a642f7826701d70ade67da132 Mon Sep 17 00:00:00 2001 From: bangerth Date: Fri, 1 Mar 2013 04:46:37 +0000 Subject: [PATCH] Remove unfinished examples. git-svn-id: https://svn.dealii.org/branches/releases/Branch-7-3@28672 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-26/Makefile | 144 - deal.II/examples/step-26/doc/builds-on | 1 - deal.II/examples/step-26/doc/intro.dox | 2 - deal.II/examples/step-26/doc/kind | 1 - deal.II/examples/step-26/doc/results.dox | 1 - deal.II/examples/step-26/doc/tooltip | 1 - deal.II/examples/step-26/step-26.cc | 688 - deal.II/examples/step-26/surface-points | 199202 --------------- deal.II/examples/step-42/Makefile | 167 - deal.II/examples/step-42/doc/builds-on | 1 - .../examples/step-42/doc/intro-step-42.tex | 405 - deal.II/examples/step-42/doc/intro.dox | 427 - deal.II/examples/step-42/doc/kind | 1 - deal.II/examples/step-42/doc/results.dox | 967 - .../step-42/doc/step-42-CellConstitution.png | Bin 52558 -> 0 bytes .../doc/step-42-CellConstitutionBall.png | Bin 52558 -> 0 bytes .../doc/step-42-CellConstitutionBall2.png | Bin 58094 -> 0 bytes .../doc/step-42-CellConstitutionLi2.png | Bin 96280 -> 0 bytes deal.II/examples/step-42/doc/tooltip | 1 - deal.II/examples/step-42/obstacle_file.pbm | 13602 - deal.II/examples/step-42/step-42.cc | 1564 - deal.II/examples/step-47/Makefile | 144 - deal.II/examples/step-47/doc/builds-on | 1 - deal.II/examples/step-47/doc/intro.dox | 3 - deal.II/examples/step-47/doc/kind | 1 - deal.II/examples/step-47/doc/points.dat | 21 - deal.II/examples/step-47/doc/results.dox | 1 - deal.II/examples/step-47/doc/tooltip | 1 - deal.II/examples/step-47/doc/vertices.dat | 20 - deal.II/examples/step-47/step-47.cc | 1128 - deal.II/examples/step-50/Makefile | 155 - deal.II/examples/step-50/doc/builds-on | 1 - deal.II/examples/step-50/doc/intro.dox | 90 - deal.II/examples/step-50/doc/kind | 1 - deal.II/examples/step-50/doc/results.dox | 98 - deal.II/examples/step-50/doc/tooltip | 1 - deal.II/examples/step-50/step-50.cc | 1052 - 37 files changed, 219893 deletions(-) delete mode 100644 deal.II/examples/step-26/Makefile delete mode 100644 deal.II/examples/step-26/doc/builds-on delete mode 100644 deal.II/examples/step-26/doc/intro.dox delete mode 100644 deal.II/examples/step-26/doc/kind delete mode 100644 deal.II/examples/step-26/doc/results.dox delete mode 100644 deal.II/examples/step-26/doc/tooltip delete mode 100644 deal.II/examples/step-26/step-26.cc delete mode 100644 deal.II/examples/step-26/surface-points delete mode 100644 deal.II/examples/step-42/Makefile delete mode 100644 deal.II/examples/step-42/doc/builds-on delete mode 100644 deal.II/examples/step-42/doc/intro-step-42.tex delete mode 100644 deal.II/examples/step-42/doc/intro.dox delete mode 100644 deal.II/examples/step-42/doc/kind delete mode 100644 deal.II/examples/step-42/doc/results.dox delete mode 100644 deal.II/examples/step-42/doc/step-42-CellConstitution.png delete mode 100644 deal.II/examples/step-42/doc/step-42-CellConstitutionBall.png delete mode 100644 deal.II/examples/step-42/doc/step-42-CellConstitutionBall2.png delete mode 100644 deal.II/examples/step-42/doc/step-42-CellConstitutionLi2.png delete mode 100644 deal.II/examples/step-42/doc/tooltip delete mode 100644 deal.II/examples/step-42/obstacle_file.pbm delete mode 100644 deal.II/examples/step-42/step-42.cc delete mode 100644 deal.II/examples/step-47/Makefile delete mode 100644 deal.II/examples/step-47/doc/builds-on delete mode 100644 deal.II/examples/step-47/doc/intro.dox delete mode 100644 deal.II/examples/step-47/doc/kind delete mode 100644 deal.II/examples/step-47/doc/points.dat delete mode 100644 deal.II/examples/step-47/doc/results.dox delete mode 100644 deal.II/examples/step-47/doc/tooltip delete mode 100644 deal.II/examples/step-47/doc/vertices.dat delete mode 100644 deal.II/examples/step-47/step-47.cc delete mode 100644 deal.II/examples/step-50/Makefile delete mode 100644 deal.II/examples/step-50/doc/builds-on delete mode 100644 deal.II/examples/step-50/doc/intro.dox delete mode 100644 deal.II/examples/step-50/doc/kind delete mode 100644 deal.II/examples/step-50/doc/results.dox delete mode 100644 deal.II/examples/step-50/doc/tooltip delete mode 100644 deal.II/examples/step-50/step-50.cc diff --git a/deal.II/examples/step-26/Makefile b/deal.II/examples/step-26/Makefile deleted file mode 100644 index 099d8280ee..0000000000 --- a/deal.II/examples/step-26/Makefile +++ /dev/null @@ -1,144 +0,0 @@ -# $Id$ - - -# For the small projects Makefile, you basically need to fill in only -# four fields. -# -# The first is the name of the application. It is assumed that the -# application name is the same as the base file name of the single C++ -# file from which the application is generated. -target = step-26 - -# The second field determines whether you want to run your program in -# debug or optimized mode. The latter is significantly faster, but no -# run-time checking of parameters and internal states is performed, so -# you should set this value to `on' while you develop your program, -# and to `off' when running production computations. -debug-mode = on - - -# As third field, we need to give the path to the top-level deal.II -# directory. You need to adjust this to your needs. Since this path is -# probably the most often needed one in the Makefile internals, it is -# designated by a single-character variable, since that can be -# reference using $D only, i.e. without the parentheses that are -# required for most other parameters, as e.g. in $(target). -D = ../../ - - -# The last field specifies the names of data and other files that -# shall be deleted when calling `make clean'. Object and backup files, -# executables and the like are removed anyway. Here, we give a list of -# files in the various output formats that deal.II supports. -clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk *ucd *.d2 - - - - -# -# -# Usually, you will not need to change anything beyond this point. -# -# -# The next statement tells the `make' program where to find the -# deal.II top level directory and to include the file with the global -# settings -include $D/common/Make.global_options - - -# Since the whole project consists of only one file, we need not -# consider difficult dependencies. We only have to declare the -# libraries which we want to link to the object file. deal.II has two -# libraries: one for the debug mode version of the -# application and one for optimized mode. -libs.g := $(lib-deal2.g) -libs.o := $(lib-deal2.o) - - -# We now use the variable defined above to switch between debug and -# optimized mode to select the set of libraries to link with. Included -# in the list of libraries is the name of the object file which we -# will produce from the single C++ file. Note that by default we use -# the extension .g.o for object files compiled in debug mode and .o for -# object files in optimized mode (or whatever local default on your -# system is instead of .o) -ifeq ($(debug-mode),on) - libraries = $(target).g.$(OBJEXT) $(libs.g) -else - libraries = $(target).$(OBJEXT) $(libs.o) -endif - - -# Now comes the first production rule: how to link the single object -# file produced from the single C++ file into the executable. Since -# this is the first rule in the Makefile, it is the one `make' selects -# if you call it without arguments. -all: $(target)$(EXEEXT) -$(target)$(EXEEXT) : $(libraries) - @echo ============================ Linking $@ - @$(CXX) -o $@ $^ $(LIBS) $(LDFLAGS) - - -# To make running the application somewhat independent of the actual -# program name, we usually declare a rule `run' which simply runs the -# program. You can then run it by typing `make run'. This is also -# useful if you want to call the executable with arguments which do -# not change frequently. You may then want to add them to the -# following rule: -run: $(target)$(EXEEXT) - @echo ============================ Running $< - @./$(target)$(EXEEXT) - - -# As a last rule to the `make' program, we define what to do when -# cleaning up a directory. This usually involves deleting object files -# and other automatically created files such as the executable itself, -# backup files, and data files. Since the latter are not usually quite -# diverse, you needed to declare them at the top of this file. -clean: - -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files) - - -# Since we have not yet stated how to make an object file from a C++ -# file, we should do so now. Since the many flags passed to the -# compiler are usually not of much interest, we suppress the actual -# command line using the `at' sign in the first column of the rules -# and write the string indicating what we do instead. -./%.g.$(OBJEXT) : - @echo "==============debug========= $( $@" - @$(CXX) $(CXXFLAGS.g) -c $< -o $@ -./%.$(OBJEXT) : - @echo "==============optimized===== $( $@" - @$(CXX) $(CXXFLAGS.o) -c $< -o $@ - - -# The following statement tells make that the rules `run' and `clean' -# are not expected to produce files of the same name as Makefile rules -# usually do. -.PHONY: all run clean - - -# Finally there is a rule which you normally need not care much about: -# since the executable depends on some include files from the library, -# besides the C++ application file of course, it is necessary to -# re-generate the executable when one of the files it depends on has -# changed. The following rule creates a dependency file -# `Makefile.dep', which `make' uses to determine when to regenerate -# the executable. This file is automagically remade whenever needed, -# i.e. whenever one of the cc-/h-files changed. Make detects whether -# to remake this file upon inclusion at the bottom of this file. -# -# If the creation of Makefile.dep fails, blow it away and fail -Makefile.dep: $(target).cc Makefile \ - $(shell echo $D/include/deal.II/*/*.h) - @echo ============================ Remaking $@ - @$D/common/scripts/make_dependencies $(INCLUDE) -B. $(target).cc \ - > $@ \ - || (rm -f $@ ; false) - @if test -s $@ ; then true ; else rm $@ ; false ; fi - -# To make the dependencies known to `make', we finally have to include -# them: -include Makefile.dep - - diff --git a/deal.II/examples/step-26/doc/builds-on b/deal.II/examples/step-26/doc/builds-on deleted file mode 100644 index 48a0f73876..0000000000 --- a/deal.II/examples/step-26/doc/builds-on +++ /dev/null @@ -1 +0,0 @@ -step-4 diff --git a/deal.II/examples/step-26/doc/intro.dox b/deal.II/examples/step-26/doc/intro.dox deleted file mode 100644 index c59cfdf609..0000000000 --- a/deal.II/examples/step-26/doc/intro.dox +++ /dev/null @@ -1,2 +0,0 @@ - -

Introduction

diff --git a/deal.II/examples/step-26/doc/kind b/deal.II/examples/step-26/doc/kind deleted file mode 100644 index c1d9154931..0000000000 --- a/deal.II/examples/step-26/doc/kind +++ /dev/null @@ -1 +0,0 @@ -techniques diff --git a/deal.II/examples/step-26/doc/results.dox b/deal.II/examples/step-26/doc/results.dox deleted file mode 100644 index f4c6feefb5..0000000000 --- a/deal.II/examples/step-26/doc/results.dox +++ /dev/null @@ -1 +0,0 @@ -

Results

diff --git a/deal.II/examples/step-26/doc/tooltip b/deal.II/examples/step-26/doc/tooltip deleted file mode 100644 index 7b3e8b447a..0000000000 --- a/deal.II/examples/step-26/doc/tooltip +++ /dev/null @@ -1 +0,0 @@ -Complicated boundary descriptions. diff --git a/deal.II/examples/step-26/step-26.cc b/deal.II/examples/step-26/step-26.cc deleted file mode 100644 index 25ebd7f654..0000000000 --- a/deal.II/examples/step-26/step-26.cc +++ /dev/null @@ -1,688 +0,0 @@ -/* Author: Wolfgang Bangerth, University of Heidelberg, 1999 */ - -/* $Id$ */ -/* */ -/* Copyright (C) 1999-2007, 2011-2012 by the deal.II authors */ -/* */ -/* This file is subject to QPL and may not be distributed */ -/* without copyright and license information. Please refer */ -/* to the file deal.II/doc/license.html for the text and */ -/* further information on this license. */ - -// @sect3{Include files} - -// The first few (many?) include files have already been used in the previous -// example, so we will not explain their meaning here again. -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include - -#include -#include -#include - -// This is new, however: in the previous example we got some unwanted output -// from the linear solvers. If we want to suppress it, we have to include this -// file and add a single line somewhere to the program (see the main() -// function below for that): -#include - - -#include -#include - -// The last step is as in all previous programs: -namespace Step26 -{ - using namespace dealii; - - class PointCloudSurface : public StraightBoundary<3> - { - public: - /** - * Constructor. - */ - PointCloudSurface (const std::string &filename); - - /** - * Let the new point be the - * arithmetic mean of the two - * vertices of the line. - * - * Refer to the general - * documentation of this class - * and the documentation of the - * base class for more - * information. - */ - virtual Point<3> - get_new_point_on_line (const Triangulation<3>::line_iterator &line) const; - - /** - * Let the new point be the - * arithmetic mean of the four - * vertices of this quad and the - * four midpoints of the lines, - * which are already created at - * the time of calling this - * function. - * - * Refer to the general - * documentation of this class - * and the documentation of the - * base class for more - * information. - */ - virtual Point<3> - get_new_point_on_quad (const Triangulation<3>::quad_iterator &quad) const; - - /** - * Gives n=points.size() - * points that splits the - * StraightBoundary line into - * $n+1$ partitions of equal - * lengths. - * - * Refer to the general - * documentation of this class - * and the documentation of the - * base class. - */ - virtual void - get_intermediate_points_on_line (const Triangulation<3>::line_iterator &line, - std::vector > &points) const; - - /** - * Gives n=points.size()=m*m - * points that splits the - * p{StraightBoundary} quad into - * (m+1)(m+1) subquads of equal - * size. - * - * Refer to the general - * documentation of this class - * and the documentation of the - * base class. - */ - virtual void - get_intermediate_points_on_quad (const Triangulation<3>::quad_iterator &quad, - std::vector > &points) const; - - /** - * A function that, given a point - * p, returns the closest - * point on the surface defined by the - * input file. For the time being, we - * simply return the closest point in the - * point cloud, rather than doing any - * sort of interpolation. - */ - Point<3> closest_point (const Point<3> &p) const; - private: - std::vector > point_list; - }; - - - PointCloudSurface::PointCloudSurface (const std::string &filename) - { - // first read in all the points - { - std::ifstream in (filename.c_str()); - AssertThrow (in, ExcIO()); - - while (in) - { - Point<3> p; - in >> p; - point_list.push_back (p); - } - - AssertThrow (point_list.size() > 1, ExcIO()); - } - - // next fit a linear model through the data cloud to rectify it in a local - // coordinate system - // - // the first step is to move the center of mass of the points to the - // origin - { - const Point<3> c_o_m = std::accumulate (point_list.begin(), - point_list.end(), - Point<3>()) / - point_list.size(); - for (unsigned int i=0; i gradient_direction - = Point<2>(a,b) / std::sqrt(a*a+b*b); - const Point<2> orthogonal_direction - = Point<2>(-b,a) / std::sqrt(a*a+b*b); - - const double stretch_factor = std::sqrt(1.+a*a+b*b); - - for (unsigned int i=0; i xy (point_list[i][0], - point_list[i][1]); - const double grad_distance = xy * gradient_direction; - const double orth_distance = xy * orthogonal_direction; - - // we then have to stretch the points in the gradient direction. the - // stretch factor is defined above (zero if the original plane was - // already the xy plane, infinity if it was vertical) - const Point<2> new_xy - = (grad_distance * stretch_factor * gradient_direction + - orth_distance * orthogonal_direction); - point_list[i][0] = new_xy[0]; - point_list[i][1] = new_xy[1]; - } - } - } - - - Point<3> - PointCloudSurface::closest_point (const Point<3> &p) const - { - double distance = p.distance (point_list[0]); - Point<3> point = point_list[0]; - - for (std::vector >::const_iterator i=point_list.begin(); - i != point_list.end(); ++i) - { - const double d = p.distance (*i); - if (d < distance) - { - distance = d; - point = *i; - } - } - - return point; - } - - - Point<3> - PointCloudSurface:: - get_new_point_on_line (const Triangulation<3>::line_iterator &line) const - { - return closest_point (StraightBoundary<3>::get_new_point_on_line (line)); - } - - - - Point<3> - PointCloudSurface:: - get_new_point_on_quad (const Triangulation<3>::quad_iterator &quad) const - { - return closest_point (StraightBoundary<3>::get_new_point_on_quad (quad)); - } - - - - void - PointCloudSurface:: - get_intermediate_points_on_line (const Triangulation<3>::line_iterator &line, - std::vector > &points) const - { - StraightBoundary<3>::get_intermediate_points_on_line (line, - points); - for (unsigned int i=0; i::quad_iterator &quad, - std::vector > &points) const - { - StraightBoundary<3>::get_intermediate_points_on_quad (quad, - points); - for (unsigned int i=0; iLaplaceProblem class template} - - // This is again the same LaplaceProblem class as in the - // previous example. The only difference is that we have now declared it as - // a class with a template parameter, and the template parameter is of - // course the spatial dimension in which we would like to solve the Laplace - // equation. Of course, several of the member variables depend on this - // dimension as well, in particular the Triangulation class, which has to - // represent quadrilaterals or hexahedra, respectively. Apart from this, - // everything is as before. - template - class LaplaceProblem - { - public: - LaplaceProblem (); - void run (); - - private: - void make_grid_and_dofs (); - void assemble_system (); - void solve (); - void output_results () const; - - Triangulation triangulation; - FE_Q fe; - DoFHandler dof_handler; - - SparsityPattern sparsity_pattern; - SparseMatrix system_matrix; - - Vector solution; - Vector system_rhs; - }; - - - // @sect3{Right hand side and boundary values} - - - - - template - class BoundaryValues : public Function - { - public: - BoundaryValues () : Function() {} - - virtual double value (const Point &p, - const unsigned int component = 0) const; - }; - - - - template - double BoundaryValues::value (const Point &p, - const unsigned int /*component*/) const - { - return std::max(p[dim-1], -5.); - } - - - - // @sect3{Implementation of the LaplaceProblem class} - - // Next for the implementation of the class template that makes use of the - // functions above. As before, we will write everything as templates that - // have a formal parameter dim that we assume unknown at the - // time we define the template functions. Only later, the compiler will find - // a declaration of LaplaceProblem@<2@> (in the - // main function, actually) and compile the entire class with - // dim replaced by 2, a process referred to as `instantiation - // of a template'. When doing so, it will also replace instances of - // RightHandSide@ by RightHandSide@<2@> and - // instantiate the latter class from the class template. - // - // In fact, the compiler will also find a declaration - // LaplaceProblem@<3@> in main(). This will cause - // it to again go back to the general LaplaceProblem@ - // template, replace all occurrences of dim, this time by 3, - // and compile the class a second time. Note that the two instantiations - // LaplaceProblem@<2@> and LaplaceProblem@<3@> are - // completely independent classes; their only common feature is that they - // are both instantiated from the same general template, but they are not - // convertible into each other, for example, and share no code (both - // instantiations are compiled completely independently). - - - // @sect4{LaplaceProblem::LaplaceProblem} - - // After this introduction, here is the constructor of the - // LaplaceProblem class. It specifies the desired polynomial - // degree of the finite elements and associates the DoFHandler to the - // triangulation just as in the previous example program, step-3: - template - LaplaceProblem::LaplaceProblem () : - fe (1), - dof_handler (triangulation) - {} - - - // @sect4{LaplaceProblem::make_grid_and_dofs} - - // Grid creation is something inherently dimension dependent. However, as - // long as the domains are sufficiently similar in 2D or 3D, the library can - // abstract for you. In our case, we would like to again solve on the square - // [-1,1]x[-1,1] in 2D, or on the cube [-1,1]x[-1,1]x[-1,1] in 3D; both can - // be termed hyper_cube, so we may use the same function in - // whatever dimension we are. Of course, the functions that create a - // hypercube in two and three dimensions are very much different, but that - // is something you need not care about. Let the library handle the - // difficult things. - // - // Likewise, associating a degree of freedom with each vertex is something - // which certainly looks different in 2D and 3D, but that does not need to - // bother you either. This function therefore looks exactly like in the - // previous example, although it performs actions that in their details are - // quite different if dim happens to be 3. The only significant - // difference from a user's perspective is the number of cells resulting, - // which is much higher in three than in two space dimensions! - template - void LaplaceProblem::make_grid_and_dofs () - { - GridGenerator::hyper_cube (triangulation, -30, 30); - - for (unsigned int f=0; f::faces_per_cell; ++f) - if (triangulation.begin()->face(f)->center()[2] > 15) - { - triangulation.begin()->face(f)->set_boundary_indicator (1); - for (unsigned int i=0; i::lines_per_face; ++i) - triangulation.begin()->face(f)->line(i)->set_boundary_indicator (1); - break; - } - triangulation.set_boundary (1, pds); - - - for (unsigned int v=0; v::vertices_per_cell; ++v) - if (triangulation.begin()->vertex(v)[2] > 0) - triangulation.begin()->vertex(v) - = pds.closest_point (Point<3>(triangulation.begin()->vertex(v)[0], - triangulation.begin()->vertex(v)[1], - 0)); - - for (unsigned int i=0; i<4; ++i) - { - for (typename Triangulation::active_cell_iterator - cell = triangulation.begin_active(); - cell != triangulation.end(); ++cell) - for (unsigned int f=0; f::faces_per_cell; ++f) - if (cell->face(f)->boundary_indicator() == 1) - cell->set_refine_flag (); - - triangulation.execute_coarsening_and_refinement (); - - std::cout << "Refinement cycle " << i << std::endl - << " Number of active cells: " - << triangulation.n_active_cells() - << std::endl - << " Total number of cells: " - << triangulation.n_cells() - << std::endl; - - } - - - dof_handler.distribute_dofs (fe); - - std::cout << " Number of degrees of freedom: " - << dof_handler.n_dofs() - << std::endl; - - sparsity_pattern.reinit (dof_handler.n_dofs(), - dof_handler.n_dofs(), - dof_handler.max_couplings_between_dofs()); - DoFTools::make_sparsity_pattern (dof_handler, sparsity_pattern); - sparsity_pattern.compress(); - - system_matrix.reinit (sparsity_pattern); - - solution.reinit (dof_handler.n_dofs()); - system_rhs.reinit (dof_handler.n_dofs()); - } - - - // @sect4{LaplaceProblem::assemble_system} - - // Unlike in the previous example, we would now like to use a non-constant - // right hand side function and non-zero boundary values. Both are tasks - // that are readily achieved with a only a few new lines of code in the - // assemblage of the matrix and right hand side. - // - // More interesting, though, is the way we assemble matrix and right hand - // side vector dimension independently: there is simply no difference to the - // two-dimensional case. Since the important objects used in this function - // (quadrature formula, FEValues) depend on the dimension by way of a - // template parameter as well, they can take care of setting up properly - // everything for the dimension for which this function is compiled. By - // declaring all classes which might depend on the dimension using a - // template parameter, the library can make nearly all work for you and you - // don't have to care about most things. - template - void LaplaceProblem::assemble_system () - { - MatrixTools::create_laplace_matrix (dof_handler, - QGauss(2), - system_matrix); - system_rhs = 0; - - std::map boundary_values; - VectorTools::interpolate_boundary_values (dof_handler, - 0, - BoundaryValues(), - boundary_values); - MatrixTools::apply_boundary_values (boundary_values, - system_matrix, - solution, - system_rhs); - } - - - // @sect4{LaplaceProblem::solve} - - // Solving the linear system of equations is something that looks almost - // identical in most programs. In particular, it is dimension independent, - // so this function is copied verbatim from the previous example. - template - void LaplaceProblem::solve () - { - // NEW - SolverControl solver_control (dof_handler.n_dofs(), - 1e-12*system_rhs.l2_norm()); - SolverCG<> cg (solver_control); - - PreconditionSSOR<> preconditioner; - preconditioner.initialize(system_matrix, 1.2); - - cg.solve (system_matrix, solution, system_rhs, - preconditioner); - } - - - // @sect4{LaplaceProblem::output_results} - - // This function also does what the respective one did in step-3. No changes - // here for dimension independence either. - // - // The only difference to the previous example is that we want to write - // output in GMV format, rather than for gnuplot (GMV is another graphics - // program that, contrary to gnuplot, shows data in nice colors, allows - // rotation of geometries with the mouse, and generates reasonable - // representations of 3d data; for ways to obtain it see the ReadMe file of - // deal.II). To write data in this format, we simply replace the - // data_out.write_gnuplot call by - // data_out.write_gmv. - // - // Since the program will run both 2d and 3d versions of the laplace solver, - // we use the dimension in the filename to generate distinct filenames for - // each run (in a better program, one would check whether `dim' can have - // other values than 2 or 3, but we neglect this here for the sake of - // brevity). - template - void LaplaceProblem::output_results () const - { - DataOut data_out; - - data_out.attach_dof_handler (dof_handler); - data_out.add_data_vector (solution, "solution"); - - data_out.build_patches (); - - std::ofstream output (dim == 2 ? - "solution-2d.gmv" : - "solution-3d.gmv"); - data_out.write_gmv (output); - } - - - - // @sect4{LaplaceProblem::run} - - // This is the function which has the top-level control over - // everything. Apart from one line of additional output, it is the same as - // for the previous example. - template - void LaplaceProblem::run () - { - std::cout << "Solving problem in " << dim << " space dimensions." << std::endl; - - make_grid_and_dofs(); - assemble_system (); - solve (); - output_results (); - } -} - - -// @sect3{The main function} - -// And this is the main function. It also looks mostly like in step-3, but if -// you look at the code below, note how we first create a variable of type -// LaplaceProblem@<2@> (forcing the compiler to compile the class -// template with dim replaced by 2) and run a 2d -// simulation, and then we do the whole thing over in 3d. -// -// In practice, this is probably not what you would do very frequently (you -// probably either want to solve a 2d problem, or one in 3d, but not both at -// the same time). However, it demonstrates the mechanism by which we can -// simply change which dimension we want in a single place, and thereby force -// the compiler to recompile the dimension independent class templates for the -// dimension we request. The emphasis here lies on the fact that we only need -// to change a single place. This makes it rather trivial to debug the program -// in 2d where computations are fast, and then switch a single place to a 3 to -// run the much more computing intensive program in 3d for `real' -// computations. -// -// Each of the two blocks is enclosed in braces to make sure that the -// laplace_problem_2d variable goes out of scope (and releases -// the memory it holds) before we move on to allocate memory for the 3d -// case. Without the additional braces, the laplace_problem_2d -// variable would only be destroyed at the end of the function, i.e. after -// running the 3d problem, and would needlessly hog memory while the 3d run -// could actually use it. -// -// Finally, the first line of the function is used to suppress some output. -// Remember that in the previous example, we had the output from the linear -// solvers about the starting residual and the number of the iteration where -// convergence was detected. This can be suppressed through the -// deallog.depth_console(0) call. -// -// The rationale here is the following: the deallog (i.e. deal-log, not -// de-allog) variable represents a stream to which some parts of the library -// write output. It redirects this output to the console and if required to a -// file. The output is nested in a way so that each function can use a prefix -// string (separated by colons) for each line of output; if it calls another -// function, that may also use its prefix which is then printed after the one -// of the calling function. Since output from functions which are nested deep -// below is usually not as important as top-level output, you can give the -// deallog variable a maximal depth of nested output for output to console and -// file. The depth zero which we gave here means that no output is written. By -// changing it you can get more information about the innards of the library. -int main () -{ - try - { - using namespace dealii; - using namespace Step26; - - deallog.depth_console (0); - - LaplaceProblem<3> laplace_problem_3d; - laplace_problem_3d.run (); - } - catch (std::exception &exc) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Exception on processing: " << std::endl - << exc.what() << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - - return 1; - } - catch (...) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Unknown exception!" << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - return 1; - } - - return 0; -} diff --git a/deal.II/examples/step-26/surface-points b/deal.II/examples/step-26/surface-points deleted file mode 100644 index 06c17bc98d..0000000000 --- a/deal.II/examples/step-26/surface-points +++ /dev/null @@ -1,199202 +0,0 @@ - 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9.997534 42.080753 -13.489339 - 99.981133 67.590904 21.090704 - 99.981483 63.586147 17.693605 - 99.981560 58.721859 15.151418 - 99.984962 65.314011 19.422552 - 99.985909 59.785046 16.037029 - 99.990257 56.900345 14.340770 - 99.995491 66.542198 20.312881 diff --git a/deal.II/examples/step-42/Makefile b/deal.II/examples/step-42/Makefile deleted file mode 100644 index 4a005b2d7f..0000000000 --- a/deal.II/examples/step-42/Makefile +++ /dev/null @@ -1,167 +0,0 @@ -# $Id$ - - -# For the small projects Makefile, you basically need to fill in only -# four fields. -# -# The first is the name of the application. It is assumed that the -# application name is the same as the base file name of the single C++ -# file from which the application is generated. -target = step-42 - -# The second field determines whether you want to run your program in -# debug or optimized mode. The latter is significantly faster, but no -# run-time checking of parameters and internal states is performed, so -# you should set this value to `on' while you develop your program, -# and to `off' when running production computations. -debug-mode = on - - -# As third field, we need to give the path to the top-level deal.II -# directory. You need to adjust this to your needs. Since this path is -# probably the most often needed one in the Makefile internals, it is -# designated by a single-character variable, since that can be -# reference using $D only, i.e. without the parentheses that are -# required for most other parameters, as e.g. in $(target). -D = ../../ - - -# The last field specifies the names of data and other files that -# shall be deleted when calling `make clean'. Object and backup files, -# executables and the like are removed anyway. Here, we give a list of -# files in the various output formats that deal.II supports. -clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk *vtu *visit *ucd *.d2 - - - - -# -# -# Usually, you will not need to change anything beyond this point. -# -# -# The next statement tells the `make' program where to find the -# deal.II top level directory and to include the file with the global -# settings -include $D/common/Make.global_options - -################################################################ -# This example program will only work if Trilinos is installed. If this -# is not the case, then simply redefine the main targets to do nothing -ifneq ($(USE_CONTRIB_TRILINOS)$(USE_CONTRIB_P4EST),yesyes) -default run clean: - @echo - @echo "===========================================================" - @echo "= This program can only be run with the following =" - @echo "= configuration: =" - @echo "= - with Trilinos =" - @echo "= - with P4EST =" - @echo "= At least one of these conditions is not satisfied. =" - @echo "===========================================================" - @echo -else -# -################################################################ - - - - -# Since the whole project consists of only one file, we need not -# consider difficult dependencies. We only have to declare the -# libraries which we want to link to the object file. deal.II has two -# libraries: one for the debug mode version of the -# application and one for optimized mode. -libs.g := $(lib-deal2.g) -libs.o := $(lib-deal2.o) - - -# We now use the variable defined above to switch between debug and -# optimized mode to select the set of libraries to link with. Included -# in the list of libraries is the name of the object file which we -# will produce from the single C++ file. Note that by default we use -# the extension .g.o for object files compiled in debug mode and .o for -# object files in optimized mode (or whatever local default on your -# system is instead of .o) -ifeq ($(debug-mode),on) - libraries = $(target).g.$(OBJEXT) $(libs.g) -else - libraries = $(target).$(OBJEXT) $(libs.o) -endif - - -# Now comes the first production rule: how to link the single object -# file produced from the single C++ file into the executable. Since -# this is the first rule in the Makefile, it is the one `make' selects -# if you call it without arguments. -all: $(target)$(EXEEXT) -$(target)$(EXEEXT) : $(libraries) - @echo ============================ Linking $@ - @$(CXX) -o $@ $^ $(LIBS) $(LDFLAGS) - - -# To make running the application somewhat independent of the actual -# program name, we usually declare a rule `run' which simply runs the -# program. You can then run it by typing `make run'. This is also -# useful if you want to call the executable with arguments which do -# not change frequently. You may then want to add them to the -# following rule: -run: $(target)$(EXEEXT) - @echo ============================ Running $< - @./$(target)$(EXEEXT) - - - -# As a last rule to the `make' program, we define what to do when -# cleaning up a directory. This usually involves deleting object files -# and other automatically created files such as the executable itself, -# backup files, and data files. Since the latter are not usually quite -# diverse, you needed to declare them at the top of this file. -clean: - -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files) - - -# Since we have not yet stated how to make an object file from a C++ -# file, we should do so now. Since the many flags passed to the -# compiler are usually not of much interest, we suppress the actual -# command line using the `at' sign in the first column of the rules -# and write the string indicating what we do instead. -./%.g.$(OBJEXT) : - @echo "==============debug========= $( $@" - @$(CXX) $(CXXFLAGS.g) -c $< -o $@ -./%.$(OBJEXT) : - @echo "==============optimized===== $( $@" - @$(CXX) $(CXXFLAGS.o) -c $< -o $@ - - - -# The following statement tells make that the rules `run' and `clean' -# are not expected to produce files of the same name as Makefile rules -# usually do. -.PHONY: all run clean - - -# Finally there is a rule which you normally need not care much about: -# since the executable depends on some include files from the library, -# besides the C++ application file of course, it is necessary to -# re-generate the executable when one of the files it depends on has -# changed. The following rule creates a dependency file -# `Makefile.dep', which `make' uses to determine when to regenerate -# the executable. This file is automagically remade whenever needed, -# i.e. whenever one of the cc-/h-files changed. Make detects whether -# to remake this file upon inclusion at the bottom of this file. -# -# If the creation of Makefile.dep fails, blow it away and fail -Makefile.dep: $(target).cc Makefile \ - $(shell echo $D/include/deal.II/*/*.h) - @echo ============================ Remaking $@ - @$D/common/scripts/make_dependencies $(INCLUDE) -B. $(target).cc \ - > $@ \ - || (rm -f $@ ; false) - @if test -s $@ ; then true ; else rm $@ ; false ; fi - -# To make the dependencies known to `make', we finally have to include -# them: -include Makefile.dep - - -endif # USE_CONTRIB_TRILINOS diff --git a/deal.II/examples/step-42/doc/builds-on b/deal.II/examples/step-42/doc/builds-on deleted file mode 100644 index 8defe7dd9e..0000000000 --- a/deal.II/examples/step-42/doc/builds-on +++ /dev/null @@ -1 +0,0 @@ -step-41 step-40 diff --git a/deal.II/examples/step-42/doc/intro-step-42.tex b/deal.II/examples/step-42/doc/intro-step-42.tex deleted file mode 100644 index 17767acd14..0000000000 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ /dev/null @@ -1,405 +0,0 @@ -\documentclass{article} - -\usepackage{amsmath} -\usepackage{amssymb} - -\title{Documentation of step-42, An obstacle problem for elasto-plastic material behavior in three dimensions} -\author{Joerg Frohne} -\date{Juni, 2012} - -\begin{document} - -\section{Introduction} - -This example is an extension of step-41, considering a contact problem with an -elasto-plastic material behavior with isotropic hardening in three dimensions. -That means that we have to take care of an additional nonlinearity: the -material behavior. Since we consider a three dimensional problem here, a -separate difference to step-41 is that the contact area is at the boundary of -the deformable body now, rather than in the interior. Finally, compared to -step-41, we also have to deal with -hanging nodes because of the adaptive mesh in both the handling of the linear -system as well as of the inequality constraints; in the latter case, we will -have to deal with prioritizing whether the constraints from the hanging nodes -or from the inequalities are more important. - -Since you can very easily reach a few million degrees of freedom in three -dimensions, even with adaptive mesh refinement, we decided to use Trilinos and -p4est to run our code in parallel, building on the framework of step-40 for -the parallelization.\\ - -\begin{huge} -{distributed} -\end{huge} - - -\section{Classical formulation} - -The classical formulation of the problem possesses the following form: -\begin{align*} - \varepsilon(u) &= A\sigma + \lambda & &\quad\text{in } \Omega,\\ - \lambda(\tau - \sigma) &\geq 0\quad\forall\tau\text{ with - }\mathcal{F}(\tau)\leq 0 & &\quad\text{in } \Omega,\\ - -\textrm{\textrm{div}}\ \sigma &= f & &\quad\text{in } \Omega,\\ - u(\mathbf x) &= 0 & &\quad\text{on }\Gamma_D,\\ - \sigma_t(u) &= 0,\quad\sigma_n(u)\leq 0 & &\quad\text{on }\Gamma_C,\\ -\sigma_n(u)(u_n - g) &= 0,\quad u_n(\mathbf x) - g(\mathbf x) \leq 0 & &\quad\text{on } \Gamma_C -\end{align*} -with $u\in H^2(\Omega),\Omega\subset\mathbb{R}^3$. The vector valued -function $u$ denotes the displacement in the deformable body. The first two lines describe the -elasto-plastic material behavior. Therein the equation shows the -strain of the deformation $\varepsilon (u)$ as the additive decomposition of the -elastic part $A\sigma$ and the plastic part $\lambda$. $A$ is defined as the -compliance tensor of fourth order which contains some material constants and -$\sigma$ as the symmetric stress tensor of second order. So we have to consider -the inequality in the second row in a pointwise sense where $\lambda(\tau - -\sigma)$ is the inner product of two symmetric tensors of second order. -Furthermore we have to distinguish two cases.\\ -The continuous and convex function $\mathcal{F}$ denotes the von Mises flow function -$$\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0¸\quad\text{with}\quad \tau^D -= \tau - \dfrac{1}{3}tr(\tau)I,$$ -$\sigma_0$ as yield stress and $\vert .\vert$ as the frobenius norm. If there -are no plastic deformations in a particular point - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < -\sigma_0$ and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. -That means if the stress is smaller than the yield stress there are only elastic -deformations in that point.\\ -To consider it the other way around if the deviator stress $\sigma^D$ is in a -norm bigger than the yield stress then $\sigma^D$ has to be projected back to the yield surface and there are plastic deformations which means $\lambda$ -would be positiv for that particular point. We refer that the stresses are -computed by Hooke's law for isotorpic materials. You can find the description at the end of section 3. Else if the norm of the deviator stress tensor is smaller or equal the yield stress then $\lambda$ is zero and there are no plastic deformations in -that point.\\ -There the index $D$ denotes the deviator part of for example the stress where -$tr(.)$ is the trace of a tensor. The definition shows an additive decomposition -of the stress $\sigma$ into a hydrostatic part (or volumetric part) $\dfrac{1}{3}tr(\tau)I$ and the deviator -part $\sigma^D$. For metal the deviator stress composes the main indicator for -plastic deformations.\\ -The third equation is called equilibrium condition with a force of volume -density $f$ which we will neglect in our example. -The boundary of $\Omega$ separates as follows $\Gamma=\Gamma_D\bigcup\Gamma_C$ and $\Gamma_D\bigcap\Gamma_C=\emptyset$. -At the boundary $\Gamma_D$ we have zero Dirichlet conditions. $\Gamma_C$ denotes the potential contact boundary.\\ -The last two lines decribe the so-called Signorini contact conditions. If there is no contact the normal stress -$$ \sigma_n = \sigma n\cdot n$$ -is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangential stress $\sigma_t = \sigma\cdot n - \sigma_n n$ -vanishes, because we consider a frictionless situation and the normal stress is -negative. The gap $g$ comes with the start configuration of the obstacle and the -deformable body. We refer that you have to ensure that the obstacle does not hit -the boundary of $\Gamma_C$ - -\section{Derivation of the variational inequality} - -As a starting point to derive the equations above, let us imagine that we want -to minimise an energy functional: -$$E(\tau) := \dfrac{1}{2}\int\limits_{\Omega}\tau A \tau d\tau,\quad \tau\in \Pi W^{\textrm{div}}$$ -with -$$W^{\textrm{div}}:=\lbrace \tau\in -L^2(\Omega,\mathbb{R}^{\textrm{dim}\times\textrm{dim}}_{\textrm{sym}}):\textrm{div}(\tau)\in L^2(\Omega,\mathbb{R}^{\textrm{dim}})\rbrace$$ and -$$\Pi \Sigma:=\lbrace \tau\in \Sigma, \mathcal{F}(\tau)\leq 0\rbrace$$ -as the set of admissible stresses which is defined -by a continious, convex flow function $\mathcal{F}$. - -With the goal of deriving the dual formulation of the minimisation -problem, we define a lagrange function: -$$L(\tau,\varphi) := E(\tau) + (\varphi, \textrm{div}(\tau)),\quad \lbrace\tau,\varphi\rbrace\in\Pi W^{\textrm{div}}\times V^+$$ -with -$$V^+ := \lbrace u\in V: u_n\leq g \text{ on } \Gamma_C \rbrace$$ -$$V:=\left[ H_0^1 \right]^{\textrm{dim}}:=\lbrace u\in \left[H^1(\Omega)\right]^{\textrm{dim}}: u -= 0 \text{ on } \Gamma_D\rbrace$$ -By building the Fr\'echet derivatives of $L$ for both components we obtain the -dual formulation for the stationary case which is known as \textbf{Hencky-Type-Model}:\\ -Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with -$$\left(A\sigma,\tau - \sigma\right) + \left(u, \textrm{div}(\tau) - \textrm{div}(\sigma)\right) \geq 0,\quad \forall \tau\in \Pi W^{\textrm{div}}$$ -$$-\left(\textrm{div}(\sigma),\varphi - u\right) \geq 0,\quad \forall \varphi\in V^+.$$ -By integrating by parts and multiplying the first inequality by the elastic -tensor $C=A^{-1}$ we achieve the primal-mixed version of our problem:\\ -Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with -$$\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) \geq 0,\quad \forall \tau\in \Pi W$$ -$$\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+.$$ -Therein $\varepsilon$ denotes the linearised deformation tensor with $\varepsilon(u) := \dfrac{1}{2}\left(\nabla u + \nabla u^T\right)$ for small deformations.\\ -Most materials - especially metals - have the property that they show some hardening effects during the forming process. -There are different constitutive laws to describe those material behaviors. The -simplest one is called linear isotropic hardening described by the flow function -$\mathcal{F}(\tau,\eta) = \vert\tau^D\vert - (\sigma_0 + \gamma^{\text{iso}}\eta)$ where -$\eta$ is the norm of the plastic strain $\eta = \vert \varepsilon - -A\sigma\vert$. -It can be considered by establishing an additional term in our primal-mixed formulation:\\ -Find a pair $\lbrace(\sigma,\xi),u\rbrace\in \Pi (W\times L^2(\Omega,\mathbb{R}))\times V^+$ with -$$\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) + \gamma^{\text{iso}}\left( \xi, \eta - \xi\right) \geq 0,\quad \forall (\tau,\eta)\in \Pi (W,L^2(\Omega,\mathbb{R}))$$ -$$\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,$$ -with the hardening parameter $\gamma^{\text{iso}} > 0$.\\ -Now we want to derive a primal problem which only depends on the displacement $u$. For that purpose we -set $\eta = \xi$ and eliminate the stress $\sigma$ by applying the projection -theorem (see Grossmann, Roos: Numerical Treatment of Partial Differential -Equations, Springer-Verlag Berlin Heidelberg, 2007 and Frohne: FEM-Simulation -der Umformtechnik metallischer Oberflächen im Mikrokosmos, Ph.D. thesis, -University of Siegen, Germany, 2011) on\\ -$$\left(\sigma - C\varepsilon(u), \tau - \sigma\right) \geq 0,\quad \forall \tau\in \Pi W,$$ -which yields with the second inequality:\\ -Find the displacement $u\in V^+$ with -$$\left(P_{\Pi}(C\varepsilon(u)),\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,$$ -with the projection: -$$P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0 + \gamma^{\text{iso}}\xi,\\ - \hat\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0 + \gamma^{\text{iso}}\xi, - \end{cases}$$ -with the radius -$$\hat\alpha := \sigma_0 + \gamma^{\text{iso}}\xi .$$ -With the relation $\xi = \vert\varepsilon(u) - A\sigma\vert$ it is possible to eliminate $\xi$ inside the projection $P_{\Pi}$:\\ -$$P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\ - \alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0, - \end{cases}$$ -$$\alpha := \sigma_0 + \dfrac{\gamma^{\text{iso}}}{2\mu+\gamma^{\text{iso}}}\left(\vert\tau^D\vert - \sigma_0\right) ,$$ -with a further material parameter $\mu>0$ called shear modulus. We refer that -this only possible for isotropic plasticity.\\ -To make things a bit easier from now on we denote -$$\gamma := \dfrac{\gamma^{\text{iso}}}{2\mu + -\gamma^{\text{iso}}}\in[0,1)\text{ with }\gamma^{\text{iso}}\in[0,\infty),$$ -$$\beta :=\dfrac{\sigma_0}{\vert\tau^D\vert}.$$ If $\gamma^{\text{iso}}$ tends to zero $\gamma$ tends also to zero. And if $\gamma^{\text{iso}}$ tends to -infinity $\gamma$ tends to one. This allows us to reformulate our problem as -follows $$P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\ - \gamma\tau^D + (1-\gamma)\beta\tau^D - + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > - \sigma_0, \end{cases}.$$ -For further details see Suttmeier: On Plasticity with Hardening: -An Adaptive Finite Element Discretisation, International Mathematical Forum, 5, -2010, no. 52, 2591-2601.\\ -So what we do is to calculate the stresses -by using Hooke's law for linear elastic, isotropic materials $$\sigma = C \varepsilon(u) = 2\mu \varepsilon^D(u) + \kappa tr(\varepsilon(u))I = \left[2\mu\left(\mathbb{I} -\dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I\right]\varepsilon(u)$$ -with the material parameter $\kappa>0$ (bulk modulus). The variables $I$ and -$\mathbb{I}$ denote the identity tensors of second and forth order. In that -notation $2\mu \varepsilon^D(u)$ is the deviatoric part and $\kappa -tr(\varepsilon(u))$ the volumetric part of the stress tensor.\\ -In the next step we test in a pointwise sense where the deviator part of the -stress in a norm is bigger than the yield stress. If there are such points we -project the deviator stress in those points back to the yield surface. Methods of this kind are called projections algorithm or radial-return-algorithm.\\ -Now we have a primal formulation of our elasto-plastic contact problem which only depends on the displacement $u$. -It consists of a nonlinear variational inequality and has a unique solution as -it satisfies the theorem of Lions and Stampaccia. A proof can be found in -Rodrigues: Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, -1987.\\ -To handle the nonlinearity of the constitutive law we use a Newton method and to deal with the contact we apply an -active set method like in step-41. To be more concrete we combine both methods to an inexact semi smooth Newton -method - inexact since we use an iterative solver for the linearised problems in each Newton step. - -\section{Linearization of the constitutive law for the Newton method} - -To apply Newton method we have to linearise the following semi-linearform -$$a(\psi;\varphi) := -\left(P_{\Pi}(C\varepsilon(\psi)),\varepsilon(\varphi)\right).$$ Because we have -to find the solution $u$ in the convex set $V^+$, we have to apply an SQP-method (SQP: sequential quadratic programming). That means we have -to solve a minimisation problem for a known $u^i$ in every SQP-step of the form -\begin{eqnarray*} - & & a(u^{i};u^{i+1} - u^i) + \dfrac{1}{2}a'(u^i;u^{i+1} - u^i,u^{i+1} - u^i)\\ - &=& a(u^i;u^{i+1}) - a(u^i;u^i) +\\ - & & \dfrac{1}{2}\left( a'(u^i;u^{i+1},u^{i+1}) - 2a'(u^i;u^i,u^{i+1}) - a'(u^i;u^i,u^i)\right)\\ - &\rightarrow& \textrm{min},\quad u^{i+1}\in V^+. -\end{eqnarray*} -Neglecting the constant terms $ a(u^i;u^i)$ and $ a'(u^i;u^i,u^i)$ we obtain the -following minimisation problem $$\dfrac{1}{2} a'(u^i;u^{i+1},u^{i+1}) - -F(u^{i+1})\rightarrow \textrm{min},\quad u^{i+1}\in V^+$$ with $$F(\varphi) := -\left(a'(u^{i};u^{i},\varphi) - a(u^{i},\varphi) \right).$$ -In the case of our constitutive law the Fr\'echet derivative of the -semi-linearform $a(.;.)$ at the point $u^i$ is - -$$a'(u^i;\psi,\varphi) = -(I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,$$ $$ -I(x) := \begin{cases} -C_{\mu} + C_{\kappa}, & -\quad \vert \tau^D \vert \leq \sigma_0\\ -\gamma C_{\mu} + (1-\gamma)\beta\left(C_{\mu} - -2\mu\dfrac{\tau^D\otimes\tau^D}{\vert\tau^D\vert^2}\right) + C_{\kappa}, &\quad -\vert \tau^D \vert > \sigma_0 -\end{cases} -$$ -with -$$C_{\mu} := 2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes -I\right)\quad\text{(shear part of the stress strain tensor)},$$ -$$C_{\kappa} := \kappa I\otimes I\quad\text{(bulk part of the stress strain -tensor)},$$ -$$\tau^D := C\varepsilon^D(u^i).$$ -Remark that $a(.;.)$ is not differentiable in the common sense but it is -slantly differentiable like the function for the contact problem and again we refer to -Hintermueller, Ito, Kunisch: The primal-dual active set strategy as a semismooth newton method, SIAM J. OPTIM., 2003, Vol. 13, No. 3, pp. 865-888. -Again the first case is for elastic and the second for plastic deformation. - -\section{Formulation as a saddle point problem} - -Just as in step-41 we compose a saddle point problem out of the minimisation -problem. Again we do so to gain a formulation that allows us to solve a linear -system of equations finally.\\ -We introduce a Lagrange multiplier $\lambda$ and the convex cone $K\subset W'$, -$W'$ dual space of the trace space $W:=\left[ H_0^{\frac{1}{2}}(\Gamma_C) -\right]^{\textrm{dim}$ of $V$ restricted to $\Gamma_C$, $$K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad -\forall v\in H_0^{\frac{1}{2}}(\Gamma_C), v \ge 0\text{ on }\Gamma_C \}$$ -of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ -denotes the duality pairing, i.e. a boundary integral, between $W'$ and $W$. -Intuitively, $K$ is the cone of all "non-positive functions", except that $ K\subset -\left( \left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}} \right)' $ and -so contains other objects besides regular functions as well. This yields:\\ - -\noindent -\textit{Find $u\in V$ and $\lambda\in K$ such that} -\begin{align*} - \hat{a}(u,v) + b(v,\lambda) &= f(v),\quad &&v\in V\\ - b(u,\mu - \lambda) &\leq \langle g,(\mu - - \lambda)n\rangle_{\Gamma_C},\quad&&\mu\in K, -\end{align*} -\textit{with} -\begin{align*} - \hat{a}(u,v) &:= a'(u^i;u,v)\\ - b(u,\mu) &:= \langle un,\mu n\rangle_{\Gamma_C},\quad &&u\in V,\quad\mu\in W'. -\end{align*} -As in the section before $u^i$ denotes the linearization point for the -semi-linearform $a(.;.)$. In contrast to step-41 we directly consider $\lambda$ -as the additional, positive force $\sigma(u)n$ that the obstacle -exerts on the boundary $\Gamma_C$ of the body.\\ - -\noindent -The existence and uniqueness of the analytical solution $(u,\lambda)\in V\times -K$ of this saddle point problem has been stated in Glowinski, Lions and Tr\'{e}moli\`{e}res: Numerical -Analysis of Variational Inequalities, North-Holland, 1981.\\ - -\noindent -NOTE: In this example as well as in the further documentation we make the -assumption that the normal vector $n$ equals to $(0,0,1)$. This comes up with -the starting condition of our deformable body. - -\section{Active Set methods to solve the saddle point problem} - -The linearized problem is essentially like a pure elastic problem with contact like -in step-41. The only difference consists in the fact that the contact area -is at the boundary instead of in the domain. But this has no further consequence -so that we refer to the documentation of step-41 with the only hint that -$\mathcal{S}$ containts all the vertices at the contact boundary $\Gamma_C$ this -time. - -\section{The primal-dual active set algorithm combined with the inexact semi smooth -Newton method} - -Now we describe an algorithm that combines the damped semismooth Newton-method, -which we use for the nonlinear constitutive law, with the semismooth Newton -method for the contact. It sums up the results of the sections before and works as follows: -\begin{itemize} - \item[(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that - $\mathcal{S} = \mathcal{A}_k \cup \mathcal{F}_k$ and $\mathcal{A}_k \cap - \mathcal{F}_k = \emptyset$ and set $k = 1$. The start value $\hat U^0 := - P_{\mathcal{A}_k}(0)$ fullfills our obstacle condition. - \item[(1)] Assemble the Newton matrix $A := a'(\hat - U^{k-1};\varphi_p,\varphi_q)$ and the right-hand-side $F(\hat U^{k-1})$. - \item[(2)] Find the primal-dual pair $(\bar U^k,\Lambda^k)$ that satisfies - \begin{align*} - A\bar U^k + B\Lambda^k & = F, &\\ - \left[B^T\bar U^k\right]_p & = G_p & \forall p\in\mathcal{A}_k,\\ - \Lambda^k_p & = 0 & \forall p\in\mathcal{F}_k. - \end{align*} - \item[(3)] Damping for $k>2$ by applying a line search and calculating a linear - combination of $U^{k-1}$ and $\bar U^k$. Find an - $\alpha_i:=2^{-i},(i=0,\ldots,10)$ so that $$U^k := \alpha_i\bar U^k + - (1-\alpha_i)U^{k-1}$$ yields $$\vert - F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.$$ - \item[(4)] Define the new active and inactive sets by - $$\mathcal{A}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + - c\left(\left[B^TU^k\right]_p - G_p\right) > 0\rbrace,$$ - $$\mathcal{F}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + - c\left(\left[B^TU^k\right]_p - G_p\right) \leq 0\rbrace.$$ - Projection $U^k$ so that it holds the second equation in (2) - $$\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).$$ - \item[(5)] If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert - F\left(U^{k}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to - step (1). -\end{itemize} -\noindent -The subscript $p$ denotes a vertex and the meaning of the -decorated and none decorated $U$ is as follows: $\bar U$ denotes the solution of the linear system of equations in (2), $U$ is the -damped solution and equals to $\bar U$ if the damping parameter $\alpha_0 = -1$ and $\hat U := P_{\mathcal{A}}(U)$ is the projection of the active -components in $\mathcal{A}$ to the gap -$$P_{\mathcal{A}}(U):=\begin{cases} -U_p, & \textrm{if}\quad p\notin\mathcal{A}\\ -g_{h,p}, & \textrm{if}\quad -p\in\mathcal{A}. -\end{cases}$$\\ -The matrix $B\in\mathbb{R}^{n\times m}$, $n>m$ describes the coupling of the -bases for the displacements and lagrange multiplier (contact forces) -and it is not quadratic in our situation since $\Lambda^k$ is only defined on -$\Gamma_C$. Due to the ansatz functions $\psi_i$ (scalar valued) of the -lagrange multiplier are fullfilling the following biorthogonal condition (see Hüeber, Wohlmuth: A primal–dual active -set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. -194, 2005, pp. 3147-3166) -$$ \int\limits_{\Gamma_C}\psi_i(x)\varphi_j(x)dx = -\delta_{ij}\int\limits_{\Gamma_C}\varphi_j(x)dx$$ -this yields -$$B_{pq} = \begin{cases} -\int\limits_{\Gamma_C}\varphi_p(x)dxI_3, & \text{if}\quad p=q,\quad -p,q\in\mathcal{S}\\ -0I_3, & \text{if}\quad p\neq q,\quad p\textrm{ or }q\notin\mathcal{S}. -\end{cases}$$ -Here $I_3$ denotes the threedimensional identity matrix. -In our programm we use the structure of a quadratic sparse for -$B\in\mathbb{R}^{n\times n}$ and for $\Lambda^k$ a vector with length $n$ where -$\Lambda^k_p = 0$ for $p\notin \mathcal{S}$. -The vector $G$ is defined by a suitable approximation $g_h$ of the gap $g$ -$$G_p = \begin{cases} -g_{h,p}\int\limits_{\Gamma_C}\varphi_p(x)dx, & \text{if}\quad p\in\mathcal{S}\\ -0, & \text{if}\quad p\notin\mathcal{S}. -\end{cases}$$\\ -Note that $G_p$ is a threedimensional vector and that again we applied the -biorthogonal property of the lagrange multiplier ansatz functions to the -integral $\int\limits_{\Gamma_C}g_h(x)\varphi_p(x)dx$ with $g_h(x)=\sum\limits_i -g_{h,p}\varphi_p(x)$ (see the reference mentioned above).\\ -Compared to step-41, step (1) is added but it should be clear from the sections -above that we only linearize the problem. In step (2) we have to solve a linear system of equations again. And now the solution has to fulfill two stopping -criteria. $\mathcal{A}_{k+1} = \mathcal{A}_k$ makes sure that the contact zones -are iterated out and the second ensures an accurate enough residual which means that the plastic zones are also iterated out.\\ -A similar method can also be found in Brunssen, Schmid, Schaefer, Wohlmuth: A -fast and robust iterative solver for nonlinear contact problems using a -primal-dual active set strategy and algebraic multigrid, Int. J. Numer. -Meth. Engng, 2007, 69, pp. 524-543. But in advance we apply a line search to -obtain a more robust method regarding the start value. Solving an elastic -problem in the very first step ($k=1$) we get a reasonable start value but as -you can see in the results damping is important if we choose for example a ball as -obstacle.\\ -Damping our Newton method is more important for the nonlinearity cause by the -constitutive law as for the contact. For this reason we start to damp our method -for $k>2$ when we have two plastic iterations. Note that $U^1$ is a solution of -an elastic problem and $U^2$ is the first plastic solution. A linear combination -between these both results in stresses which are not in the convex set -of the feasible stresses. - -\section{Adaptive mesh refinement} - -Since we run our program in 3d, there is a good reason to use adaptive -mesh refined. To make life a bit easier we are choosing the -KellyErrorEstimator that is already implemented in deal.II. We hand the -solution vector to it which contains the displacement $u$. As we will see in the -results it yields a quite reasonable adaptive mesh for the contact zone. - -\section{Implementation} - -This tutorial is essentailly a mixture of step-40 and step-41 but instead of -PETSc we let the Trilinos library deal with parallelizing the linear algebra -(like in step-32). Since we are trying to solve a similar problem like in -step-41 we will use the same methods but now in parallel. - -Another difficulty is the handling of the different constraints from -(the dirichlet conditons), the hanging nodes and the inequality condition that -arises from the contact. For this purpose we create three objects of type -ConstraintMatrix. - -Beside the ConstitutiveLaw class there is another new class called Input. This -class allows us to read in an obstacle from a file. In our example the file -'obstacle\_file.dat' containts data which describe an Chinese, Japanese or -Korean symbol for force or power. (See www.orientaloutpost.com/: ``This word can be used for motivation - it -can also mean power / motion / propulsion / force. It can be anything -internal or external that keeps you going. This is the safest way to express -motivation in Chinese. If your audience is Japanese, please see the other entry -for motivation. This is a word in Japanese and Korean, but it means "motive -power" or "kinetic energy" (without the motivation meaning that you are -probably looking for)''). - -\section{Results} - - - -\end{document} diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox deleted file mode 100644 index f33e5b31fd..0000000000 --- a/deal.II/examples/step-42/doc/intro.dox +++ /dev/null @@ -1,427 +0,0 @@ -
- -This program was contributed by Jörg Frohne (University of Siegen, -Germany) while on a long-term visit to Texas A&M University. -
- - - - - -

Introduction

- -This example is an extension of step-41, considering a contact problem with an -elasto-plastic material behavior with isotropic hardening in three dimensions. -That means that we have to take care of an additional nonlinearity: the -material behavior. Since we consider a three dimensional problem here, a -separate difference to step-41 is that the contact area is at the boundary of -the deformable body now, rather than in the interior. Finally, compared to -step-41, we also have to deal with hanging nodes because of the adaptive mesh -in both the handling of the linear -system as well as of the inequality constraints; in the latter case, we will -have to deal with prioritizing whether the constraints from the hanging nodes -or from the inequalities are more important. - -Since you can very easily reach a few million degrees of freedom in three -dimensions, even with adaptive mesh refinement, we decided to use Trilinos and -p4est to run our code in parallel, building on the framework of step-40 for -the parallelization. - -@f{huge} -{distributed} -@f} - - -

Classical formulation

- -The classical formulation of the problem possesses the following form: -@f{align*} - \varepsilon(u) &= A\sigma + \lambda & &\quad\text{in } \Omega,\\ - \lambda(\tau - \sigma) &\geq 0\quad\forall\tau\text{ with - }\mathcal{F}(\tau)\leq 0 & &\quad\text{in } \Omega,\\ - -\textrm{\textrm{div}}\ \sigma &= f & &\quad\text{in } \Omega,\\ - u(\mathbf x) &= 0 & &\quad\text{on }\Gamma_D,\\ - \sigma_t(u) &= 0,\quad\sigma_n(u)\leq 0 & &\quad\text{on }\Gamma_C,\\ -\sigma_n(u)(u_n - g) &= 0,\quad u_n(\mathbf x) - g(\mathbf x) \leq 0 & &\quad\text{on } \Gamma_C -@f} -with $u\in H^2(\Omega),\Omega\subset\mathbb{R}^3$. The vector valued -function $u$ denotes the displacement in the deformable body. The first two lines describe the -elasto-plastic material behavior. Therein the equation shows the -strain of the deformation $\varepsilon (u)$ as the additive decomposition of the -elastic part $A\sigma$ and the plastic part $\lambda$. $A$ is defined as the compliance tensor of fourth order which contains some material constants and $\sigma$ as the -symmetric stress tensor of second order. So we have to consider -the inequality in the second row in a pointwise sense where $\lambda(\tau - -\sigma)$ is the inner product of two symmetric tensors of second order. Furthermore we have to -distinguish two cases. - -The continuous and convex function $\mathcal{F}$ denotes the von Mises flow function -@f{gather*}\mathcal{F}(\tau) = \vert\tau^D\vert - \sigma_0�\quad\text{with}\quad \tau^D -= \tau - \dfrac{1}{3}tr(\tau)I,@f} -$\sigma_0$ as yield stress and $\vert .\vert$ as the Frobenius norm. If there -are no plastic deformations in a particular point - that is $\lambda=0$ - this yields $\vert\sigma^D\vert < -\sigma_0$ and otherwise if $\lambda > 0$ it follows that $\vert\sigma^D\vert = \sigma_0$. -That means if the stress is smaller than the yield stress there are only elastic -deformations in that point. - -To consider it the other way around if the deviator stress $\sigma^D$ is in a -norm bigger than the yield stress then $\sigma^D$ has to be projected back to the yield surface and there are plastic deformations which means $\lambda$ -would be positiv for that particular point. We refer that the stresses are -computed by Hooke's law for isotorpic materials. You can find the description at the end of section 3. Else if the norm of the deviator stress tensor is smaller or equal the yield stress then $\lambda$ is zero and there are no plastic deformations in -that point. - -There the index $D$ denotes the deviator part of for example the stress where -$tr(.)$ is the trace of a tensor. The definition shows an additive decomposition -of the stress $\sigma$ into a hydrostatic part (or volumetric part) $\dfrac{1}{3}tr(\tau)I$ and the deviator -part $\sigma^D$. For metal the deviator stress composes the main indicator for -plastic deformations. - -The third equation is called equilibrium condition with a force of volume -density $f$ which we will neglect in our example. -The boundary of $\Omega$ separates as follows $\Gamma=\Gamma_D\bigcup\Gamma_C$ and $\Gamma_D\bigcap\Gamma_C=\emptyset$. -At the boundary $\Gamma_D$ we have zero Dirichlet conditions. $\Gamma_C$ denotes the potential contact boundary. - -The last two lines decribe the so-called Signorini contact conditions. If there is no contact the normal stress -@f{gather*} \sigma_n = \sigma n\cdot n@f} -is zero with the outward normal $n$. If there is contact ($u_n = g$) the tangential stress $\sigma_t = \sigma\cdot n - \sigma_n n$ -vanishes, because we consider a frictionless situation and the normal stress is -negative. The gap $g$ comes with the start configuration of the obstacle and the -deformable body. - - -

Derivation of the variational inequality

- -As a starting point to derive the equations above, let us imagine that we want -to minimise an energy functional: -@f{gather*}E(\tau) := \dfrac{1}{2}\int\limits_{\Omega}\tau A \tau d\tau,\quad \tau\in \Pi W^{\textrm{div}}@f} -with -@f{gather*}W^{\textrm{div}}:=\lbrace \tau\in -L^2(\Omega,\mathbb{R}^{\textrm{dim}\times\textrm{dim}}_{\textrm{sym}}):\textrm{div}(\tau)\in L^2(\Omega,\mathbb{R}^{\textrm{dim}})\rbrace@f} and -@f{gather*}\Pi \Sigma:=\lbrace \tau\in \Sigma, \mathcal{F}(\tau)\leq 0\rbrace@f} -as the set of admissible stresses which is defined -by a continious, convex flow function $\mathcal{F}$. - -With the goal of deriving the dual formulation of the minimisation -problem, we define a lagrange function: -@f{gather*}L(\tau,\varphi) := E(\tau) + (\varphi, \textrm{div}(\tau)),\quad \lbrace\tau,\varphi\rbrace\in\Pi W^{\textrm{div}}\times V^+@f} -with -@f{gather*}V^+ := \lbrace u\in V: u_n\leq g \text{ on } \Gamma_C \rbrace@f} -@f{gather*}V:=\left[ H_0^1 \right]^{\textrm{dim}}:=\lbrace u\in \left[H^1(\Omega)\right]^{\textrm{dim}}: u -= 0 \text{ on } \Gamma_D\rbrace@f} -By building the Fréchet derivatives of $L$ for both components we obtain the -dual formulation for the stationary case which is known as Hencky-Type-Model:\\ -Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with -@f{gather*}\left(A\sigma,\tau - \sigma\right) + \left(u, \textrm{div}(\tau) - \textrm{div}(\sigma)\right) \geq 0,\quad \forall \tau\in \Pi W^{\textrm{div}}@f} -@f{gather*}-\left(\textrm{div}(\sigma),\varphi - u\right) \geq 0,\quad \forall \varphi\in V^+.@f} -By integrating by parts and multiplying the first inequality by the elastic -tensor $C=A^{-1}$ we achieve the primal-mixed version of our problem: -Find a pair $\lbrace\sigma,u\rbrace\in \Pi W\times V^+$ with -@f{gather*}\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) \geq 0,\quad \forall \tau\in \Pi W@f} -@f{gather*}\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+.@f} -Therein $\varepsilon$ denotes the linearised deformation tensor with $\varepsilon(u) := \dfrac{1}{2}\left(\nabla u + \nabla u^T\right)$ for small deformations. - -Most materials - especially metals - have the property that they show some hardening effects during the forming process. -There are different constitutive laws to describe those material behaviors. The -simplest one is called linear isotropic hardening described by the flow function -$\mathcal{F}(\tau,\eta) = \vert\tau^D\vert - (\sigma_0 + \gamma^{\text{iso}}\eta)$ where -$\eta$ is the norm of the plastic strain $\eta = \vert \varepsilon - -A\sigma\vert$. -It can be considered by establishing an additional term in our primal-mixed formulation: -Find a pair $\lbrace(\sigma,\xi),u\rbrace\in \Pi (W\times L^2(\Omega,\mathbb{R}))\times V^+$ with -@f{gather*}\left(\sigma,\tau - \sigma\right) - \left(C\varepsilon(u), \tau - \sigma\right) + \gamma^{\text{iso}}\left( \xi, \eta - \xi\right) \geq 0,\quad \forall (\tau,\eta)\in \Pi (W,L^2(\Omega,\mathbb{R}))@f} -@f{gather*}\left(\sigma,\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,@f} -with the hardening parameter $\gamma^{\text{iso}} > 0$. - -Now we want to derive a primal problem which only depends on the displacement $u$. For that purpose we -set $\eta = \xi$ and eliminate the stress $\sigma$ by applying the projection -theorem (see Grossmann, Roos: Numerical Treatment of Partial Differential -Equations, Springer-Verlag Berlin Heidelberg, 2007 and Frohne: FEM-Simulation -der Umformtechnik metallischer Oberflächen im Mikrokosmos, Ph.D. thesis, -University of Siegen, Germany, 2011) on -@f{gather*}\left(\sigma - C\varepsilon(u), \tau - \sigma\right) \geq 0,\quad \forall \tau\in \Pi W,@f} -which yields with the second inequality:\\ -Find the displacement $u\in V^+$ with -@f{gather*}\left(P_{\Pi}(C\varepsilon(u)),\varepsilon(\varphi) - \varepsilon(u)\right) \geq 0,\quad \forall \varphi\in V^+,@f} -with the projection: -@f{gather*}P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0 + \gamma^{\text{iso}}\xi,\\ - \hat\alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0 + \gamma^{\text{iso}}\xi, - \end{cases}@f} -with the radius -@f{gather*}\hat\alpha := \sigma_0 + \gamma^{\text{iso}}\xi .@f} -With the relation $\xi = \vert\varepsilon(u) - A\sigma\vert$ it is possible to eliminate $\xi$ inside the projection $P_{\Pi}$:\\ -@f{gather*}P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\ - \alpha\dfrac{\tau^D}{\vert\tau^D\vert} + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > \sigma_0, - \end{cases}@f} -@f{gather*}\alpha := \sigma_0 + \dfrac{\gamma^{\text{iso}}}{2\mu+\gamma^{\text{iso}}}\left(\vert\tau^D\vert - \sigma_0\right) ,@f} -with a further material parameter $\mu>0$ called shear modulus. We refer that -this only possible for isotropic plasticity. - -To make things a bit easier from now on we denote -@f{gather*}\gamma := \dfrac{\gamma^{\text{iso}}}{2\mu + -\gamma^{\text{iso}}}\in[0,1)\text{ with }\gamma^{\text{iso}}\in[0,\infty),@f} -@f{gather*}\beta :=\dfrac{\sigma_0}{\vert\tau^D\vert}.@f} -If $\gamma^{\text{iso}}$ tends to zero $\gamma$ tends also to zero. And if $\gamma^{\text{iso}}$ tends to -infinity $\gamma$ tends to one. This allows us to reformulate our problem as -follows -@f{gather*}P_{\Pi}(\tau):=\begin{cases} - \tau, & \text{if }\vert\tau^D\vert \leq \sigma_0,\\ - \gamma\tau^D + (1-\gamma)\beta\tau^D - + \dfrac{1}{3}tr(\tau), & \text{if }\vert\tau^D\vert > - \sigma_0, \end{cases}.@f} -For further details e.g., see Suttmeier: On Plasticity with Hardening: -An Adaptive Finite Element Discretisation, International Mathematical Forum, 5, -2010, no. 52, 2591-2601. - -So what we do is to calculate the stresses by using Hooke's law for linear elastic, isotropic materials -@f{gather*}\sigma = C \varepsilon(u) = 2\mu \varepsilon^D(u) + \kappa tr(\varepsilon(u))I = \left[2\mu\left(\mathbb{I} -\dfrac{1}{3} I\otimes I\right) + \kappa I\otimes I\right]\varepsilon(u)@f} -with the material parameter $\kappa>0$ (bulk modulus). The variables $I$ and -$\mathbb{I}$ denote the identity tensors of second and forth order. In that -notation $2\mu \varepsilon^D(u)$ is the deviatoric part and $\kappa -tr(\varepsilon(u))$ the volumetric part of the stress tensor. - -In the next step we test in a pointwise sense where the deviator part of the -stress in a norm is bigger than the yield stress. If there are such points we -project the deviator stress in those points back to the yield surface. Methods of this kind are called projections algorithm or radial-return-algorithm. - -Now we have a primal formulation of our elasto-plastic contact problem which only depends on the displacement $u$. -It consists of a nonlinear variational inequality and has a unique solution as -it satisfies the theorem of Lions and Stampaccia. A proof can be found in -Rodrigues: Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, -1987. - -To handle the nonlinearity of the constitutive law we use a Newton method and to deal with the contact we apply an -active set method like in step-41. To be more concrete we combine both methods to an inexact semi smooth Newton -method - inexact since we use an iterative solver for the linearised problems in each Newton step. - - -

Linearisation of the constitutive law for the Newton method

- -For the Newton method we have to linearise the following semi-linearform -@f{gather*}a(\psi;\varphi) := \left(P_{\Pi}(C\varepsilon(\psi)),\varepsilon(\varphi)\right).@f} -Because we have to find the solution $u$ in the convex set $V^+$, we have to -apply an SQP-method (SQP: sequential quadratic programming). That means we have -to solve a minimisation problem for a known $u^i$ in every SQP-step of the form -@f{eqnarray*} - & & a(u^{i};u^{i+1} - u^i) + \dfrac{1}{2}a'(u^i;u^{i+1} - u^i,u^{i+1} - u^i)\\ - &=& a(u^i;u^{i+1}) - a(u^i;u^i) +\\ - & & \dfrac{1}{2}\left( a'(u^i;u^{i+1},u^{i+1}) - 2a'(u^i;u^i,u^{i+1}) - a'(u^i;u^i,u^i)\right)\\ - &\rightarrow& \textrm{min},\quad u^{i+1}\in V^+. -@f} -Neglecting the constant terms $ a(u^i;u^i)$ and $ a'(u^i;u^i,u^i)$ we obtain the -following minimisation problem @f{gather*}\dfrac{1}{2} a'(u^i;u^{i+1},u^{i+1}) - F(u^i)\rightarrow \textrm{min},\quad u^{i+1}\in V^+@f} with -@f{gather*}F(\varphi) := \left(a'(\varphi;\varphi,u^{i+1}) - a(\varphi;u^{i+1}) \right).@f} -In the case of our constitutive law the Fréchet derivative of the -semi-linearform $a(.;.)$ at the point $u^i$ is - -@f{gather*}a'(u^i;\psi,\varphi) = -(I(x)\varepsilon(\psi),\varepsilon(\varphi)),\quad x\in\Omega,@f} -@f{gather*} -I(x) := \begin{cases} -C_{\mu} + C_{\kappa}, & -\quad \vert \tau^D \vert \leq \sigma_0\\ -\gamma C_{\mu} + (1-\gamma)\beta\left(C_{\mu} - -2\mu\dfrac{\tau^D\otimes\tau^D}{\vert\tau^D\vert^2}\right) + C_{\kappa}, &\quad -\vert \tau^D \vert > \sigma_0 -\end{cases} -@f} -with -@f{gather*}C_{\mu} := 2\mu\left(\mathbb{I} - \dfrac{1}{3} I\otimes -I\right)\quad\text{(shear part of the stress strain tensor)},@f} -@f{gather*}C_{\kappa} := \kappa I\otimes I\quad\text{(bulk part of the stress strain -tensor)},@f} -@f{gather*}\tau^D := C\varepsilon^D(u^i).@f} -Remark that $a(.;.)$ is not differentiable in the common sense but it is -slantly differentiable like the function for the contact problem and again we refer to -Hintermueller, Ito, Kunisch: The primal-dual active set strategy as a semismooth newton method, SIAM J. OPTIM., 2003, Vol. 13, No. 3, pp. 865-888. -Again the first case is for elastic and the second for plastic deformation. - - -

Formulation as a saddle point problem

- -Just as in step-41 we compose a saddle point problem out of the minimisation -problem. Again we do so to gain a formulation that allows us to solve a linear -system of equations finally. - -We introduce a Lagrange multiplier $\lambda$ and the convex cone $K\subset W'$, -$W'$ dual space of the trace space $W:=\left[ H_0^{\frac{1}{2}}(\Gamma_C) -\right]^{\textrm{dim}$ of $V$ restricted to $\Gamma_C$, -@f{gather*}K:=\{\mu\in W':\mu_T = 0,\quad\langle\mu n,v\rangle_{\Gamma_C}\geq 0,\quad -\forall v\in H_0^{\frac{1}{2}}(\Gamma_C), v \ge 0\text{ on }\Gamma_C \}@f} -of Lagrange multipliers, where $\langle\cdot,\cdot\rangle$ -denotes the duality pairing, i.e. a boundary integral, between $W'$ and $W$. -Intuitively, $K$ is the cone of all "non-positive functions", except that $ K\subset -\left( \left[ H_0^{\frac{1}{2}}(\Gamma_C) \right]^{\textrm{dim}} \right)' $ and so contains other -objects besides regular functions as well. This yields: - -Find $u\in V$ and $\lambda\in K$ such that -@f{align*} - \hat{a}(u,v) + b(v,\lambda) &= f(v),\quad &&v\in V\\ - b(u,\mu - \lambda) &\leq \langle g,(\mu - - \lambda)n\rangle_{\Gamma_C},\quad&&\mu\in K, -@f} -with -@f{align*} - \hat{a}(u,v) &:= a'(u^i;u,v)\\ - b(u,\mu) &:= \langle un,\mu n\rangle_{\Gamma_C},\quad &&u\in V,\quad\mu\in W'. -@f} -As in the section before $u^i$ denotes the linearization point for the -semi-linearform $a(.;.)$. In contrast to step-41 we directly consider $\lambda$ -as the additional, positive force $\sigma(u)n$ that the obstacle -exerts on the boundary $\Gamma_C$ of the body. - -The existence and uniqueness of the analytical solution $(u,\lambda)\in V\times -K$ of this saddle point problem has been stated in Glowinski, Lions and Tr\'{e}moli\`{e}res: Numerical -Analysis of Variational Inequalities, North-Holland, 1981. - -NOTE: In this example as well as in the further documentation we make the -assumption that the normal vector $n$ equals to $(0,0,1)$. This comes up with -the starting condition of our deformable body. - - -

Active Set methods to solve the saddle point problem

- -The linearized problem is essentially like a pure elastic problem with contact like -in step-41. The only difference consists in the fact that the contact area -is at the boundary instead of in the domain. But this has no further consequence -so that we refer to the documentation of step-41 with the only hint that -$\mathcal{S}$ containts all the vertices at the contact boundary $\Gamma_C$ this -time. - - -

The primal-dual active set algorithm combined with the inexact semi smooth -Newton method

- -Now we describe an algorithm that combines the damped semismooth Newton-method, -which we use for the nonlinear constitutive law, with the semismooth Newton -method for the contact. It sums up the results of the sections before and works as follows: -
    -
  1. Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that - $\mathcal{S} = \mathcal{A}_k \cup \mathcal{F}_k$ and $\mathcal{A}_k \cap - \mathcal{F}_k = \emptyset$ and set $k = 1$. The start value $\hat U^0 := - P_{\mathcal{A}_k}(0)$ fullfills our obstacle condition. -
  2. Assemble the Newton matrix $A := a'(\hat - U^{k-1};\varphi_p,\varphi_q)$ and the right-hand-side $F(\hat U^{k-1})$. -
  3. Find the primal-dual pair $(\bar U^k,\Lambda^k)$ that satisfies - @f{align*} - A\bar U^k + B\Lambda^k & = F, &\\ - \left[B^T\bar U^k\right]_p & = G_p & \forall p\in\mathcal{A}_k,\\ - \Lambda^k_p & = 0 & \forall p\in\mathcal{F}_k. - @f} -
  4. Damping for $k>2$ by applying a line search and calculating a linear - combination of $U^{k-1}$ and $\bar U^k$. Find an - $\alpha_i:=2^{-i},(i=0,\ldots,10)$ so that - @f{gather*}U^k := \alpha_i\bar U^k + - (1-\alpha_i)U^{k-1}@f} - yields - @f{gather*}\vert F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.\f} -
  5. Define the new active and inactive sets by - @f{gather*}\mathcal{A}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + - c\left(\left[B^TU^k\right]_p - G_p\right) > 0\rbrace,@f} - @f{gather*}\mathcal{F}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + - c\left(\left[B^TU^k\right]_p - G_p\right) \leq 0\rbrace.@f} - Projection $U^k$ so that it holds the second equation in (2) - @f{gather*}\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).@f} -
  6. If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert - F\left(U^{k}\right) \vert < \delta$ then stop, else set $k=k+1$ and go to - step (1). -
- -The subscript $p$ denotes a vertex and the meaning of the -decorated and none decorated $U$ is as follows: $\bar U$ denotes the solution of the linear system of equations in (2), $U$ is the -damped solution and equals to $\bar U$ if the damping parameter $\alpha_0 = -1$ and $\hat U := P_{\mathcal{A}}(U)$ is the projection of the active -components in $\mathcal{A}$ to the gap - -@f{gather*}P_{\mathcal{A}}(U):=\begin{cases} -U_p, & \textrm{if}\quad p\notin\mathcal{A}\\ -g_{h,p}, & \textrm{if}\quad -p\in\mathcal{A}. -\end{cases}@f}\\ -The matrix $B\in\mathbb{R}^{n\times m}$, $n>m$ describes the coupling of the -bases for the displacements and lagrange multiplier (contact forces) -and it is not quadratic in our situation since $\Lambda^k$ is only defined on -$\Gamma_C$. Due to the ansatz functions $\psi_i$ (scalar valued) of the -lagrange multiplier are fullfilling the following biorthogonal condition (see Hüeber, Wohlmuth: A primal–dual active -set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. -194, 2005, pp. 3147-3166) -@f{gather} \int\limits_{\Gamma_C}\psi_i(x)\varphi_j(x)dx = -\delta_{ij}\int\limits_{\Gamma_C}\varphi_j(x)dx@f} -this yields -@f{gather*}B_{pq} = \begin{cases} -\int\limits_{\Gamma_C}\varphi_p(x)dxI_3, & \text{if}\quad p=q,\quad -p,q\in\mathcal{S}\\ -0I_3, & \text{if}\quad p\neq q,\quad p\textrm{ or }q\notin\mathcal{S}. -\end{cases}@f} -Here $I_3$ denotes the threedimensional identity matrix. -In our programm we use the structure of a quadratic sparse for -$B\in\mathbb{R}^{n\times n}$ and for $\Lambda^k$ a vector with length $n$ where -$\Lambda^k_p = 0$ for $p\notin \mathcal{S}$. -The vector $G$ is defined by a suitable approximation $g_h$ of the gap $g$ -@f{gather*}G_p = \begin{cases} -g_{h,p}\int\limits_{\Gamma_C}\varphi_p(x)dx, & \text{if}\quad p\in\mathcal{S}\\ -0, & \text{if}\quad p\notin\mathcal{S}. -\end{cases}@f} - -Note that $G_p$ is a threedimensional vector and that again we applied the -biorthogonal property of the lagrange multiplier ansatz functions to the -integral $\int\limits_{\Gamma_C}g_h(x)\varphi_p(x)dx$ with $g_h(x)=\sum\limits_i -g_{h,p}\varphi_p(x)$ (see the reference mentioned above). - -Compared to step-41, step (1) is added but it should be clear -from the sections above that we only linearize the problem. In step (2) we have -to solve a linear system of equations again. And now the solution has to fulfill two stopping -criteria. $\mathcal{A}_{k+1} = \mathcal{A}_k$ makes sure that the contact zones -are iterated out and the second ensures an accurate enough residual which means -that the plastic zones are also iterated out. - -A similar method can also be found in Brunssen, Schmid, Schäfer, -Wohlmuth: A fast and robust iterative solver for nonlinear contact problems -using a primal-dual active set strategy and algebraic multigrid, Int. J. Numer. -Meth. Engng, 2007, 69, pp. 524-543. But in advance we apply a line search to -obtain a more robust method regarding the start value. Solving an elastic -problem in the very first step ($k=1$) we get a reasonable start value but as -you can see in the results damping is important if we choose for example a ball as -obstacle. - -Damping our Newton method is more important for the nonlinearity cause by the -constitutive law as for the contact. For this reason we start to damp our method -for $k>2$ when we have two plastic iterations. Note that $U^1$ is a solution of -an elastic problem and $U^2$ is the first plastic solution. A linear combination -between these both results in stresses which are not in the convex set -of the feasible stresses. - - -

Adaptive mesh refinement

- -Since we run our program in 3d, there is a good reason to use adaptive -mesh refinement. To make things a bit easier we are choosing the -KellyErrorEstimator that is already implemented in deal.II. We hand the -solution vector to it which contains the displacement $u$. As we will see in the -results it yields a quite reasonable adaptive mesh for the contact zone as well -as for plasticity! - - -

Implementation

- -This tutorial is essentially a mixture of step-40 and step-41 but instead of -PETSc we let the Trilinos library deal with parallelizing the linear algebra -(like in step-32). Since we are trying to solve a similar problem like in -step-41 we will use the same methods but now in parallel. - -Another difficulty is the handling of the different constraints from -(the dirichlet conditons), the hanging nodes and the inequality condition that -arises from the contact. For this purpose we create three objects of type -ConstraintMatrix. - -Beside the ConstitutiveLaw class there is another new class called Input. This -class allows us to read in an obstacle from a file. In our example the file -'obstacle_file.dat' containts data which describe an Chinese, Japanese or -Korean symbol for force or power. (See http://www.orientaloutpost.com/ : -"This word can be used for motivation - it -can also mean power/motion/propulsion/force. It can be anything -internal or external that keeps you going. This is the safest way to express -motivation in Chinese. If your audience is Japanese, please see the other entry -for motivation. This is a word in Japanese and Korean, but it means "motive -power" or "kinetic energy" (without the motivation meaning that you are -probably looking for)".) diff --git a/deal.II/examples/step-42/doc/kind b/deal.II/examples/step-42/doc/kind deleted file mode 100644 index 56e049c91a..0000000000 --- a/deal.II/examples/step-42/doc/kind +++ /dev/null @@ -1 +0,0 @@ -solids diff --git a/deal.II/examples/step-42/doc/results.dox b/deal.II/examples/step-42/doc/results.dox deleted file mode 100644 index 9616f65892..0000000000 --- a/deal.II/examples/step-42/doc/results.dox +++ /dev/null @@ -1,967 +0,0 @@ -

Results

- -Running the program (with a ball as obstacle) on 12 cores produces output like this: -@code -Cycle 0: - Number of active cells: 218 - Total number of cells: 249 - Number of degrees of freedom: 2187 - - Newton iteration 0 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 173.076 -> 0.157131 in 8 FGMRES iterations. - Number of elastic quadrature points: 4096 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.157131 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 57.3852 -> 0.0426777 in 9 FGMRES iterations. - Number of elastic quadrature points: 3944 and plastic quadrature points: 152 - Residual of the non-contact part of the system: 24.9209 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 24.9209 -> 0.0157852 in 8 FGMRES iterations. - Number of elastic quadrature points: 3960 and plastic quadrature points: 136 - Residual of the non-contact part of the system: 1.63601 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 1.63601 -> 0.00131281 in 8 FGMRES iterations. - Number of elastic quadrature points: 3960 and plastic quadrature points: 136 - Residual of the non-contact part of the system: 0.00479668 - with a damping parameter alpha = 1 - - Newton iteration 4 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 0.00479668 -> 2.27976e-06 in 9 FGMRES iterations. - Number of elastic quadrature points: 3960 and plastic quadrature points: 136 - Residual of the non-contact part of the system: 2.29051e-06 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 1 - Assembling system... - Solving system... - Error: 2.29051e-06 -> 1.31066e-09 in 9 FGMRES iterations. - Number of elastic quadrature points: 3960 and plastic quadrature points: 136 - Residual of the non-contact part of the system: 1.31066e-09 - with a damping parameter alpha = 1 - - Number of assembled systems = 6 - Number of Solver-Iterations = 51 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 0.736s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 6 | 0.0779s | 11% | -| Graphical output | 1 | 0.219s | 30% | -| Mesh refinement and setup system| 1 | 0.0274s | 3.7% | -| Preconditioning | 6 | 0.147s | 20% | -| Residual and lambda | 6 | 0.0167s | 2.3% | -| Solving | 6 | 0.0365s | 5% | -| Solving and Preconditioning | 6 | 0.197s | 27% | -| Update solution and constraints | 6 | 0.0154s | 2.1% | -+---------------------------------+-----------+------------+------------+ - - -Cycle 1: - Number of active cells: 463 - Total number of cells: 529 - Number of degrees of freedom: 6294 - - Newton iteration 0 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 254.373 -> 0.190863 in 9 FGMRES iterations. - Number of elastic quadrature points: 12384 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.190863 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 43.8161 -> 0.0315499 in 12 FGMRES iterations. - Number of elastic quadrature points: 11092 and plastic quadrature points: 1292 - Residual of the non-contact part of the system: 24.5462 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 24.5462 -> 0.0222984 in 11 FGMRES iterations. - Number of elastic quadrature points: 11152 and plastic quadrature points: 1232 - Residual of the non-contact part of the system: 5.47753 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 5.47753 -> 0.00277991 in 11 FGMRES iterations. - Number of elastic quadrature points: 11160 and plastic quadrature points: 1224 - Residual of the non-contact part of the system: 1.21575 - with a damping parameter alpha = 1 - - Newton iteration 4 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 1.21575 -> 0.00104015 in 9 FGMRES iterations. - Number of elastic quadrature points: 11144 and plastic quadrature points: 1240 - Residual of the non-contact part of the system: 0.0594461 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 0.0594461 -> 2.73182e-05 in 10 FGMRES iterations. - Number of elastic quadrature points: 11144 and plastic quadrature points: 1240 - Residual of the non-contact part of the system: 5.50309e-05 - with a damping parameter alpha = 1 - - Newton iteration 6 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 5.50309e-05 -> 2.99676e-08 in 13 FGMRES iterations. - Number of elastic quadrature points: 11144 and plastic quadrature points: 1240 - Residual of the non-contact part of the system: 2.99656e-08 - with a damping parameter alpha = 1 - - Newton iteration 7 - Updating active set... - Size of active set: 9 - Assembling system... - Solving system... - Error: 2.99656e-08 -> 1.70821e-11 in 14 FGMRES iterations. - Number of elastic quadrature points: 11144 and plastic quadrature points: 1240 - Residual of the non-contact part of the system: 1.70843e-11 - with a damping parameter alpha = 1 - - Number of assembled systems = 8 - Number of Solver-Iterations = 89 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 1.89s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 14 | 0.354s | 19% | -| Graphical output | 2 | 0.394s | 21% | -| Mesh refinement and setup system| 2 | 0.109s | 5.8% | -| Preconditioning | 14 | 0.302s | 16% | -| Residual and lambda | 14 | 0.0587s | 3.1% | -| Solving | 14 | 0.142s | 7.6% | -| Solving and Preconditioning | 14 | 0.48s | 25% | -| Update solution and constraints | 14 | 0.0452s | 2.4% | -+---------------------------------+-----------+------------+------------+ - - -Cycle 2: - Number of active cells: 1170 - Total number of cells: 1337 - Number of degrees of freedom: 18207 - - Newton iteration 0 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 284.124 -> 0.219038 in 12 FGMRES iterations. - Number of elastic quadrature points: 38144 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.219038 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 25 - Assembling system... - Solving system... - Error: 29.3138 -> 0.0233274 in 18 FGMRES iterations. - Number of elastic quadrature points: 26802 and plastic quadrature points: 11342 - Residual of the non-contact part of the system: 18.6169 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 40.4533 -> 0.0289656 in 17 FGMRES iterations. - Number of elastic quadrature points: 26314 and plastic quadrature points: 11830 - Residual of the non-contact part of the system: 13.0957 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 13.0957 -> 0.00985332 in 18 FGMRES iterations. - Number of elastic quadrature points: 27074 and plastic quadrature points: 11070 - Residual of the non-contact part of the system: 13.4255 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 30111 and plastic quadrature points: 8033 - Residual of the non-contact part of the system: 4.53997 - with a damping parameter alpha = 0.5 - - Newton iteration 4 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 4.53997 -> 0.00288706 in 16 FGMRES iterations. - Number of elastic quadrature points: 29624 and plastic quadrature points: 8520 - Residual of the non-contact part of the system: 0.376174 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 0.376174 -> 0.000319472 in 15 FGMRES iterations. - Number of elastic quadrature points: 29500 and plastic quadrature points: 8644 - Residual of the non-contact part of the system: 0.0174717 - with a damping parameter alpha = 1 - - Newton iteration 6 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 0.0174717 -> 1.41647e-05 in 17 FGMRES iterations. - Number of elastic quadrature points: 29512 and plastic quadrature points: 8632 - Residual of the non-contact part of the system: 6.13152e-05 - with a damping parameter alpha = 1 - - Newton iteration 7 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 6.13152e-05 -> 5.43905e-08 in 20 FGMRES iterations. - Number of elastic quadrature points: 29512 and plastic quadrature points: 8632 - Residual of the non-contact part of the system: 5.4391e-08 - with a damping parameter alpha = 1 - - Newton iteration 8 - Updating active set... - Size of active set: 37 - Assembling system... - Solving system... - Error: 5.4391e-08 -> 4.00879e-11 in 22 FGMRES iterations. - Number of elastic quadrature points: 29512 and plastic quadrature points: 8632 - Residual of the non-contact part of the system: 4.0089e-11 - with a damping parameter alpha = 1 - - Number of assembled systems = 9 - Number of Solver-Iterations = 155 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 4.76s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 23 | 1.24s | 26% | -| Graphical output | 3 | 0.61s | 13% | -| Mesh refinement and setup system| 3 | 0.215s | 4.5% | -| Preconditioning | 23 | 0.651s | 14% | -| Residual and lambda | 24 | 0.19s | 4% | -| Solving | 23 | 0.513s | 11% | -| Solving and Preconditioning | 23 | 1.24s | 26% | -| Update solution and constraints | 23 | 0.104s | 2.2% | -+---------------------------------+-----------+------------+------------+ - - -Cycle 3: - Number of active cells: 2955 - Total number of cells: 3377 - Number of degrees of freedom: 52497 - - Newton iteration 0 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 296.309 -> 0.274641 in 16 FGMRES iterations. - Number of elastic quadrature points: 117216 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.274641 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 101 - Assembling system... - Solving system... - Error: 16.3381 -> 0.0141064 in 28 FGMRES iterations. - Number of elastic quadrature points: 43467 and plastic quadrature points: 73749 - Residual of the non-contact part of the system: 16.0978 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 169 - Assembling system... - Solving system... - Error: 37.9212 -> 0.0359077 in 23 FGMRES iterations. - Number of elastic quadrature points: 49376 and plastic quadrature points: 67840 - Residual of the non-contact part of the system: 13.2713 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 13.2966 -> 0.0122436 in 31 FGMRES iterations. - Number of elastic quadrature points: 39170 and plastic quadrature points: 78046 - Residual of the non-contact part of the system: 15.8407 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 57541 and plastic quadrature points: 59675 - Residual of the non-contact part of the system: 5.37003 - with a damping parameter alpha = 0.5 - - Newton iteration 4 - Updating active set... - Size of active set: 153 - Assembling system... - Solving system... - Error: 5.37242 -> 0.00503175 in 21 FGMRES iterations. - Number of elastic quadrature points: 53138 and plastic quadrature points: 64078 - Residual of the non-contact part of the system: 1.91646 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 1.9277 -> 0.00180035 in 25 FGMRES iterations. - Number of elastic quadrature points: 52162 and plastic quadrature points: 65054 - Residual of the non-contact part of the system: 1.12614 - with a damping parameter alpha = 1 - - Newton iteration 6 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 1.12614 -> 0.000885797 in 19 FGMRES iterations. - Number of elastic quadrature points: 51667 and plastic quadrature points: 65549 - Residual of the non-contact part of the system: 0.817432 - with a damping parameter alpha = 1 - - Newton iteration 7 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 0.817432 -> 0.000802786 in 20 FGMRES iterations. - Number of elastic quadrature points: 51659 and plastic quadrature points: 65557 - Residual of the non-contact part of the system: 0.557455 - with a damping parameter alpha = 1 - - Newton iteration 8 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 0.557455 -> 0.000527895 in 18 FGMRES iterations. - Number of elastic quadrature points: 51572 and plastic quadrature points: 65644 - Residual of the non-contact part of the system: 0.235355 - with a damping parameter alpha = 1 - - Newton iteration 9 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 0.235355 -> 0.000197953 in 23 FGMRES iterations. - Number of elastic quadrature points: 51482 and plastic quadrature points: 65734 - Residual of the non-contact part of the system: 0.0275413 - with a damping parameter alpha = 1 - - Newton iteration 10 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 0.0275413 -> 2.37996e-05 in 25 FGMRES iterations. - Number of elastic quadrature points: 51464 and plastic quadrature points: 65752 - Residual of the non-contact part of the system: 0.000526512 - with a damping parameter alpha = 1 - - Newton iteration 11 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 0.000526512 -> 4.98197e-07 in 27 FGMRES iterations. - Number of elastic quadrature points: 51464 and plastic quadrature points: 65752 - Residual of the non-contact part of the system: 5.35391e-07 - with a damping parameter alpha = 1 - - Newton iteration 12 - Updating active set... - Size of active set: 145 - Assembling system... - Solving system... - Error: 5.35391e-07 -> 5.141e-10 in 45 FGMRES iterations. - Number of elastic quadrature points: 51464 and plastic quadrature points: 65752 - Residual of the non-contact part of the system: 5.14106e-10 - with a damping parameter alpha = 1 - - Number of assembled systems = 13 - Number of Solver-Iterations = 321 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 18.6s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 36 | 5.02s | 27% | -| Graphical output | 4 | 1.02s | 5.5% | -| Mesh refinement and setup system| 4 | 0.453s | 2.4% | -| Preconditioning | 36 | 1.94s | 10% | -| Residual and lambda | 38 | 0.682s | 3.7% | -| Solving | 36 | 3.57s | 19% | -| Solving and Preconditioning | 36 | 5.68s | 30% | -| Update solution and constraints | 36 | 0.254s | 1.4% | -+---------------------------------+-----------+------------+------------+ - - -Cycle 4: - Number of active cells: 7400 - Total number of cells: 8457 - Number of degrees of freedom: 154647 - - Newton iteration 0 - Updating active set... - Size of active set: 609 - Assembling system... - Solving system... - Error: 299.883 -> 0.198142 in 22 FGMRES iterations. - Number of elastic quadrature points: 362944 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.198142 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 437 - Assembling system... - Solving system... - Error: 10.4425 -> 0.00864082 in 54 FGMRES iterations. - Number of elastic quadrature points: 50243 and plastic quadrature points: 312701 - Residual of the non-contact part of the system: 13.0573 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 629 - Assembling system... - Solving system... - Error: 25.4296 -> 0.0250775 in 32 FGMRES iterations. - Number of elastic quadrature points: 52693 and plastic quadrature points: 310251 - Residual of the non-contact part of the system: 4.31469 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 609 - Assembling system... - Solving system... - Error: 4.42111 -> 0.0042128 in 29 FGMRES iterations. - Number of elastic quadrature points: 53842 and plastic quadrature points: 309102 - Residual of the non-contact part of the system: 3.2446 - with a damping parameter alpha = 1 - - Newton iteration 4 - Updating active set... - Size of active set: 593 - Assembling system... - Solving system... - Error: 3.2476 -> 0.00316464 in 29 FGMRES iterations. - Number of elastic quadrature points: 53576 and plastic quadrature points: 309368 - Residual of the non-contact part of the system: 2.71394 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 579 - Assembling system... - Solving system... - Error: 2.71408 -> 0.00247937 in 27 FGMRES iterations. - Number of elastic quadrature points: 53936 and plastic quadrature points: 309008 - Residual of the non-contact part of the system: 2.42929 - with a damping parameter alpha = 1 - - Newton iteration 6 - Updating active set... - Size of active set: 579 - Assembling system... - Solving system... - Error: 2.42929 -> 0.00234882 in 29 FGMRES iterations. - Number of elastic quadrature points: 53201 and plastic quadrature points: 309743 - Residual of the non-contact part of the system: 2.52783 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 54507 and plastic quadrature points: 308437 - Residual of the non-contact part of the system: 1.22978 - with a damping parameter alpha = 0.5 - - Newton iteration 7 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 1.23087 -> 0.00117414 in 27 FGMRES iterations. - Number of elastic quadrature points: 53932 and plastic quadrature points: 309012 - Residual of the non-contact part of the system: 0.913556 - with a damping parameter alpha = 1 - - Newton iteration 8 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.913556 -> 0.000846686 in 29 FGMRES iterations. - Number of elastic quadrature points: 53676 and plastic quadrature points: 309268 - Residual of the non-contact part of the system: 0.608655 - with a damping parameter alpha = 1 - - Newton iteration 9 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.608915 -> 0.000590468 in 36 FGMRES iterations. - Number of elastic quadrature points: 53769 and plastic quadrature points: 309175 - Residual of the non-contact part of the system: 0.572935 - with a damping parameter alpha = 1 - - Newton iteration 10 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.574492 -> 0.000516963 in 41 FGMRES iterations. - Number of elastic quadrature points: 53419 and plastic quadrature points: 309525 - Residual of the non-contact part of the system: 0.575742 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 53568 and plastic quadrature points: 309376 - Residual of the non-contact part of the system: 0.351537 - with a damping parameter alpha = 0.5 - - Newton iteration 11 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.351691 -> 0.000350487 in 32 FGMRES iterations. - Number of elastic quadrature points: 53596 and plastic quadrature points: 309348 - Residual of the non-contact part of the system: 0.498347 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 53574 and plastic quadrature points: 309370 - Residual of the non-contact part of the system: 0.287064 - with a damping parameter alpha = 0.5 - - Newton iteration 12 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.287174 -> 0.000275609 in 32 FGMRES iterations. - Number of elastic quadrature points: 53675 and plastic quadrature points: 309269 - Residual of the non-contact part of the system: 0.248843 - with a damping parameter alpha = 1 - - Newton iteration 13 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.248843 -> 0.000232694 in 29 FGMRES iterations. - Number of elastic quadrature points: 53682 and plastic quadrature points: 309262 - Residual of the non-contact part of the system: 0.341443 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 53686 and plastic quadrature points: 309258 - Residual of the non-contact part of the system: 0.232367 - with a damping parameter alpha = 0.5 - - Newton iteration 14 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.232367 -> 0.000231641 in 35 FGMRES iterations. - Number of elastic quadrature points: 53691 and plastic quadrature points: 309253 - Residual of the non-contact part of the system: 0.263644 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 53694 and plastic quadrature points: 309250 - Residual of the non-contact part of the system: 0.185686 - with a damping parameter alpha = 0.5 - - Newton iteration 15 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.185686 -> 0.000171579 in 29 FGMRES iterations. - Number of elastic quadrature points: 53721 and plastic quadrature points: 309223 - Residual of the non-contact part of the system: 0.273968 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 53726 and plastic quadrature points: 309218 - Residual of the non-contact part of the system: 0.189887 - with a damping parameter alpha = 0.5 - Number of elastic quadrature points: 53713 and plastic quadrature points: 309231 - Residual of the non-contact part of the system: 0.0832966 - with a damping parameter alpha = 0.25 - - Newton iteration 16 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.0832966 -> 8.32624e-05 in 38 FGMRES iterations. - Number of elastic quadrature points: 53749 and plastic quadrature points: 309195 - Residual of the non-contact part of the system: 0.0306643 - with a damping parameter alpha = 1 - - Newton iteration 17 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.0306643 -> 3.01898e-05 in 38 FGMRES iterations. - Number of elastic quadrature points: 53766 and plastic quadrature points: 309178 - Residual of the non-contact part of the system: 0.00261731 - with a damping parameter alpha = 1 - - Newton iteration 18 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 0.00261731 -> 2.33193e-06 in 51 FGMRES iterations. - Number of elastic quadrature points: 53768 and plastic quadrature points: 309176 - Residual of the non-contact part of the system: 4.64159e-05 - with a damping parameter alpha = 1 - - Newton iteration 19 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 4.64159e-05 -> 4.62653e-08 in 62 FGMRES iterations. - Number of elastic quadrature points: 53768 and plastic quadrature points: 309176 - Residual of the non-contact part of the system: 5.07264e-08 - with a damping parameter alpha = 1 - - Newton iteration 20 - Updating active set... - Size of active set: 585 - Assembling system... - Solving system... - Error: 5.07264e-08 -> 4.59618e-11 in 111 FGMRES iterations. - Number of elastic quadrature points: 53768 and plastic quadrature points: 309176 - Residual of the non-contact part of the system: 4.59615e-11 - with a damping parameter alpha = 1 - - Number of assembled systems = 21 - Number of Solver-Iterations = 812 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 131s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 57 | 23.5s | 18% | -| Graphical output | 5 | 2.34s | 1.8% | -| Mesh refinement and setup system| 5 | 1.07s | 0.82% | -| Preconditioning | 57 | 8.46s | 6.5% | -| Residual and lambda | 66 | 3.61s | 2.8% | -| Solving | 57 | 41s | 31% | -| Solving and Preconditioning | 57 | 50s | 38% | -| Update solution and constraints | 57 | 0.736s | 0.56% | -+---------------------------------+-----------+------------+------------+ - - -Cycle 5: - Number of active cells: 19279 - Total number of cells: 22033 - Number of degrees of freedom: 461106 - - Newton iteration 0 - Updating active set... - Size of active set: 2449 - Assembling system... - Solving system... - Error: 300.845 -> 0.249318 in 28 FGMRES iterations. - Number of elastic quadrature points: 1122752 and plastic quadrature points: 0 - Residual of the non-contact part of the system: 0.249318 - with a damping parameter alpha = 1 - - Newton iteration 1 - Updating active set... - Size of active set: 1863 - Assembling system... - Solving system... - Error: 6.17714 -> 0.0061549 in 82 FGMRES iterations. - Number of elastic quadrature points: 53510 and plastic quadrature points: 1069242 - Residual of the non-contact part of the system: 10.188 - with a damping parameter alpha = 1 - - Newton iteration 2 - Updating active set... - Size of active set: 2348 - Assembling system... - Solving system... - Error: 15.5579 -> 0.0151548 in 41 FGMRES iterations. - Number of elastic quadrature points: 54822 and plastic quadrature points: 1067930 - Residual of the non-contact part of the system: 1.5258 - with a damping parameter alpha = 1 - - Newton iteration 3 - Updating active set... - Size of active set: 2485 - Assembling system... - Solving system... - Error: 3.2747 -> 0.00308114 in 55 FGMRES iterations. - Number of elastic quadrature points: 61107 and plastic quadrature points: 1061645 - Residual of the non-contact part of the system: 0.816662 - with a damping parameter alpha = 1 - - Newton iteration 4 - Updating active set... - Size of active set: 2294 - Assembling system... - Solving system... - Error: 0.846485 -> 0.000840713 in 105 FGMRES iterations. - Number of elastic quadrature points: 54660 and plastic quadrature points: 1068092 - Residual of the non-contact part of the system: 0.651754 - with a damping parameter alpha = 1 - - Newton iteration 5 - Updating active set... - Size of active set: 2375 - Assembling system... - Solving system... - Error: 1.07345 -> 0.0010383 in 95 FGMRES iterations. - Number of elastic quadrature points: 59982 and plastic quadrature points: 1062770 - Residual of the non-contact part of the system: 0.81415 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 57363 and plastic quadrature points: 1065389 - Residual of the non-contact part of the system: 0.432614 - with a damping parameter alpha = 0.5 - - Newton iteration 6 - Updating active set... - Size of active set: 2294 - Assembling system... - Solving system... - Error: 0.600941 -> 0.000577431 in 51 FGMRES iterations. - Number of elastic quadrature points: 56754 and plastic quadrature points: 1065998 - Residual of the non-contact part of the system: 0.434547 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 57005 and plastic quadrature points: 1065747 - Residual of the non-contact part of the system: 0.209227 - with a damping parameter alpha = 0.5 - - Newton iteration 7 - Updating active set... - Size of active set: 2327 - Assembling system... - Solving system... - Error: 0.299306 -> 0.00029369 in 66 FGMRES iterations. - Number of elastic quadrature points: 56775 and plastic quadrature points: 1065977 - Residual of the non-contact part of the system: 0.0890688 - with a damping parameter alpha = 1 - - Newton iteration 8 - Updating active set... - Size of active set: 2317 - Assembling system... - Solving system... - Error: 0.113493 -> 0.000112941 in 65 FGMRES iterations. - Number of elastic quadrature points: 56792 and plastic quadrature points: 1065960 - Residual of the non-contact part of the system: 0.0846807 - with a damping parameter alpha = 1 - - Newton iteration 9 - Updating active set... - Size of active set: 2311 - Assembling system... - Solving system... - Error: 0.0847051 -> 8.37451e-05 in 38 FGMRES iterations. - Number of elastic quadrature points: 56797 and plastic quadrature points: 1065955 - Residual of the non-contact part of the system: 0.0892047 - with a damping parameter alpha = 1 - Number of elastic quadrature points: 56793 and plastic quadrature points: 1065959 - Residual of the non-contact part of the system: 0.0226437 - with a damping parameter alpha = 0.5 - - Newton iteration 10 - Updating active set... - Size of active set: 2313 - Assembling system... - Solving system... - Error: 0.0240356 -> 2.37508e-05 in 72 FGMRES iterations. - Number of elastic quadrature points: 56816 and plastic quadrature points: 1065936 - Residual of the non-contact part of the system: 0.00540846 - with a damping parameter alpha = 1 - - Newton iteration 11 - Updating active set... - Size of active set: 2313 - Assembling system... - Solving system... - Error: 0.00540846 -> 5.39193e-06 in 62 FGMRES iterations. - Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924 - Residual of the non-contact part of the system: 0.000268289 - with a damping parameter alpha = 1 - - Newton iteration 12 - Updating active set... - Size of active set: 2313 - Assembling system... - Solving system... - Error: 0.000268289 -> 2.64338e-07 in 83 FGMRES iterations. - Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924 - Residual of the non-contact part of the system: 8.91195e-07 - with a damping parameter alpha = 1 - - Newton iteration 13 - Updating active set... - Size of active set: 2313 - Assembling system... - Solving system... - Error: 8.91195e-07 -> 8.85869e-10 in 255 FGMRES iterations. - Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924 - Residual of the non-contact part of the system: 8.86261e-10 - with a damping parameter alpha = 1 - - Number of assembled systems = 14 - Number of Solver-Iterations = 1098 - Writing graphical output... - - -+---------------------------------------------+------------+------------+ -| Total wallclock time elapsed since start | 465s | | -| | | | -| Section | no. calls | wall time | % of total | -+---------------------------------+-----------+------------+------------+ -| Assembling | 71 | 61.6s | 13% | -| Graphical output | 6 | 10.5s | 2.2% | -| Mesh refinement and setup system| 6 | 2.71s | 0.58% | -| Preconditioning | 71 | 19.9s | 4.3% | -| Residual and lambda | 83 | 8.97s | 1.9% | -| Solving | 71 | 170s | 36% | -| Solving and Preconditioning | 71 | 191s | 41% | -| Update solution and constraints | 71 | 1.46s | 0.31% | -+---------------------------------+-----------+------------+------------+ -@endcode - -For each adaptive refinement cycle the iterations end if the active set and -the residual for the non-linear problem don't change anymore. In the tables -at the end of each cycle you find informations about computing time and the -number of calls of different parts of the program like Assembling or calculating -the residuals. After the last cycle there are 83 calls for the residual function. -12 calls as the Solving or Assembling what means that 12 damping steps were -necessary summed up over the cycles. - -In every refinement step you can observe that the acitve set - the contact points - -are iterated out at first. After that the Newton method has only to struggle with the -plasticity. For the finer meshes there is quadratic convergence not until the -last 4 or 5 newton iterations. - - - - - - - - -
- 0   - - - - 0   - - -
- -The picture shows the adaptive refinement and as well how much a cell is -plastified druing the contact with the ball. In the middle - where the -mesh is finest - you can see the hollow caused by the ball. The blue -color means that this cell contains only elastic quadrature points in -contrast to the red cells in which all quadrature points are plastified. -Since we used trilinear ansatz functions each cell contains eight -of these points which may or may not be elastic or plastic. - - -

Possibilities for extensions

- -Extend the program from a static to a quasi-static problem, perhaps by choosing a -backward-euler-scheme for the time discretization (for theoretical results see Frohne: FEM-Simulation -der Umformtechnik metallischer Oberflächen im Mikrokosmos, Ph.D. thesis, -University of Siegen, Germany, 2011). - -Like mentioned in step-41 it would be an interesting advance to consider a contact -problem with friction. In almost every mechanical process friction has a big influence. -For the modelling we have to take into account tangential stresses at the contact -surface. Also we have to observe that friction adds another nonlinearity to -our problem. - -If we simulate a frictional contact the next step to consider heat development -over the contact zone is not too far away. The heat that is caused by friction -between two bodies rises up the temperature in the deformable body and entails an -change of some material parameters. - -It might be of interest to implement more accurate, problem-adapted error -estimators for contact as well for the plasticity. 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This data will be bilinear - // interpolated and provides in this way - // a function which describes an obstacle. - // - // The data which we read in by the - // function read_obstacle () from the file - // "obstacle_file.pbm" will be stored - // in a double std::vector named - // obstacle_data. - // This vector composes the base - // to calculate a piecewise bilinear - // function as a polynomial interpolation. - // This will be done by obstacle_function (). - // - // In the function run () of the class - // PlasticityContactProblem we create - // an object of the class Input which will - // be used in the class Obstacle to - // supply the obstacle function in - // update_solution_and_constraints () of - // the class PlasticityContactProblem. - - template - class Input - { - public: - Input (const char* _name) : - name (_name), - mpi_communicator (MPI_COMM_WORLD), - pcout (std::cout, - (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)), - obstacle_data (0), - lx (1.0), // length of the cube in x direction - ly (1.0), // length of the cube in y direction - hx (0), // - hy (0), - nx (0), - ny (0) - {read_obstacle (name);} - - double hv (int i, int j); - - double obstacle_function (double x,double y); - - void read_obstacle (const char* name); - - private: - const char* name; - MPI_Comm mpi_communicator; - ConditionalOStream pcout; - std::vector obstacle_data; - double lx, ly; - double hx, hy; - int nx, ny; - }; - - template - double Input::hv (int i, int j) - { - assert(i>=0 && i=0 && j - double Input::obstacle_function (double x,double y) - { - int ix = (int)(x/hx); - int iy = (int)(y/hy); - - if (ix<0) - ix = 0; - - if (iy<0) - iy = 0; - - if (ix>=nx-1) - ix = nx-2; - - if (iy>=ny-1) - iy = ny-2; - - double val = 0.0; - { - FullMatrix H(4,4); - Vector X(4); - Vector b(4); - - double xx = 0.0; - double yy = 0.0; - - xx = ix*hx; - yy = iy*hy; - H(0,0) = xx; - H(0,1) = yy; - H(0,2) = xx*yy; - H(0,3) = 1.0; - b(0) = hv (ix, iy); - - xx = (ix + 1)*hx; - yy = iy*hy; - H(1,0) = xx; - H(1,1) = yy; - H(1,2) = xx*yy; - H(1,3) = 1.0; - b(1) = hv (ix + 1, iy); - - xx = (ix + 1)*hx; - yy = (iy + 1)*hy; - H(2,0) = xx; - H(2,1) = yy; - H(2,2) = xx*yy; - H(2,3) = 1.0; - b(2) = hv (ix + 1, iy + 1); - - xx = ix*hx; - yy = (iy + 1)*hy; - H(3,0) = xx; - H(3,1) = yy; - H(3,2) = xx*yy; - H(3,3) = 1.0; - b(3) = hv (ix, iy + 1); - - H.gauss_jordan (); - H.vmult (X, b); - - val = X(0)*x + X(1)*y + X(2)*x*y + X(3); - } - - return val; - } - - template - void Input::read_obstacle (const char* name) - { - int SZ = 100000; - FILE* fp = fopen (name, "r"); - char* hlp_str = new char[SZ]; - double hlp; - - fscanf (fp, "%s", hlp_str); - fscanf (fp, "%d", &nx); - fscanf (fp, "%d", &ny); - - assert(nx>0 && ny>0); - - for (int k=0; kConstitutiveLaw class template} - - // This class provides an interface - // for a constitutive law. In this - // example we are using an elastic - // plastic material with linear, - // isotropic hardening. - - template - class ConstitutiveLaw - { - public: - ConstitutiveLaw (double _E, - double _nu, - double _sigma_0, - double _gamma, - MPI_Comm _mpi_communicator, - ConditionalOStream _pcout); - - void plast_linear_hardening (SymmetricTensor<4,dim> &stress_strain_tensor, - const SymmetricTensor<2,dim> &strain_tensor, - unsigned int &elast_points, - unsigned int &plast_points, - double &yield); - void linearized_plast_linear_hardening (SymmetricTensor<4,dim> &stress_strain_tensor_linearized, - SymmetricTensor<4,dim> &stress_strain_tensor, - const SymmetricTensor<2,dim> &strain_tensor); - inline SymmetricTensor<2,dim> get_strain (const FEValues &fe_values, - const unsigned int shape_func, - const unsigned int q_point) const; - void set_sigma_0 (double sigma_hlp) {sigma_0 = sigma_hlp;} - - private: - SymmetricTensor<4,dim> stress_strain_tensor_mu; - SymmetricTensor<4,dim> stress_strain_tensor_kappa; - double E; - double nu; - double sigma_0; - double gamma; - double mu; - double kappa; - MPI_Comm mpi_communicator; - ConditionalOStream pcout; - }; - - template - ConstitutiveLaw::ConstitutiveLaw(double _E, double _nu, double _sigma_0, double _gamma, MPI_Comm _mpi_communicator, ConditionalOStream _pcout) - :E (_E), - nu (_nu), - sigma_0 (_sigma_0), - gamma (_gamma), - mpi_communicator (_mpi_communicator), - pcout (_pcout) - { - mu = E/(2*(1+nu)); - kappa = E/(3*(1-2*nu)); - stress_strain_tensor_kappa = kappa*outer_product(unit_symmetric_tensor(), unit_symmetric_tensor()); - stress_strain_tensor_mu = 2*mu*(identity_tensor() - outer_product(unit_symmetric_tensor(), unit_symmetric_tensor())/3.0); - } - - template - inline - SymmetricTensor<2,dim> ConstitutiveLaw::get_strain (const FEValues &fe_values, - const unsigned int shape_func, - const unsigned int q_point) const - { - const FEValuesExtractors::Vector displacement (0); - SymmetricTensor<2,dim> tmp; - - tmp = fe_values[displacement].symmetric_gradient (shape_func,q_point); - - return tmp; - } - - template - void ConstitutiveLaw::plast_linear_hardening (SymmetricTensor<4,dim> &stress_strain_tensor, - const SymmetricTensor<2,dim> &strain_tensor, - unsigned int &elast_points, - unsigned int &plast_points, - double &yield) - { - if (dim == 3) - { - SymmetricTensor<2,dim> stress_tensor; - stress_tensor = (stress_strain_tensor_kappa + stress_strain_tensor_mu)*strain_tensor; - - SymmetricTensor<2,dim> deviator_stress_tensor = deviator(stress_tensor); - - double deviator_stress_tensor_norm = deviator_stress_tensor.norm (); - - yield = 0; - stress_strain_tensor = stress_strain_tensor_mu; - double beta = 1.0; - if (deviator_stress_tensor_norm > sigma_0) - { - beta = sigma_0/deviator_stress_tensor_norm; - stress_strain_tensor *= (gamma + (1 - gamma)*beta); - yield = 1; - plast_points += 1; - } - else - elast_points += 1; - - stress_strain_tensor += stress_strain_tensor_kappa; - } - } - - template - void ConstitutiveLaw::linearized_plast_linear_hardening (SymmetricTensor<4,dim> &stress_strain_tensor_linearized, - SymmetricTensor<4,dim> &stress_strain_tensor, - const SymmetricTensor<2,dim> &strain_tensor) - { - if (dim == 3) - { - SymmetricTensor<2,dim> stress_tensor; - stress_tensor = (stress_strain_tensor_kappa + stress_strain_tensor_mu)*strain_tensor; - - SymmetricTensor<2,dim> deviator_stress_tensor = deviator(stress_tensor); - - double deviator_stress_tensor_norm = deviator_stress_tensor.norm (); - - stress_strain_tensor = stress_strain_tensor_mu; - stress_strain_tensor_linearized = stress_strain_tensor_mu; - double beta = 1.0; - if (deviator_stress_tensor_norm > sigma_0) - { - beta = sigma_0/deviator_stress_tensor_norm; - stress_strain_tensor *= (gamma + (1 - gamma)*beta); - stress_strain_tensor_linearized *= (gamma + (1 - gamma)*beta); - deviator_stress_tensor /= deviator_stress_tensor_norm; - stress_strain_tensor_linearized -= (1 - gamma)*beta*2*mu*outer_product(deviator_stress_tensor, deviator_stress_tensor); - } - - stress_strain_tensor += stress_strain_tensor_kappa; - stress_strain_tensor_linearized += stress_strain_tensor_kappa; - } - } - - namespace EquationData - { - template - class RightHandSide : public Function - { - public: - RightHandSide () : Function(dim) {} - - virtual double value (const Point &p, - const unsigned int component = 0) const; - - virtual void vector_value (const Point &p, - Vector &values) const; - }; - - template - double RightHandSide::value (const Point &p, - const unsigned int component) const - { - double return_value = 0.0; - - if (component == 0) - return_value = 0.0; - if (component == 1) - return_value = 0.0; - if (component == 2) - // if ((p(0)-0.5)*(p(0)-0.5)+(p(1)-0.5)*(p(1)-0.5) < 0.2) - // return_value = -5000; - // else - return_value = 0.0; - // for (unsigned int i=0; i - void RightHandSide::vector_value (const Point &p, - Vector &values) const - { - for (unsigned int c=0; cn_components; ++c) - values(c) = RightHandSide::value (p, c); - } - - - template - class BoundaryValues : public Function - { - public: - BoundaryValues () : Function(dim) {}; - - virtual double value (const Point &p, - const unsigned int component = 0) const; - - virtual void vector_value (const Point &p, - Vector &values) const; - }; - - template - double BoundaryValues::value (const Point &p, - const unsigned int component) const - { - double return_value = 0; - - if (component == 0) - return_value = 0.0; - if (component == 1) - return_value = 0.0; - if (component == 2) - return_value = 0.0; - - return return_value; - } - - template - void BoundaryValues::vector_value (const Point &p, - Vector &values) const - { - for (unsigned int c=0; cn_components; ++c) - values(c) = BoundaryValues::value (p, c); - } - - - template - class Obstacle : public Function - { - public: - Obstacle (std::unique_ptr > const &_input) : - Function(dim), - input_obstacle_copy(std::move (_input)) {}; - - virtual double value (const Point &p, - const unsigned int component = 0) const; - - virtual void vector_value (const Point &p, - Vector &values) const; - - private: - std::unique_ptr > const &input_obstacle_copy; - }; - - template - double Obstacle::value (const Point &p, - const unsigned int component) const - { - double R = 0.03; - double return_value = 100.0; - if (component == 0) - return_value = p(0); - if (component == 1) - return_value = p(1); - if (component == 2) - { - return_value = -std::sqrt (0.36 - (p(0)-0.5)*(p(0)-0.5) - (p(1)-0.5)*(p(1)-0.5)) + 1.59; - // return_value = 1.999 - input_obstacle_copy->obstacle_function (p(0), p(1)); - } - return return_value; - } - - template - void Obstacle::vector_value (const Point &p, - Vector &values) const - { - for (unsigned int c=0; cn_components; ++c) - values(c) = Obstacle::value (p, c); - } - } - - // @sect3{The PlasticityContactProblem class template} - - // This class supplies all function - // and variables needed to describe - // the nonlinear contact problem. It is - // close to step-41 but with some additional - // features like: handling hanging nodes, - // a newton method, using Trilinos and p4est - // for parallel distributed computing. - // To deal with hanging nodes makes - // life a bit more complicated since - // we need an other ConstraintMatrix now. - // We create a newton method for the - // active set method for the contact - // situation and to handle the nonlinear - // operator for the constitutive law. - - template - class PlasticityContactProblem - { - public: - PlasticityContactProblem (int _n_refinements_global); - void run (); - - private: - void make_grid (); - void setup_system(); - void assemble_nl_system (TrilinosWrappers::MPI::Vector &u); - void residual_nl_system (TrilinosWrappers::MPI::Vector &u); - void assemble_mass_matrix_diagonal (TrilinosWrappers::SparseMatrix &mass_matrix); - void update_solution_and_constraints (); - void dirichlet_constraints (); - void solve (); - void solve_newton (); - void refine_grid (); - void move_mesh (const TrilinosWrappers::MPI::Vector &_complete_displacement) const; - void output_results (const std::string &title) const; - - unsigned int n_refinements_global; - unsigned int cycle; - - MPI_Comm mpi_communicator; - - parallel::distributed::Triangulation triangulation; - - FESystem fe; - DoFHandler dof_handler; - - std::unique_ptr > soltrans; - - IndexSet locally_owned_dofs; - IndexSet locally_relevant_dofs; - - unsigned int number_iterations; - - ConstraintMatrix constraints; - ConstraintMatrix constraints_hanging_nodes; - ConstraintMatrix constraints_dirichlet_hanging_nodes; - - TrilinosWrappers::SparseMatrix system_matrix_newton; - - TrilinosWrappers::MPI::Vector solution; - TrilinosWrappers::MPI::Vector system_rhs_newton; - TrilinosWrappers::MPI::Vector resid_vector; - TrilinosWrappers::MPI::Vector diag_mass_matrix_vector; - Vector cell_constitution; - IndexSet active_set; - - ConditionalOStream pcout; - - TrilinosWrappers::PreconditionAMG::AdditionalData additional_data; - TrilinosWrappers::PreconditionAMG preconditioner_u; - - std::unique_ptr > input_obstacle; - std::unique_ptr > plast_lin_hard; - - double sigma_0; // Yield stress - double gamma; // Parameter for the linear isotropic hardening - double e_modul; // E-Modul - double nu; // Poisson ratio - - TimerOutput computing_timer; - }; - - // @sect3{Implementation of the PlasticityContactProblem class} - - // Next for the implementation of the class - // template that makes use of the functions - // above. As before, we will write everything - - template - PlasticityContactProblem::PlasticityContactProblem (int _n_refinements_global) - : - n_refinements_global (_n_refinements_global), - mpi_communicator (MPI_COMM_WORLD), - triangulation (mpi_communicator), - fe (FE_Q(1), dim), - dof_handler (triangulation), - pcout (std::cout, - (Utilities::MPI::this_mpi_process(mpi_communicator) == 0)), - sigma_0 (400), - gamma (0.01), - e_modul (2.0e+5), - nu (0.3), - computing_timer (pcout, - TimerOutput::summary, - TimerOutput::wall_times) - { - // double _E, double _nu, double _sigma_0, double _gamma - plast_lin_hard.reset (new ConstitutiveLaw (e_modul, nu, sigma_0, gamma, mpi_communicator, pcout)); - } - - template - void PlasticityContactProblem::make_grid () - { - std::vector repet(3); - repet[0] = 1;//20; - repet[1] = 1; - repet[2] = 1; - - Point p1 (0,0,0); - Point p2 (1.0, 1.0, 1.0); - GridGenerator::subdivided_hyper_rectangle (triangulation, repet, p1, p2); - - Triangulation<3>::active_cell_iterator - cell = triangulation.begin_active(), - endc = triangulation.end(); - - /* boundary_indicators: - _______ - / 9 /| - /______ / | - 8| | 8| - | 8 | / - |_______|/ - 6 - */ - - for (; cell!=endc; ++cell) - for (unsigned int face=0; face::faces_per_cell; ++face) - { - if (cell->face (face)->center ()[2] == p2(2)) - cell->face (face)->set_boundary_indicator (9); - if (cell->face (face)->center ()[0] == p1(0) || - cell->face (face)->center ()[0] == p2(0) || - cell->face (face)->center ()[1] == p1(1) || - cell->face (face)->center ()[1] == p2(1)) - cell->face (face)->set_boundary_indicator (8); - if (cell->face (face)->center ()[2] == p1(2)) - cell->face (face)->set_boundary_indicator (6); - } - - triangulation.refine_global (n_refinements_global); - } - - template - void PlasticityContactProblem::setup_system () - { - // setup dofs - { - dof_handler.distribute_dofs (fe); - - locally_owned_dofs = dof_handler.locally_owned_dofs (); - locally_relevant_dofs.clear(); - DoFTools::extract_locally_relevant_dofs (dof_handler, - locally_relevant_dofs); - } - - // setup hanging nodes and dirichlet constraints - { - constraints_hanging_nodes.clear (); - constraints_hanging_nodes.reinit (locally_relevant_dofs); - DoFTools::make_hanging_node_constraints (dof_handler, - constraints_hanging_nodes); - constraints_hanging_nodes.close (); - - pcout << " Number of active cells: " - << triangulation.n_active_cells() - << std::endl - << " Total number of cells: " - << triangulation.n_cells() - << std::endl - << " Number of degrees of freedom: " - << dof_handler.n_dofs () - << std::endl; - - dirichlet_constraints (); - } - - // Initialzation for matrices and vectors - { - solution.reinit (locally_relevant_dofs, mpi_communicator); - system_rhs_newton.reinit (locally_owned_dofs, mpi_communicator); - resid_vector.reinit (system_rhs_newton); - diag_mass_matrix_vector.reinit (system_rhs_newton); - cell_constitution.reinit (triangulation.n_active_cells ()); - active_set.clear (); - active_set.set_size (locally_relevant_dofs.size ()); - } - - // setup sparsity pattern - { - TrilinosWrappers::SparsityPattern sp (locally_owned_dofs, - mpi_communicator); - - DoFTools::make_sparsity_pattern (dof_handler, sp, constraints_dirichlet_hanging_nodes, false, - Utilities::MPI::this_mpi_process(mpi_communicator)); - - sp.compress(); - - system_matrix_newton.reinit (sp); - - TrilinosWrappers::SparseMatrix mass_matrix; - mass_matrix.reinit (sp); - assemble_mass_matrix_diagonal (mass_matrix); - const unsigned int - start = (system_rhs_newton.local_range().first), - end = (system_rhs_newton.local_range().second); - for (unsigned int j=start; j - void PlasticityContactProblem::assemble_nl_system (TrilinosWrappers::MPI::Vector &u) - { - computing_timer.enter_section("Assembling"); - - QGauss quadrature_formula(2); - QGauss face_quadrature_formula(2); - - FEValues fe_values (fe, quadrature_formula, - UpdateFlags(update_values | - update_gradients | - update_q_points | - update_JxW_values)); - - FEFaceValues fe_values_face (fe, face_quadrature_formula, - update_values | update_quadrature_points | - update_JxW_values); - - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_q_points = quadrature_formula.size (); - const unsigned int n_face_q_points = face_quadrature_formula.size(); - - const EquationData::RightHandSide right_hand_side; - std::vector > right_hand_side_values (n_q_points, - Vector(dim)); - std::vector > right_hand_side_values_face (n_face_q_points, - Vector(dim)); - - FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); - Vector cell_rhs (dofs_per_cell); - - std::vector local_dof_indices (dofs_per_cell); - - typename DoFHandler::active_cell_iterator cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - const FEValuesExtractors::Vector displacement (0); - - const double kappa = 1.0; - for (; cell!=endc; ++cell) - if (cell->is_locally_owned()) - { - fe_values.reinit (cell); - cell_matrix = 0; - cell_rhs = 0; - - right_hand_side.vector_value_list (fe_values.get_quadrature_points(), - right_hand_side_values); - - std::vector > strain_tensor (n_q_points); - fe_values[displacement].get_function_symmetric_gradients (u, strain_tensor); - - for (unsigned int q_point=0; q_point stress_strain_tensor_linearized; - SymmetricTensor<4,dim> stress_strain_tensor; - SymmetricTensor<2,dim> stress_tensor; - - plast_lin_hard->linearized_plast_linear_hardening (stress_strain_tensor_linearized, - stress_strain_tensor, - strain_tensor[q_point]); - - for (unsigned int i=0; iget_strain(fe_values, i, q_point); - - for (unsigned int j=0; jget_strain(fe_values, j, q_point) * - fe_values.JxW (q_point)); - } - - // the linearized part a(v^i;v^i,v) of the rhs - cell_rhs(i) += (stress_tensor * - strain_tensor[q_point] * - fe_values.JxW (q_point)); - - // the residual part a(v^i;v) of the rhs - cell_rhs(i) -= (strain_tensor[q_point] * stress_strain_tensor * - plast_lin_hard->get_strain(fe_values, i, q_point) * - fe_values.JxW (q_point)); - - // the residual part F(v) of the rhs - Tensor<1,dim> rhs_values; - rhs_values = 0; - cell_rhs(i) += (fe_values[displacement].value (i, q_point) * - rhs_values * - fe_values.JxW (q_point)); - } - } - - for (unsigned int face=0; face::faces_per_cell; ++face) - { - if (cell->face (face)->at_boundary() - && cell->face (face)->boundary_indicator () == 9) - { - fe_values_face.reinit (cell, face); - - right_hand_side.vector_value_list (fe_values_face.get_quadrature_points(), - right_hand_side_values_face); - - for (unsigned int q_point=0; q_point rhs_values; - rhs_values = 0; - for (unsigned int i=0; iget_dof_indices (local_dof_indices); - constraints.distribute_local_to_global (cell_matrix, cell_rhs, - local_dof_indices, - system_matrix_newton, system_rhs_newton, true); - }; - - system_matrix_newton.compress (VectorOperation::add); - system_rhs_newton.compress (VectorOperation::add); - - computing_timer.exit_section("Assembling"); - } - - template - void PlasticityContactProblem::residual_nl_system (TrilinosWrappers::MPI::Vector &u) - { - QGauss quadrature_formula(2); - QGauss face_quadrature_formula(2); - - FEValues fe_values (fe, quadrature_formula, - UpdateFlags(update_values | - update_gradients | - update_q_points | - update_JxW_values)); - - FEFaceValues fe_values_face (fe, face_quadrature_formula, - update_values | update_quadrature_points | - update_JxW_values); - - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_q_points = quadrature_formula.size (); - const unsigned int n_face_q_points = face_quadrature_formula.size(); - - const EquationData::RightHandSide right_hand_side; - std::vector > right_hand_side_values (n_q_points, - Vector(dim)); - std::vector > right_hand_side_values_face (n_face_q_points, - Vector(dim)); - - Vector cell_rhs (dofs_per_cell); - - std::vector local_dof_indices (dofs_per_cell); - - const FEValuesExtractors::Vector displacement (0); - - typename DoFHandler::active_cell_iterator cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - unsigned int elast_points = 0; - unsigned int plast_points = 0; - double yield = 0; - unsigned int cell_number = 0; - cell_constitution = 0; - - for (; cell!=endc; ++cell) - if (cell->is_locally_owned()) - { - fe_values.reinit (cell); - cell_rhs = 0; - - right_hand_side.vector_value_list (fe_values.get_quadrature_points(), - right_hand_side_values); - - std::vector > strain_tensor (n_q_points); - fe_values[displacement].get_function_symmetric_gradients (u, strain_tensor); - - for (unsigned int q_point=0; q_point stress_strain_tensor; - SymmetricTensor<2,dim> stress_tensor; - - plast_lin_hard->plast_linear_hardening (stress_strain_tensor, strain_tensor[q_point], - elast_points, plast_points, yield); - - cell_constitution (cell_number) += yield; - for (unsigned int i=0; iget_strain(fe_values, i, q_point) * - fe_values.JxW (q_point)); - - Tensor<1,dim> rhs_values; - rhs_values = 0; - cell_rhs(i) += ((fe_values[displacement].value (i, q_point) * - rhs_values) * - fe_values.JxW (q_point)); - }; - }; - - for (unsigned int face=0; face::faces_per_cell; ++face) - { - if (cell->face (face)->at_boundary() - && cell->face (face)->boundary_indicator () == 9) - { - fe_values_face.reinit (cell, face); - - right_hand_side.vector_value_list (fe_values_face.get_quadrature_points(), - right_hand_side_values_face); - - for (unsigned int q_point=0; q_point rhs_values; - rhs_values = 0; - for (unsigned int i=0; iget_dof_indices (local_dof_indices); - constraints_dirichlet_hanging_nodes.distribute_local_to_global (cell_rhs, - local_dof_indices, - system_rhs_newton); - - cell_number += 1; - } - else - { - cell_constitution (cell_number) = 0; - cell_number += 1; - }; - - cell_constitution /= n_q_points; - cell_constitution.compress (VectorOperation::add); - system_rhs_newton.compress (VectorOperation::add); - - unsigned int sum_elast_points = Utilities::MPI::sum(elast_points, mpi_communicator); - unsigned int sum_plast_points = Utilities::MPI::sum(plast_points, mpi_communicator); - pcout << " Number of elastic quadrature points: " << sum_elast_points - << " and plastic quadrature points: " << sum_plast_points < - void PlasticityContactProblem::assemble_mass_matrix_diagonal (TrilinosWrappers::SparseMatrix &mass_matrix) - { - QTrapez face_quadrature_formula; - - FEFaceValues fe_values_face (fe, face_quadrature_formula, - update_values | - update_quadrature_points | - update_JxW_values); - - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_face_q_points = face_quadrature_formula.size(); - - FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); - Tensor<1,dim,double> ones (dim); - for (unsigned i=0; i local_dof_indices (dofs_per_cell); - - const FEValuesExtractors::Vector displacement (0); - - typename DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - for (; cell!=endc; ++cell) - if (cell->is_locally_owned()) - for (unsigned int face=0; face::faces_per_cell; ++face) - if (cell->face (face)->at_boundary() - && cell->face (face)->boundary_indicator () == 9) - { - fe_values_face.reinit (cell, face); - cell_matrix = 0; - - for (unsigned int q_point=0; q_pointget_dof_indices (local_dof_indices); - - constraints_dirichlet_hanging_nodes.distribute_local_to_global (cell_matrix, - local_dof_indices, - mass_matrix); - } - - mass_matrix.compress (VectorOperation::add); - } - - // @sect4{PlasticityContactProblem::update_solution_and_constraints} - - // Projection and updating of the active set - // for the dofs which penetrates the obstacle. - template - void PlasticityContactProblem::update_solution_and_constraints () - { - computing_timer.enter_section("Update solution and constraints"); - - const EquationData::Obstacle obstacle (input_obstacle); - std::vector vertex_touched (dof_handler.n_dofs (), false); - - typename DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - TrilinosWrappers::MPI::Vector distributed_solution (system_rhs_newton); - distributed_solution = solution; - TrilinosWrappers::MPI::Vector lambda (solution); - lambda = resid_vector; - TrilinosWrappers::MPI::Vector diag_mass_matrix_vector_relevant (solution); - diag_mass_matrix_vector_relevant = diag_mass_matrix_vector; - - constraints.reinit(locally_relevant_dofs); - active_set.clear (); - IndexSet active_set_locally_owned; - active_set_locally_owned.set_size (locally_owned_dofs.size ()); - const double c = 100.0*e_modul; - - for (; cell!=endc; ++cell) - if (cell->is_locally_owned()) - for (unsigned int face=0; face::faces_per_cell; ++face) - if (cell->face (face)->at_boundary() - && cell->face (face)->boundary_indicator () == 9) - for (unsigned int v=0; v::vertices_per_cell; ++v) - { - unsigned int index_z = cell->face (face)->vertex_dof_index (v,2); - - if (vertex_touched[cell->face (face)->vertex_index(v)] == false) - vertex_touched[cell->face (face)->vertex_index(v)] = true; - else - continue; - - // the local row where - Point point (cell->face (face)->vertex (v)[0], - cell->face (face)->vertex (v)[1], - cell->face (face)->vertex (v)[2]); - - double obstacle_value = obstacle.value (point, 2); - double solution_index_z = solution (index_z); - double gap = obstacle_value - point (2); - - -// std::cout << "lambda = " << lambda (index_z) -// << ", solution_index_z - gap = " << solution_index_z - gap -// << ", diag_mass_matrix_vector_relevant = " << diag_mass_matrix_vector_relevant (index_z) -// << std::endl; - - if (lambda (index_z) + - c * - diag_mass_matrix_vector_relevant (index_z) * - (solution_index_z - gap) - > 0 && - !(constraints_hanging_nodes.is_constrained(index_z))) - { - constraints.add_line (index_z); - constraints.set_inhomogeneity (index_z, gap); - - distributed_solution (index_z) = gap; - - if (locally_relevant_dofs.is_element (index_z)) - active_set.add_index (index_z); - - if (locally_owned_dofs.is_element (index_z)) - active_set_locally_owned.add_index (index_z); - } - } - distributed_solution.compress (VectorOperation::insert); - - unsigned int sum_contact_constraints = Utilities::MPI::sum(active_set_locally_owned.n_elements (), - mpi_communicator); - pcout << " Size of active set: " << sum_contact_constraints < - void PlasticityContactProblem::dirichlet_constraints () - { - /* boundary_indicators: - _______ - / 9 /| - /______ / | - 8| | 8| - | 8 | / - |_______|/ - 6 - */ - - constraints_dirichlet_hanging_nodes.reinit (locally_relevant_dofs); - constraints_dirichlet_hanging_nodes.merge (constraints_hanging_nodes); - - // interpolate all components of the solution - VectorTools::interpolate_boundary_values (dof_handler, - 6, - EquationData::BoundaryValues(), - constraints_dirichlet_hanging_nodes, - ComponentMask()); - - // interpolate x- and y-components of the - // solution (this is a bit mask, so apply - // operator| ) - FEValuesExtractors::Scalar x_displacement(0); - FEValuesExtractors::Scalar y_displacement(1); - VectorTools::interpolate_boundary_values (dof_handler, - 8, - EquationData::BoundaryValues(), - constraints_dirichlet_hanging_nodes, - (fe.component_mask(x_displacement) - | - fe.component_mask(y_displacement))); - constraints_dirichlet_hanging_nodes.close (); - } - - // @sect4{PlasticityContactProblem::solve} - - // In addition to step-41 we have - // to deal with the hanging node - // constraints. Again we also consider - // the locally_owned_dofs only by - // creating the vector distributed_solution. - // - // For the hanging nodes we have to apply - // the set_zero function to system_rhs_newton. - // This is necessary if a hanging node value x_0 - // has one neighbor which is in contact with - // value x_0 and one neighbor which is not with - // value x_1. This leads to an inhomogeneity - // constraint with value x_1/2 = gap/2 in the - // ConstraintMatrix. - // So the corresponding entries in the - // ride-hang-side are non-zero with a - // meaningless value. These values have to - // to set to zero. - - // The rest of the funtion is smiliar to - // step-41 except that we use a FGMRES-solver - // instead of CG. For a very small hardening - // value gamma the linear system becomes - // almost semi definite but still symmetric. - template - void PlasticityContactProblem::solve () - { - computing_timer.enter_section ("Solving and Preconditioning"); - - TrilinosWrappers::MPI::Vector distributed_solution (system_rhs_newton); - distributed_solution = solution; - - constraints_hanging_nodes.set_zero (distributed_solution); - constraints_hanging_nodes.set_zero (system_rhs_newton); - distributed_solution.compress(VectorOperation::insert); - system_rhs_newton.compress(VectorOperation::insert); - - computing_timer.enter_section("Preconditioning"); - - preconditioner_u.initialize (system_matrix_newton, additional_data); - - computing_timer.exit_section("Preconditioning"); - - computing_timer.enter_section("Solving"); - - PrimitiveVectorMemory mem; - TrilinosWrappers::MPI::Vector tmp (system_rhs_newton); - const double solver_tolerance = 1e-3 * - system_matrix_newton.residual (tmp, distributed_solution, system_rhs_newton); - - SolverControl solver_control (system_matrix_newton.m(), solver_tolerance); - SolverFGMRES - solver(solver_control, mem, - SolverFGMRES:: - AdditionalData(30, true)); - solver.solve(system_matrix_newton, distributed_solution, system_rhs_newton, preconditioner_u); - - pcout << " Error: " << solver_control.initial_value() - << " -> " << solver_control.last_value() - << " in " << solver_control.last_step() - << " FGMRES iterations." - << std::endl; - - computing_timer.exit_section("Solving"); - - number_iterations += solver_control.last_step(); - - constraints.distribute (distributed_solution); - - solution = distributed_solution; - - computing_timer.exit_section("Solving and Preconditioning"); - } - - - - template - void PlasticityContactProblem::solve_newton () - { - double resid=0; - double resid_old=100000; - TrilinosWrappers::MPI::Vector old_solution (system_rhs_newton); - TrilinosWrappers::MPI::Vector res (system_rhs_newton); - TrilinosWrappers::MPI::Vector tmp_vector (system_rhs_newton); - - std::vector > constant_modes; - DoFTools::extract_constant_modes (dof_handler, - ComponentMask(), - constant_modes); - - double sigma_hlp = sigma_0; - - additional_data.elliptic = true; - additional_data.n_cycles = 1; - additional_data.w_cycle = false; - additional_data.output_details = false; - additional_data.smoother_sweeps = 2; - additional_data.aggregation_threshold = 1e-2; - - IndexSet active_set_old (active_set); - unsigned int j = 1; - unsigned int number_assemble_system = 0; - for (; j<=100; j++) - { - // Solve an elastic problem to obtain a better start solution - if (j == 1 && cycle == 0) - plast_lin_hard->set_sigma_0 (1e+10); - else if (j == 2 || cycle > 0) - plast_lin_hard->set_sigma_0 (sigma_hlp); - - pcout<< " " <(i)); - old_solution = tmp_vector; - old_solution.sadd(1-a,a, distributed_solution); -// constraints.distribute (old_solution); - old_solution.compress (VectorOperation::add); - - computing_timer.enter_section("Residual and lambda"); - - system_rhs_newton = 0; - - solution = old_solution; - residual_nl_system (solution); - res = system_rhs_newton; - - const unsigned int - start_res = (res.local_range().first), - end_res = (res.local_range().second); - for (unsigned int n=start_res; n::type_dof_data, - data_component_interpretation); - data_out.add_data_vector (lambda, std::vector(dim, "Residual"), - DataOut::type_dof_data, - data_component_interpretation); - data_out.add_data_vector (active_set, std::vector(dim, "ActiveSet"), - DataOut::type_dof_data, - data_component_interpretation); - - Vector subdomain (triangulation.n_active_cells()); - for (unsigned int i=0; i filenames; - for (unsigned int i=0; - i - void PlasticityContactProblem::run () - { - pcout << "Read the obstacle from a file." << std::endl; - input_obstacle.reset (new Input("obstacle_file.pbm")); - pcout << "Ostacle is available now." << std::endl; - - const unsigned int n_cycles = 6; - for (cycle=0; cycle(dof_handler)); - refine_grid (); - } - - setup_system (); - - if (cycle > 0) - { - TrilinosWrappers::MPI::Vector distributed_solution (system_rhs_newton); - distributed_solution = solution; - soltrans->interpolate(distributed_solution); - solution = distributed_solution; - } - - computing_timer.exit_section("Mesh refinement and setup system"); - - solve_newton (); - - pcout<< " Writing graphical output..." <main function} - -int main (int argc, char *argv[]) -{ - using namespace dealii; - using namespace Step42; - - deallog.depth_console (0); - - clock_t start, end; - - start = clock(); - - Utilities::MPI::MPI_InitFinalize mpi_initialization (argc, argv); - { - int _n_refinements_global = 3; - - if (argc == 3) - { - _n_refinements_global = atoi(argv[1]); - } - - PlasticityContactProblem<3> laplace_problem_3d (_n_refinements_global); - laplace_problem_3d.run (); - } - - end = clock(); - cout<< "%%%%%% Rechenzeit overall = " << (double)(end-start)/CLOCKS_PER_SEC < $@" - @$(CXX) -w $(CXXFLAGS.g) -c $< -o $@ -./%.$(OBJEXT) : - @echo "==============optimized===== $( $@" - @$(CXX) $(CXXFLAGS.o) -c $< -o $@ - - -# The following statement tells make that the rules `run' and `clean' -# are not expected to produce files of the same name as Makefile rules -# usually do. -.PHONY: all run clean - - -# Finally there is a rule which you normally need not care much about: -# since the executable depends on some include files from the library, -# besides the C++ application file of course, it is necessary to -# re-generate the executable when one of the files it depends on has -# changed. The following rule creates a dependency file -# `Makefile.dep', which `make' uses to determine when to regenerate -# the executable. This file is automagically remade whenever needed, -# i.e. whenever one of the cc-/h-files changed. Make detects whether -# to remake this file upon inclusion at the bottom of this file. -# -# If the creation of Makefile.dep fails, blow it away and fail -Makefile.dep: $(target).cc Makefile \ - $(shell echo $D/include/deal.II/*/*.h) - @echo ============================ Remaking $@ - @$D/common/scripts/make_dependencies $(INCLUDE) -B. $(target).cc \ - > $@ \ - || (rm -f $@ ; false) - @if test -s $@ ; then true ; else rm $@ ; false ; fi - -# To make the dependencies known to `make', we finally have to include -# them: -include Makefile.dep - - diff --git a/deal.II/examples/step-47/doc/builds-on b/deal.II/examples/step-47/doc/builds-on deleted file mode 100644 index 9fdd726254..0000000000 --- a/deal.II/examples/step-47/doc/builds-on +++ /dev/null @@ -1 +0,0 @@ -step-5 diff --git a/deal.II/examples/step-47/doc/intro.dox b/deal.II/examples/step-47/doc/intro.dox deleted file mode 100644 index 1518606e43..0000000000 --- a/deal.II/examples/step-47/doc/intro.dox +++ /dev/null @@ -1,3 +0,0 @@ - -

Introduction

- diff --git a/deal.II/examples/step-47/doc/kind b/deal.II/examples/step-47/doc/kind deleted file mode 100644 index 15a13db451..0000000000 --- a/deal.II/examples/step-47/doc/kind +++ /dev/null @@ -1 +0,0 @@ -basic diff --git a/deal.II/examples/step-47/doc/points.dat b/deal.II/examples/step-47/doc/points.dat deleted file mode 100644 index 01ce16ad3c..0000000000 --- a/deal.II/examples/step-47/doc/points.dat +++ /dev/null @@ -1,21 +0,0 @@ -#xfem quadrature Points -0.781998 0.781998 -0.941587 0.72508 -0.72508 0.941587 -0.926335 0.926335 -0.397329 0.926335 -0.583333 0.72508 -0.583333 0.941587 -0.686004 0.781998 -0.72508 0.583333 -0.926335 0.397329 -0.781998 0.686004 -0.941587 0.583333 -0.115331 0.315492 -0.430422 0.60008 -0.069578 0.816587 -0.259669 0.892842 -0.315492 0.115331 -0.816587 0.069578 -0.60008 0.430422 -0.892842 0.259669 diff --git a/deal.II/examples/step-47/doc/results.dox b/deal.II/examples/step-47/doc/results.dox deleted file mode 100644 index f4c6feefb5..0000000000 --- a/deal.II/examples/step-47/doc/results.dox +++ /dev/null @@ -1 +0,0 @@ -

Results

diff --git a/deal.II/examples/step-47/doc/tooltip b/deal.II/examples/step-47/doc/tooltip deleted file mode 100644 index 00262bfbfb..0000000000 --- a/deal.II/examples/step-47/doc/tooltip +++ /dev/null @@ -1 +0,0 @@ -Adaptive local refinement. Higher order elements. \ No newline at end of file diff --git a/deal.II/examples/step-47/doc/vertices.dat b/deal.II/examples/step-47/doc/vertices.dat deleted file mode 100644 index 8d1ef87323..0000000000 --- a/deal.II/examples/step-47/doc/vertices.dat +++ /dev/null @@ -1,20 +0,0 @@ -#vertices of xfem subcells -0 0 -1 0 -1 1 -0 1 - -0.25 1 -1 0.25 - -0.625 0.625 -0.75 0.75 - -0.75 0.75 -0.625 1 - -0.75 0.75 -1 0.625 - -0 0 -0.625 0.625 diff --git a/deal.II/examples/step-47/step-47.cc b/deal.II/examples/step-47/step-47.cc deleted file mode 100644 index aded835b7c..0000000000 --- a/deal.II/examples/step-47/step-47.cc +++ /dev/null @@ -1,1128 +0,0 @@ -/* Author: Wolfgang Bangerth, University of Heidelberg, 2000 */ - -/* $Id$ */ -/* */ -/* Copyright (C) 2011-2012 by the deal.II authors */ -/* */ -/* This file is subject to QPL and may not be distributed */ -/* without copyright and license information. Please refer */ -/* to the file deal.II/doc/license.html for the text and */ -/* further information on this license. */ - - -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include -#include - -#include -#include - -#include -#include - - -#include - -#include - -#include - -namespace Step47 -{ - using namespace dealii; - - - - double sign (double d) - { - if (d > 0) - return 1; - else if (d < 0) - return -1; - else - return 0; - } - - - template - class LaplaceProblem - { - public: - LaplaceProblem (); - ~LaplaceProblem (); - - void run (); - - private: - bool interface_intersects_cell (const typename Triangulation::cell_iterator &cell) const; - std::pair > compute_quadrature(const Quadrature &plain_quadrature, const typename hp::DoFHandler::active_cell_iterator &cell, const std::vector &level_set_values); - void append_quadrature(const Quadrature &plain_quadrature, - const std::vector > &v , - std::vector > &xfem_points, - std::vector &xfem_weights); - - void setup_system (); - void assemble_system (); - void solve (); - void refine_grid (); - void output_results (const unsigned int cycle) const; - void compute_error () const; - - Triangulation triangulation; - - hp::DoFHandler dof_handler; - hp::FECollection fe_collection; - - ConstraintMatrix constraints; - - SparsityPattern sparsity_pattern; - SparseMatrix system_matrix; - - Vector solution; - Vector system_rhs; - }; - - - - - template - class Coefficient : public Function - { - public: - Coefficient () : Function() {} - - virtual double value (const Point &p, - const unsigned int component = 0) const; - - virtual void value_list (const std::vector > &points, - std::vector &values, - const unsigned int component = 0) const; - }; - - - - template - double Coefficient::value (const Point &p, - const unsigned int) const - { - if (p.square() < 0.5*0.5) - return 20; - else - return 1; - } - - - - template - void Coefficient::value_list (const std::vector > &points, - std::vector &values, - const unsigned int component) const - { - const unsigned int n_points = points.size(); - - Assert (values.size() == n_points, - ExcDimensionMismatch (values.size(), n_points)); - - Assert (component == 0, - ExcIndexRange (component, 0, 1)); - - for (unsigned int i=0; i - double exact_solution (const Point &p) - { - const double r = p.norm(); - - return (r < 0.5 - ? - 1./20 * (-1./4*r*r + 61./16) - : - 1./4 * (1-r*r)); - } - - - template - LaplaceProblem::LaplaceProblem () - : - dof_handler (triangulation) - { - fe_collection.push_back (FESystem (FE_Q(1), 1, - FE_Nothing(), 1)); - fe_collection.push_back (FESystem (FE_Q(1), 1, - FE_Q(1), 1)); - } - - - - template - LaplaceProblem::~LaplaceProblem () - { - dof_handler.clear (); - } - - - - template - double - level_set (const Point &p) - { - return p.norm() - 0.5; - } - - - - template - Tensor<1,dim> - grad_level_set (const Point &p) - { - return p / p.norm(); - } - - - - template - bool - LaplaceProblem:: - interface_intersects_cell (const typename Triangulation::cell_iterator &cell) const - { - for (unsigned int v=0; v::vertices_per_cell-1; ++v) - if (level_set(cell->vertex(v)) * level_set(cell->vertex(v+1)) < 0) - return true; - - // we get here only if all vertices have the same sign, which means that - // the cell is not intersected - return false; - } - - - - template - void LaplaceProblem::setup_system () - { - for (typename hp::DoFHandler::cell_iterator cell - = dof_handler.begin_active(); - cell != dof_handler.end(); ++cell) - if (interface_intersects_cell(cell) == false) - cell->set_active_fe_index(0); - else - cell->set_active_fe_index(1); - - dof_handler.distribute_dofs (fe_collection); - - solution.reinit (dof_handler.n_dofs()); - system_rhs.reinit (dof_handler.n_dofs()); - - - constraints.clear (); -//TODO: fix this, it currently crashes - // DoFTools::make_hanging_node_constraints (dof_handler, constraints); - -//TODO: component 1 must satisfy zero boundary conditions - constraints.close(); - - - CompressedSparsityPattern c_sparsity(dof_handler.n_dofs()); - DoFTools::make_sparsity_pattern (dof_handler, c_sparsity); - - constraints.condense (c_sparsity); - - sparsity_pattern.copy_from(c_sparsity); - - system_matrix.reinit (sparsity_pattern); - } - - - template - void LaplaceProblem::assemble_system () - { - const QGauss quadrature_formula(3); - - - FEValues plain_fe_values (fe_collection[0], quadrature_formula, - update_values | update_gradients | - update_quadrature_points | update_JxW_values); - FEValues enriched_fe_values (fe_collection[1], quadrature_formula, - update_values | update_gradients | - update_quadrature_points | update_JxW_values); - - const unsigned int n_q_points = quadrature_formula.size(); - - FullMatrix cell_matrix; - Vector cell_rhs; - - std::vector local_dof_indices; - - const Coefficient coefficient; - std::vector coefficient_values (n_q_points); - - typename hp::DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - for (; cell!=endc; ++cell) - { - const unsigned int dofs_per_cell = cell->get_fe().dofs_per_cell; - cell_matrix.reinit (dofs_per_cell, dofs_per_cell); - cell_rhs.reinit (dofs_per_cell); - - cell_matrix = 0; - cell_rhs = 0; - - if (cell->active_fe_index() == 0) - { - plain_fe_values.reinit (cell); - - coefficient_values.resize (plain_fe_values.n_quadrature_points); - coefficient.value_list (plain_fe_values.get_quadrature_points(), - coefficient_values); - - for (unsigned int q_point=0; q_pointactive_fe_index() == 1, ExcInternalError()); - Assert (interface_intersects_cell(cell) == true, ExcInternalError()); - - std::vector level_set_values (GeometryInfo::vertices_per_cell); - for (unsigned int v=0; v::vertices_per_cell; ++v) - level_set_values[v] = level_set (cell->vertex(v)); - - FEValues this_fe_values (fe_collection[1], - compute_quadrature(quadrature_formula, cell, - level_set_values).second, - update_values | update_gradients | - update_quadrature_points | update_JxW_values ); - - this_fe_values.reinit (cell); - - coefficient_values.resize (this_fe_values.n_quadrature_points); - coefficient.value_list (this_fe_values.get_quadrature_points(), - coefficient_values); - - for (unsigned int q_point=0; q_pointget_fe().system_to_component_index(i).first == 0) - { - for (unsigned int j=0; jget_fe().system_to_component_index(j).first == 0) - cell_matrix(i,j) += (coefficient_values[q_point] * - this_fe_values.shape_grad(i,q_point) * - this_fe_values.shape_grad(j,q_point) * - this_fe_values.JxW(q_point)); - else - cell_matrix(i,j) += (coefficient_values[q_point] * - this_fe_values.shape_grad(i,q_point) - * - ((std::fabs(level_set(this_fe_values.quadrature_point(q_point))) - - - std::fabs(level_set(cell->vertex(cell->get_fe().system_to_component_index(j).second))))* - this_fe_values.shape_grad(j,q_point) - + - grad_level_set(this_fe_values.quadrature_point(q_point)) * - sign(level_set(this_fe_values.quadrature_point(q_point))) * - this_fe_values.shape_value(j,q_point)) * - this_fe_values.JxW(q_point)); - - cell_rhs(i) += (this_fe_values.shape_value(i,q_point) * - 1.0 * - this_fe_values.JxW(q_point)); - } - else - { - for (unsigned int j=0; jget_fe().system_to_component_index(j).first == 0) - cell_matrix(i,j) += (coefficient_values[q_point] * - ((std::fabs(level_set(this_fe_values.quadrature_point(q_point))) - - - std::fabs(level_set(cell->vertex(cell->get_fe().system_to_component_index(i).second))))* - this_fe_values.shape_grad(i,q_point) - + - grad_level_set(this_fe_values.quadrature_point(q_point)) * - sign(level_set(this_fe_values.quadrature_point(q_point))) * - this_fe_values.shape_value(i,q_point)) * - this_fe_values.shape_grad(j,q_point) * - this_fe_values.JxW(q_point)); - else - cell_matrix(i,j) += (coefficient_values[q_point] * - ((std::fabs(level_set(this_fe_values.quadrature_point(q_point))) - - - std::fabs(level_set(cell->vertex(cell->get_fe().system_to_component_index(i).second))))* - this_fe_values.shape_grad(i,q_point) - + - grad_level_set(this_fe_values.quadrature_point(q_point)) * - sign(level_set(this_fe_values.quadrature_point(q_point))) * - this_fe_values.shape_value(i,q_point)) * - ((std::fabs(level_set(this_fe_values.quadrature_point(q_point))) - - - std::fabs(level_set(cell->vertex(cell->get_fe().system_to_component_index(j).second))))* - this_fe_values.shape_grad(j,q_point) - + - grad_level_set(this_fe_values.quadrature_point(q_point)) * - sign(level_set(this_fe_values.quadrature_point(q_point))) * - this_fe_values.shape_value(j,q_point)) * - this_fe_values.JxW(q_point)); - - cell_rhs(i) += ((std::fabs(level_set(this_fe_values.quadrature_point(q_point))) - - - std::fabs(level_set(cell->vertex(cell->get_fe().system_to_component_index(i).second))))* - this_fe_values.shape_value(i,q_point) * - 1.0 * - this_fe_values.JxW(q_point)); - } - } - - local_dof_indices.resize (dofs_per_cell); - cell->get_dof_indices (local_dof_indices); - constraints.distribute_local_to_global (cell_matrix, cell_rhs, - local_dof_indices, - system_matrix, system_rhs); - } - - - std::map boundary_values; - VectorTools::interpolate_boundary_values (dof_handler, - 0, - ZeroFunction(2), - boundary_values); - MatrixTools::apply_boundary_values (boundary_values, - system_matrix, - solution, - system_rhs); - - } - -// To integrate the enriched elements we have to find the geometrical -// decomposition of the original element in subelements. The subelements are -// used to integrate the elements on both sides of the discontinuity. The -// disontinuity line is approximated by a piece-wise linear interpolation -// between the intersection of the discontinuity with the edges of the -// elements. The vector level_set_values has the values of the level set -// function at the vertices of the elements. From these values can be found by -// linear interpolation the intersections. There are three kind of -// decomposition that are considered. Type 1: there is not cut. Type 2: a -// corner of the element is cut. Type 3: two corners are cut. - - template - std::pair > - LaplaceProblem::compute_quadrature (const Quadrature &plain_quadrature, - const typename hp::DoFHandler::active_cell_iterator &cell, - const std::vector &level_set_values ) - { - - unsigned int type = 0; - - // find the type of cut - int sign_ls[GeometryInfo::vertices_per_cell]; - for (unsigned int v=0; v::vertices_per_cell; ++v) - { - if (level_set_values[v] > 0) sign_ls[v] = 1; - else if (level_set_values[v] < 0) sign_ls[v] = -1; - else sign_ls[v] = 0; - } - - // the sign of the level set function at the 4 nodes of the elements can - // be positive + or negative - depending on the sign of the level set - // function we have the folloing three classes of decomposition type 1: - // ++++, ---- type 2: -+++, +-++, ++-+, +++-, +---, -+--, --+-, ---+ type - // 3: +--+, ++--, +-+-, -++-, --++, -+-+ - - if ( sign_ls[0]==sign_ls[1] & sign_ls[0]==sign_ls[2] & sign_ls[0]==sign_ls[3] ) type =1; - else if ( sign_ls[0]*sign_ls[1]*sign_ls[2]*sign_ls[3] < 0 ) type = 2; - else type = 3; - - unsigned int Pos = 100; - - Point v0(0,0); - Point v1(1,0); - Point v2(0,1); - Point v3(1,1); - - Point A(0,0); - Point B(0,0); - Point C(0,0); - Point D(0,0); - Point E(0,0); - Point F(0,0); - - if (type == 1) - return std::pair >(1, plain_quadrature); - - if (type==2) - { - const unsigned int n_q_points = plain_quadrature.size(); - - // loop over all subelements for integration in type 2 there are 5 - // subelements - - Quadrature xfem_quadrature(5*n_q_points); - - std::vector > v(GeometryInfo::vertices_per_cell); - - if (sign_ls[0]!=sign_ls[1] && sign_ls[0]!=sign_ls[2] && sign_ls[0]!=sign_ls[3]) Pos = 0; - else if (sign_ls[1]!=sign_ls[0] && sign_ls[1]!=sign_ls[2] && sign_ls[1]!=sign_ls[3]) Pos = 1; - else if (sign_ls[2]!=sign_ls[0] && sign_ls[2]!=sign_ls[1] && sign_ls[2]!=sign_ls[3]) Pos = 2; - else if (sign_ls[3]!=sign_ls[0] && sign_ls[3]!=sign_ls[1] && sign_ls[3]!=sign_ls[2]) Pos = 3; - else assert(0); // error message - - // Find cut coordinates - - // deal.ii local coordinates - - // 2-------3 | | | | | | 0-------1 - - if (Pos == 0) - { - A[0] = 1. - level_set_values[1]/(level_set_values[1]-level_set_values[0]); - B[1] = 1. - level_set_values[2]/(level_set_values[2]-level_set_values[0]); - A(1) = 0.; - B(0) = 0.; - C(0) = 0.5*( A(0) + B(0) ); - C(1) = 0.5*( A(1) + B(1) ); - D(0) = 2./3. * C(0); - D(1) = 2./3. * C(1); - E(0) = 0.5*A(0); - E(1) = 0.; - F(0) = 0.; - F(1) = 0.5*B(1); - } - else if (Pos == 1) - { - A[0] = level_set_values[0]/(level_set_values[0]-level_set_values[1]); - B[1] = 1 - level_set_values[3]/(level_set_values[3]-level_set_values[1]); - A(1) = 0.; - B(0) = 1.; - C(0) = 0.5*( A(0) + B(0) ); - C(1) = 0.5*( A(1) + B(1) ); - D(0) = 1./3. + 2./3. * C(0); - D(1) = 2./3. * C(1); - E(0) = 0.5*(1 + A(0)); - E(1) = 0.; - F(0) = 1.; - F(1) = 0.5*B(1); - } - else if (Pos == 2) - { - A[0] = 1 - level_set_values[3]/(level_set_values[3]-level_set_values[2]); - B[1] = level_set_values[0]/(level_set_values[0]-level_set_values[2]); - A(1) = 1.; - B(0) = 0.; - C(0) = 0.5*( A(0) + B(0) ); - C(1) = 0.5*( A(1) + B(1) ); - D(0) = 2./3. * C(0); - D(1) = 1./3. + 2./3. * C(1); - E(0) = 0.5* A(0); - E(1) = 1.; - F(0) = 0.; - F(1) = 0.5*( 1. + B(1) ); - } - else if (Pos == 3) - { - A[0] = level_set_values[2]/(level_set_values[2]-level_set_values[3]); - B[1] = level_set_values[1]/(level_set_values[1]-level_set_values[3]); - A(1) = 1.; - B(0) = 1.; - C(0) = 0.5*( A(0) + B(0) ); - C(1) = 0.5*( A(1) + B(1) ); - D(0) = 1./3. + 2./3. * C(0); - D(1) = 1./3. + 2./3. * C(1); - E(0) = 0.5*( 1. + A(0) ); - E(1) = 1.; - F(0) = 1.; - F(1) = 0.5*( 1. + B(1) ); - } - - //std::cout << A << std::endl; std::cout << B << std::endl; std::cout - //<< C << std::endl; std::cout << D << std::endl; std::cout << E << - //std::endl; std::cout << F << std::endl; - - std::string filename = "vertices.dat"; - std::ofstream output (filename.c_str()); - output << "#vertices of xfem subcells" << std::endl; - output << v0(0) << " " << v0(1) << std::endl; - output << v1(0) << " " << v1(1) << std::endl; - output << v3(0) << " " << v3(1) << std::endl; - output << v2(0) << " " << v2(1) << std::endl; - output << std::endl; - output << A(0) << " " << A(1) << std::endl; - output << B(0) << " " << B(1) << std::endl; - output << std::endl; - output << C(0) << " " << C(1) << std::endl; - output << D(0) << " " << D(1) << std::endl; - output << std::endl; - output << D(0) << " " << D(1) << std::endl; - output << E(0) << " " << E(1) << std::endl; - output << std::endl; - output << D(0) << " " << D(1) << std::endl; - output << F(0) << " " << F(1) << std::endl; - output << std::endl; - - if (Pos==0) - output << v3(0) << " " << v3(1) << std::endl; - else if (Pos==1) - output << v2(0) << " " << v2(1) << std::endl; - else if (Pos==2) - output << v1(0) << " " << v1(1) << std::endl; - else if (Pos==3) - output << v0(0) << " " << v0(1) << std::endl; - output << C(0) << " " << C(1) << std::endl; - - Point subcell_vertices[10]; - subcell_vertices[0] = v0; - subcell_vertices[1] = v1; - subcell_vertices[2] = v2; - subcell_vertices[3] = v3; - subcell_vertices[4] = A; - subcell_vertices[5] = B; - subcell_vertices[6] = C; - subcell_vertices[7] = D; - subcell_vertices[8] = E; - subcell_vertices[9] = F; - - std::vector > xfem_points; - std::vector xfem_weights; - - // lookup table for the decomposition - - if (dim==2) - { - unsigned int subcell_v_indices[4][5][4] = - { - {{0,8,9,7}, {9,7,5,6}, {8,4,7,6}, {5,6,2,3}, {6,4,3,1}}, - {{8,1,7,9}, {4,8,6,7}, {6,7,5,9}, {0,4,2,6}, {2,6,3,5}}, - {{9,7,2,8}, {5,6,9,7}, {6,4,7,8}, {0,1,5,6}, {6,1,4,3}}, - {{7,9,8,3}, {4,6,8,7}, {6,5,7,9}, {0,6,2,4}, {0,1,6,5}} - }; - - for (unsigned int subcell = 0; subcell<5; subcell++) - { - //std::cout << "subcell : " << subcell << std::endl; - std::vector > vertices; - for (unsigned int i=0; i<4; i++) - { - vertices.push_back( subcell_vertices[subcell_v_indices[Pos][subcell][i]] ); - //std::cout << "i : " << i << std::endl; std::cout << - //"subcell v : " << subcell_v_indices[Pos][subcell][i] << - //std::endl; std::cout << vertices[i](0) << " " << - //vertices[i](1) << std::endl; - } - //std::cout << std::endl; create quadrature rule - append_quadrature( plain_quadrature, - vertices, - xfem_points, - xfem_weights); - //initialize xfem_quadrature with quadrature points of all - //subelements - xfem_quadrature.initialize(xfem_points, xfem_weights); - } - } - - Assert (xfem_quadrature.size() == plain_quadrature.size() * 5, ExcInternalError()); - return std::pair >(2, xfem_quadrature); - } - - // Type three decomposition (+--+, ++--, +-+-, -++-, --++, -+-+) - - if (type==3) - { - const unsigned int n_q_points = plain_quadrature.size(); - - // loop over all subelements for integration in type 2 there are 5 - // subelements - - Quadrature xfem_quadrature(5*n_q_points); - - std::vector > v(GeometryInfo::vertices_per_cell); - - if ( sign_ls[0]==sign_ls[1] && sign_ls[2]==sign_ls[3] ) - { - Pos = 0; - A(0) = 0.; - A(1) = level_set_values[0]/((level_set_values[0]-level_set_values[2])); - B(0) = 1.; - B(1) = level_set_values[1]/((level_set_values[1]-level_set_values[3])); - } - else if ( sign_ls[0]==sign_ls[2] && sign_ls[1]==sign_ls[3] ) - { - Pos = 1; - A(0) = level_set_values[0]/((level_set_values[0]-level_set_values[1])); - A(1) = 0.; - B(0) = level_set_values[2]/((level_set_values[2]-level_set_values[3])); - B(1) = 1.; - } - else if ( sign_ls[0]==sign_ls[3] && sign_ls[1]==sign_ls[2] ) - { - std::cout << "Error: the element has two cut lines and this is not allowed" << std::endl; - assert(0); - } - else - { - std::cout << "Error: the level set function has not the right values" << std::endl; - assert(0); - } - - //std::cout << "Pos " << Pos << std::endl; std::cout << A << - //std::endl; std::cout << B << std::endl; - std::string filename = "vertices.dat"; - std::ofstream output (filename.c_str()); - output << "#vertices of xfem subcells" << std::endl; - output << A(0) << " " << A(1) << std::endl; - output << B(0) << " " << B(1) << std::endl; - - //fill xfem_quadrature - Point subcell_vertices[6]; - subcell_vertices[0] = v0; - subcell_vertices[1] = v1; - subcell_vertices[2] = v2; - subcell_vertices[3] = v3; - subcell_vertices[4] = A; - subcell_vertices[5] = B; - - std::vector > xfem_points; - std::vector xfem_weights; - - if (dim==2) - { - unsigned int subcell_v_indices[2][2][4] = - { - {{0,1,4,5}, {4,5,2,3}}, - {{0,4,2,5}, {4,1,5,3}} - }; - - //std::cout << "Pos : " << Pos << std::endl; - for (unsigned int subcell = 0; subcell<2; subcell++) - { - //std::cout << "subcell : " << subcell << std::endl; - std::vector > vertices; - for (unsigned int i=0; i<4; i++) - { - vertices.push_back( subcell_vertices[subcell_v_indices[Pos][subcell][i]] ); - //std::cout << "i : " << i << std::endl; std::cout << - //"subcell v : " << subcell_v_indices[Pos][subcell][i] << - //std::endl; std::cout << vertices[i](0) << " " << - //vertices[i](1) << std::endl; - } - //std::cout << std::endl; create quadrature rule - append_quadrature( plain_quadrature, - vertices, - xfem_points, - xfem_weights); - //initialize xfem_quadrature with quadrature points of all - //subelements - xfem_quadrature.initialize(xfem_points, xfem_weights); - } - } - Assert (xfem_quadrature.size() == plain_quadrature.size() * 2, ExcInternalError()); - return std::pair >(3, xfem_quadrature); - } - - return std::pair >(0, plain_quadrature);; - - } - - template - void LaplaceProblem::append_quadrature ( const Quadrature &plain_quadrature, - const std::vector > &v, - std::vector > &xfem_points, - std::vector &xfem_weights) - - { - // Project integration points into sub-elements. This maps quadrature - // points from a reference element to a subelement of a reference element. - // To implement the action of this map the coordinates of the subelements - // have been calculated (A(0)...F(0),A(1)...F(1)) the coordinates of the - // quadrature points are given by the bi-linear map defined by the form - // functions $x^\prime_i = \sum_j v^\prime \phi_j(x^hat_i)$, where the - // $\phi_j$ are the shape functions of the FEQ. - - unsigned int n_v = GeometryInfo::vertices_per_cell; - - std::vector > q_points = plain_quadrature.get_points(); - std::vector > q_transf(q_points.size()); - std::vector W = plain_quadrature.get_weights(); - std::vector phi(n_v); - std::vector > grad_phi(n_v); - - const unsigned int n_q_points = plain_quadrature.size(); - - std::vector JxW(n_q_points); - - for ( unsigned int i = 0; i < n_q_points; i++) - { - switch (dim) - { - case 2: - { - double xi = q_points[i](0); - double eta = q_points[i](1); - - // Define shape functions on reference element we consider a - // bi-linear mapping - phi[0] = (1. - xi) * (1. - eta); - phi[1] = xi * (1. - eta); - phi[2] = (1. - xi) * eta; - phi[3] = xi * eta; - - grad_phi[0][0] = (-1. + eta); - grad_phi[1][0] = (1. - eta); - grad_phi[2][0] = -eta; - grad_phi[3][0] = eta; - - grad_phi[0][1] = (-1. + xi); - grad_phi[1][1] = -xi; - grad_phi[2][1] = 1-xi; - grad_phi[3][1] = xi; - - break; - } - - default: - Assert (false, ExcNotImplemented()); - } - - - Tensor<2,dim> jacobian; - - // Calculate Jacobian of transformation - for (unsigned int d=0; d::vertices_per_cell; j++) - { - jacobian[d][e] += grad_phi[j][e] * v[j](d); - } - } - - double detJ = determinant(jacobian); - xfem_weights.push_back (W[i] * detJ); - - // Map integration points from reference element to subcell of - // reference element - Point q_prime; - for (unsigned int d=0; d::vertices_per_cell; j++) - q_prime[d] += v[j](d) * phi[j]; - xfem_points.push_back(q_prime); - } - - } - - - template - void LaplaceProblem::solve () - { - SolverControl solver_control (1000, 1e-12); - SolverCG<> solver (solver_control); - - PreconditionSSOR<> preconditioner; - preconditioner.initialize(system_matrix, 1.2); - - solver.solve (system_matrix, solution, system_rhs, - preconditioner); - - constraints.distribute (solution); - } - - - - template - void LaplaceProblem::refine_grid () - { - Vector estimated_error_per_cell (triangulation.n_active_cells()); - - KellyErrorEstimator::estimate (dof_handler, - QGauss(3), - typename FunctionMap::type(), - solution, - estimated_error_per_cell); - - GridRefinement::refine_and_coarsen_fixed_number (triangulation, - estimated_error_per_cell, - 0.3, 0.03); - - triangulation.execute_coarsening_and_refinement (); - } - - - - template - class Postprocessor : public DataPostprocessor - { - public: - virtual - void - compute_derived_quantities_vector (const std::vector > &uh, - const std::vector > > &duh, - const std::vector > > &dduh, - const std::vector > &normals, - const std::vector > &evaluation_points, - std::vector > &computed_quantities) const; - - virtual std::vector get_names () const; - - virtual - std::vector - get_data_component_interpretation () const; - - virtual UpdateFlags get_needed_update_flags () const; - }; - - - template - std::vector - Postprocessor::get_names() const - { - std::vector solution_names (1, "total_solution"); - solution_names.push_back ("error"); - return solution_names; - } - - - template - std::vector - Postprocessor:: - get_data_component_interpretation () const - { - std::vector - interpretation (2, - DataComponentInterpretation::component_is_scalar); - return interpretation; - } - - - template - UpdateFlags - Postprocessor::get_needed_update_flags() const - { - return update_values | update_q_points; - } - - - template - void - Postprocessor:: - compute_derived_quantities_vector (const std::vector > &uh, - const std::vector > > & /*duh*/, - const std::vector > > & /*dduh*/, - const std::vector > & /*normals*/, - const std::vector > &evaluation_points, - std::vector > &computed_quantities) const - { - const unsigned int n_quadrature_points = uh.size(); - Assert (computed_quantities.size() == n_quadrature_points, ExcInternalError()); - Assert (uh[0].size() == 2, ExcInternalError()); - - for (unsigned int q=0; q - void LaplaceProblem::output_results (const unsigned int cycle) const - { - Assert (cycle < 10, ExcNotImplemented()); - - std::string filename = "solution-"; - filename += ('0' + cycle); - filename += ".vtk"; - - std::ofstream output (filename.c_str()); - - Postprocessor postprocessor; - DataOut > data_out; - - data_out.attach_dof_handler (dof_handler); - data_out.add_data_vector (solution, "solution"); - data_out.add_data_vector (solution, postprocessor); - data_out.build_patches (5); - - data_out.write_vtk (output); - } - - - - template - void LaplaceProblem::compute_error () const - { - hp::QCollection q_collection; - q_collection.push_back (QGauss(2)); - q_collection.push_back (QIterated(QGauss<1>(2), 4)); - - hp::FEValues hp_fe_values (fe_collection, q_collection, - update_values | update_q_points | update_JxW_values); - - double l2_error_square = 0; - - std::vector > solution_values; - - typename hp::DoFHandler::active_cell_iterator - cell = dof_handler.begin_active(), - endc = dof_handler.end(); - - for (; cell!=endc; ++cell) - { - hp_fe_values.reinit (cell); - - const FEValues &fe_values = hp_fe_values.get_present_fe_values (); - - solution_values.resize (fe_values.n_quadrature_points, - Vector(2)); - fe_values.get_function_values (solution, - solution_values); - - for (unsigned int q=0; q boundary; - triangulation.set_boundary (0, boundary); - - triangulation.refine_global (2); - } - else - triangulation.refine_global (1); -// refine_grid (); - - - std::cout << " Number of active cells: " - << triangulation.n_active_cells() - << std::endl; - - setup_system (); - - std::cout << " Number of degrees of freedom: " - << dof_handler.n_dofs() - << std::endl; - - assemble_system (); - solve (); - compute_error (); - output_results (cycle); - } - } -} - - - -int main () -{ - - try - { - using namespace dealii; - using namespace Step47; - - deallog.depth_console (0); - - LaplaceProblem<2> laplace_problem_2d; - laplace_problem_2d.run (); - } - catch (std::exception &exc) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Exception on processing: " << std::endl - << exc.what() << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - - return 1; - } - catch (...) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Unknown exception!" << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - return 1; - } - - return 0; -} diff --git a/deal.II/examples/step-50/Makefile b/deal.II/examples/step-50/Makefile deleted file mode 100644 index 724ba8ac31..0000000000 --- a/deal.II/examples/step-50/Makefile +++ /dev/null @@ -1,155 +0,0 @@ -# $Id$ - -# For the small projects Makefile, you basically need to fill in only -# four fields. -# -# The first is the name of the application. It is assumed that the -# application name is the same as the base file name of the single C++ -# file from which the application is generated. -target = step-50 - -# The second field determines whether you want to run your program in -# debug or optimized mode. The latter is significantly faster, but no -# run-time checking of parameters and internal states is performed, so -# you should set this value to `on' while you develop your program, -# and to `off' when running production computations. -debug-mode = on - - -# As third field, we need to give the path to the top-level deal.II -# directory. You need to adjust this to your needs. Since this path is -# probably the most often needed one in the Makefile internals, it is -# designated by a single-character variable, since that can be -# reference using $D only, i.e. without the parentheses that are -# required for most other parameters, as e.g. in $(target). -D = ../../ - - -# The last field specifies the names of data and other files that -# shall be deleted when calling `make clean'. Object and backup files, -# executables and the like are removed anyway. Here, we give a list of -# files in the various output formats that deal.II supports. -clean-up-files = *gmv *gnuplot *gpl *eps *pov *vtk *ucd *.d2 - - - - -# -# -# Usually, you will not need to change anything beyond this point. -# -# -# The next statement tells the `make' program where to find the -# deal.II top level directory and to include the file with the global -# settings -include $D/common/Make.global_options - -ifneq ($(USE_CONTRIB_TRILINOS)$(USE_CONTRIB_P4EST),yesyes) -default run clean: - @echo - @echo "===========================================================" - @echo "= This program can only be run with the following =" - @echo "= configuration: =" - @echo "= - with Trilinos =" - @echo "= - with P4EST =" - @echo "= At least one of these conditions is not satisfied. =" - @echo "===========================================================" - @echo -else -# Since the whole project consists of only one file, we need not -# consider difficult dependencies. We only have to declare the -# libraries which we want to link to the object file. deal.II has two -# libraries: one for the debug mode version of the -# application and one for optimized mode. -libs.g := $(lib-deal2.g) -libs.o := $(lib-deal2.o) - - -# We now use the variable defined above to switch between debug and -# optimized mode to select the set of libraries to link with. Included -# in the list of libraries is the name of the object file which we -# will produce from the single C++ file. Note that by default we use -# the extension .g.o for object files compiled in debug mode and .o for -# object files in optimized mode (or whatever local default on your -# system is instead of .o) -ifeq ($(debug-mode),on) - libraries = $(target).g.$(OBJEXT) $(libs.g) -else - libraries = $(target).$(OBJEXT) $(libs.o) -endif - - -# Now comes the first production rule: how to link the single object -# file produced from the single C++ file into the executable. Since -# this is the first rule in the Makefile, it is the one `make' selects -# if you call it without arguments. -all: $(target)$(EXEEXT) -$(target)$(EXEEXT) : $(libraries) - @echo ============================ Linking $@ - @$(CXX) -o $@ $^ $(LIBS) $(LDFLAGS) - - -# To make running the application somewhat independent of the actual -# program name, we usually declare a rule `run' which simply runs the -# program. You can then run it by typing `make run'. This is also -# useful if you want to call the executable with arguments which do -# not change frequently. You may then want to add them to the -# following rule: -run: $(target)$(EXEEXT) - @echo ============================ Running $< - @./$(target)$(EXEEXT) - - -# As a last rule to the `make' program, we define what to do when -# cleaning up a directory. This usually involves deleting object files -# and other automatically created files such as the executable itself, -# backup files, and data files. Since the latter are not usually quite -# diverse, you needed to declare them at the top of this file. -clean: - -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files) - - -# Since we have not yet stated how to make an object file from a C++ -# file, we should do so now. Since the many flags passed to the -# compiler are usually not of much interest, we suppress the actual -# command line using the `at' sign in the first column of the rules -# and write the string indicating what we do instead. -./%.g.$(OBJEXT) : - @echo "==============debug========= $( $@" - @$(CXX) $(CXXFLAGS.g) -c $< -o $@ -./%.$(OBJEXT) : - @echo "==============optimized===== $( $@" - @$(CXX) $(CXXFLAGS.o) -c $< -o $@ - - -# The following statement tells make that the rules `run' and `clean' -# are not expected to produce files of the same name as Makefile rules -# usually do. -.PHONY: all run clean - - -# Finally there is a rule which you normally need not care much about: -# since the executable depends on some include files from the library, -# besides the C++ application file of course, it is necessary to -# re-generate the executable when one of the files it depends on has -# changed. The following rule creates a dependency file -# `Makefile.dep', which `make' uses to determine when to regenerate -# the executable. This file is automagically remade whenever needed, -# i.e. whenever one of the cc-/h-files changed. Make detects whether -# to remake this file upon inclusion at the bottom of this file. -# -# If the creation of Makefile.dep fails, blow it away and fail -Makefile.dep: $(target).cc Makefile \ - $(shell echo $D/include/deal.II/*/*.h) - @echo ============================ Remaking $@ - @$D/common/scripts/make_dependencies $(INCLUDE) -B. $(target).cc \ - > $@ \ - || (rm -f $@ ; false) - @if test -s $@ ; then true ; else rm $@ ; false ; fi - -# To make the dependencies known to `make', we finally have to include -# them: -include Makefile.dep - - -endif # USE_CONTRIB_TRILINOS diff --git a/deal.II/examples/step-50/doc/builds-on b/deal.II/examples/step-50/doc/builds-on deleted file mode 100644 index 17402734c7..0000000000 --- a/deal.II/examples/step-50/doc/builds-on +++ /dev/null @@ -1 +0,0 @@ -step-6 diff --git a/deal.II/examples/step-50/doc/intro.dox b/deal.II/examples/step-50/doc/intro.dox deleted file mode 100644 index 99666726d3..0000000000 --- a/deal.II/examples/step-50/doc/intro.dox +++ /dev/null @@ -1,90 +0,0 @@ -
- -This program has evolved from a version originally written by Guido -Kanschat in 2003. It has undergone significant revisions by Bärbel -Janssen, Guido Kanschat and Wolfgang Bangerth in 2009 and 2010 to demonstrate -multigrid algorithms on adaptively refined meshes. - - - - -

Introduction

- - -This example shows the basic usage of the multilevel functions in -deal.II. It solves the same problem as used in step-6, -but demonstrating the things one has to provide when using multigrid -as a preconditioner. In particular, this requires that we define a -hierarchy of levels, provide transfer operators from one level to the -next and back, and provide representations of the Laplace operator on -each level. - -In order to allow sufficient flexibility in conjunction with systems of -differential equations and block preconditioners, quite a few different objects -have to be created before starting the multilevel method, although -most of what needs to be done is provided by deal.II itself. These are -
    -
  • An the object handling transfer between grids; we use the - MGTransferPrebuilt class for this that does almost all of the work - inside the library. -
  • The solver on the coarsest level; here, we use MGCoarseGridHouseholder. -
  • The smoother on all other levels, which in our case will be the - MGSmootherRelaxation class using SOR as the underlying method -
  • And MGMatrix, a class having a special level multiplication, i.e. we - basically store one matrix per grid level and allow multiplication - with it. -
-Most of these objects will only be needed inside the function that -actually solves the linear system. There, these objects are combined -in an object of type Multigrid, containing the implementation of the -V-cycle, which is in turn used by the preconditioner PreconditionMG, -ready for plug-in into a linear solver of the LAC library. - -The multilevel method in deal.II follows in many respects the outlines -of the various publications by James Bramble, Joseph Pasciak and -Jinchao Xu (i.e. the "BPX" framework). In order to understand many of -the options, a rough familiarity with their work is quite helpful. - -However, in comparison to this framework, the implementation in -deal.II has to take into account the fact that we want to solve linear -systems on adaptively refined meshes. This leads to the complication -that it isn't quite as clear any more what exactly a "level" in a -multilevel hierarchy of a mesh is. The following image shows what we -consider to be a "level": - -

- @image html "hanging_nodes.png" "" -

- -In other words, the fine level in this mesh consists only of the -degrees of freedom that are defined on the refined cells, but does not -extend to that part of the domain that is not refined. While this -guarantees that the overall effort grows as ${\cal O}(N)$ as necessary -for optimal multigrid complexity, it leads to problems when defining -where to smooth and what boundary conditions to pose for the operators -defined on individual levels if the level boundary is not an external -boundary. These questions are discussed in detail in the -@ref mg_paper "Multigrid paper by Janssen and Kanschat" that describes -the implementation in deal.II. - - - -

The testcase

- -The problem we solve here is exactly the same as in -step-6, the only difference being the solver we use -here. You may want to look there for a definition of what we solve, -right hand side and boundary conditions. Obviously, the program would -also work if we changed the geometry and other pieces of data that -defines this particular problem. - -The things that are new are all those parts that concern the -multigrid. In particular, this includes the following members of the -main class: -- LaplaceProblem::mg_dof_handler -- LaplaceProblem::mg_sparsity -- LaplaceProblem::mg_matrices -- LaplaceProblem::mg_interface_matrices_up -- LaplaceProblem::assemble_multigrid () -- LaplaceProblem::solve () -Take a look at these functions. diff --git a/deal.II/examples/step-50/doc/kind b/deal.II/examples/step-50/doc/kind deleted file mode 100644 index c1d9154931..0000000000 --- a/deal.II/examples/step-50/doc/kind +++ /dev/null @@ -1 +0,0 @@ -techniques diff --git a/deal.II/examples/step-50/doc/results.dox b/deal.II/examples/step-50/doc/results.dox deleted file mode 100644 index 2d5fe0deb0..0000000000 --- a/deal.II/examples/step-50/doc/results.dox +++ /dev/null @@ -1,98 +0,0 @@ -

Results

- -The output that this program generates is, of course, the same as that -of step-6, so you may see there for more results. On the -other hand, since no tutorial program is a good one unless it has at -least one colorful picture, here is, again, the solution: - - -When run, the output of this program is -
-Cycle 0:
-   Number of active cells:       20
-   Number of degrees of freedom: 25 (by level: 8, 25)
-   7 CG iterations needed to obtain convergence.
-Cycle 1:
-   Number of active cells:       44
-   Number of degrees of freedom: 57 (by level: 8, 25, 48)
-   8 CG iterations needed to obtain convergence.
-Cycle 2:
-   Number of active cells:       92
-   Number of degrees of freedom: 117 (by level: 8, 25, 80, 60)
-   9 CG iterations needed to obtain convergence.
-Cycle 3:
-   Number of active cells:       188
-   Number of degrees of freedom: 221 (by level: 8, 25, 80, 200)
-   12 CG iterations needed to obtain convergence.
-Cycle 4:
-   Number of active cells:       416
-   Number of degrees of freedom: 485 (by level: 8, 25, 89, 288, 280)
-   13 CG iterations needed to obtain convergence.
-Cycle 5:
-   Number of active cells:       800
-   Number of degrees of freedom: 925 (by level: 8, 25, 89, 288, 784, 132)
-   14 CG iterations needed to obtain convergence.
-Cycle 6:
-   Number of active cells:       1628
-   Number of degrees of freedom: 1865 (by level: 8, 25, 89, 304, 1000, 1164, 72)
-   14 CG iterations needed to obtain convergence.
-Cycle 7:
-   Number of active cells:       3194
-   Number of degrees of freedom: 3603 (by level: 8, 25, 89, 328, 1032, 2200, 1392)
-   16 CG iterations needed to obtain convergence.
-
-That's not perfect — we would have hoped for a constant number -of iterations rather than one that increases as we get more and more -degrees of freedom — but it is also not far away. The reason for -this is easy enough to understand, however: since we have a strongly -varying coefficient, the operators that we assembly by quadrature on -the lower levels become worse and worse approximations of the operator -on the finest level. Consequently, even if we had perfect solvers on -the coarser levels, they would not be good preconditioners on the -finest level. This theory is easily tested by comparing results when -we use a constant coefficient: in that case, the number of iterations -remains constant at 9 after the first three or four refinement steps. - -We can also compare what this program produces with how @ref step_5 -"step-5" performed. To solve the same problem as in step-5, the only -two changes that are necessary are (i) to replace the body of the -function LaplaceProblem::refine_grid by a call to -triangulation.refine_global(1), and (ii) to use the same -SolverControl object and tolerance as in step-5 — the rest of the -program remains unchanged. In that case, here is how the solvers used -in step-5 and the multigrid solver used in the current program -compare: - - - - - - - - -
cellsstep-5step-16
20 13 6
80 17 7
320 29 9
1280 51 10
5120 94 11
2048018013
-This isn't only fewer iterations than in step-5 (each of which -is, however, much more expensive) but more importantly, the number of -iterations also grows much more slowly under mesh refinement (again, -it would be almost constant if the coefficient was constant rather -than strongly varying as chosen here). This justifies the common -observation that, whenever possible, multigrid methods should be used -for second order problems. - - -

Possible extensions

- -A close inspection of this program's performance shows that it is mostly -dominated by matrix-vector operations. step-37 shows one way -how this can be avoided by working with matrix-free methods. - -Another avenue would be to use algebraic multigrid methods. The -geometric multigrid method used here can at times be a bit awkward to -implement because it needs all those additional data structures, and -it becomes even more difficult if the program is to run in %parallel on -machines coupled through MPI, for example. In that case, it would be -simpler if one could use a black-box preconditioner that uses some -sort of multigrid hierarchy for good performance but can figure out -level matrices and similar things out by itself. Algebraic multigrid -methods do exactly this, and we will use them in -step-31 for the solution of a Stokes problem. diff --git a/deal.II/examples/step-50/doc/tooltip b/deal.II/examples/step-50/doc/tooltip deleted file mode 100644 index 2fd65590a0..0000000000 --- a/deal.II/examples/step-50/doc/tooltip +++ /dev/null @@ -1 +0,0 @@ -Multigrid on adaptive meshes. diff --git a/deal.II/examples/step-50/step-50.cc b/deal.II/examples/step-50/step-50.cc deleted file mode 100644 index 640c4b38be..0000000000 --- a/deal.II/examples/step-50/step-50.cc +++ /dev/null @@ -1,1052 +0,0 @@ -/* Author: Guido Kanschat, University of Heidelberg, 2003 */ -/* Baerbel Janssen, University of Heidelberg, 2010 */ -/* Wolfgang Bangerth, Texas A&M University, 2010 */ - -/* $Id$ */ -/* */ -/* Copyright (C) 2003, 2004, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 by the deal.II authors */ -/* */ -/* This file is subject to QPL and may not be distributed */ -/* without copyright and license information. Please refer */ -/* to the file deal.II/doc/license.html for the text and */ -/* further information on this license. */ - -// As discussed in the introduction, most of -// this program is copied almost verbatim -// from step-6, which itself is only a slight -// modification of step-5. Consequently, a -// significant part of this program is not -// new if you've read all the material up to -// step-6, and we won't comment on that part -// of the functionality that is -// unchanged. Rather, we will focus on those -// aspects of the program that have to do -// with the multigrid functionality which -// forms the new aspect of this tutorial -// program. - -// @sect3{Include files} - -// Again, the first few include files -// are already known, so we won't -// comment on them: -#include -#include -#include -#include - -#include -#include -#include -#include -#include -#include -#include - -#include -#include -#include -#include -#include -#include - -#include -#include - -#include -#include - -#include -#include -#include - -#include -#include -#include - -#include -#include -#include - -// These, now, are the include necessary for -// the multi-level methods. The first two -// declare classes that allow us to enumerate -// degrees of freedom not only on the finest -// mesh level, but also on intermediate -// levels (that's what the MGDoFHandler class -// does) as well as allow to access this -// information (iterators and accessors over -// these cells). -// -// The rest of the include files deals with -// the mechanics of multigrid as a linear -// operator (solver or preconditioner). -#include -#include -#include -#include -#include -#include -#include -#include - -// This is C++: -#include -#include - -// The last step is as in all -// previous programs: -namespace Step50 -{ - using namespace dealii; - - - // @sect3{The LaplaceProblem class template} - - // This main class is basically the same - // class as in step-6. As far as member - // functions is concerned, the only addition - // is the assemble_multigrid - // function that assembles the matrices that - // correspond to the discrete operators on - // intermediate levels: - template - class LaplaceProblem - { - public: - LaplaceProblem (const unsigned int deg); - void run (); - - private: - void setup_system (); - void assemble_system (); - void assemble_multigrid (); - void solve (); - void refine_grid (); - void output_results (const unsigned int cycle) const; - - parallel::distributed::Triangulation triangulation; - FE_Q fe; - DoFHandler mg_dof_handler; - - typedef TrilinosWrappers::SparseMatrix matrix_t; - typedef TrilinosWrappers::MPI::Vector vector_t; - - SparsityPattern sparsity_pattern; - matrix_t system_matrix; - - // We need an additional object for the - // hanging nodes constraints. They are - // handed to the transfer object in the - // multigrid. Since we call a compress - // inside the multigrid these constraints - // are not allowed to be inhomogeneous so - // we store them in different ConstraintMatrix - // objects. - ConstraintMatrix hanging_node_constraints; - ConstraintMatrix constraints; - - vector_t solution; - vector_t system_rhs; - - const unsigned int degree; - - // The following four objects are the - // only additional member variables, - // compared to step-6. They first three - // represent the - // operators that act on individual - // levels of the multilevel hierarchy, - // rather than on the finest mesh as do - // the objects above while the last object - // stores information about the boundary - // indices on each level and information - // about indices lying on a refinement - // edge between two different refinement - // levels. - // - // To facilitate having objects on each - // level of a multilevel hierarchy, - // deal.II has the MGLevelObject class - // template that provides storage for - // objects on each level. What we need - // here are matrices on each level, which - // implies that we also need sparsity - // patterns on each level. As outlined in - // the @ref mg_paper, the operators - // (matrices) that we need are actually - // twofold: one on the interior of each - // level, and one at the interface - // between each level and that part of - // the domain where the mesh is - // coarser. In fact, we will need the - // latter in two versions: for the - // direction from coarse to fine mesh and - // from fine to coarse. Fortunately, - // however, we here have a self-adjoint - // problem for which one of these is the - // transpose of the other, and so we only - // have to build one; we choose the one - // from coarse to fine. - MGLevelObject mg_sparsity_patterns; - MGLevelObject mg_matrices; - MGLevelObject mg_interface_matrices; - MGConstrainedDoFs mg_constrained_dofs; - }; - - - - // @sect3{Nonconstant coefficients} - - // The implementation of nonconstant - // coefficients is copied verbatim - // from step-5 and step-6: - - template - class Coefficient : public Function - { - public: - Coefficient () : Function() {} - - virtual double value (const Point &p, - const unsigned int component = 0) const; - - virtual void value_list (const std::vector > &points, - std::vector &values, - const unsigned int component = 0) const; - }; - - - - template - double Coefficient::value (const Point &p, - const unsigned int) const - { - if (p.square() < 0.5*0.5) - return 20; - else - return 1; - } - - - - template - void Coefficient::value_list (const std::vector > &points, - std::vector &values, - const unsigned int component) const - { - const unsigned int n_points = points.size(); - - Assert (values.size() == n_points, - ExcDimensionMismatch (values.size(), n_points)); - - Assert (component == 0, - ExcIndexRange (component, 0, 1)); - - for (unsigned int i=0; i::value (points[i]); - } - - - // @sect3{The LaplaceProblem class implementation} - - // @sect4{LaplaceProblem::LaplaceProblem} - - // The constructor is left mostly - // unchanged. We take the polynomial degree - // of the finite elements to be used as a - // constructor argument and store it in a - // member variable. - // - // By convention, all adaptively refined - // triangulations in deal.II never change by - // more than one level across a face between - // cells. For our multigrid algorithms, - // however, we need a slightly stricter - // guarantee, namely that the mesh also does - // not change by more than refinement level - // across vertices that might connect two - // cells. In other words, we must prevent the - // following situation: - // - // @image html limit_level_difference_at_vertices.png "" - // - // This is achieved by passing the - // Triangulation::limit_level_difference_at_vertices - // flag to the constructor of the - // triangulation class. - template - LaplaceProblem::LaplaceProblem (const unsigned int degree) - : - triangulation (MPI_COMM_WORLD,Triangulation:: - limit_level_difference_at_vertices), - fe (degree), - mg_dof_handler (triangulation), - degree(degree) - {} - - - - // @sect4{LaplaceProblem::setup_system} - - // The following function extends what the - // corresponding one in step-6 did. The top - // part, apart from the additional output, - // does the same: - template - void LaplaceProblem::setup_system () - { - mg_dof_handler.distribute_dofs (fe); - mg_dof_handler.distribute_mg_dofs (fe); - - - // Here we output not only the - // degrees of freedom on the finest - // level, but also in the - // multilevel structure - deallog << "Number of degrees of freedom: " - << mg_dof_handler.n_dofs(); - - for (unsigned int l=0; l::type dirichlet_boundary; - ZeroFunction homogeneous_dirichlet_bc (1); - dirichlet_boundary[0] = &homogeneous_dirichlet_bc; - VectorTools::interpolate_boundary_values (static_cast&>(mg_dof_handler), - dirichlet_boundary, - constraints); - constraints.close (); - hanging_node_constraints.close (); - constraints.condense (sparsity_pattern); - sparsity_pattern.compress(); - system_matrix.reinit (sparsity_pattern); - - // The multigrid constraints have to be - // initialized. They need to know about - // the boundary values as well, so we - // pass the dirichlet_boundary - // here as well. - mg_constrained_dofs.clear(); - mg_constrained_dofs.initialize(mg_dof_handler, dirichlet_boundary); - - - // Now for the things that concern the - // multigrid data structures. First, we - // resize the multi-level objects to hold - // matrices and sparsity patterns for every - // level. The coarse level is zero (this is - // mandatory right now but may change in a - // future revision). Note that these - // functions take a complete, inclusive - // range here (not a starting index and - // size), so the finest level is - // n_levels-1. We first have - // to resize the container holding the - // SparseMatrix classes, since they have to - // release their SparsityPattern before the - // can be destroyed upon resizing. - const unsigned int n_levels = triangulation.n_levels(); - - mg_interface_matrices.resize(0, n_levels-1); - mg_interface_matrices.clear (); - mg_matrices.resize(0, n_levels-1); - mg_matrices.clear (); - mg_sparsity_patterns.resize(0, n_levels-1); - - // Now, we have to provide a matrix on each - // level. To this end, we first use the - // MGTools::make_sparsity_pattern function - // to first generate a preliminary - // compressed sparsity pattern on each - // level (see the @ref Sparsity module for - // more information on this topic) and then - // copy it over to the one we really - // want. The next step is to initialize - // both kinds of level matrices with these - // sparsity patterns. - // - // It may be worth pointing out that the - // interface matrices only have entries for - // degrees of freedom that sit at or next - // to the interface between coarser and - // finer levels of the mesh. They are - // therefore even sparser than the matrices - // on the individual levels of our - // multigrid hierarchy. If we were more - // concerned about memory usage (and - // possibly the speed with which we can - // multiply with these matrices), we should - // use separate and different sparsity - // patterns for these two kinds of - // matrices. - for (unsigned int level=0; level - void LaplaceProblem::assemble_system () - { - const QGauss quadrature_formula(degree+1); - - FEValues fe_values (fe, quadrature_formula, - update_values | update_gradients | - update_quadrature_points | update_JxW_values); - - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_q_points = quadrature_formula.size(); - - FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); - Vector cell_rhs (dofs_per_cell); - - std::vector local_dof_indices (dofs_per_cell); - - const Coefficient coefficient; - std::vector coefficient_values (n_q_points); - - typename DoFHandler::active_cell_iterator - cell = mg_dof_handler.begin_active(), - endc = mg_dof_handler.end(); - for (; cell!=endc; ++cell) - if (cell->is_locally_owned()) - { - cell_matrix = 0; - cell_rhs = 0; - - fe_values.reinit (cell); - - coefficient.value_list (fe_values.get_quadrature_points(), - coefficient_values); - - for (unsigned int q_point=0; q_pointget_dof_indices (local_dof_indices); - constraints.distribute_local_to_global (cell_matrix, cell_rhs, - local_dof_indices, - system_matrix, system_rhs); - } - } - - - // @sect4{LaplaceProblem::assemble_multigrid} - - // The next function is the one that builds - // the linear operators (matrices) that - // define the multigrid method on each level - // of the mesh. The integration core is the - // same as above, but the loop below will go - // over all existing cells instead of just - // the active ones, and the results must be - // entered into the correct matrix. Note also - // that since we only do multi-level - // preconditioning, no right-hand side needs - // to be assembled here. - // - // Before we go there, however, we have to - // take care of a significant amount of book - // keeping: - template - void LaplaceProblem::assemble_multigrid () - { - QGauss quadrature_formula(1+degree); - - FEValues fe_values (fe, quadrature_formula, - update_values | update_gradients | - update_quadrature_points | update_JxW_values); - - const unsigned int dofs_per_cell = fe.dofs_per_cell; - const unsigned int n_q_points = quadrature_formula.size(); - - FullMatrix cell_matrix (dofs_per_cell, dofs_per_cell); - - std::vector local_dof_indices (dofs_per_cell); - - const Coefficient coefficient; - std::vector coefficient_values (n_q_points); - - // Next a few things that are specific to - // building the multigrid data structures - // (since we only need them in the current - // function, rather than also elsewhere, we - // build them here instead of the - // setup_system - // function). Some of the following may be - // a bit obscure if you're not familiar - // with the algorithm actually implemented - // in deal.II to support multilevel - // algorithms on adaptive meshes; if some - // of the things below seem strange, take a - // look at the @ref mg_paper. - // - // Our first job is to identify those - // degrees of freedom on each level that - // are located on interfaces between - // adaptively refined levels, and those - // that lie on the interface but also on - // the exterior boundary of the domain. As - // in many other parts of the library, we - // do this by using boolean masks, - // i.e. vectors of booleans each element of - // which indicates whether the - // corresponding degree of freedom index is - // an interface DoF or not. The MGConstraints - // already computed the information for us - // when we called initialize in setup_system(). - std::vector > interface_dofs - = mg_constrained_dofs.get_refinement_edge_indices (); - std::vector > boundary_interface_dofs - = mg_constrained_dofs.get_refinement_edge_boundary_indices (); - - // The indices just identified will later - // be used to decide where the assembled value - // has to be added into on each level. - // On the other hand, - // we also have to impose zero boundary - // conditions on the external boundary of - // each level. But this the MGConstraints - // knows it. So we simply ask for them by calling - // get_boundary_indices (). - // The third step is to construct - // constraints on all those degrees of - // freedom: their value should be zero - // after each application of the level - // operators. To this end, we construct - // ConstraintMatrix objects for each level, - // and add to each of these constraints for - // each degree of freedom. Due to the way - // the ConstraintMatrix stores its data, - // the function to add a constraint on a - // single degree of freedom and force it to - // be zero is called - // Constraintmatrix::add_line(); doing so - // for several degrees of freedom at once - // can be done using - // Constraintmatrix::add_lines(): - std::vector boundary_constraints (triangulation.n_levels()); - std::vector boundary_interface_constraints (triangulation.n_levels()); - for (unsigned int level=0; levelassemble_system, with two - // exceptions: (i) we don't need a right - // hand side, and more significantly (ii) we - // don't just loop over all active cells, - // but in fact all cells, active or - // not. Consequently, the correct iterator - // to use is MGDoFHandler::cell_iterator - // rather than - // MGDoFHandler::active_cell_iterator. Let's - // go about it: - typename DoFHandler::cell_iterator cell = mg_dof_handler.begin(), - endc = mg_dof_handler.end(); - - for (; cell!=endc; ++cell) - if (cell->level_subdomain_id()==triangulation.locally_owned_subdomain()) - { - cell_matrix = 0; - fe_values.reinit (cell); - - coefficient.value_list (fe_values.get_quadrature_points(), - coefficient_values); - - for (unsigned int q_point=0; q_pointget_mg_dof_indices (local_dof_indices); - - // Next, we need to copy local - // contributions into the level - // objects. We can do this in the same - // way as in the global assembly, using - // a constraint object that takes care - // of constrained degrees (which here - // are only boundary nodes, as the - // individual levels have no hanging - // node constraints). Note that the - // boundary_constraints - // object makes sure that the level - // matrices contains no contributions - // from degrees of freedom at the - // interface between cells of different - // refinement level. - boundary_constraints[cell->level()] - .distribute_local_to_global (cell_matrix, - local_dof_indices, - mg_matrices[cell->level()]); - - // The next step is again slightly more - // obscure (but explained in the @ref - // mg_paper): We need the remainder of - // the operator that we just copied - // into the mg_matrices - // object, namely the part on the - // interface between cells at the - // current level and cells one level - // coarser. This matrix exists in two - // directions: for interior DoFs (index - // $i$) of the current level to those - // sitting on the interface (index - // $j$), and the other way around. Of - // course, since we have a symmetric - // operator, one of these matrices is - // the transpose of the other. - // - // The way we assemble these matrices - // is as follows: since the are formed - // from parts of the local - // contributions, we first delete all - // those parts of the local - // contributions that we are not - // interested in, namely all those - // elements of the local matrix for - // which not $i$ is an interface DoF - // and $j$ is not. The result is one of - // the two matrices that we are - // interested in, and we then copy it - // into the - // mg_interface_matrices - // object. The - // boundary_interface_constraints - // object at the same time makes sure - // that we delete contributions from - // all degrees of freedom that are not - // only on the interface but also on - // the external boundary of the domain. - // - // The last part to remember is how to - // get the other matrix. Since it is - // only the transpose, we will later - // (in the solve() - // function) be able to just pass the - // transpose matrix where necessary. - for (unsigned int i=0; ilevel()][local_dof_indices[i]]==true && - interface_dofs[cell->level()][local_dof_indices[j]]==false)) - cell_matrix(i,j) = 0; - - boundary_interface_constraints[cell->level()] - .distribute_local_to_global (cell_matrix, - local_dof_indices, - mg_interface_matrices[cell->level()]); - } - } - - - - // @sect4{LaplaceProblem::solve} - - // This is the other function that is - // significantly different in support of the - // multigrid solver (or, in fact, the - // preconditioner for which we use the - // multigrid method). - // - // Let us start out by setting up two of the - // components of multilevel methods: transfer - // operators between levels, and a solver on - // the coarsest level. In finite element - // methods, the transfer operators are - // derived from the finite element function - // spaces involved and can often be computed - // in a generic way independent of the - // problem under consideration. In that case, - // we can use the MGTransferPrebuilt class - // that, given the constraints on the global - // level and an MGDoFHandler object computes - // the matrices corresponding to these - // transfer operators. - // - // The second part of the following lines - // deals with the coarse grid solver. Since - // our coarse grid is very coarse indeed, we - // decide for a direct solver (a Householder - // decomposition of the coarsest level - // matrix), even if its implementation is not - // particularly sophisticated. If our coarse - // mesh had many more cells than the five we - // have here, something better suited would - // obviously be necessary here. - template - void LaplaceProblem::solve () - { - - // Create the object that deals with the transfer - // between different refinement levels. We need to - // pass it the hanging node constraints. - MGTransferPrebuilt mg_transfer(hanging_node_constraints, mg_constrained_dofs); - // Now the prolongation matrix has to be built. - // This matrix needs to take the boundary values on - // each level into account and needs to know about - // the indices at the refinement egdes. The - // MGConstraints knows about that so - // pass it as an argument. - mg_transfer.build_matrices(mg_dof_handler); - - matrix_t coarse_matrix; - coarse_matrix.copy_from (mg_matrices[0]); - //MGCoarseGridHouseholder coarse_grid_solver; - //coarse_grid_solver.initialize (coarse_matrix); - - SolverControl coarse_solver_control (1000, 1e-10, false, false); - SolverGMRES coarse_solver(coarse_solver_control); - PreconditionIdentity id; - MGCoarseGridLACIteration,vector_t> coarse_grid_solver(coarse_solver, - coarse_matrix, - id); - - // The next component of a multilevel - // solver or preconditioner is that we need - // a smoother on each level. A common - // choice for this is to use the - // application of a relaxation method (such - // as the SOR, Jacobi or Richardson method). The - // MGSmootherPrecondition class provides - // support for this kind of - // smoother. Here, we opt for the - // application of a single SOR - // iteration. To this end, we define an - // appropriate typedef and - // then setup a smoother object. - // - // The last step is to initialize the - // smoother object with our level matrices - // and to set some smoothing parameters. - // The initialize() function - // can optionally take additional arguments - // that will be passed to the smoother - // object on each level. In the current - // case for the SOR smoother, this could, - // for example, include a relaxation - // parameter. However, we here leave these - // at their default values. The call to - // set_steps() indicates that - // we will use two pre- and two - // post-smoothing steps on each level; to - // use a variable number of smoother steps - // on different levels, more options can be - // set in the constructor call to the - // mg_smoother object. - // - // The last step results from the fact that - // we use the SOR method as a smoother - - // which is not symmetric - but we use the - // conjugate gradient iteration (which - // requires a symmetric preconditioner) - // below, we need to let the multilevel - // preconditioner make sure that we get a - // symmetric operator even for nonsymmetric - // smoothers: - typedef TrilinosWrappers::PreconditionSOR Smoother; - MGSmootherPrecondition mg_smoother; - mg_smoother.initialize(mg_matrices); - mg_smoother.set_steps(2); - mg_smoother.set_symmetric(true); - - // The next preparatory step is that we - // must wrap our level and interface - // matrices in an object having the - // required multiplication functions. We - // will create two objects for the - // interface objects going from coarse to - // fine and the other way around; the - // multigrid algorithm will later use the - // transpose operator for the latter - // operation, allowing us to initialize - // both up and down versions of the - // operator with the matrices we already - // built: - MGMatrix mg_matrix(&mg_matrices); - MGMatrix mg_interface_up(&mg_interface_matrices); - MGMatrix mg_interface_down(&mg_interface_matrices); - - // Now, we are ready to set up the - // V-cycle operator and the - // multilevel preconditioner. - Multigrid mg(mg_dof_handler, - mg_matrix, - coarse_grid_solver, - mg_transfer, - mg_smoother, - mg_smoother); - mg.set_edge_matrices(mg_interface_down, mg_interface_up); - - PreconditionMG > - preconditioner(mg_dof_handler, mg, mg_transfer); - - // With all this together, we can finally - // get about solving the linear system in - // the usual way: - SolverControl solver_control (1000, 1e-12); - SolverCG cg (solver_control); - - solution = 0; - - cg.solve (system_matrix, solution, system_rhs, - preconditioner); - constraints.distribute (solution); - } - - - - // @sect4{Postprocessing} - - // The following two functions postprocess a - // solution once it is computed. In - // particular, the first one refines the mesh - // at the beginning of each cycle while the - // second one outputs results at the end of - // each such cycle. The functions are almost - // unchanged from those in step-6, with the - // exception of two minor differences: The - // KellyErrorEstimator::estimate function - // wants an argument of type DoFHandler, not - // MGDoFHandler, and so we have to cast from - // derived to base class; and we generate - // output in VTK format, to use the more - // modern visualization programs available - // today compared to those that were - // available when step-6 was written. - template - void LaplaceProblem::refine_grid () - { - Vector estimated_error_per_cell (triangulation.n_active_cells()); - - KellyErrorEstimator::estimate (static_cast&>(mg_dof_handler), - QGauss(3), - typename FunctionMap::type(), - solution, - estimated_error_per_cell); - GridRefinement::refine_and_coarsen_fixed_number (triangulation, - estimated_error_per_cell, - 0.3, 0.03); - triangulation.execute_coarsening_and_refinement (); - } - - - - template - void LaplaceProblem::output_results (const unsigned int cycle) const - { - DataOut data_out; - - data_out.attach_dof_handler (mg_dof_handler); - data_out.add_data_vector (solution, "solution"); - data_out.build_patches (); - - std::ostringstream filename; - filename << "solution-" - << cycle - << ".vtk"; - - std::ofstream output (filename.str().c_str()); - data_out.write_vtk (output); - } - - - // @sect4{LaplaceProblem::run} - - // Like several of the functions above, this - // is almost exactly a copy of of the - // corresponding function in step-6. The only - // difference is the call to - // assemble_multigrid that takes - // care of forming the matrices on every - // level that we need in the multigrid - // method. - template - void LaplaceProblem::run () - { - for (unsigned int cycle=0; cycle<8; ++cycle) - { - deallog << "Cycle " << cycle << ':' << std::endl; - - if (cycle == 0) - { - GridGenerator::hyper_ball (triangulation); - - static const HyperBallBoundary boundary; - triangulation.set_boundary (0, boundary); - - triangulation.refine_global (1); - } - else - refine_grid (); - - - deallog << " Number of active cells: " - << triangulation.n_active_cells() - << std::endl; - - setup_system (); - - deallog << " Number of degrees of freedom: " - << mg_dof_handler.n_dofs() - << " (by level: "; - for (unsigned int level=0; level laplace_problem(1); - laplace_problem.run (); - } - catch (std::exception &exc) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Exception on processing: " << std::endl - << exc.what() << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - - return 1; - } - catch (...) - { - std::cerr << std::endl << std::endl - << "----------------------------------------------------" - << std::endl; - std::cerr << "Unknown exception!" << std::endl - << "Aborting!" << std::endl - << "----------------------------------------------------" - << std::endl; - return 1; - } - - return 0; -} -- 2.39.5