Commit fc83c49d authored by Thomas Steinreiter's avatar Thomas Steinreiter
Browse files

replaced existing simple example with a slightly modified version of libMesh FEM-example 4

parent 24ab72de
Language: Cpp
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AllowShortIfStatementsOnASingleLine: true
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......@@ -2,6 +2,7 @@
cmake_minimum_required(VERSION 2.8.10 FATAL_ERROR)
find_package(MPI) # Built-in in CMake
......@@ -22,15 +23,15 @@ if (MPI_FOUND)
include_directories(${MPI_INCLUDE_PATH} ${LIBMESH_INCLUDE_PATH})
add_executable(${NAME} main.cpp)
add_executable(${NAME} fem_system_ex4.C heatsystem.C)
set(CMAKE_BUILD_TYPE RelWithDebInfo)
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -march=native")
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -xHost -std=c++14")
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -xHost -std=c++11")
target_link_libraries(${NAME} ${LIBMESH_LIBRARY} ${MPI_LIBRARIES})
message("** Enabling '${NAME}': with MPI")
# README - libMesh Example
## Description
The libMesh library provides a framework for the numerical simulation of partial differential equations using arbitrary unstructured discretizations on serial and parallel platforms. (
This example is a slightly modified version of the official libMesh example "fem_system_ex4" ( In this example, a heat transfer equation is solved with an FEM System.
The example demonstrates:
* Initialization of libMesh
* Definition of a equation system
* Usage of adaptive mesh refinement
* IO: reading and exporting meshes in different file formats
The example is structured as followed:
* `fem_system_ex4.C`: The main program, Initialization, IO, refinement
* `heatsystem.C|h`: Heat system with laplace heat equation
* `meshes/bridge.e|xda`: FEM-Mesh of a bridge
Screenshot of the result:
![Screenshot of Result](hpc_kernel_samples/unstructured_grids/libmesh/meshes/bridge_screenshot.PNG)
## Release Date
## Version History
* 2016-09-08 Initial Release on PRACE CodeVault repository
## Contributors
* Thomas Steinreiter - [](
## Copyright
This code is available under LGPL, Version 2.1 - see also the license file in the CodeVault root directory.
## Languages
This sample is written in C++ 11.
## Parallelisation
This sample uses PETSCs internal parallelisation.
## Level of the code sample complexity
## Compiling
Follow the compilation instructions given in the main directory of the kernel samples directory (`/hpc_kernel_samples/`).
Note: libMesh needs to be built against the correct version of MPI. PETSC is needed for parallelism.
## Running
To run the program, use something similar to
mpiexec -n [nprocs] ./8_unstructured_libmesh
either on the command line or in your batch script, where `nprocs` specifies the number of processes used. Node: if `nprocs` > 1, libMesh must be built with PETSC enabled.
### Example
If you run
mpiexec -n 8 ./8_unstructured_libmesh
the output should look similar to
Mesh Information:
elem_dimensions()={2, 3}
System #0, "Heat"
Type "Implicit"
Finite Element Types="LAGRANGE"
Approximation Orders="FIRST"
DofMap Sparsity
Average On-Processor Bandwidth <= 13.6686
Average Off-Processor Bandwidth <= 0.533635
Maximum On-Processor Bandwidth <= 27
Maximum Off-Processor Bandwidth <= 15
DofMap Constraints
Number of DoF Constraints = 1887
Average DoF Constraint Length= 0
Assembling the System
*** Warning, This code is deprecated, and likely to be removed in future library versions! /usr/local/include/libmesh/libmesh_common.h, line 497, compiled Sep 1 2016 at 15:28:09 ***
Nonlinear Residual: 8495.66
Linear solve starting, tolerance 0.001
Linear solve finished, step 55, residual 3.10027
Trying full Newton step
Current Residual: 14.2871
Nonlinear step: |du|/|u| = 1, |du| = 510394
Assembling the System
Nonlinear Residual: 14.2871
Linear solve starting, tolerance 0.001
Linear solve finished, step 58, residual 0.00263574
Trying full Newton step
Current Residual: 0.012002
Nonlinear step: |du|/|u| = 0.000564056, |du| = 287.935
Assembling the System
Nonlinear Residual: 0.012002
Linear solve starting, tolerance 1.2002e-05
Linear solve finished, step 94, residual 2.75721e-08
Trying full Newton step
Current Residual: 1.29858e-07
Nonlinear solver converged, step 2, residual reduction 1.52852e-11 < 1e-07
Nonlinear solver relative step size 5.66919e-07 > 1e-07
L2-Error is: 59155
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// The libMesh Finite Element Library.
// Copyright (C) 2002-2016 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
// 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
// <h1>FEMSystem Example 4 - Mixed-dimension heat transfer equation
// with FEMSystem</h1>
// \author Roy Stogner
// \date 2015
// This example shows how elements of multiple dimensions can be
// linked and computed upon simultaneously within the
// DifferentiableSystem class framework
// C++ includes
#include <iomanip>
// Basic include files
#include "libmesh/elem.h"
#include "libmesh/equation_systems.h"
#include "libmesh/error_vector.h"
#include "libmesh/exact_solution.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/getpot.h"
#include "libmesh/gmv_io.h"
#include "libmesh/kelly_error_estimator.h"
#include "libmesh/mesh.h"
#include "libmesh/mesh_generation.h"
#include "libmesh/mesh_refinement.h"
#include "libmesh/parsed_function.h"
#include "libmesh/uniform_refinement_estimator.h"
// The systems and solvers we may use
#include "heatsystem.h"
#include "libmesh/diff_solver.h"
#include "libmesh/euler_solver.h"
#include "libmesh/steady_solver.h"
// Bring in everything from the libMesh namespace
using namespace libMesh;
// The main program.
int main(int argc, char** argv) {
// Initialize libMesh.
LibMeshInit init(argc, argv);
libmesh_example_requires(false, "--enable-amr");
// This doesn't converge with Eigen BICGSTAB for some reason...
libmesh_example_requires(libMesh::default_solver_package() != EIGEN_SOLVERS,
// This doesn't converge without at least double precision
libmesh_example_requires(sizeof(Real) > 4, "--disable-singleprecision");
// Parse the input file
GetPot infile("../");
// Read in parameters from the input file
const Real global_tolerance = infile("global_tolerance", 0.);
const unsigned int nelem_target = infile("n_elements", 400);
const Real deltat = infile("deltat", 0.005);
const unsigned int coarsegridsize = infile("coarsegridsize", 20);
const unsigned int coarserefinements = infile("coarserefinements", 0);
const unsigned int max_adaptivesteps = infile("max_adaptivesteps", 10);
const unsigned int dim = infile("dimension", 2);
// Skip higher-dimensional examples on a lower-dimensional libMesh build
libmesh_example_requires(dim <= LIBMESH_DIM, "2D/3D support");
// We have only defined 2 and 3 dimensional problems
libmesh_assert(dim == 2 || dim == 3);
// Create a mesh, with dimension to be overridden later, distributed
// across the default MPI communicator.
Mesh mesh(init.comm());
// And an object to refine it
MeshRefinement mesh_refinement(mesh);
mesh_refinement.coarsen_by_parents() = true;
mesh_refinement.absolute_global_tolerance() = global_tolerance;
mesh_refinement.nelem_target() = nelem_target;
mesh_refinement.refine_fraction() = 0.3;
mesh_refinement.coarsen_fraction() = 0.3;
mesh_refinement.coarsen_threshold() = 0.1;
// Use the MeshTools::Generation mesh generator to create a uniform
// grid on the square or cube. We crop the domain at y=2/3 to allow
// for a homogeneous Neumann BC in our benchmark there.
boundary_id_type bcid = 3; // +y in 3D"../meshes/bridge.xda");
// Add boundary elements corresponding to the +y boundary of our
// volume mesh
std::set<boundary_id_type> bcids;
mesh.get_boundary_info().add_elements(bcids, mesh);
// To work around ExodusII file format limitations, we need elements
// of different dimensionality to belong to different subdomains.
// Our interior elements defaulted to subdomain id 0, so we'll set
// boundary elements to subdomain 2.
const MeshBase::element_iterator end_el = mesh.elements_end();
for (MeshBase::element_iterator el = mesh.elements_begin();
el != end_el; ++el) {
Elem* elem = *el;
if (elem->dim() < dim) elem->subdomain_id() = 2;
// Print information about the mesh to the screen.
// Create an equation systems object.
EquationSystems equation_systems(mesh);
// Declare the system "Heat" and its variables.
HeatSystem& system = equation_systems.add_system<HeatSystem>("Heat");
// Solve this as a steady system
system.time_solver = UniquePtr<TimeSolver>(new SteadySolver(system));
// Initialize the system
// Set the time stepping options
system.deltat = deltat;
// And the nonlinear solver options
DiffSolver& solver = *(system.time_solver->diff_solver().get());
solver.quiet = infile("solver_quiet", true);
solver.verbose = !solver.quiet;
solver.max_nonlinear_iterations = infile("max_nonlinear_iterations", 15);
solver.relative_step_tolerance = infile("relative_step_tolerance", 1.e-3);
solver.relative_residual_tolerance =
infile("relative_residual_tolerance", 0.0);
solver.absolute_residual_tolerance =
infile("absolute_residual_tolerance", 0.0);
// And the linear solver options
solver.max_linear_iterations = infile("max_linear_iterations", 50000);
solver.initial_linear_tolerance = infile("initial_linear_tolerance", 1.e-3);
// Print information about the system to the screen.
// Adaptively solve the steady solution
unsigned int a_step = max_adaptivesteps;
for (; a_step != max_adaptivesteps; ++a_step) {
ErrorVector error;
UniquePtr<ErrorEstimator> error_estimator;
// To solve to a tolerance in this problem we
// need a better estimator than Kelly
if (global_tolerance != 0.) {
// We can't adapt to both a tolerance and a mesh
// size at once
libmesh_assert_equal_to(nelem_target, 0);
UniformRefinementEstimator* u = new UniformRefinementEstimator;
// The lid-driven cavity problem isn't in H1, so
// lets estimate L2 error
u->error_norm = L2;
} else {
// If we aren't adapting to a tolerance we need a
// target mesh size
libmesh_assert_greater(nelem_target, 0);
// Kelly is a lousy estimator to use for a problem
// not in H1 - if we were doing more than a few
// timesteps we'd need to turn off or limit the
// maximum level of our adaptivity eventually
error_estimator.reset(new KellyErrorEstimator);
error_estimator->estimate_error(system, error);
// Print out status at each adaptive step.
Real global_error = error.l2_norm();
libMesh::out << "Adaptive step " << a_step << ": " << std::endl;
if (global_tolerance != 0.)
libMesh::out << "Global_error = " << global_error << std::endl;
if (global_tolerance != 0.)
libMesh::out << "Worst element error = " << error.maximum()
<< ", mean = " << error.mean() << std::endl;
if (global_tolerance != 0.) {
// If we've reached our desired tolerance, we
// don't need any more adaptive steps
if (global_error < global_tolerance) break;
} else {
// If flag_elements_by_nelem_target returns true, this
// should be our last adaptive step.
if (mesh_refinement.flag_elements_by_nelem_target(error)) {
a_step = max_adaptivesteps;
// Carry out the adaptive mesh refinement/coarsening
libMesh::out << "Refined mesh to " << mesh.n_active_elem()
<< " active elements and "
<< equation_systems.n_active_dofs() << " active dofs."
<< std::endl;
// Do one last solve if necessary
if (a_step == max_adaptivesteps) {
ExodusII_IO(mesh).write_equation_systems("out.e", equation_systems);
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
GMVIO(mesh).write_equation_systems("out.gmv", equation_systems);
#endif // #ifdef LIBMESH_HAVE_GMV
// Check that we got close to the analytic solution
ExactSolution exact_sol(equation_systems);
const std::string exact_str =
(dim == 2) ? "sin(pi*x)*sin(pi*y)" : "sin(pi*x)*sin(pi*y)*sin(pi*z)";
ParsedFunction<Number> exact_func(exact_str);
exact_sol.attach_exact_value(0, &exact_func);
exact_sol.compute_error("Heat", "T");
Number err = exact_sol.l2_error("Heat", "T");
// Print out the error value
libMesh::out << "L2-Error is: " << err << std::endl;
libmesh_assert_less(libmesh_real(err), 2e-3);
#endif // #ifdef LIBMESH_HAVE_FPARSER
#endif // #ifndef LIBMESH_ENABLE_AMR
// All done.
return 0;
# The global FEM error tolerance at each timestep
# Make this nonzero to solve to a specified tolerance
# This will probably break with KellyErrorIndicator
# const Real global_tolerance = 1.e-3;
global_tolerance = 0.0
# The desired number of active mesh elements
# Make this nonzero to solve to a specified mesh size
n_elements = 1200
# The coarse grid size from which to start adaptivity
coarsegridsize = 20
# The maximum number of adaptive steps per timestep
max_adaptivesteps = 2
# Turn this off to see the NewtonSolver chatter
solver_quiet = false
# Solve the 2D or 3D problem
dimension = 3
# Nonlinear solver tolerance
relative_step_tolerance = 1e-7
relative_residual_tolerance = 1.e-7
#include "heatsystem.h"
#include "libmesh/dirichlet_boundaries.h"
#include "libmesh/dof_map.h"
#include "libmesh/elem.h"
#include "libmesh/fe_base.h"
#include "libmesh/fe_interface.h"
#include "libmesh/fem_context.h"
#include "libmesh/getpot.h"
#include "libmesh/mesh.h"
#include "libmesh/quadrature.h"
#include "libmesh/string_to_enum.h"
#include "libmesh/zero_function.h"
using namespace libMesh;
void HeatSystem::init_data() {
T_var = this->add_variable("T", static_cast<Order>(_fe_order),
std::set<boundary_id_type> bdys{0, 1, 2};
std::vector<unsigned int> T_only(1, T_var);
ZeroFunction<Number> zero;
DirichletBoundary(bdys, T_only, &zero));
// Do the parent's initialization after variables are defined
void HeatSystem::init_context(DiffContext& context) {
FEMContext& c = libmesh_cast_ref<FEMContext&>(context);
const std::set<unsigned char>& elem_dims = c.elem_dimensions();
for (std::set<unsigned char>::const_iterator dim_it = elem_dims.begin();
dim_it != elem_dims.end(); ++dim_it) {
const unsigned char dim = *dim_it;
FEBase* fe = libmesh_nullptr;
c.get_element_fe(T_var, fe, dim);
fe->get_JxW(); // For integration
fe->get_dphi(); // For bilinear form
fe->get_xyz(); // For forcing
fe->get_phi(); // For forcing
bool HeatSystem::element_time_derivative(bool request_jacobian,
DiffContext& context) {
FEMContext& c = libmesh_cast_ref<FEMContext&>(context);
const unsigned int mesh_dim = c.get_system().get_mesh().mesh_dimension();
// First we get some references to cell-specific data that
// will be used to assemble the linear system.
const unsigned int dim = c.get_elem().dim();
FEBase* fe = libmesh_nullptr;
c.get_element_fe(T_var, fe, dim);
// Element Jacobian * quadrature weights for interior integration
const std::vector<Real>& JxW = fe->get_JxW();
const std::vector<Point>& xyz = fe->get_xyz();
const std::vector<std::vector<Real>>& phi = fe->get_phi();
const std::vector<std::vector<RealGradient>>& dphi = fe->get_dphi();
// The number of local degrees of freedom in each variable
const unsigned int n_T_dofs = c.get_dof_indices(T_var).size();
// The subvectors and submatrices we need to fill:
DenseSubMatrix<Number>& K = c.get_elem_jacobian(T_var, T_var);
DenseSubVector<Number>& F = c.get_elem_residual(T_var);
// Now we will build the element Jacobian and residual.
// Constructing the residual requires the solution and its
// gradient from the previous timestep. This must be
// calculated at each quadrature point by summing the
// solution degree-of-freedom values by the appropriate
// weight functions.
unsigned int n_qpoints = c.get_element_qrule().n_points();
for (unsigned int qp = 0; qp != n_qpoints; qp++) {
// Compute the solution gradient at the Newton iterate
Gradient grad_T = c.interior_gradient(T_var, qp);
const Number k = _k[dim];
const Point& p = xyz[qp];
const Number u_exact = +1.0;
// Only apply forcing to interior elements
const Number forcing =
(dim == mesh_dim) ? -k * u_exact * (dim * libMesh::pi * libMesh::pi)
: 0;
const Number JxWxNK = JxW[qp] * -k;
for (unsigned int i = 0; i != n_T_dofs; i++)
F(i) += JxWxNK * (grad_T * dphi[i][qp] + forcing * phi[i][qp]);
if (request_jacobian) {
const Number JxWxNKxD =
JxWxNK * context.get_elem_solution_derivative();
for (unsigned int i = 0; i != n_T_dofs; i++)
for (unsigned int j = 0; j != n_T_dofs; ++j)
K(i, j) += JxWxNKxD * (dphi[i][qp] * dphi[j][qp]);
} // end of the quadrature point qp-loop
return request_jacobian;
#include "libmesh/enum_fe_family.h"
#include "libmesh/fem_system.h"
// FEMSystem, TimeSolver and NewtonSolver will handle most tasks,
// but we must specify element residuals
class HeatSystem : public libMesh::FEMSystem {
// Constructor
HeatSystem(libMesh::EquationSystems& es, const std::string& name,
const unsigned int number)
: libMesh::FEMSystem(es, name, number), _fe_family("LAGRANGE"),
_fe_order(1) {
// Get the conductivity ratios right for both 2D and 3D
// benchmarks
_k[1] = 1 / libMesh::pi / std::sqrt(libMesh::Real(3));
_k[2] = 1;