mfem/examples/ex15p.cpp

//                       MFEM Example 15 - Parallel Version
//
// Compile with: make ex15p
//
// Sample runs:  mpirun -np 4 ex15p
//               mpirun -np 4 ex15p -o 1 -y 0.2
//               mpirun -np 4 ex15p -o 4 -y 0.1
//               mpirun -np 4 ex15p -n 5
//               mpirun -np 4 ex15p -p 1 -n 3
//
//               Other meshes:
//
//               mpirun -np 4 ex15p -m ../data/square-disc-nurbs.mesh
//               mpirun -np 4 ex15p -m ../data/disc-nurbs.mesh
//               mpirun -np 4 ex15p -m ../data/fichera.mesh -tf 0.5
//               mpirun -np 4 ex15p -m ../data/fichera-mixed.mesh -tf 0.5
//               mpirun -np 4 ex15p -m ../data/ball-nurbs.mesh -tf 0.5
//               mpirun -np 4 ex15p -m ../data/mobius-strip.mesh
//               mpirun -np 4 ex15p -m ../data/amr-quad.mesh
//               mpirun -np 4 ex15p -m ../data/square-disc.mesh
//               mpirun -np 4 ex15p -m ../data/escher.mesh -r 2 -tf 0.3
//
//               Different estimators:
//
//               mpirun -np 4 ex15p -est 0 -e 1e-4
//               mpirun -np 4 ex15p -est 1 -e 1e-6
//               mpirun -np 4 ex15p -est 1 -o 3 -tf 0.3
//               mpirun -np 4 ex15p -est 2 -o 2
//
// Description:  Building on Example 6, this example demonstrates dynamic AMR.
//               The mesh is adapted to a time-dependent solution by refinement
//               as well as by derefinement. For simplicity, the solution is
//               prescribed and no time integration is done. However, the error
//               estimation and refinement/derefinement decisions are realistic.
//
//               At each outer iteration the right hand side function is changed
//               to mimic a time dependent problem.  Within each inner iteration
//               the problem is solved on a sequence of meshes which are locally
//               refined according to a chosen error estimator. Currently there
//               are three error estimators supported: A L2 formulation of the
//               Zienkiewicz-Zhu error estimator (0), a Kelly error indicator (1)
//               and a traditional Zienkiewicz-Zhu error estimator (2). At the
//               end of the inner iteration the error estimates are also used to
//               identify any elements which may be over-refined and a single
//               derefinement step is performed.  After each refinement or
//               derefinement step a rebalance operation is performed to keep
//               the mesh evenly distributed among the available processors.
//
//               The example demonstrates MFEM's capability to refine, derefine
//               and load balance nonconforming meshes, in 2D and 3D, and on
//               linear, curved and surface meshes. Interpolation of functions
//               between coarse and fine meshes, persistent GLVis visualization,
//               and saving of time-dependent fields for external visualization
//               with VisIt (visit.llnl.gov) are also illustrated.
//
//               We recommend viewing Examples 1, 6 and 9 before viewing this
//               example.

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

using namespace std;
using namespace mfem;

// Choices for the problem setup. Affect bdr_func and rhs_func.
int problem;
int nfeatures;

// Prescribed time-dependent boundary and right-hand side functions.
real_t bdr_func(const Vector &pt, real_t t);
real_t rhs_func(const Vector &pt, real_t t);

// Update the finite element space, interpolate the solution and perform
// parallel load balancing.
void UpdateAndRebalance(ParMesh &pmesh, ParFiniteElementSpace &fespace,
                        ParGridFunction &x, ParBilinearForm &a,
                        ParLinearForm &b);


int main(int argc, char *argv[])
{
   // 1. Initialize MPI and HYPRE.
   Mpi::Init(argc, argv);
   int num_procs = Mpi::WorldSize();
   int myid = Mpi::WorldRank();
   Hypre::Init();

   // 2. Parse command-line options.
   problem = 0;
   nfeatures = 1;
   const char *mesh_file = "../data/star-hilbert.mesh";
   int order = 2;
   real_t t_final = 1.0;
   real_t max_elem_error = 1.0e-4;
   real_t hysteresis = 0.25; // derefinement safety coefficient
   int ref_levels = 0;
   int nc_limit = 3;         // maximum level of hanging nodes
   bool visualization = true;
   bool visit = false;
   int which_estimator = 0;

   OptionsParser args(argc, argv);
   args.AddOption(&mesh_file, "-m", "--mesh",
                  "Mesh file to use.");
   args.AddOption(&problem, "-p", "--problem",
                  "Problem setup to use: 0 = spherical front, 1 = ball.");
   args.AddOption(&nfeatures, "-n", "--nfeatures",
                  "Number of solution features (fronts/balls).");
   args.AddOption(&order, "-o", "--order",
                  "Finite element order (polynomial degree).");
   args.AddOption(&max_elem_error, "-e", "--max-err",
                  "Maximum element error");
   args.AddOption(&hysteresis, "-y", "--hysteresis",
                  "Derefinement safety coefficient.");
   args.AddOption(&ref_levels, "-r", "--ref-levels",
                  "Number of initial uniform refinement levels.");
   args.AddOption(&nc_limit, "-l", "--nc-limit",
                  "Maximum level of hanging nodes.");
   args.AddOption(&t_final, "-tf", "--t-final",
                  "Final time; start time is 0.");
   args.AddOption(&which_estimator, "-est", "--estimator",
                  "Which estimator to use: "
                  "0 = L2ZZ, 1 = Kelly, 2 = ZZ. Defaults to L2ZZ.");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.AddOption(&visit, "-visit", "--visit-datafiles", "-no-visit",
                  "--no-visit-datafiles",
                  "Save data files for VisIt (visit.llnl.gov) visualization.");
   args.Parse();
   if (!args.Good())
   {
      if (myid == 0)
      {
         args.PrintUsage(cout);
      }
      return 1;
   }
   if (myid == 0)
   {
      args.PrintOptions(cout);
   }

   // 3. Read the (serial) mesh from the given mesh file on all processors.  We
   //    can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
   //    and volume meshes with the same code.
   Mesh *mesh = new Mesh(mesh_file, 1, 1);
   int dim = mesh->Dimension();
   int sdim = mesh->SpaceDimension();

   // 4. Project a NURBS mesh to a piecewise-quadratic curved mesh. Make sure
   //    that the mesh is non-conforming if it has quads or hexes and refine it.
   if (mesh->NURBSext)
   {
      mesh->UniformRefinement();
      if (ref_levels > 0) { ref_levels--; }
      mesh->SetCurvature(2);
   }
   mesh->EnsureNCMesh(true);
   for (int l = 0; l < ref_levels; l++)
   {
      mesh->UniformRefinement();
   }
   // Make sure tet-only meshes are marked for local refinement.
   mesh->Finalize(true);

   // 5. Define a parallel mesh by partitioning the serial mesh.  Once the
   //    parallel mesh is defined, the serial mesh can be deleted.
   ParMesh pmesh(MPI_COMM_WORLD, *mesh);
   delete mesh;

   MFEM_VERIFY(pmesh.bdr_attributes.Size() > 0,
               "Boundary attributes required in the mesh.");
   Array<int> ess_bdr(pmesh.bdr_attributes.Max());
   ess_bdr = 1;

   // 6. 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);
   ParFiniteElementSpace fespace(&pmesh, &fec);

   // 7. As in Example 1p, we set up bilinear and linear forms corresponding to
   //    the Laplace problem -\Delta u = 1. We don't assemble the discrete
   //    problem yet, this will be done in the inner loop.
   ParBilinearForm a(&fespace);
   ParLinearForm b(&fespace);

   ConstantCoefficient one(1.0);
   FunctionCoefficient bdr(bdr_func);
   FunctionCoefficient rhs(rhs_func);

   BilinearFormIntegrator *integ = new DiffusionIntegrator(one);
   a.AddDomainIntegrator(integ);
   b.AddDomainIntegrator(new DomainLFIntegrator(rhs));

   // 8. The solution vector x and the associated finite element grid function
   //    will be maintained over the AMR iterations.
   ParGridFunction x(&fespace);

   // 9. Connect to GLVis. Prepare for VisIt output.
   char vishost[] = "localhost";
   int  visport   = 19916;

   socketstream sout;
   if (visualization)
   {
      sout.open(vishost, visport);
      if (!sout)
      {
         if (myid == 0)
         {
            cout << "Unable to connect to GLVis server at "
                 << vishost << ':' << visport << endl;
            cout << "GLVis visualization disabled.\n";
         }
         visualization = false;
      }
      sout.precision(8);
   }

   VisItDataCollection visit_dc("Example15-Parallel", &pmesh);
   visit_dc.RegisterField("solution", &x);
   int vis_cycle = 0;

   // 10. As in Example 6p, we set up an estimator that will be used to obtain
   //     element error indicators. The integrator needs to provide the method
   //     ComputeElementFlux. We supply an L2 space for the discontinuous flux
   //     and an H(div) space for the smoothed flux.
   L2_FECollection flux_fec(order, dim);
   RT_FECollection smooth_flux_fec(order-1, dim);
   ErrorEstimator* estimator{nullptr};

   switch (which_estimator)
   {
      case 1:
      {
         auto flux_fes = new ParFiniteElementSpace(&pmesh, &flux_fec, sdim);
         estimator = new KellyErrorEstimator(*integ, x, flux_fes);
         break;
      }
      case 2:
      {
         auto flux_fes = new ParFiniteElementSpace(&pmesh, &fec, sdim);
         estimator = new ZienkiewiczZhuEstimator(*integ, x, flux_fes);
         break;
      }

      default:
         if (myid == 0)
         {
            std::cout << "Unknown estimator. Falling back to L2ZZ." << std::endl;
         }
      case 0:
      {
         auto flux_fes = new ParFiniteElementSpace(&pmesh, &flux_fec, sdim);
         auto smooth_flux_fes = new ParFiniteElementSpace(&pmesh, &smooth_flux_fec);
         estimator = new L2ZienkiewiczZhuEstimator(*integ, x, flux_fes, smooth_flux_fes);
         break;
      }
   }

   // 11. As in Example 6p, we also need a refiner. This time the refinement
   //     strategy is based on a fixed threshold that is applied locally to each
   //     element. The global threshold is turned off by setting the total error
   //     fraction to zero. We also enforce a maximum refinement ratio between
   //     adjacent elements.
   ThresholdRefiner refiner(*estimator);
   refiner.SetTotalErrorFraction(0.0); // use purely local threshold
   refiner.SetLocalErrorGoal(max_elem_error);
   refiner.PreferConformingRefinement();
   refiner.SetNCLimit(nc_limit);

   // 12. A derefiner selects groups of elements that can be coarsened to form
   //     a larger element. A conservative enough threshold needs to be set to
   //     prevent derefining elements that would immediately be refined again.
   ThresholdDerefiner derefiner(*estimator);
   derefiner.SetThreshold(hysteresis * max_elem_error);
   derefiner.SetNCLimit(nc_limit);

   // 13. The outer time loop. In each iteration we update the right hand side,
   //     solve the problem on the current mesh, visualize the solution and
   //     refine the mesh as many times as necessary. Then we derefine any
   //     elements which have very small errors.
   for (real_t time = 0.0; time < t_final + 1e-10; time += 0.01)
   {
      if (myid == 0)
      {
         cout << "\nTime " << time << "\n\nRefinement:" << endl;
      }

      // Set the current time in the coefficients
      bdr.SetTime(time);
      rhs.SetTime(time);

      // Make sure errors will be recomputed in the following.
      refiner.Reset();
      derefiner.Reset();

      // 14. The inner refinement loop. At the end we want to have the current
      //     time step resolved to the prescribed tolerance in each element.
      for (int ref_it = 1; ; ref_it++)
      {
         HYPRE_BigInt global_dofs = fespace.GlobalTrueVSize();
         if (myid == 0)
         {
            cout << "Iteration: " << ref_it << ", number of unknowns: "
                 << global_dofs << flush;
         }

         // 15. Recompute the field on the current mesh: assemble the stiffness
         //     matrix and the right-hand side.
         a.Assemble();
         b.Assemble();

         // 16. Project the exact solution to the essential DOFs.
         x.ProjectBdrCoefficient(bdr, ess_bdr);

         // 17. Create and solve the parallel linear system.
         Array<int> ess_tdof_list;
         fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);

         HypreParMatrix A;
         Vector B, X;
         a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);

         HypreBoomerAMG amg(A);
         amg.SetPrintLevel(0);
         HyprePCG pcg(A);
         pcg.SetTol(1e-12);
         pcg.SetMaxIter(200);
         pcg.SetPrintLevel(0);
         pcg.SetPreconditioner(amg);
         pcg.Mult(B, X);

         // 18. Extract the local solution on each processor.
         a.RecoverFEMSolution(X, b, x);

         // 19. Send the solution by socket to a GLVis server and optionally
         //     save it in VisIt format.
         if (visualization)
         {
            sout << "parallel " << num_procs << " " << myid << "\n";
            sout << "solution\n" << pmesh << x << flush;
         }
         if (visit)
         {
            visit_dc.SetCycle(vis_cycle++);
            visit_dc.SetTime(time);
            visit_dc.Save();
         }

         // 20. Apply the refiner on 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
         //     determines if all elements satisfy the local threshold.
         refiner.Apply(pmesh);
         if (myid == 0)
         {
            cout << ", total error: " << estimator->GetTotalError() << endl;
         }

         // 21. Quit the AMR loop if the termination criterion has been met
         if (refiner.Stop())
         {
            a.Update(); // Free the assembled data
            break;
         }

         // 22. Update the space, interpolate the solution, rebalance the mesh.
         UpdateAndRebalance(pmesh, fespace, x, a, b);
      }

      // 23. Use error estimates from the last inner iteration to check for
      //     possible derefinements. The derefiner works similarly as the
      //     refiner. The errors are not recomputed because the mesh did not
      //     change (and also the estimator was not Reset() at this time).
      if (derefiner.Apply(pmesh))
      {
         if (myid == 0)
         {
            cout << "\nDerefined elements." << endl;
         }

         // 24. Update the space and the solution, rebalance the mesh.
         UpdateAndRebalance(pmesh, fespace, x, a, b);
      }
   }

   delete estimator;

   // 25. Exit
   return 0;
}


void UpdateAndRebalance(ParMesh &pmesh, ParFiniteElementSpace &fespace,
                        ParGridFunction &x, ParBilinearForm &a,
                        ParLinearForm &b)
{
   // Update the space: recalculate the number of DOFs and construct a matrix
   // that will adjust any GridFunctions to the new mesh state.
   fespace.Update();

   // Interpolate the solution on the new mesh by applying the transformation
   // matrix computed in the finite element space. Multiple GridFunctions could
   // be updated here.
   x.Update();

   if (pmesh.Nonconforming())
   {
      // Load balance the mesh.
      pmesh.Rebalance();

      // Update the space again, this time a GridFunction redistribution matrix
      // is created. Apply it to the solution.
      fespace.Update();
      x.Update();
   }

   // Inform the linear and bilinear forms that the space has changed.
   a.Update();
   b.Update();

   // Free any transformation matrices to save memory.
   fespace.UpdatesFinished();
}


const real_t alpha = 0.02;

// Spherical front with a Gaussian cross section and radius t
real_t front(real_t x, real_t y, real_t z, real_t t, int)
{
   real_t r = sqrt(x*x + y*y + z*z);
   return exp(-0.5*pow((r - t)/alpha, 2));
}

real_t front_laplace(real_t x, real_t y, real_t z, real_t t, int dim)
{
   real_t x2 = x*x, y2 = y*y, z2 = z*z, t2 = t*t;
   real_t r = sqrt(x2 + y2 + z2);
   real_t a2 = alpha*alpha, a4 = a2*a2;
   return -exp(-0.5*pow((r - t)/alpha, 2)) / a4 *
          (-2*t*(x2 + y2 + z2 - (dim-1)*a2/2)/r + x2 + y2 + z2 + t2 - dim*a2);
}

// Smooth spherical step function with radius t
real_t ball(real_t x, real_t y, real_t z, real_t t, int)
{
   real_t r = sqrt(x*x + y*y + z*z);
   return -atan(2*(r - t)/alpha);
}

real_t ball_laplace(real_t x, real_t y, real_t z, real_t t, int dim)
{
   real_t x2 = x*x, y2 = y*y, z2 = z*z, t2 = 4*t*t;
   real_t r = sqrt(x2 + y2 + z2);
   real_t a2 = alpha*alpha;
   real_t den = pow(-a2 - 4*(x2 + y2 + z2 - 2*r*t) - t2, 2.0);
   return (dim == 2) ? 2*alpha*(a2 + t2 - 4*x2 - 4*y2)/r/den
          /*      */ : 4*alpha*(a2 + t2 - 4*r*t)/r/den;
}

// Composes several features into one function
template<typename F0, typename F1>
real_t composite_func(const Vector &pt, real_t t, F0 f0, F1 f1)
{
   int dim = pt.Size();
   real_t x = pt(0), y = pt(1), z = 0.0;
   if (dim == 3) { z = pt(2); }

   if (problem == 0)
   {
      if (nfeatures <= 1)
      {
         return f0(x, y, z, t, dim);
      }
      else
      {
         real_t sum = 0.0;
         for (int i = 0; i < nfeatures; i++)
         {
            real_t x0 = 0.5*cos(2*M_PI * i / nfeatures);
            real_t y0 = 0.5*sin(2*M_PI * i / nfeatures);
            sum += f0(x - x0, y - y0, z, t, dim);
         }
         return sum;
      }
   }
   else
   {
      real_t sum = 0.0;
      for (int i = 0; i < nfeatures; i++)
      {
         real_t x0 = 0.5*cos(2*M_PI * i / nfeatures + M_PI*t);
         real_t y0 = 0.5*sin(2*M_PI * i / nfeatures + M_PI*t);
         sum += f1(x - x0, y - y0, z, 0.25, dim);
      }
      return sum;
   }
}

// Exact solution, used for the Dirichlet BC.
real_t bdr_func(const Vector &pt, real_t t)
{
   return composite_func(pt, t, front, ball);
}

// Laplace of the exact solution, used for the right hand side.
real_t rhs_func(const Vector &pt, real_t t)
{
   return composite_func(pt, t, front_laplace, ball_laplace);
}