380 lines
12 KiB
C++
380 lines
12 KiB
C++
// Serial hp-refinement example
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//
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// Compile with: make hpref
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//
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// Sample runs: hpref -dim 2 -n 1000
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// hpref -dim 3 -n 500
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// hpref -m ../../data/star-mixed.mesh -pref -n 100
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// hpref -m ../../data/fichera-mixed.mesh -pref -n 30
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//
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// Description: This example demonstrates h- and p-refinement in a serial
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// finite element discretization of the Poisson problem (cf. ex1)
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// -Delta u = 1 with homogeneous Dirichlet boundary conditions.
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// Refinements are performed iteratively, each iteration having h-
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// or p-refinements. For simplicity, we randomly choose the
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// elements and the type of refinement, for each iteration. In
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// practice, these choices may be made in a problem-dependent way,
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// but this example serves only to illustrate the capabilities of
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// hp-refinement.
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//
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// We recommend viewing Example 1 before viewing this example.
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#include "mfem.hpp"
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#include <fstream>
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#include <iostream>
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using namespace std;
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using namespace mfem;
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real_t CheckH1Continuity(GridFunction & x);
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// Deterministic function for "random" integers.
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int DetRand(int & seed)
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{
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seed++;
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return int(std::abs(1.0e5 * sin(seed * 1.1234 * M_PI)));
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}
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void f_exact(const Vector &x, Vector &f);
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int main(int argc, char *argv[])
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{
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// 1. Parse command-line options.
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const char *mesh_file = "";
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int order = 1;
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const char *device_config = "cpu";
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bool visualization = true;
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int numIter = 0;
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int dim = 2;
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bool deterministic = true;
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bool projectSolution = false;
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bool onlyPref = false;
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OptionsParser args(argc, argv);
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args.AddOption(&mesh_file, "-m", "--mesh",
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"Mesh file to use.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree) or -1 for"
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" isoparametric space.");
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args.AddOption(&device_config, "-d", "--device",
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"Device configuration string, see Device::Configure().");
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args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
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"--no-visualization",
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"Enable or disable GLVis visualization.");
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args.AddOption(&numIter, "-n", "--num-iter", "Number of hp-ref iterations");
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args.AddOption(&dim, "-dim", "--dim", "Mesh dimension (2 or 3)");
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args.AddOption(&deterministic, "-det", "--deterministic", "-not-det",
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"--not-deterministic",
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"Use deterministic random refinements");
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args.AddOption(&projectSolution, "-proj", "--project-solution", "-no-proj",
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"--no-project",
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"Project a coefficient to solution");
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args.AddOption(&onlyPref, "-pref", "--only-p-refinement", "-no-pref",
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"--hp-refinement",
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"Use only p-refinement");
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args.Parse();
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if (!args.Good())
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{
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args.PrintUsage(cout);
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return 1;
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}
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args.PrintOptions(cout);
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// 2. Enable hardware devices such as GPUs, and programming models such as
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// CUDA, OCCA, RAJA and OpenMP based on command line options.
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Device device(device_config);
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device.Print();
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// 3. Construct or load a coarse mesh.
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std::string mesh_filename(mesh_file);
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Mesh mesh;
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if (!mesh_filename.empty())
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{
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mesh = Mesh::LoadFromFile(mesh_filename, 1, 1);
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dim = mesh.Dimension();
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}
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else if (dim == 3)
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{
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mesh = Mesh::MakeCartesian3D(2, 2, 2, Element::HEXAHEDRON);
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}
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else
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{
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mesh = Mesh::MakeCartesian2D(2, 2, Element::QUADRILATERAL, true);
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}
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mesh.EnsureNCMesh();
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// 4. Define a finite element space on the mesh. Here we use continuous
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// Lagrange finite elements of the specified order. If order < 1, we
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// instead use an isoparametric/isogeometric space.
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FiniteElementCollection *fec;
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bool delete_fec;
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if (order > 0)
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{
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fec = new H1_FECollection(order, dim);
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delete_fec = true;
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}
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else if (mesh.GetNodes())
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{
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fec = mesh.GetNodes()->OwnFEC();
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delete_fec = false;
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cout << "Using isoparametric FEs: " << fec->Name() << endl;
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}
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else
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{
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fec = new H1_FECollection(order = 1, dim);
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delete_fec = true;
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}
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const int fespaceDim = projectSolution ? dim : 1;
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FiniteElementSpace fespace(&mesh, fec, fespaceDim);
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// 5. Iteratively perform h- and p-refinements.
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int numH = 0;
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int numP = 0;
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int seed = 0;
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const std::vector<char> hp_char = {'h', 'p'};
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for (int iter=0; iter<numIter; ++iter)
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{
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const int r1 = deterministic ? DetRand(seed) : rand();
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const int r2 = deterministic ? DetRand(seed) : rand();
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const int elem = r1 % mesh.GetNE();
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const int hp = onlyPref ? 1 : r2 % 2;
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cout << "hp-refinement iteration " << iter << ": "
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<< hp_char[hp] << "-refinement" << endl;
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if (hp == 1)
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{
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// p-ref
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Array<pRefinement> refs;
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refs.Append(pRefinement(elem, 1)); // Increase the element order by 1
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fespace.PRefineAndUpdate(refs);
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numP++;
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}
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else
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{
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// h-ref
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Array<Refinement> refs;
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refs.Append(Refinement(elem));
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mesh.GeneralRefinement(refs);
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fespace.Update(false);
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numH++;
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}
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}
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const int size = fespace.GetTrueVSize();
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cout << "Number of finite element unknowns: " << size << endl;
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const int maxP = fespace.GetMaxElementOrder();
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cout << "Total number of h-refinements: " << numH
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<< "\nTotal number of p-refinements: " << numP
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<< "\nMaximum order " << maxP << "\n";
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GridFunction x(&fespace);
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Vector X;
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if (projectSolution)
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{
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VectorFunctionCoefficient vec_coef(dim, f_exact);
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x.ProjectCoefficient(vec_coef);
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X.SetSize(fespace.GetTrueVSize());
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fespace.GetHpRestrictionMatrix()->Mult(x, X);
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fespace.GetProlongationMatrix()->Mult(X, x);
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// Compute and print the L^2 norm of the error.
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const real_t error = x.ComputeL2Error(vec_coef);
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cout << "\n|| E_h - E ||_{L^2} = " << error << '\n' << endl;
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}
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else
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{
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// 6. Determine the list of essential boundary dofs. In this example, the
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// boundary conditions are defined by marking all the boundary attributes
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// from the mesh as essential (Dirichlet) and converting them to a list of
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// true dofs.
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Array<int> ess_tdof_list;
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if (mesh.bdr_attributes.Size())
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{
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Array<int> ess_bdr(mesh.bdr_attributes.Max());
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ess_bdr = 1;
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fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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}
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// 7. Set up the linear form b(.) which corresponds to the right-hand side of
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// the FEM linear system, which in this case is (1,phi_i) where phi_i are
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// the basis functions in fespace.
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LinearForm b(&fespace);
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ConstantCoefficient one(1.0);
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b.AddDomainIntegrator(new DomainLFIntegrator(one));
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b.Assemble();
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// 8. Define the solution vector x as a finite element grid function
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// corresponding to fespace. Initialize x with initial guess of zero,
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// which satisfies the boundary conditions.
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x = 0.0;
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// 9. Set up the bilinear form a(.,.) on the finite element space
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// corresponding to the Laplacian operator -Delta, by adding the diffusion
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// domain integrator.
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BilinearForm a(&fespace);
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a.AddDomainIntegrator(new DiffusionIntegrator(one));
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// 10. Assemble the bilinear form and the corresponding linear system,
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// applying any necessary transformations such as: assembly, eliminating
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// boundary conditions, applying conforming constraints for non-conforming
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// AMR, static condensation, etc.
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a.Assemble();
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OperatorPtr A;
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Vector B;
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a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
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// 11. Solve the linear system A X = B.
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{
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#ifndef MFEM_USE_SUITESPARSE
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// Use a simple symmetric Gauss-Seidel preconditioner with PCG.
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GSSmoother M((SparseMatrix&)(*A));
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PCG(*A, M, B, X, 1, 200, 1e-12, 0.0);
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#else
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// If MFEM was compiled with SuiteSparse, use UMFPACK to solve the system.
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UMFPackSolver umf_solver;
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umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
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umf_solver.SetOperator(*A);
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umf_solver.Mult(B, X);
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#endif
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}
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// 12. Recover the grid function corresponding to X.
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a.RecoverFEMSolution(X, b, x);
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}
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if (fespaceDim == 1)
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{
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const real_t h1error = CheckH1Continuity(x);
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cout << "H1 continuity error " << h1error << endl;
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MFEM_VERIFY(h1error < 1.0e-12, "H1 continuity is not satisfied");
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}
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L2_FECollection fecL2(0, dim);
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FiniteElementSpace l2fespace(&mesh, &fecL2);
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GridFunction xo(&l2fespace);
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xo = 0.0;
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for (int e=0; e<mesh.GetNE(); ++e)
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{
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const int p_elem = fespace.GetElementOrder(e);
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Array<int> dofs;
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l2fespace.GetElementDofs(e, dofs);
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MFEM_VERIFY(dofs.Size() == 1, "");
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xo[dofs[0]] = p_elem;
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}
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// 13. Save the refined mesh and the solution. This output can be viewed later
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// using GLVis: "glvis -m refined.mesh -g sol.gf".
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std::unique_ptr<GridFunction> vis_x = x.ProlongateToMaxOrder();
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ofstream mesh_ofs("refined.mesh");
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mesh_ofs.precision(8);
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mesh.Print(mesh_ofs);
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ofstream sol_ofs("sol.gf");
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sol_ofs.precision(8);
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vis_x->Save(sol_ofs);
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ofstream order_ofs("order.gf");
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order_ofs.precision(8);
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xo.Save(order_ofs);
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// 14. Send the solution by socket to a GLVis server.
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if (visualization)
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{
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char vishost[] = "localhost";
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int visport = 19916;
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socketstream sol_sock(vishost, visport);
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sol_sock.precision(8);
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sol_sock << "solution\n" << mesh << *vis_x << flush;
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}
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// 15. Free the used memory.
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if (delete_fec)
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{
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delete fec;
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}
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return 0;
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}
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real_t CheckH1Continuity(GridFunction & x)
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{
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const FiniteElementSpace *fes = x.FESpace();
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Mesh *mesh = fes->GetMesh();
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const int dim = mesh->Dimension();
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// Following the example of KellyErrorEstimator::ComputeEstimates(),
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// we loop over interior faces and compute their error contributions.
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real_t errorMax = 0.0;
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for (int f = 0; f < mesh->GetNumFaces(); f++)
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{
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if (mesh->FaceIsInterior(f))
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{
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int Inf1, Inf2, NCFace;
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mesh->GetFaceInfos(f, &Inf1, &Inf2, &NCFace);
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auto FT = mesh->GetFaceElementTransformations(f);
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const int faceOrder = dim == 3 ? fes->GetFaceOrder(f) :
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fes->GetEdgeOrder(f);
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auto &int_rule = IntRules.Get(FT->FaceGeom, 2 * faceOrder);
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const auto nip = int_rule.GetNPoints();
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// Convention
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// * Conforming face: Face side with smaller element id handles
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// the integration
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// * Non-conforming face: The slave handles the integration.
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// See FaceInfo documentation for details.
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bool isNCSlave = FT->Elem2No >= 0 && NCFace >= 0;
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bool isConforming = FT->Elem2No >= 0 && NCFace == -1;
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if ((FT->Elem1No < FT->Elem2No && isConforming) || isNCSlave)
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{
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for (int i = 0; i < nip; i++)
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{
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const auto &fip = int_rule.IntPoint(i);
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IntegrationPoint ip;
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FT->Loc1.Transform(fip, ip);
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const real_t v1 = x.GetValue(FT->Elem1No, ip);
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FT->Loc2.Transform(fip, ip);
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const real_t v2 = x.GetValue(FT->Elem2No, ip);
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const real_t err_i = std::abs(v1 - v2);
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errorMax = std::max(errorMax, err_i);
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}
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}
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}
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}
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return errorMax;
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}
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void f_exact(const Vector &x, Vector &f)
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{
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constexpr real_t freq = 1.0;
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constexpr real_t kappa = freq * M_PI;
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if (x.Size() == 3)
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{
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f(0) = (1. + kappa * kappa) * sin(kappa * x(1));
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f(1) = (1. + kappa * kappa) * sin(kappa * x(2));
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f(2) = (1. + kappa * kappa) * sin(kappa * x(0));
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}
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else
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{
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f(0) = (1. + kappa * kappa) * sin(kappa * x(1));
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f(1) = (1. + kappa * kappa) * sin(kappa * x(0));
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if (x.Size() == 3) { f(2) = 0.0; }
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}
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}
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