ex5p.cpp

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//                       MFEM Example 5 - Parallel Version
//
// Compile with: make ex5p
//
// Sample runs:  mpirun -np 4 ex5p -m ../data/square-disc.mesh
//               mpirun -np 4 ex5p -m ../data/star.mesh
//               mpirun -np 4 ex5p -m ../data/star.mesh -r 2 -pa
//               mpirun -np 4 ex5p -m ../data/beam-tet.mesh
//               mpirun -np 4 ex5p -m ../data/beam-hex.mesh
//               mpirun -np 4 ex5p -m ../data/beam-hex.mesh -pa
//               mpirun -np 4 ex5p -m ../data/escher.mesh
//               mpirun -np 4 ex5p -m ../data/fichera.mesh
//
// Device sample runs:
//               mpirun -np 4 ex5p -m ../data/star.mesh -r 2 -pa -d cuda
//               mpirun -np 4 ex5p -m ../data/star.mesh -r 2 -pa -d raja-cuda
//               mpirun -np 4 ex5p -m ../data/star.mesh -r 2 -pa -d raja-omp
//               mpirun -np 4 ex5p -m ../data/beam-hex.mesh -pa -d cuda
//
// Description:  This example code solves a simple 2D/3D mixed Darcy problem
//               corresponding to the saddle point system
//
//                                 k*u + grad p = f
//                                 - div u      = g
//
//               with natural boundary condition -p = <given pressure>.
//               Here, we use a given exact solution (u,p) and compute the
//               corresponding r.h.s. (f,g).  We discretize with Raviart-Thomas
//               finite elements (velocity u) and piecewise discontinuous
//               polynomials (pressure p).
//
//               The example demonstrates the use of the BlockMatrix class, as
//               well as the collective saving of several grid functions in
//               VisIt (visit.llnl.gov) and ParaView (paraview.org) formats.
//               Optional saving with ADIOS2 (adios2.readthedocs.io) streams is
//               also illustrated.
//
//               We recommend viewing examples 1-4 before viewing this example.

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

using namespace std;
using namespace mfem;

// Define the analytical solution and forcing terms / boundary conditions
void uFun_ex(const Vector & x, Vector & u);
double pFun_ex(const Vector & x);
void fFun(const Vector & x, Vector & f);
double gFun(const Vector & x);
double f_natural(const Vector & x);

int main(int argc, char *argv[])
{
   StopWatch chrono;

   // 1. Initialize MPI.
   int num_procs, myid;
   MPI_Init(&argc, &argv);
   MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
   MPI_Comm_rank(MPI_COMM_WORLD, &myid);
   bool verbose = (myid == 0);

   // 2. Parse command-line options.
   const char *mesh_file = "../data/star.mesh";
   int ref_levels = -1;
   int order = 1;
   bool par_format = false;
   bool pa = false;
   const char *device_config = "cpu";
   bool visualization = 1;
   bool adios2 = false;

   OptionsParser args(argc, argv);
   args.AddOption(&mesh_file, "-m", "--mesh",
                  "Mesh file to use.");
   args.AddOption(&ref_levels, "-r", "--refine",
                  "Number of times to refine the mesh uniformly.");
   args.AddOption(&order, "-o", "--order",
                  "Finite element order (polynomial degree).");
   args.AddOption(&par_format, "-pf", "--parallel-format", "-sf",
                  "--serial-format",
                  "Format to use when saving the results for VisIt.");
   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
                  "--no-partial-assembly", "Enable Partial Assembly.");
   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.AddOption(&adios2, "-adios2", "--adios2-streams", "-no-adios2",
                  "--no-adios2-streams",
                  "Save data using adios2 streams.");
   args.Parse();
   if (!args.Good())
   {
      if (verbose)
      {
         args.PrintUsage(cout);
      }
      MPI_Finalize();
      return 1;
   }
   if (verbose)
   {
      args.PrintOptions(cout);
   }

   // 3. 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);
   if (myid == 0) { device.Print(); }

   // 4. 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();

   // 5. Refine the serial mesh on all processors 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 10,000 elements, unless the user specifies it as input.
   {
      if (ref_levels == -1)
      {
         ref_levels = (int)floor(log(10000./mesh->GetNE())/log(2.)/dim);
      }

      for (int l = 0; l < ref_levels; l++)
      {
         mesh->UniformRefinement();
      }
   }

   // 6. Define a parallel mesh by a partitioning of the serial mesh. Refine
   //    this mesh further in parallel to increase the resolution. Once the
   //    parallel mesh is defined, the serial mesh can be deleted.
   ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
   delete mesh;
   {
      int par_ref_levels = 2;
      for (int l = 0; l < par_ref_levels; l++)
      {
         pmesh->UniformRefinement();
      }
   }

   // 7. Define a parallel finite element space on the parallel mesh. Here we
   //    use the Raviart-Thomas finite elements of the specified order.
   FiniteElementCollection *hdiv_coll(new RT_FECollection(order, dim));
   FiniteElementCollection *l2_coll(new L2_FECollection(order, dim));

   ParFiniteElementSpace *R_space = new ParFiniteElementSpace(pmesh, hdiv_coll);
   ParFiniteElementSpace *W_space = new ParFiniteElementSpace(pmesh, l2_coll);

   HYPRE_Int dimR = R_space->GlobalTrueVSize();
   HYPRE_Int dimW = W_space->GlobalTrueVSize();

   if (verbose)
   {
      std::cout << "***********************************************************\n";
      std::cout << "dim(R) = " << dimR << "\n";
      std::cout << "dim(W) = " << dimW << "\n";
      std::cout << "dim(R+W) = " << dimR + dimW << "\n";
      std::cout << "***********************************************************\n";
   }

   // 8. Define the two BlockStructure of the problem.  block_offsets is used
   //    for Vector based on dof (like ParGridFunction or ParLinearForm),
   //    block_trueOffstes is used for Vector based on trueDof (HypreParVector
   //    for the rhs and solution of the linear system).  The offsets computed
   //    here are local to the processor.
   Array<int> block_offsets(3); // number of variables + 1
   block_offsets[0] = 0;
   block_offsets[1] = R_space->GetVSize();
   block_offsets[2] = W_space->GetVSize();
   block_offsets.PartialSum();

   Array<int> block_trueOffsets(3); // number of variables + 1
   block_trueOffsets[0] = 0;
   block_trueOffsets[1] = R_space->TrueVSize();
   block_trueOffsets[2] = W_space->TrueVSize();
   block_trueOffsets.PartialSum();

   // 9. Define the coefficients, analytical solution, and rhs of the PDE.
   ConstantCoefficient k(1.0);

   VectorFunctionCoefficient fcoeff(dim, fFun);
   FunctionCoefficient fnatcoeff(f_natural);
   FunctionCoefficient gcoeff(gFun);

   VectorFunctionCoefficient ucoeff(dim, uFun_ex);
   FunctionCoefficient pcoeff(pFun_ex);

   // 10. Define the parallel grid function and parallel linear forms, solution
   //     vector and rhs.
   MemoryType mt = device.GetMemoryType();
   BlockVector x(block_offsets, mt), rhs(block_offsets, mt);
   BlockVector trueX(block_trueOffsets, mt), trueRhs(block_trueOffsets, mt);

   ParLinearForm *fform(new ParLinearForm);
   fform->Update(R_space, rhs.GetBlock(0), 0);
   fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff));
   fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fnatcoeff));
   fform->Assemble();
   fform->SyncAliasMemory(rhs);
   fform->ParallelAssemble(trueRhs.GetBlock(0));
   trueRhs.GetBlock(0).SyncAliasMemory(trueRhs);

   ParLinearForm *gform(new ParLinearForm);
   gform->Update(W_space, rhs.GetBlock(1), 0);
   gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff));
   gform->Assemble();
   gform->SyncAliasMemory(rhs);
   gform->ParallelAssemble(trueRhs.GetBlock(1));
   trueRhs.GetBlock(1).SyncAliasMemory(trueRhs);

   // 11. Assemble the finite element matrices for the Darcy operator
   //
   //                            D = [ M  B^T ]
   //                                [ B   0  ]
   //     where:
   //
   //     M = \int_\Omega k u_h \cdot v_h d\Omega   u_h, v_h \in R_h
   //     B   = -\int_\Omega \div u_h q_h d\Omega   u_h \in R_h, q_h \in W_h
   ParBilinearForm *mVarf(new ParBilinearForm(R_space));
   ParMixedBilinearForm *bVarf(new ParMixedBilinearForm(R_space, W_space));

   HypreParMatrix *M = NULL;
   HypreParMatrix *B = NULL;

   if (pa) { mVarf->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
   mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(k));
   mVarf->Assemble();
   if (!pa) { mVarf->Finalize(); }

   if (pa) { bVarf->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
   bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator);
   bVarf->Assemble();
   if (!pa) { bVarf->Finalize(); }

   BlockOperator *darcyOp = new BlockOperator(block_trueOffsets);

   Array<int> empty_tdof_list;  // empty
   OperatorPtr opM, opB;

   TransposeOperator *Bt = NULL;

   if (pa)
   {
      mVarf->FormSystemMatrix(empty_tdof_list, opM);
      bVarf->FormRectangularSystemMatrix(empty_tdof_list, empty_tdof_list, opB);
      Bt = new TransposeOperator(opB.Ptr());

      darcyOp->SetBlock(0,0, opM.Ptr());
      darcyOp->SetBlock(0,1, Bt, -1.0);
      darcyOp->SetBlock(1,0, opB.Ptr(), -1.0);
   }
   else
   {
      M = mVarf->ParallelAssemble();
      B = bVarf->ParallelAssemble();
      (*B) *= -1;
      Bt = new TransposeOperator(B);

      darcyOp->SetBlock(0,0, M);
      darcyOp->SetBlock(0,1, Bt);
      darcyOp->SetBlock(1,0, B);
   }

   // 12. Construct the operators for preconditioner
   //
   //                 P = [ diag(M)         0         ]
   //                     [  0       B diag(M)^-1 B^T ]
   //
   //     Here we use Symmetric Gauss-Seidel to approximate the inverse of the
   //     pressure Schur Complement.
   HypreParMatrix *MinvBt = NULL;
   HypreParVector *Md = NULL;
   HypreParMatrix *S = NULL;
   Vector Md_PA;
   Solver *invM, *invS;

   if (pa)
   {
      Md_PA.SetSize(R_space->GetTrueVSize());
      mVarf->AssembleDiagonal(Md_PA);
      auto Md_host = Md_PA.HostRead();
      Vector invMd(Md_PA.Size());
      for (int i=0; i<Md_PA.Size(); ++i)
      {
         invMd(i) = 1.0 / Md_host[i];
      }

      Vector BMBt_diag(W_space->GetTrueVSize());
      bVarf->AssembleDiagonal_ADAt(invMd, BMBt_diag);

      Array<int> ess_tdof_list;  // empty

      invM = new OperatorJacobiSmoother(Md_PA, ess_tdof_list);
      invS = new OperatorJacobiSmoother(BMBt_diag, ess_tdof_list);
   }
   else
   {
      Md = new HypreParVector(MPI_COMM_WORLD, M->GetGlobalNumRows(),
                              M->GetRowStarts());
      M->GetDiag(*Md);

      MinvBt = B->Transpose();
      MinvBt->InvScaleRows(*Md);
      S = ParMult(B, MinvBt);

      invM = new HypreDiagScale(*M);
      invS = new HypreBoomerAMG(*S);
   }

   invM->iterative_mode = false;
   invS->iterative_mode = false;

   BlockDiagonalPreconditioner *darcyPr = new BlockDiagonalPreconditioner(
      block_trueOffsets);
   darcyPr->SetDiagonalBlock(0, invM);
   darcyPr->SetDiagonalBlock(1, invS);

   // 13. Solve the linear system with MINRES.
   //     Check the norm of the unpreconditioned residual.
   int maxIter(pa ? 1000 : 500);
   double rtol(1.e-6);
   double atol(1.e-10);

   chrono.Clear();
   chrono.Start();
   MINRESSolver solver(MPI_COMM_WORLD);
   solver.SetAbsTol(atol);
   solver.SetRelTol(rtol);
   solver.SetMaxIter(maxIter);
   solver.SetOperator(*darcyOp);
   solver.SetPreconditioner(*darcyPr);
   solver.SetPrintLevel(verbose);
   trueX = 0.0;
   solver.Mult(trueRhs, trueX);
   if (device.IsEnabled()) { trueX.HostRead(); }
   chrono.Stop();

   if (verbose)
   {
      if (solver.GetConverged())
         std::cout << "MINRES converged in " << solver.GetNumIterations()
                   << " iterations with a residual norm of " << solver.GetFinalNorm() << ".\n";
      else
         std::cout << "MINRES did not converge in " << solver.GetNumIterations()
                   << " iterations. Residual norm is " << solver.GetFinalNorm() << ".\n";
      std::cout << "MINRES solver took " << chrono.RealTime() << "s. \n";
   }

   // 14. Extract the parallel grid function corresponding to the finite element
   //     approximation X. This is the local solution on each processor. Compute
   //     L2 error norms.
   ParGridFunction *u(new ParGridFunction);
   ParGridFunction *p(new ParGridFunction);
   u->MakeRef(R_space, x.GetBlock(0), 0);
   p->MakeRef(W_space, x.GetBlock(1), 0);
   u->Distribute(&(trueX.GetBlock(0)));
   p->Distribute(&(trueX.GetBlock(1)));

   int order_quad = max(2, 2*order+1);
   const IntegrationRule *irs[Geometry::NumGeom];
   for (int i=0; i < Geometry::NumGeom; ++i)
   {
      irs[i] = &(IntRules.Get(i, order_quad));
   }

   double err_u  = u->ComputeL2Error(ucoeff, irs);
   double norm_u = ComputeGlobalLpNorm(2, ucoeff, *pmesh, irs);
   double err_p  = p->ComputeL2Error(pcoeff, irs);
   double norm_p = ComputeGlobalLpNorm(2, pcoeff, *pmesh, irs);

   if (verbose)
   {
      std::cout << "|| u_h - u_ex || / || u_ex || = " << err_u / norm_u << "\n";
      std::cout << "|| p_h - p_ex || / || p_ex || = " << err_p / norm_p << "\n";
   }

   // 15. Save the refined mesh and the solution in parallel. This output can be
   //     viewed later using GLVis: "glvis -np <np> -m mesh -g sol_*".
   {
      ostringstream mesh_name, u_name, p_name;
      mesh_name << "mesh." << setfill('0') << setw(6) << myid;
      u_name << "sol_u." << setfill('0') << setw(6) << myid;
      p_name << "sol_p." << setfill('0') << setw(6) << myid;

      ofstream mesh_ofs(mesh_name.str().c_str());
      mesh_ofs.precision(8);
      pmesh->Print(mesh_ofs);

      ofstream u_ofs(u_name.str().c_str());
      u_ofs.precision(8);
      u->Save(u_ofs);

      ofstream p_ofs(p_name.str().c_str());
      p_ofs.precision(8);
      p->Save(p_ofs);
   }

   // 16. Save data in the VisIt format
   VisItDataCollection visit_dc("Example5-Parallel", pmesh);
   visit_dc.RegisterField("velocity", u);
   visit_dc.RegisterField("pressure", p);
   visit_dc.SetFormat(!par_format ?
                      DataCollection::SERIAL_FORMAT :
                      DataCollection::PARALLEL_FORMAT);
   visit_dc.Save();

   // 17. Save data in the ParaView format
   ParaViewDataCollection paraview_dc("Example5P", pmesh);
   paraview_dc.SetPrefixPath("ParaView");
   paraview_dc.SetLevelsOfDetail(order);
   paraview_dc.SetDataFormat(VTKFormat::BINARY);
   paraview_dc.SetHighOrderOutput(true);
   paraview_dc.SetCycle(0);
   paraview_dc.SetTime(0.0);
   paraview_dc.RegisterField("velocity",u);
   paraview_dc.RegisterField("pressure",p);
   paraview_dc.Save();

   // 18. Optionally output a BP (binary pack) file using ADIOS2. This can be
   //     visualized with the ParaView VTX reader.
#ifdef MFEM_USE_ADIOS2
   if (adios2)
   {
      std::string postfix(mesh_file);
      postfix.erase(0, std::string("../data/").size() );
      postfix += "_o" + std::to_string(order);
      const std::string collection_name = "ex5-p_" + postfix + ".bp";

      ADIOS2DataCollection adios2_dc(MPI_COMM_WORLD, collection_name, pmesh);
      adios2_dc.SetLevelsOfDetail(1);
      adios2_dc.SetCycle(1);
      adios2_dc.SetTime(0.0);
      adios2_dc.RegisterField("velocity",u);
      adios2_dc.RegisterField("pressure",p);
      adios2_dc.Save();
   }
#endif

   // 19. Send the solution by socket to a GLVis server.
   if (visualization)
   {
      char vishost[] = "localhost";
      int  visport   = 19916;
      socketstream u_sock(vishost, visport);
      u_sock << "parallel " << num_procs << " " << myid << "\n";
      u_sock.precision(8);
      u_sock << "solution\n" << *pmesh << *u << "window_title 'Velocity'"
             << endl;
      // Make sure all ranks have sent their 'u' solution before initiating
      // another set of GLVis connections (one from each rank):
      MPI_Barrier(pmesh->GetComm());
      socketstream p_sock(vishost, visport);
      p_sock << "parallel " << num_procs << " " << myid << "\n";
      p_sock.precision(8);
      p_sock << "solution\n" << *pmesh << *p << "window_title 'Pressure'"
             << endl;
   }

   // 20. Free the used memory.
   delete fform;
   delete gform;
   delete u;
   delete p;
   delete darcyOp;
   delete darcyPr;
   delete invM;
   delete invS;
   delete S;
   delete Md;
   delete MinvBt;
   delete Bt;
   delete B;
   delete M;
   delete mVarf;
   delete bVarf;
   delete W_space;
   delete R_space;
   delete l2_coll;
   delete hdiv_coll;
   delete pmesh;

   MPI_Finalize();

   return 0;
}


void uFun_ex(const Vector & x, Vector & u)
{
   double xi(x(0));
   double yi(x(1));
   double zi(0.0);
   if (x.Size() == 3)
   {
      zi = x(2);
   }

   u(0) = - exp(xi)*sin(yi)*cos(zi);
   u(1) = - exp(xi)*cos(yi)*cos(zi);

   if (x.Size() == 3)
   {
      u(2) = exp(xi)*sin(yi)*sin(zi);
   }
}

// Change if needed
double pFun_ex(const Vector & x)
{
   double xi(x(0));
   double yi(x(1));
   double zi(0.0);

   if (x.Size() == 3)
   {
      zi = x(2);
   }

   return exp(xi)*sin(yi)*cos(zi);
}

void fFun(const Vector & x, Vector & f)
{
   f = 0.0;
}

double gFun(const Vector & x)
{
   if (x.Size() == 3)
   {
      return -pFun_ex(x);
   }
   else
   {
      return 0;
   }
}

double f_natural(const Vector & x)
{
   return (-pFun_ex(x));
}