351 lines
11 KiB
C++
351 lines
11 KiB
C++
// MFEM Example 29
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//
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// Compile with: make ex29
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//
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// Sample runs: ex29
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// ex29 -r 2 -sc
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// ex29 -mt 3 -o 4 -sc
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// ex29 -mt 3 -r 2 -o 4 -sc
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//
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// Description: This example code demonstrates the use of MFEM to define a
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// finite element discretization of a PDE on a 2 dimensional
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// surface embedded in a 3 dimensional domain. In this case we
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// solve the Poisson problem -Div(sigma Grad u) = 1, with
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// homogeneous Dirichlet boundary conditions, where sigma is an
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// anisotropic diffusion constant defined as a 3x3 matrix
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// coefficient.
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//
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// This example demonstrates the use of finite element integrators
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// on 2D domains with 3D coefficients.
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//
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// We recommend viewing examples 1 and 7 before viewing this
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// 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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Mesh * GetMesh(int type);
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void trans(const Vector &x, Vector &r);
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void sigmaFunc(const Vector &x, DenseMatrix &s);
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real_t uExact(const Vector &x)
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{
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return (0.25 * (2.0 + x[0]) - x[2]) * (x[2] + 0.25 * (2.0 + x[0]));
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}
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void duExact(const Vector &x, Vector &du)
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{
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du.SetSize(3);
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du[0] = 0.125 * (2.0 + x[0]) * x[1] * x[1];
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du[1] = -0.125 * (2.0 + x[0]) * x[0] * x[1];
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du[2] = -2.0 * x[2];
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}
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void fluxExact(const Vector &x, Vector &f)
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{
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f.SetSize(3);
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DenseMatrix s(3);
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sigmaFunc(x, s);
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Vector du(3);
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duExact(x, du);
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s.Mult(du, f);
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f *= -1.0;
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}
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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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int order = 3;
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int mesh_type = 4; // Default to Quadrilateral mesh
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int mesh_order = 3;
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int ref_levels = 0;
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bool static_cond = false;
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bool visualization = true;
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OptionsParser args(argc, argv);
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args.AddOption(&mesh_type, "-mt", "--mesh-type",
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"Mesh type: 3 - Triangular, 4 - Quadrilateral.");
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args.AddOption(&mesh_order, "-mo", "--mesh-order",
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"Geometric order of the curved mesh.");
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args.AddOption(&ref_levels, "-r", "--refine",
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"Number of times to refine the mesh uniformly in serial.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
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"--no-static-condensation", "Enable static condensation.");
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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.ParseCheck();
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// 2. Construct a quadrilateral or triangular mesh with the topology of a
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// cylindrical surface.
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Mesh *mesh = GetMesh(mesh_type);
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int dim = mesh->Dimension();
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// 3. Refine the mesh to increase the resolution. In this example we do
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// 'ref_levels' of uniform refinement.
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for (int l = 0; l < ref_levels; l++)
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{
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mesh->UniformRefinement();
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}
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// 4. Transform the mesh so that it has a more interesting geometry.
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mesh->SetCurvature(mesh_order);
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mesh->Transform(trans);
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// 5. Define a finite element space on the mesh. Here we use continuous
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// Lagrange finite elements of the specified order.
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H1_FECollection fec(order, dim);
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FiniteElementSpace fespace(mesh, &fec);
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cout << "Number of finite element unknowns: "
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<< fespace.GetTrueVSize() << endl;
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// 6. Determine the list of true (i.e. conforming) essential boundary dofs.
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// In this example, the boundary conditions are defined by marking all
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// the boundary attributes from the mesh as essential (Dirichlet) and
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// converting them to a list of 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 the finite element 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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GridFunction x(&fespace);
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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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MatrixFunctionCoefficient sigma(3, sigmaFunc);
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BilinearFormIntegrator *integ = new DiffusionIntegrator(sigma);
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a.AddDomainIntegrator(integ);
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// 10. Assemble the bilinear form and the corresponding linear system,
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// applying any necessary transformations such as: eliminating boundary
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// conditions, applying conforming constraints for non-conforming AMR,
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// static condensation, etc.
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if (static_cond) { a.EnableStaticCondensation(); }
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a.Assemble();
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OperatorPtr A;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
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cout << "Size of linear system: " << A->Height() << endl;
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// 11. Solve the linear system A X = B.
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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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// 12. Recover the solution as a finite element grid function.
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a.RecoverFEMSolution(X, b, x);
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// 13. Compute error in the solution and its flux
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FunctionCoefficient uCoef(uExact);
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real_t error = x.ComputeL2Error(uCoef);
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cout << "|u - u_h|_2 = " << error << endl;
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FiniteElementSpace flux_fespace(mesh, &fec, 3);
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GridFunction flux(&flux_fespace);
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x.ComputeFlux(*integ, flux); flux *= -1.0;
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VectorFunctionCoefficient fluxCoef(3, fluxExact);
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real_t flux_err = flux.ComputeL2Error(fluxCoef);
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cout << "|f - f_h|_2 = " << flux_err << endl;
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// 14. Save the refined mesh and the solution. This output can be viewed
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// later using GLVis: "glvis -m refined.mesh -g sol.gf".
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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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x.Save(sol_ofs);
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// 15. 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 << x
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<< "window_title 'Solution'\n" << flush;
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socketstream flux_sock(vishost, visport);
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flux_sock.precision(8);
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flux_sock << "solution\n" << *mesh << flux
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<< "keys vvv\n"
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<< "window_geometry 402 0 400 350\n"
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<< "window_title 'Flux'\n" << flush;
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}
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// 16. Free the used memory.
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delete mesh;
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return 0;
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}
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// Defines a mesh consisting of four flat rectangular surfaces connected to form
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// a loop.
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Mesh * GetMesh(int type)
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{
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Mesh * mesh = NULL;
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if (type == 3)
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{
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mesh = new Mesh(2, 12, 16, 8, 3);
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mesh->AddVertex(-1.0, -1.0, 0.0);
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mesh->AddVertex( 1.0, -1.0, 0.0);
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mesh->AddVertex( 1.0, 1.0, 0.0);
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mesh->AddVertex(-1.0, 1.0, 0.0);
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mesh->AddVertex(-1.0, -1.0, 1.0);
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mesh->AddVertex( 1.0, -1.0, 1.0);
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mesh->AddVertex( 1.0, 1.0, 1.0);
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mesh->AddVertex(-1.0, 1.0, 1.0);
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mesh->AddVertex( 0.0, -1.0, 0.5);
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mesh->AddVertex( 1.0, 0.0, 0.5);
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mesh->AddVertex( 0.0, 1.0, 0.5);
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mesh->AddVertex(-1.0, 0.0, 0.5);
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mesh->AddTriangle(0, 1, 8);
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mesh->AddTriangle(1, 5, 8);
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mesh->AddTriangle(5, 4, 8);
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mesh->AddTriangle(4, 0, 8);
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mesh->AddTriangle(1, 2, 9);
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mesh->AddTriangle(2, 6, 9);
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mesh->AddTriangle(6, 5, 9);
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mesh->AddTriangle(5, 1, 9);
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mesh->AddTriangle(2, 3, 10);
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mesh->AddTriangle(3, 7, 10);
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mesh->AddTriangle(7, 6, 10);
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mesh->AddTriangle(6, 2, 10);
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mesh->AddTriangle(3, 0, 11);
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mesh->AddTriangle(0, 4, 11);
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mesh->AddTriangle(4, 7, 11);
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mesh->AddTriangle(7, 3, 11);
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mesh->AddBdrSegment(0, 1, 1);
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mesh->AddBdrSegment(1, 2, 1);
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mesh->AddBdrSegment(2, 3, 1);
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mesh->AddBdrSegment(3, 0, 1);
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mesh->AddBdrSegment(5, 4, 2);
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mesh->AddBdrSegment(6, 5, 2);
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mesh->AddBdrSegment(7, 6, 2);
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mesh->AddBdrSegment(4, 7, 2);
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}
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else if (type == 4)
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{
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mesh = new Mesh(2, 8, 4, 8, 3);
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mesh->AddVertex(-1.0, -1.0, 0.0);
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mesh->AddVertex( 1.0, -1.0, 0.0);
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mesh->AddVertex( 1.0, 1.0, 0.0);
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mesh->AddVertex(-1.0, 1.0, 0.0);
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mesh->AddVertex(-1.0, -1.0, 1.0);
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mesh->AddVertex( 1.0, -1.0, 1.0);
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mesh->AddVertex( 1.0, 1.0, 1.0);
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mesh->AddVertex(-1.0, 1.0, 1.0);
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mesh->AddQuad(0, 1, 5, 4);
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mesh->AddQuad(1, 2, 6, 5);
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mesh->AddQuad(2, 3, 7, 6);
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mesh->AddQuad(3, 0, 4, 7);
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mesh->AddBdrSegment(0, 1, 1);
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mesh->AddBdrSegment(1, 2, 1);
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mesh->AddBdrSegment(2, 3, 1);
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mesh->AddBdrSegment(3, 0, 1);
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mesh->AddBdrSegment(5, 4, 2);
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mesh->AddBdrSegment(6, 5, 2);
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mesh->AddBdrSegment(7, 6, 2);
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mesh->AddBdrSegment(4, 7, 2);
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}
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else
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{
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MFEM_ABORT("Unrecognized mesh type " << type << "!");
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}
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mesh->FinalizeTopology();
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return mesh;
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}
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// Transforms the four-sided loop into a curved cylinder with skewed top and
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// base.
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void trans(const Vector &x, Vector &r)
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{
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r.SetSize(3);
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real_t tol = 1e-6;
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real_t theta = 0.0;
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if (fabs(x[1] + 1.0) < tol)
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{
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theta = 0.25 * M_PI * (x[0] - 2.0);
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}
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else if (fabs(x[0] - 1.0) < tol)
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{
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theta = 0.25 * M_PI * x[1];
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}
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else if (fabs(x[1] - 1.0) < tol)
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{
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theta = 0.25 * M_PI * (2.0 - x[0]);
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}
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else if (fabs(x[0] + 1.0) < tol)
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{
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theta = 0.25 * M_PI * (4.0 - x[1]);
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}
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else
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{
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cerr << "side not recognized "
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<< x[0] << " " << x[1] << " " << x[2] << endl;
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}
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r[0] = cos(theta);
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r[1] = sin(theta);
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r[2] = 0.25 * (2.0 * x[2] - 1.0) * (r[0] + 2.0);
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}
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// Anisotropic diffusion coefficient
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void sigmaFunc(const Vector &x, DenseMatrix &s)
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{
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s.SetSize(3);
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real_t a = 17.0 - 2.0 * x[0] * (1.0 + x[0]);
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s(0,0) = 0.5 + x[0] * x[0] * (8.0 / a - 0.5);
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s(0,1) = x[0] * x[1] * (8.0 / a - 0.5);
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s(0,2) = 0.0;
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s(1,0) = s(0,1);
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s(1,1) = 0.5 * x[0] * x[0] + 8.0 * x[1] * x[1] / a;
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s(1,2) = 0.0;
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s(2,0) = 0.0;
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s(2,1) = 0.0;
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s(2,2) = a / 32.0;
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}
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