ex15.cpp

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//                                MFEM Example 15
//
// Compile with: make ex15
//
// Sample runs:  ex15
//               ex15 -o 1 -y 0.4
//               ex15 -o 4 -y 0.1
//               ex15 -n 5
//               ex15 -p 1 -n 3
//
//               Other meshes:
//
//               ex15 -m ../data/square-disc-nurbs.mesh
//               ex15 -m ../data/disc-nurbs.mesh
//               ex15 -m ../data/fichera.mesh -tf 0.3
//               ex15 -m ../data/ball-nurbs.mesh -tf 0.3
//               ex15 -m ../data/mobius-strip.mesh
//               ex15 -m ../data/amr-quad.mesh
//               ex15 -m ../data/square-disc.mesh
//               ex15 -m ../data/escher.mesh -r 2 -tf 0.3
//
//               Kelly estimator:
//
//               ex15 -est 1 -e 0.0001
//               ex15 -est 1 -o 1 -y 0.4
//               ex15 -est 1 -o 4 -y 0.1
//               ex15 -est 1 -n 5
//               ex15 -est 1 -p 1 -n 3
//
// 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 simple ZZ or Kelly error estimator.  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.
//
//               The example demonstrates MFEM's capability to refine and
//               derefine 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 UpdateProblem(Mesh &mesh, FiniteElementSpace &fespace,
                   GridFunction &x, BilinearForm &a, LinearForm &b);
 
 
int main(int argc, char *argv[])
{
   // 1. 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 = 5.0e-3;
   real_t hysteresis = 0.15; // 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 = ZZ, 1 = Kelly. Defaults to ZZ.");
   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())
   {
      args.PrintUsage(cout);
      return 1;
   }
   args.PrintOptions(cout);
 
   // 2. Read the 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(mesh_file, 1, 1);
   int dim = mesh.Dimension();
   int sdim = mesh.SpaceDimension();
 
   // 3. 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);
 
   // 4. All boundary attributes will be used for essential (Dirichlet) BC.
   MFEM_VERIFY(mesh.bdr_attributes.Size() > 0,
               "Boundary attributes required in the mesh.");
   Array<int> ess_bdr(mesh.bdr_attributes.Max());
   ess_bdr = 1;
 
   // 5. 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);
 
   // 6. 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.
   BilinearForm a(&fespace);
   LinearForm 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));
 
   // 7. The solution vector x and the associated finite element grid function
   //    will be maintained over the AMR iterations.
   GridFunction x(&fespace);
 
   // 8. Connect to GLVis. Prepare for VisIt output.
   char vishost[] = "localhost";
   int  visport   = 19916;
 
   socketstream sout;
   if (visualization)
   {
      sout.open(vishost, visport);
      if (!sout)
      {
         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", &mesh);
   visit_dc.RegisterField("solution", &x);
   int vis_cycle = 0;
 
   // 9. As in Example 6, we set up an estimator that will be used to obtain
   //    element error indicators. The integrator needs to provide the method
   //    ComputeElementFlux. The smoothed flux space is a vector valued H1 (ZZ)
   //    or L2 (Kelly) space here.
   L2_FECollection flux_fec(order, dim);
   ErrorEstimator* estimator{nullptr};
 
   switch (which_estimator)
   {
      case 1:
      {
         auto flux_fes = new FiniteElementSpace(&mesh, &flux_fec, sdim);
         estimator = new KellyErrorEstimator(*integ, x, flux_fes);
         break;
      }
 
      default:
         std::cout << "Unknown estimator. Falling back to ZZ." << std::endl;
      case 0:
      {
         auto flux_fes = new FiniteElementSpace(&mesh, &fec, sdim);
         estimator = new ZienkiewiczZhuEstimator(*integ, x, flux_fes);
         break;
      }
   }
 
   // 10. As in Example 6, 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);
 
   // 11. 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);
 
   // 12. 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.
   x = 0.0;
   for (real_t time = 0.0; time < t_final + 1e-10; time += 0.01)
   {
      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();
 
      // 13. 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++)
      {
         cout << "Iteration: " << ref_it << ", number of unknowns: "
              << fespace.GetVSize() << endl;
 
         // 14. Recompute the field on the current mesh: assemble the stiffness
         //     matrix and the right-hand side.
         a.Assemble();
         b.Assemble();
 
         // 15. Project the exact solution to the essential boundary DOFs.
         x.ProjectBdrCoefficient(bdr, ess_bdr);
 
         // 16. Create and solve the linear system.
         Array<int> ess_tdof_list;
         fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
 
         SparseMatrix A;
         Vector B, X;
         a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
 
#ifndef MFEM_USE_SUITESPARSE
         GSSmoother M(A);
         PCG(A, M, B, X, 0, 500, 1e-12, 0.0);
#else
         UMFPackSolver umf_solver;
         umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
         umf_solver.SetOperator(A);
         umf_solver.Mult(B, X);
#endif
 
         // 17. Extract the local solution on each processor.
         a.RecoverFEMSolution(X, b, x);
 
         // 18. Send the solution by socket to a GLVis server and optionally
         //     save it in VisIt format.
         if (visualization)
         {
            sout.precision(8);
            sout << "solution\n" << mesh << x << flush;
         }
         if (visit)
         {
            visit_dc.SetCycle(vis_cycle++);
            visit_dc.SetTime(time);
            visit_dc.Save();
         }
 
         // 19. 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(mesh);
         if (refiner.Stop())
         {
            break;
         }
 
         // 20. Update the space and interpolate the solution.
         UpdateProblem(mesh, fespace, x, a, b);
      }
 
      // 21. 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(mesh))
      {
         cout << "\nDerefined elements." << endl;
 
         // 22. Update the space and interpolate the solution.
         UpdateProblem(mesh, fespace, x, a, b);
      }
 
      a.Update();
      b.Update();
   }
 
   delete estimator;
 
   return 0;
}
 
 
void UpdateProblem(Mesh &mesh, FiniteElementSpace &fespace,
                   GridFunction &x, BilinearForm &a, LinearForm &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();
 
   // Free any transformation matrices to save memory.
   fespace.UpdatesFinished();
 
   // Inform the linear and bilinear forms that the space has changed.
   a.Update();
   b.Update();
}
 
 
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);
}