ex3.cpp

//                                MFEM Example 3
//
// Compile with: make ex3
//
// Sample runs:  ex3 -m ../data/star.mesh
//               ex3 -m ../data/beam-tri.mesh -o 2
//               ex3 -m ../data/beam-tet.mesh
//               ex3 -m ../data/beam-hex.mesh
//               ex3 -m ../data/beam-hex.mesh -o 2 -pa
//               ex3 -m ../data/escher.mesh
//               ex3 -m ../data/escher.mesh -o 2
//               ex3 -m ../data/fichera.mesh
//               ex3 -m ../data/fichera-q2.vtk
//               ex3 -m ../data/fichera-q3.mesh
//               ex3 -m ../data/square-disc-nurbs.mesh
//               ex3 -m ../data/beam-hex-nurbs.mesh
//               ex3 -m ../data/amr-hex.mesh
//               ex3 -m ../data/fichera-amr.mesh
//               ex3 -m ../data/ref-prism.mesh -o 1
//               ex3 -m ../data/octahedron.mesh -o 1
//               ex3 -m ../data/star-surf.mesh -o 1
//               ex3 -m ../data/mobius-strip.mesh -f 0.1
//               ex3 -m ../data/klein-bottle.mesh -f 0.1
//
// Device sample runs:
//               ex3 -m ../data/star.mesh -pa -d cuda
//               ex3 -m ../data/star.mesh -pa -d raja-cuda
//               ex3 -m ../data/star.mesh -pa -d raja-omp
//               ex3 -m ../data/beam-hex.mesh -pa -d cuda
//
// Description:  This example code solves a simple electromagnetic diffusion
//               problem corresponding to the second order definite Maxwell
//               equation curl curl E + E = f with boundary condition
//               E x n = <given tangential field>. Here, we use a given exact
//               solution E and compute the corresponding r.h.s. f.
//               We discretize with Nedelec finite elements in 2D or 3D.
//
//               The example demonstrates the use of H(curl) finite element
//               spaces with the curl-curl and the (vector finite element) mass
//               bilinear form, as well as the computation of discretization
//               error when the exact solution is known. Static condensation is
//               also illustrated.
//
//               We recommend viewing examples 1-2 before viewing this example.

#include "mfem.hpp"
#include <fstream>
#include <iostream>

using namespace std;
using namespace mfem;

// Exact solution, E, and r.h.s., f. See below for implementation.
void E_exact(const Vector &, Vector &);
void f_exact(const Vector &, Vector &);
real_t freq = 1.0, kappa;
int dim;

int main(int argc, char *argv[])
{
   // 1. Parse command-line options.
   const char *mesh_file = "../data/beam-tet.mesh";
   int order = 1;
   bool static_cond = false;
   bool pa = false;
   bool nc = false;
   const char *device_config = "cpu";
   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(&freq, "-f", "--frequency", "Set the frequency for the exact"
                  " solution.");
   args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
                  "--no-static-condensation", "Enable static condensation.");
   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
                  "--no-partial-assembly", "Enable Partial Assembly.");
   args.AddOption(&nc, "-nc", "--non-conforming", "-c",
                  "--conforming",
                  "Mark the mesh as nonconforming before partitioning.");
   args.AddOption(&device_config, "-d", "--device",
                  "Device configuration string, see Device::Configure().");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(cout);
      return 1;
   }
   args.PrintOptions(cout);
   kappa = freq * M_PI;

   // 2. Enable hardware devices such as GPUs, and programming models such as
   //    CUDA, OCCA, RAJA and OpenMP based on command line options.
   Device device(device_config);
   device.Print();

   // 3. Read the mesh from the given mesh file. We can handle triangular,
   //    quadrilateral, tetrahedral, hexahedral, surface and volume meshes with
   //    the same code.
   Mesh *mesh = new Mesh(mesh_file, 1, 1);
   dim = mesh->Dimension();
   int sdim = mesh->SpaceDimension();
   if (nc)
   {
      // Can set to false to use conformal refinement for simplices.
      mesh->EnsureNCMesh(true);
   }

   // 4. Refine the mesh to increase the resolution. In this example we do
   //    'ref_levels' of uniform refinement. We choose 'ref_levels' to be the
   //    largest number that gives a final mesh with no more than 50,000
   //    elements.
   {
      int ref_levels =
         (int)floor(log(50000./mesh->GetNE())/log(2.)/dim);
      for (int l = 0; l < ref_levels; l++)
      {
         mesh->UniformRefinement();
      }
   }

   // 5. Define a finite element space on the mesh. Here we use the Nedelec
   //    finite elements of the specified order.
   FiniteElementCollection *fec = new ND_FECollection(order, dim);
   FiniteElementSpace *fespace = new FiniteElementSpace(mesh, fec);
   cout << "Number of finite element unknowns: "
        << fespace->GetTrueVSize() << endl;

   // 6. Determine the list of true (i.e. conforming) essential boundary dofs.
   //    In this example, the boundary conditions are defined by marking all
   //    the boundary attributes from the mesh as essential (Dirichlet) and
   //    converting them to a list of true dofs.
   Array<int> ess_tdof_list;
   if (mesh->bdr_attributes.Size())
   {
      Array<int> ess_bdr(mesh->bdr_attributes.Max());
      ess_bdr = 1;
      fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
   }

   // 7. Set up the linear form b(.) which corresponds to the right-hand side
   //    of the FEM linear system, which in this case is (f,phi_i) where f is
   //    given by the function f_exact and phi_i are the basis functions in the
   //    finite element fespace.
   VectorFunctionCoefficient f(sdim, f_exact);
   LinearForm *b = new LinearForm(fespace);
   b->AddDomainIntegrator(new VectorFEDomainLFIntegrator(f));
   b->Assemble();

   // 8. Define the solution vector x as a finite element grid function
   //    corresponding to fespace. Initialize x by projecting the exact
   //    solution. Note that only values from the boundary edges will be used
   //    when eliminating the non-homogeneous boundary condition to modify the
   //    r.h.s. vector b.
   GridFunction x(fespace);
   VectorFunctionCoefficient E(sdim, E_exact);
   x.ProjectCoefficient(E);

   // 9. Set up the bilinear form corresponding to the EM diffusion operator
   //    curl muinv curl + sigma I, by adding the curl-curl and the mass domain
   //    integrators.
   Coefficient *muinv = new ConstantCoefficient(1.0);
   Coefficient *sigma = new ConstantCoefficient(1.0);
   BilinearForm *a = new BilinearForm(fespace);
   if (pa) { a->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
   a->AddDomainIntegrator(new CurlCurlIntegrator(*muinv));
   a->AddDomainIntegrator(new VectorFEMassIntegrator(*sigma));

   // 10. Assemble the bilinear form and the corresponding linear system,
   //     applying any necessary transformations such as: eliminating boundary
   //     conditions, applying conforming constraints for non-conforming AMR,
   //     static condensation, etc.
   if (static_cond) { a->EnableStaticCondensation(); }
   a->Assemble();

   OperatorPtr A;
   Vector B, X;
   a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);

   cout << "Size of linear system: " << A->Height() << endl;

   // 11. Solve the linear system A X = B.
   if (pa) // Jacobi preconditioning in partial assembly mode
   {
      OperatorJacobiSmoother M(*a, ess_tdof_list);
      PCG(*A, M, B, X, 1, 1000, 1e-12, 0.0);
   }
   else
   {
#ifndef MFEM_USE_SUITESPARSE
      // 11. Define a simple symmetric Gauss-Seidel preconditioner and use it to
      //     solve the system Ax=b with PCG.
      GSSmoother M((SparseMatrix&)(*A));
      PCG(*A, M, B, X, 1, 500, 1e-12, 0.0);
#else
      // 11. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the
      //     system.
      UMFPackSolver umf_solver;
      umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
      umf_solver.SetOperator(*A);
      umf_solver.Mult(B, X);
#endif
   }

   // 12. Recover the solution as a finite element grid function.
   a->RecoverFEMSolution(X, *b, x);

   // 13. Compute and print the L^2 norm of the error.
   cout << "\n|| E_h - E ||_{L^2} = " << x.ComputeL2Error(E) << '\n' << endl;

   // 14. Save the refined mesh and the solution. This output can be viewed
   //     later using GLVis: "glvis -m refined.mesh -g sol.gf".
   {
      ofstream mesh_ofs("refined.mesh");
      mesh_ofs.precision(8);
      mesh->Print(mesh_ofs);
      ofstream sol_ofs("sol.gf");
      sol_ofs.precision(8);
      x.Save(sol_ofs);
   }

   // 15. Send the solution by socket to a GLVis server.
   if (visualization)
   {
      char vishost[] = "localhost";
      int  visport   = 19916;
      socketstream sol_sock(vishost, visport);
      sol_sock.precision(8);
      sol_sock << "solution\n" << *mesh << x << flush;
   }

   // 16. Free the used memory.
   delete a;
   delete sigma;
   delete muinv;
   delete b;
   delete fespace;
   delete fec;
   delete mesh;

   return 0;
}


void E_exact(const Vector &x, Vector &E)
{
   if (dim == 3)
   {
      E(0) = sin(kappa * x(1));
      E(1) = sin(kappa * x(2));
      E(2) = sin(kappa * x(0));
   }
   else
   {
      E(0) = sin(kappa * x(1));
      E(1) = sin(kappa * x(0));
      if (x.Size() == 3) { E(2) = 0.0; }
   }
}

void f_exact(const Vector &x, Vector &f)
{
   if (dim == 3)
   {
      f(0) = (1. + kappa * kappa) * sin(kappa * x(1));
      f(1) = (1. + kappa * kappa) * sin(kappa * x(2));
      f(2) = (1. + kappa * kappa) * sin(kappa * x(0));
   }
   else
   {
      f(0) = (1. + kappa * kappa) * sin(kappa * x(1));
      f(1) = (1. + kappa * kappa) * sin(kappa * x(0));
      if (x.Size() == 3) { f(2) = 0.0; }
   }
}