diff --git a/configure b/configure index 00a058dc01..3b6b7ba3ba 100755 --- a/configure +++ b/configure @@ -74647,7 +74647,7 @@ printf "%s\n" "#define BUILD_DATE __DATE__ \" \" __TIME__" >>confdefs.h if test "x$enableexamples" = "xyes" then : - ac_config_files="$ac_config_files examples/introduction/introduction_ex1/Makefile examples/introduction/introduction_ex2/Makefile examples/introduction/introduction_ex3/Makefile examples/introduction/introduction_ex4/Makefile examples/introduction/introduction_ex5/Makefile examples/adaptivity/adaptivity_ex1/Makefile examples/adaptivity/adaptivity_ex2/Makefile examples/adaptivity/adaptivity_ex3/Makefile examples/adaptivity/adaptivity_ex4/Makefile examples/adaptivity/adaptivity_ex5/Makefile examples/adjoints/adjoints_ex1/Makefile examples/adjoints/adjoints_ex2/Makefile examples/adjoints/adjoints_ex3/Makefile examples/adjoints/adjoints_ex4/Makefile examples/adjoints/adjoints_ex5/Makefile examples/adjoints/adjoints_ex6/Makefile examples/adjoints/adjoints_ex7/Makefile examples/eigenproblems/eigenproblems_ex1/Makefile examples/eigenproblems/eigenproblems_ex2/Makefile examples/eigenproblems/eigenproblems_ex3/Makefile examples/eigenproblems/eigenproblems_ex4/Makefile examples/fem_system/fem_system_ex1/Makefile examples/fem_system/fem_system_ex2/Makefile examples/fem_system/fem_system_ex3/Makefile examples/fem_system/fem_system_ex4/Makefile examples/fem_system/fem_system_ex5/Makefile examples/fem_system/fem_system_ex6/Makefile examples/solution_transfer/solution_transfer_ex1/Makefile examples/miscellaneous/miscellaneous_ex1/Makefile examples/miscellaneous/miscellaneous_ex2/Makefile examples/miscellaneous/miscellaneous_ex3/Makefile examples/miscellaneous/miscellaneous_ex4/Makefile examples/miscellaneous/miscellaneous_ex5/Makefile examples/miscellaneous/miscellaneous_ex6/Makefile examples/miscellaneous/miscellaneous_ex7/Makefile examples/miscellaneous/miscellaneous_ex8/Makefile examples/miscellaneous/miscellaneous_ex9/Makefile examples/miscellaneous/miscellaneous_ex10/Makefile examples/miscellaneous/miscellaneous_ex11/Makefile examples/miscellaneous/miscellaneous_ex12/Makefile examples/miscellaneous/miscellaneous_ex13/Makefile examples/miscellaneous/miscellaneous_ex14/Makefile examples/miscellaneous/miscellaneous_ex15/Makefile examples/miscellaneous/miscellaneous_ex16/Makefile examples/miscellaneous/miscellaneous_ex17/Makefile examples/optimization/optimization_ex1/Makefile examples/optimization/optimization_ex2/Makefile examples/subdomains/subdomains_ex1/Makefile examples/subdomains/subdomains_ex2/Makefile examples/subdomains/subdomains_ex3/Makefile examples/systems_of_equations/systems_of_equations_ex1/Makefile examples/systems_of_equations/systems_of_equations_ex2/Makefile examples/systems_of_equations/systems_of_equations_ex3/Makefile examples/systems_of_equations/systems_of_equations_ex4/Makefile examples/systems_of_equations/systems_of_equations_ex5/Makefile examples/systems_of_equations/systems_of_equations_ex6/Makefile examples/systems_of_equations/systems_of_equations_ex7/Makefile examples/systems_of_equations/systems_of_equations_ex8/Makefile examples/systems_of_equations/systems_of_equations_ex9/Makefile examples/reduced_basis/reduced_basis_ex1/Makefile examples/reduced_basis/reduced_basis_ex2/Makefile examples/reduced_basis/reduced_basis_ex3/Makefile examples/reduced_basis/reduced_basis_ex4/Makefile examples/reduced_basis/reduced_basis_ex5/Makefile examples/reduced_basis/reduced_basis_ex6/Makefile examples/reduced_basis/reduced_basis_ex7/Makefile examples/transient/transient_ex1/Makefile examples/transient/transient_ex2/Makefile examples/transient/transient_ex3/Makefile examples/vector_fe/vector_fe_ex1/Makefile examples/vector_fe/vector_fe_ex2/Makefile examples/vector_fe/vector_fe_ex3/Makefile examples/vector_fe/vector_fe_ex4/Makefile examples/vector_fe/vector_fe_ex5/Makefile examples/vector_fe/vector_fe_ex6/Makefile examples/vector_fe/vector_fe_ex7/Makefile examples/vector_fe/vector_fe_ex8/Makefile examples/vector_fe/vector_fe_ex9/Makefile examples/vector_fe/vector_fe_ex10/Makefile examples/Makefile" + ac_config_files="$ac_config_files examples/introduction/introduction_ex1/Makefile examples/introduction/introduction_ex2/Makefile examples/introduction/introduction_ex3/Makefile examples/introduction/introduction_ex4/Makefile examples/introduction/introduction_ex5/Makefile examples/adaptivity/adaptivity_ex1/Makefile examples/adaptivity/adaptivity_ex2/Makefile examples/adaptivity/adaptivity_ex3/Makefile examples/adaptivity/adaptivity_ex4/Makefile examples/adaptivity/adaptivity_ex5/Makefile examples/adjoints/adjoints_ex1/Makefile examples/adjoints/adjoints_ex2/Makefile examples/adjoints/adjoints_ex3/Makefile examples/adjoints/adjoints_ex4/Makefile examples/adjoints/adjoints_ex5/Makefile examples/adjoints/adjoints_ex6/Makefile examples/adjoints/adjoints_ex7/Makefile examples/eigenproblems/eigenproblems_ex1/Makefile examples/eigenproblems/eigenproblems_ex2/Makefile examples/eigenproblems/eigenproblems_ex3/Makefile examples/eigenproblems/eigenproblems_ex4/Makefile examples/fem_system/fem_system_ex1/Makefile examples/fem_system/fem_system_ex2/Makefile examples/fem_system/fem_system_ex3/Makefile examples/fem_system/fem_system_ex4/Makefile examples/fem_system/fem_system_ex5/Makefile examples/fem_system/fem_system_ex6/Makefile examples/finite_volume/finite_volume_ex1/Makefile examples/solution_transfer/solution_transfer_ex1/Makefile examples/miscellaneous/miscellaneous_ex1/Makefile examples/miscellaneous/miscellaneous_ex2/Makefile examples/miscellaneous/miscellaneous_ex3/Makefile examples/miscellaneous/miscellaneous_ex4/Makefile examples/miscellaneous/miscellaneous_ex5/Makefile examples/miscellaneous/miscellaneous_ex6/Makefile examples/miscellaneous/miscellaneous_ex7/Makefile examples/miscellaneous/miscellaneous_ex8/Makefile examples/miscellaneous/miscellaneous_ex9/Makefile examples/miscellaneous/miscellaneous_ex10/Makefile examples/miscellaneous/miscellaneous_ex11/Makefile examples/miscellaneous/miscellaneous_ex12/Makefile examples/miscellaneous/miscellaneous_ex13/Makefile examples/miscellaneous/miscellaneous_ex14/Makefile examples/miscellaneous/miscellaneous_ex15/Makefile examples/miscellaneous/miscellaneous_ex16/Makefile examples/miscellaneous/miscellaneous_ex17/Makefile examples/optimization/optimization_ex1/Makefile examples/optimization/optimization_ex2/Makefile examples/subdomains/subdomains_ex1/Makefile examples/subdomains/subdomains_ex2/Makefile examples/subdomains/subdomains_ex3/Makefile examples/systems_of_equations/systems_of_equations_ex1/Makefile examples/systems_of_equations/systems_of_equations_ex2/Makefile examples/systems_of_equations/systems_of_equations_ex3/Makefile examples/systems_of_equations/systems_of_equations_ex4/Makefile examples/systems_of_equations/systems_of_equations_ex5/Makefile examples/systems_of_equations/systems_of_equations_ex6/Makefile examples/systems_of_equations/systems_of_equations_ex7/Makefile examples/systems_of_equations/systems_of_equations_ex8/Makefile examples/systems_of_equations/systems_of_equations_ex9/Makefile examples/reduced_basis/reduced_basis_ex1/Makefile examples/reduced_basis/reduced_basis_ex2/Makefile examples/reduced_basis/reduced_basis_ex3/Makefile examples/reduced_basis/reduced_basis_ex4/Makefile examples/reduced_basis/reduced_basis_ex5/Makefile examples/reduced_basis/reduced_basis_ex6/Makefile examples/reduced_basis/reduced_basis_ex7/Makefile examples/transient/transient_ex1/Makefile examples/transient/transient_ex2/Makefile examples/transient/transient_ex3/Makefile examples/vector_fe/vector_fe_ex1/Makefile examples/vector_fe/vector_fe_ex2/Makefile examples/vector_fe/vector_fe_ex3/Makefile examples/vector_fe/vector_fe_ex4/Makefile examples/vector_fe/vector_fe_ex5/Makefile examples/vector_fe/vector_fe_ex6/Makefile examples/vector_fe/vector_fe_ex7/Makefile examples/vector_fe/vector_fe_ex8/Makefile examples/vector_fe/vector_fe_ex9/Makefile examples/vector_fe/vector_fe_ex10/Makefile examples/Makefile" fi @@ -77133,6 +77133,7 @@ do "examples/fem_system/fem_system_ex4/Makefile") CONFIG_FILES="$CONFIG_FILES examples/fem_system/fem_system_ex4/Makefile" ;; "examples/fem_system/fem_system_ex5/Makefile") CONFIG_FILES="$CONFIG_FILES examples/fem_system/fem_system_ex5/Makefile" ;; "examples/fem_system/fem_system_ex6/Makefile") CONFIG_FILES="$CONFIG_FILES examples/fem_system/fem_system_ex6/Makefile" ;; + "examples/finite_volume/finite_volume_ex1/Makefile") CONFIG_FILES="$CONFIG_FILES examples/finite_volume/finite_volume_ex1/Makefile" ;; "examples/solution_transfer/solution_transfer_ex1/Makefile") CONFIG_FILES="$CONFIG_FILES examples/solution_transfer/solution_transfer_ex1/Makefile" ;; "examples/miscellaneous/miscellaneous_ex1/Makefile") CONFIG_FILES="$CONFIG_FILES examples/miscellaneous/miscellaneous_ex1/Makefile" ;; "examples/miscellaneous/miscellaneous_ex2/Makefile") CONFIG_FILES="$CONFIG_FILES examples/miscellaneous/miscellaneous_ex2/Makefile" ;; diff --git a/configure.ac b/configure.ac index 9ef5d6e502..5fe361d4ba 100644 --- a/configure.ac +++ b/configure.ac @@ -441,6 +441,7 @@ AS_IF([test "x$enableexamples" = "xyes"], examples/fem_system/fem_system_ex4/Makefile examples/fem_system/fem_system_ex5/Makefile examples/fem_system/fem_system_ex6/Makefile + examples/finite_volume/finite_volume_ex1/Makefile examples/solution_transfer/solution_transfer_ex1/Makefile examples/miscellaneous/miscellaneous_ex1/Makefile examples/miscellaneous/miscellaneous_ex2/Makefile diff --git a/examples/Makefile.am b/examples/Makefile.am index 9fd801d4ab..a1775284d0 100644 --- a/examples/Makefile.am +++ b/examples/Makefile.am @@ -26,6 +26,7 @@ SUBDIRS = \ fem_system/fem_system_ex4 \ fem_system/fem_system_ex5 \ fem_system/fem_system_ex6 \ + finite_volume/finite_volume_ex1 \ solution_transfer/solution_transfer_ex1 \ miscellaneous/miscellaneous_ex1 \ miscellaneous/miscellaneous_ex2 \ diff --git a/examples/Makefile.in b/examples/Makefile.in index 2ae42f4ea9..643453ea50 100644 --- a/examples/Makefile.in +++ b/examples/Makefile.in @@ -638,6 +638,7 @@ SUBDIRS = \ fem_system/fem_system_ex4 \ fem_system/fem_system_ex5 \ fem_system/fem_system_ex6 \ + finite_volume/finite_volume_ex1 \ solution_transfer/solution_transfer_ex1 \ miscellaneous/miscellaneous_ex1 \ miscellaneous/miscellaneous_ex2 \ diff --git a/examples/finite_volume/finite_volume_ex1/Makefile.am b/examples/finite_volume/finite_volume_ex1/Makefile.am new file mode 100644 index 0000000000..a6625a18a8 --- /dev/null +++ b/examples/finite_volume/finite_volume_ex1/Makefile.am @@ -0,0 +1,21 @@ +example_name = finite_volume_ex1 +check_SCRIPTS = run.sh +install_dir = $(examples_install_path)/finite_volume/ex1 +data = 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If we are careful, common +# rules like 'make doc' or 'make run' can be defined here, once and +# for all. +# +AUTOMAKE_OPTIONS = subdir-objects + +# We might have turned on -Werror and/or paranoid warnings. 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+@LIBMESH_DEVEL_MODE_TRUE@example_devel_LDADD = $(top_builddir)/libmesh_devel.la +@LIBMESH_OPT_MODE_TRUE@example_opt_SOURCES = $(sources) +@LIBMESH_OPT_MODE_TRUE@example_opt_CPPFLAGS = $(CPPFLAGS_OPT) $(AM_CPPFLAGS) +@LIBMESH_OPT_MODE_TRUE@example_opt_CXXFLAGS = $(CXXFLAGS_OPT) +@LIBMESH_OPT_MODE_TRUE@example_opt_LDADD = $(top_builddir)/libmesh_opt.la +@LIBMESH_PROF_MODE_TRUE@example_prof_SOURCES = $(sources) +@LIBMESH_PROF_MODE_TRUE@example_prof_CPPFLAGS = $(CPPFLAGS_PROF) $(AM_CPPFLAGS) +@LIBMESH_PROF_MODE_TRUE@example_prof_CXXFLAGS = $(CXXFLAGS_PROF) +@LIBMESH_PROF_MODE_TRUE@example_prof_LDADD = $(top_builddir)/libmesh_prof.la +@LIBMESH_OPROF_MODE_TRUE@example_oprof_SOURCES = $(sources) +@LIBMESH_OPROF_MODE_TRUE@example_oprof_CPPFLAGS = $(CPPFLAGS_OPROF) $(AM_CPPFLAGS) +@LIBMESH_OPROF_MODE_TRUE@example_oprof_CXXFLAGS = $(CXXFLAGS_OPROF) +@LIBMESH_OPROF_MODE_TRUE@example_oprof_LDADD = $(top_builddir)/libmesh_oprof.la +@LIBMESH_DBG_MODE_TRUE@@LIBMESH_DEVEL_MODE_FALSE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_PROGRAM = example-dbg +@LIBMESH_DEVEL_MODE_TRUE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_PROGRAM = example-devel +@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_TRUE@DOC_PROGRAM = example-prof +@LIBMESH_OPROF_MODE_TRUE@@LIBMESH_OPT_MODE_FALSE@DOC_PROGRAM = example-oprof + +###################################################################### +# +# Choose a flavor to run when making documentation +# +@LIBMESH_OPT_MODE_TRUE@DOC_PROGRAM = example-opt +@LIBMESH_DBG_MODE_TRUE@@LIBMESH_DEVEL_MODE_FALSE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_METHOD = dbg +@LIBMESH_DEVEL_MODE_TRUE@@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_FALSE@DOC_METHOD = devel +@LIBMESH_OPROF_MODE_FALSE@@LIBMESH_OPT_MODE_FALSE@@LIBMESH_PROF_MODE_TRUE@DOC_METHOD = prof +@LIBMESH_OPROF_MODE_TRUE@@LIBMESH_OPT_MODE_FALSE@DOC_METHOD = oprof +@LIBMESH_OPT_MODE_TRUE@DOC_METHOD = opt + +###################################################################### +# +# Running the tests +# +TESTS_ENVIRONMENT = LIBMESH_DIR=$(abs_top_srcdir); export LIBMESH_DIR; \ + METHODS="$(METHODS)"; export METHODS; petscmajor=@petscmajor@ \ + petscminor=@petscminor@ enableeigen=@enableeigen@ +TESTS = $(check_SCRIPTS) +all: $(BUILT_SOURCES) + $(MAKE) $(AM_MAKEFLAGS) all-am + +.SUFFIXES: +.SUFFIXES: .C .lo .o .obj +$(srcdir)/Makefile.in: @MAINTAINER_MODE_TRUE@ $(srcdir)/Makefile.am $(top_srcdir)/examples/Make.common $(am__configure_deps) + @for dep in $?; do \ + case '$(am__configure_deps)' in \ + *$$dep*) \ + ( cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh ) \ + && { if test -f $@; then exit 0; else break; fi; }; \ + exit 1;; \ + esac; \ + done; \ + echo ' cd $(top_srcdir) && $(AUTOMAKE) --gnu examples/finite_volume/finite_volume_ex1/Makefile'; \ + $(am__cd) $(top_srcdir) && \ + $(AUTOMAKE) --gnu examples/finite_volume/finite_volume_ex1/Makefile +Makefile: $(srcdir)/Makefile.in $(top_builddir)/config.status + @case '$?' in \ + *config.status*) \ + cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh;; \ + *) \ + echo ' cd $(top_builddir) && $(SHELL) ./config.status $(subdir)/$@ $(am__maybe_remake_depfiles)'; \ + cd $(top_builddir) && $(SHELL) ./config.status $(subdir)/$@ $(am__maybe_remake_depfiles);; \ + esac; +$(top_srcdir)/examples/Make.common $(am__empty): + +$(top_builddir)/config.status: $(top_srcdir)/configure $(CONFIG_STATUS_DEPENDENCIES) + cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh + +$(top_srcdir)/configure: @MAINTAINER_MODE_TRUE@ $(am__configure_deps) + cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh +$(ACLOCAL_M4): @MAINTAINER_MODE_TRUE@ $(am__aclocal_m4_deps) + cd $(top_builddir) && $(MAKE) $(AM_MAKEFLAGS) am--refresh +$(am__aclocal_m4_deps): + +clean-checkPROGRAMS: + $(am__rm_f) $(check_PROGRAMS) 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a/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.C b/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.C new file mode 100644 index 0000000000..56d48bec24 --- /dev/null +++ b/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.C @@ -0,0 +1,425 @@ +// The libMesh Finite Element Library. +// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner + +// This library is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 2.1 of the License, or (at your option) any later version. + +// This library is distributed in the hope that it will be useful, +// but WITHOUT ANY WARRANTY; without even the implied warranty of +// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +// Lesser General Public License for more details. + +// You should have received a copy of the GNU Lesser General Public +// License along with this library; if not, write to the Free Software +// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + +//

Finite Volume Example 1 - Cell-Centered Advection-Diffusion on Polygonal Meshes

+// \author Alexander Lindsay +// \date 2026 +// +// This example solves the steady advection-diffusion equation +// +// -div(D grad u) + div(v u) = f +// +// on the unit square with a cell-centered finite volume method on a +// mesh of C0Polygon elements. The solution is a CONSTANT MONOMIAL +// variable, so each cell carries a single degree of freedom, +// interpreted as the solution value at the cell centroid. +// +// Integrating the equation over a cell and applying the divergence +// theorem gives a balance of fluxes through the cell faces. Cell +// gradients, reconstructed by least squares from the face neighbors +// and the boundary data, enter these fluxes in two places: +// +// * The diffusive flux uses a two-point difference along the vector +// joining adjacent cell centroids, plus a correction for the angle +// between that vector and the face normal. On this mesh the two +// are not parallel across the slanted hexagon faces, and without +// the correction the scheme would not converge to the exact +// solution. +// +// * The advective flux is upwinded, with the upwind cell value +// linearly extrapolated to the face centroid, giving second-order +// accuracy. +// +// A face flux therefore depends on the two cells sharing the face and +// on their face neighbors, so the matrix row of a cell couples it to +// two layers of side neighbors. The DofMap's DefaultCoupling functor +// is configured for two layers before the mesh is built, so that the +// required ghost elements are retained when a distributed mesh deletes +// remote elements. +// +// The example solves a manufactured-solution problem on a sequence of +// uniformly refined meshes and reports the centroid-sampled L2 error. +// Refinement continues until a least-squares fit over the last three +// meshes gives a convergence rate close to 2, and the example fails if +// that does not happen within the allowed number of refinements. + +// libMesh includes +#include "libmesh/libmesh.h" +#include "libmesh/mesh.h" +#include "libmesh/mesh_generation.h" +#include "libmesh/equation_systems.h" +#include "libmesh/linear_implicit_system.h" +#include "libmesh/dof_map.h" +#include "libmesh/default_coupling.h" +#include "libmesh/elem.h" +#include "libmesh/sparse_matrix.h" +#include "libmesh/numeric_vector.h" +#include "libmesh/dense_matrix.h" +#include "libmesh/dense_vector.h" +#include "libmesh/getpot.h" +#include "libmesh/enum_solver_package.h" +#include "libmesh/int_range.h" +#include "libmesh/exodusII_io.h" + +// C++ includes +#include +#include +#include +#include + +using namespace libMesh; + +namespace +{ + +// The manufactured solution, which also supplies the Dirichlet data +// on the whole boundary. +Real +exact_solution(const Point & p) +{ + return std::sin(libMesh::pi * p(0)) * std::cos(libMesh::pi * p(1)); +} + +// The source term -D lap(u) + v.grad(u) for the manufactured solution +// and a constant, hence divergence-free, velocity v. +Real +forcing(const Point & p, const Real diffusivity, const RealVectorValue & velocity) +{ + const Real pi = libMesh::pi; + const Real x = p(0); + const Real y = p(1); + return 2 * pi * pi * diffusivity * exact_solution(p) + + velocity(0) * pi * std::cos(pi * x) * std::cos(pi * y) - + velocity(1) * pi * std::sin(pi * x) * std::sin(pi * y); +} + +// An affine function of the cell solution values, +// sum_i coefficients[i] * u_i + constant, +// whose value type T is Real for scalar quantities such as face fluxes +// and RealVectorValue for reconstructed gradients. +template +struct AffineStencil +{ + std::map coefficients; + T constant = T(); +}; + +typedef AffineStencil ScalarStencil; +typedef AffineStencil GradientStencil; + +// Adds scale * (g . a) to out. +void +add_dot(ScalarStencil & out, const GradientStencil & g, const RealVectorValue & a, const Real scale) +{ + for (const auto & [dof, coef] : g.coefficients) + out.coefficients[dof] += scale * (coef * a); + out.constant += scale * (g.constant * a); +} + +// Returns 0.5 * (a + b). +GradientStencil +average(const GradientStencil & a, const GradientStencil & b) +{ + GradientStencil avg; + for (const auto & [dof, coef] : a.coefficients) + avg.coefficients[dof] += 0.5 * coef; + for (const auto & [dof, coef] : b.coefficients) + avg.coefficients[dof] += 0.5 * coef; + avg.constant = 0.5 * (a.constant + b.constant); + return avg; +} + +struct FaceGeometry +{ + Point centroid; + Real area; + Point normal; +}; + +// Computes the centroid, area, and outward unit normal of a planar +// element side. Polygons and polyhedra have no reference element, so +// the geometry is taken from the physical side directly. +FaceGeometry +face_geometry(const Elem & elem, const unsigned int side) +{ + const std::unique_ptr side_elem = elem.build_side_ptr(side); + return {side_elem->true_centroid(), side_elem->volume(), elem.side_vertex_average_normal(side)}; +} + +// Returns the least-squares gradient of u in elem as an affine function +// of cell values. Each face contributes one sample: the neighbor's +// centroid value for interior faces and the Dirichlet value at the +// face centroid for boundary faces. The gradient g minimizes +// sum_k ((x_k - x_elem) . g - (u_k - u_elem))^2, +// which reproduces linear functions exactly. +GradientStencil +gradient_stencil(const Elem & elem, const unsigned int sys_num) +{ + struct Sample + { + RealVectorValue offset; + dof_id_type dof; + Real value; + }; + + const unsigned int dim = elem.dim(); + const Point x_elem = elem.true_centroid(); + + std::vector samples; + DenseMatrix normal_matrix(dim, dim); + for (const auto s : elem.side_index_range()) + { + Sample sample; + if (const Elem * neighbor = elem.neighbor_ptr(s)) + sample = {neighbor->true_centroid() - x_elem, neighbor->dof_number(sys_num, 0, 0), 0}; + else + { + const FaceGeometry face = face_geometry(elem, s); + sample = {face.centroid - x_elem, DofObject::invalid_id, exact_solution(face.centroid)}; + } + + for (const auto i : make_range(dim)) + for (const auto j : make_range(dim)) + normal_matrix(i, j) += sample.offset(i) * sample.offset(j); + samples.push_back(sample); + } + + // g = sum_k w_k (u_k - u_elem), with weights w_k solving + // normal_matrix w_k = x_k - x_elem. + GradientStencil g; + const dof_id_type elem_dof = elem.dof_number(sys_num, 0, 0); + for (const auto & sample : samples) + { + DenseVector rhs(dim), w; + for (const auto i : make_range(dim)) + rhs(i) = sample.offset(i); + normal_matrix.lu_solve(rhs, w); + + RealVectorValue weight; + for (const auto i : make_range(dim)) + weight(i) = w(i); + + g.coefficients[elem_dof] -= weight; + if (sample.dof == DofObject::invalid_id) + g.constant += sample.value * weight; + else + g.coefficients[sample.dof] += weight; + } + + return g; +} + +} // anonymous namespace + +// Assembles one matrix row per local cell from the balance +// sum_faces (-D grad(u) + v u) . n |face| = f(x_elem) |elem|. +void +assemble_advection_diffusion(EquationSystems & es, const std::string & system_name) +{ + const MeshBase & mesh = es.get_mesh(); + auto & system = es.get_system(system_name); + const unsigned int sys_num = system.number(); + + const Real diffusivity = es.parameters.get("diffusivity"); + const RealVectorValue velocity = es.parameters.get("velocity"); + + for (const Elem * elem : mesh.active_local_element_ptr_range()) + { + const dof_id_type elem_dof = elem->dof_number(sys_num, 0, 0); + const Point x_elem = elem->true_centroid(); + const GradientStencil grad_elem = gradient_stencil(*elem, sys_num); + + // The net outward flux through the cell boundary + ScalarStencil net_flux; + + for (const auto s : elem->side_index_range()) + { + const FaceGeometry face = face_geometry(*elem, s); + const Elem * neighbor = elem->neighbor_ptr(s); + + // On interior faces, the neighbor centroid and gradient; on + // boundary faces, the face centroid, where the Dirichlet + // value is known, and the gradient of this cell. + const Point x_across = neighbor ? neighbor->true_centroid() : face.centroid; + const GradientStencil grad_neighbor = + neighbor ? gradient_stencil(*neighbor, sys_num) : GradientStencil(); + const GradientStencil grad_face = neighbor ? average(grad_elem, grad_neighbor) : grad_elem; + + // Diffusion: with d the offset from x_elem to x_across, + // n.grad(u) ~= (u_across - u_elem) / (d.n) + (n - d / (d.n)).grad(u)_face, + // which is exact for linear u. The second term vanishes when d + // is parallel to n. + const RealVectorValue d = x_across - x_elem; + const Real d_dot_n = d * face.normal; + const Real diffusive_scale = -diffusivity * face.area; + + net_flux.coefficients[elem_dof] -= diffusive_scale / d_dot_n; + if (neighbor) + net_flux.coefficients[neighbor->dof_number(sys_num, 0, 0)] += diffusive_scale / d_dot_n; + else + net_flux.constant += diffusive_scale / d_dot_n * exact_solution(face.centroid); + add_dot(net_flux, grad_face, face.normal - d / d_dot_n, diffusive_scale); + + // Advection: the upwind cell value, extrapolated to the face + // centroid with that cell's gradient. Inflow boundary faces + // take the Dirichlet value. + const Real mass_flux = (velocity * face.normal) * face.area; + if (mass_flux >= 0) + { + net_flux.coefficients[elem_dof] += mass_flux; + add_dot(net_flux, grad_elem, face.centroid - x_elem, mass_flux); + } + else if (neighbor) + { + net_flux.coefficients[neighbor->dof_number(sys_num, 0, 0)] += mass_flux; + add_dot(net_flux, grad_neighbor, face.centroid - x_across, mass_flux); + } + else + net_flux.constant += mass_flux * exact_solution(face.centroid); + } + + std::vector row_dofs = {elem_dof}; + std::vector column_dofs; + DenseMatrix Ke(1, net_flux.coefficients.size()); + for (const auto & [dof, coef] : net_flux.coefficients) + { + Ke(0, column_dofs.size()) = coef; + column_dofs.push_back(dof); + } + + DenseVector Fe(1); + Fe(0) = forcing(x_elem, diffusivity, velocity) * elem->volume() - net_flux.constant; + + system.matrix->add_matrix(Ke, row_dofs, column_dofs); + system.rhs->add_vector(Fe, row_dofs); + } +} + +// Returns the L2 norm of the difference between the cell values and +// the exact solution sampled at cell centroids. +Real +centroid_l2_error(const System & system) +{ + const unsigned int sys_num = system.number(); + + Real error_sq = 0; + for (const Elem * elem : system.get_mesh().active_local_element_ptr_range()) + { + const Real u_h = libmesh_real(system.current_solution(elem->dof_number(sys_num, 0, 0))); + const Real diff = u_h - exact_solution(elem->true_centroid()); + error_sq += elem->volume() * diff * diff; + } + system.comm().sum(error_sq); + + return std::sqrt(error_sq); +} + +int +main(int argc, char ** argv) +{ + // Initialize libMesh. + LibMeshInit init(argc, argv); + + // This example requires a linear solver package. + libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE, + "--enable-petsc, --enable-trilinos, or --enable-eigen"); + + // The mesh is two-dimensional. + libmesh_example_requires(2 <= LIBMESH_DIM, "2D support"); + + // Parse the input file, allowing the command line to override it. + GetPot infile("finite_volume_ex1.in"); + infile.parse_command_line(argc, argv); + + const unsigned int coarse_grid_size = infile("coarse_grid_size", 4); + const unsigned int max_refinements = infile("max_refinements", 6); + const Real diffusivity = infile("diffusivity", 1.); + const RealVectorValue velocity(infile("velocity_x", 1.), infile("velocity_y", 0.5)); + + // The scheme is second order, and a measured rate within this + // distance of 2 is taken as confirmation of that order. The bound is + // loose enough to absorb the pre-asymptotic behavior of the error on + // the coarsest grids and the round-off of any supported Real + // precision, and tight enough to catch a loss of second-order + // accuracy. Dropping the advection reconstruction reduces the scheme + // to first order, and dropping the non-orthogonal diffusion + // correction stops the error from converging at all. + const Real expected_rate = 2; + const Real rate_tolerance = 0.1; + + std::vector errors; + bool converged = false; + for (const auto level : make_range(max_refinements + 1)) + { + const unsigned int grid_size = coarse_grid_size << level; + + Mesh mesh(init.comm()); + EquationSystems equation_systems(mesh); + auto & system = equation_systems.add_system("AdvectionDiffusion"); + system.add_variable("u", CONSTANT, MONOMIAL); + system.attach_assemble_function(assemble_advection_diffusion); + + // Couple each cell to two layers of side neighbors. The DofMap + // registers its coupling functor with the mesh, so building the + // mesh after this call also retains the corresponding ghost + // elements on a distributed mesh. + system.get_dof_map().default_coupling().set_n_levels(2); + + MeshTools::Generation::build_square(mesh, grid_size, grid_size, 0., 1., 0., 1., C0POLYGON); + + equation_systems.parameters.set("diffusivity") = diffusivity; + equation_systems.parameters.set("velocity") = velocity; + equation_systems.init(); + + system.solve(); + + const Real error = centroid_l2_error(system); + errors.push_back(error); + libMesh::out << "Grid size " << grid_size << ": " << mesh.n_active_elem() + << " cells, centroid L2 error " << error; + if (level) + libMesh::out << ", convergence rate " << std::log2(errors[level - 1] / error); + + // Fit a line to log2(error) against log2(h) over the last three + // grids. The grid spacing h halves at each level, so the sample + // points are equally spaced and the least-squares slope reduces + // to the difference between the first and last points divided by + // their separation of two levels. + if (level >= 2) + { + const Real fit_rate = std::log2(errors[level - 2] / error) / 2; + libMesh::out << ", three-grid fit rate " << fit_rate; + converged = std::abs(fit_rate - expected_rate) < rate_tolerance; + } + libMesh::out << std::endl; + + if (converged) + { +#ifdef LIBMESH_HAVE_EXODUS_API + ExodusII_IO(mesh).write_equation_systems("out.e", equation_systems); +#endif + break; + } + } + + libmesh_error_msg_if(!converged, + "The three-grid fit of the L2 error convergence rate did not reach " + << expected_rate << " +/- " << rate_tolerance << " within " + << max_refinements << " refinements"); + + return 0; +} diff --git a/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.in b/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.in new file mode 100644 index 0000000000..4cf109ab18 --- /dev/null +++ b/examples/finite_volume/finite_volume_ex1/finite_volume_ex1.in @@ -0,0 +1,13 @@ +# The number of hexagons along each direction of the coarsest mesh +coarse_grid_size = 4 + +# The maximum number of times the coarse grid size is doubled while +# waiting for the measured convergence rate to approach 2 +max_refinements = 6 + +# The diffusion coefficient +diffusivity = 1 + +# The components of the constant advecting velocity +velocity_x = 1 +velocity_y = 0.5 diff --git a/examples/finite_volume/finite_volume_ex1/run.sh b/examples/finite_volume/finite_volume_ex1/run.sh new file mode 100755 index 0000000000..a9555831f5 --- /dev/null +++ b/examples/finite_volume/finite_volume_ex1/run.sh @@ -0,0 +1,9 @@ +#!/bin/sh + +#set -x + +. "$LIBMESH_DIR"/examples/run_common.sh + +example_name=finite_volume_ex1 + +run_example "$example_name"