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// MFEM Example 21
// Compile with: make ex21
// Sample runs: ex21
// ex21 -o 3
// ex21 -m ../data/beam-quad.mesh
// ex21 -m ../data/beam-quad.mesh -o 3
// ex21 -m ../data/beam-quad.mesh -o 3 -f 1
// ex21 -m ../data/beam-tet.mesh
// ex21 -m ../data/beam-tet.mesh -o 2
// ex21 -m ../data/beam-hex.mesh
// ex21 -m ../data/beam-hex.mesh -o 2
// Description: This is a version of Example 2 with a simple adaptive mesh
// refinement loop. The problem being solved is again the linear
// elasticity describing a multi-material cantilever beam.
// The problem is solved on a sequence of meshes which
// are locally refined in a conforming (triangles, tetrahedrons)
// or non-conforming (quadrilaterals, hexahedra) manner according
// to a simple ZZ error estimator.
// The example demonstrates MFEM's capability to work with both
// conforming and nonconforming refinements, in 2D and 3D, on
// linear and curved meshes. Interpolation of functions from
// coarse to fine meshes, as well as persistent GLVis
// visualization are also illustrated.
// We recommend viewing Examples 2 and 6 before viewing this
// example.
#include "mfem.hpp"
#include <fstream>
#include <iostream>
using namespace std;
using namespace mfem;
int main(int argc, char *argv[])
// 1. Parse command-line options.
const char *mesh_file = "../data/beam-tri.mesh";
int order = 1;
bool static_cond = false;
int flux_averaging = 0;
bool visualization = 1;
OptionsParser args(argc, argv);
args.AddOption(&mesh_file, "-m", "--mesh",
"Mesh file to use.");
args.AddOption(&order, "-o", "--order",
"Finite element order (polynomial degree).");
args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
"--no-static-condensation", "Enable static condensation.");
args.AddOption(&flux_averaging, "-f", "--flux-averaging",
"Flux averaging: 0 - global, 1 - by mesh attribute.");
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
"Enable or disable GLVis visualization.");
if (!args.Good())
return 1;
// 2. Read the mesh from the given mesh file. We can handle triangular,
// quadrilateral, tetrahedral, and hexahedral meshes with the same code.
Mesh mesh(mesh_file, 1, 1);
int dim = mesh.Dimension();
MFEM_VERIFY(mesh.SpaceDimension() == dim, "invalid mesh");
if (mesh.attributes.Max() < 2 || mesh.bdr_attributes.Max() < 2)
cerr << "\nInput mesh should have at least two materials and "
<< "two boundary attributes! (See schematic in ex2.cpp)\n"
<< endl;
return 3;
// 3. Since a NURBS mesh can currently only be refined uniformly, we need to
// convert it to a piecewise-polynomial curved mesh. First we refine the
// NURBS mesh a bit more and then project the curvature to quadratic Nodes.
if (mesh.NURBSext)
for (int i = 0; i < 2; i++)
// 4. Define a finite element space on the mesh. The polynomial order is
// one (linear) by default, but this can be changed on the command line.
H1_FECollection fec(order, dim);
FiniteElementSpace fespace(&mesh, &fec, dim);
// 5. As in Example 2, we set up the linear form b(.) which corresponds to
// the right-hand side of the FEM linear system. In this case, b_i equals
// the boundary integral of f*phi_i where f represents a "pull down"
// force on the Neumann part of the boundary and phi_i are the basis
// functions in the finite element fespace. The force is defined by the
// VectorArrayCoefficient object f, which is a vector of Coefficient
// objects. The fact that f is non-zero on boundary attribute 2 is
// indicated by the use of piece-wise constants coefficient for its last
// component. We don't assemble the discrete problem yet, this will be
// done in the main loop.
VectorArrayCoefficient f(dim);
for (int i = 0; i < dim-1; i++)
f.Set(i, new ConstantCoefficient(0.0));
Vector pull_force(mesh.bdr_attributes.Max());
pull_force = 0.0;
pull_force(1) = -1.0e-2;
f.Set(dim-1, new PWConstCoefficient(pull_force));
LinearForm b(&fespace);
b.AddDomainIntegrator(new VectorBoundaryLFIntegrator(f));
// 6. Set up the bilinear form a(.,.) on the finite element space
// corresponding to the linear elasticity integrator with piece-wise
// constants coefficient lambda and mu.
Vector lambda(mesh.attributes.Max());
lambda = 1.0;
lambda(0) = lambda(1)*50;
PWConstCoefficient lambda_func(lambda);
Vector mu(mesh.attributes.Max());
mu = 1.0;
mu(0) = mu(1)*50;
PWConstCoefficient mu_func(mu);
BilinearForm a(&fespace);
BilinearFormIntegrator *integ =
new ElasticityIntegrator(lambda_func,mu_func);
if (static_cond) { a.EnableStaticCondensation(); }
// 7. The solution vector x and the associated finite element grid function
// will be maintained over the AMR iterations. We initialize it to zero.
Vector zero_vec(dim);
zero_vec = 0.0;
VectorConstantCoefficient zero_vec_coeff(zero_vec);
GridFunction x(&fespace);
x = 0.0;
// 8. Determine the list of true (i.e. conforming) essential boundary dofs.
// In this example, the boundary conditions are defined by marking only
// boundary attribute 1 from the mesh as essential and converting it to a
// list of true dofs. The conversion to true dofs will be done in the
// main loop.
Array<int> ess_bdr(mesh.bdr_attributes.Max());
ess_bdr = 0;
ess_bdr[0] = 1;
// 9. Connect to GLVis.
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock;
if (visualization)
{, visport);
// 10. Set up an error estimator. Here we use the Zienkiewicz-Zhu estimator
// that uses the ComputeElementFlux method of the ElasticityIntegrator to
// recover a smoothed flux (stress) that is subtracted from the element
// flux to get an error indicator. We need to supply the space for the
// smoothed flux: an (H1)^tdim (i.e., vector-valued) space is used here.
// Here, tdim represents the number of components for a symmetric (dim x
// dim) tensor.
const int tdim = dim*(dim+1)/2;
FiniteElementSpace flux_fespace(&mesh, &fec, tdim);
ZienkiewiczZhuEstimator estimator(*integ, x, flux_fespace);
// 11. A refiner selects and refines elements based on a refinement strategy.
// The strategy here is to refine elements with errors larger than a
// fraction of the maximum element error. Other strategies are possible.
// The refiner will call the given error estimator.
ThresholdRefiner refiner(estimator);
// 12. The main AMR loop. In each iteration we solve the problem on the
// current mesh, visualize the solution, and refine the mesh.
const int max_dofs = 50000;
const int max_amr_itr = 20;
for (int it = 0; it <= max_amr_itr; it++)
int cdofs = fespace.GetTrueVSize();
cout << "\nAMR iteration " << it << endl;
cout << "Number of unknowns: " << cdofs << endl;
// 13. Assemble the stiffness matrix and the right-hand side.
// 14. Set Dirichlet boundary values in the GridFunction x.
// Determine the list of Dirichlet true DOFs in the linear system.
Array<int> ess_tdof_list;
x.ProjectBdrCoefficient(zero_vec_coeff, ess_bdr);
fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
// 15. Create the linear system: eliminate boundary conditions, constrain
// hanging nodes and possibly apply other transformations. The system
// will be solved for true (unconstrained) DOFs only.
SparseMatrix A;
Vector B, X;
const int copy_interior = 1;
a.FormLinearSystem(ess_tdof_list, x, b, A, X, B, copy_interior);
// 16. Define a simple symmetric Gauss-Seidel preconditioner and use it to
// solve the linear system with PCG.
GSSmoother M(A);
PCG(A, M, B, X, 3, 2000, 1e-12, 0.0);
// 16. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the
// the linear system.
UMFPackSolver umf_solver;
umf_solver.Mult(B, X);
// 17. After solving the linear system, reconstruct the solution as a
// finite element GridFunction. Constrained nodes are interpolated
// from true DOFs (it may therefore happen that x.Size() >= X.Size()).
a.RecoverFEMSolution(X, b, x);
// 18. Send solution by socket to the GLVis server.
if (visualization && sol_sock.good())
GridFunction nodes(&fespace), *nodes_p = &nodes;
nodes += x;
int own_nodes = 0;
mesh.SwapNodes(nodes_p, own_nodes);
x.Neg(); // visualize the backward displacement
sol_sock << "solution\n" << mesh << x << flush;
mesh.SwapNodes(nodes_p, own_nodes);
if (it == 0)
sol_sock << "keys '" << ((dim == 2) ? "Rjl" : "") << "m'" << endl;
sol_sock << "window_title 'AMR iteration: " << it << "'\n"
<< "pause" << endl;
cout << "Visualization paused. "
"Press <space> in the GLVis window to continue." << endl;
if (cdofs > max_dofs)
cout << "Reached the maximum number of dofs. Stop." << endl;
// 19. Call the refiner to modify the mesh. The refiner calls the error
// estimator to obtain element errors, then it selects elements to be
// refined and finally it modifies the mesh. The Stop() method can be
// used to determine if a stopping criterion was met.
if (refiner.Stop())
cout << "Stopping criterion satisfied. Stop." << endl;
// 20. Update the space to reflect the new state of the mesh. Also,
// interpolate the solution x so that it lies in the new space but
// represents the same function. This saves solver iterations later
// since we'll have a good initial guess of x in the next step.
// Internally, FiniteElementSpace::Update() calculates an
// interpolation matrix which is then used by GridFunction::Update().
// 21. Inform also the bilinear and linear forms that the space has
// changed.
ofstream mesh_ref_out("ex21_reference.mesh");
ofstream mesh_out("ex21_deformed.mesh");
GridFunction nodes(&fespace), *nodes_p = &nodes;
nodes += x;
int own_nodes = 0;
mesh.SwapNodes(nodes_p, own_nodes);
mesh.SwapNodes(nodes_p, own_nodes);
ofstream x_out("ex21_displacement.sol");
return 0;