mesh-explorer.cpp

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// Copyright (c) 2010-2021, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
//
//            -----------------------------------------------------
//            Mesh Explorer Miniapp:  Explore and manipulate meshes
//            -----------------------------------------------------
//
// This miniapp is a handy tool to examine, visualize and manipulate a given
// mesh. Some of its features are:
//
//    - visualizing of mesh materials and individual mesh elements
//    - mesh scaling, randomization, and general transformation
//    - manipulation of the mesh curvature
//    - the ability to simulate parallel partitioning
//    - quantitative and visual reports of mesh quality
//
// Compile with: make mesh-explorer
//
// Sample runs:  mesh-explorer
//               mesh-explorer -m ../../data/beam-tri.mesh
//               mesh-explorer -m ../../data/star-q2.mesh
//               mesh-explorer -m ../../data/disc-nurbs.mesh
//               mesh-explorer -m ../../data/escher-p3.mesh
//               mesh-explorer -m ../../data/mobius-strip.mesh

#include "mfem.hpp"
#include <fstream>
#include <limits>
#include <cstdlib>

using namespace mfem;
using namespace std;

// This transformation can be applied to a mesh with the 't' menu option.
void transformation(const Vector &p, Vector &v)
{
   // simple shear transformation
   double s = 0.1;

   if (p.Size() == 3)
   {
      v(0) = p(0) + s*p(1) + s*p(2);
      v(1) = p(1) + s*p(2) + s*p(0);
      v(2) = p(2);
   }
   else if (p.Size() == 2)
   {
      v(0) = p(0) + s*p(1);
      v(1) = p(1) + s*p(0);
   }
   else
   {
      v = p;
   }
}

// This function is used with the 'r' menu option, sub-option 'l' to refine a
// mesh locally in a region, defined by return values <= region_eps.
double region_eps = 1e-8;
double region(const Vector &p)
{
   const double x = p(0), y = p(1);
   // here we describe the region: (x <= 1/4) && (y >= 0) && (y <= 1)
   return std::max(std::max(x - 0.25, -y), y - 1.0);
}

// The projection of this function can be plotted with the 'l' menu option
double f(const Vector &p)
{
   double x = p(0);
   double y = p.Size() > 1 ? p(1) : 0.0;
   double z = p.Size() > 2 ? p(2) : 0.0;

   if (1)
   {
      // torus in the xy-plane
      const double r_big = 2.0;
      const double r_small = 1.0;
      return hypot(r_big - hypot(x, y), z) - r_small;
   }
   if (0)
   {
      // sphere at the origin:
      const double r = 1.0;
      return hypot(hypot(x, y), z) - r;
   }
}

Mesh *read_par_mesh(int np, const char *mesh_prefix, Array<int>& partitioning,
                    Array<int>& bdr_partitioning)
{
   Mesh *mesh;
   Array<Mesh *> mesh_array;

   mesh_array.SetSize(np);
   for (int p = 0; p < np; p++)
   {
      ostringstream fname;
      fname << mesh_prefix << '.' << setfill('0') << setw(6) << p;
      ifgzstream meshin(fname.str().c_str());
      if (!meshin)
      {
         cerr << "Can not open mesh file: " << fname.str().c_str()
              << '!' << endl;
         for (p--; p >= 0; p--)
         {
            delete mesh_array[p];
         }
         return NULL;
      }
      mesh_array[p] = new Mesh(meshin, 1, 0);
      // Assign corresponding processor number to element + boundary partitions
      for (int i = 0; i < mesh_array[p]->GetNE(); i++)
      {
         partitioning.Append(p);
      }
      for (int i = 0; i < mesh_array[p]->GetNBE(); i++)
      {
         bdr_partitioning.Append(p);
      }
   }
   mesh = new Mesh(mesh_array, np);

   for (int p = 0; p < np; p++)
   {
      delete mesh_array[np-1-p];
   }
   mesh_array.DeleteAll();

   return mesh;
}

// Given a 3D mesh, produce a 2D mesh consisting of its boundary elements.
// We guarantee that the skin preserves the boundary index order.
Mesh *skin_mesh(Mesh *mesh)
{
   // Determine mapping from vertex to boundary vertex
   Array<int> v2v(mesh->GetNV());
   v2v = -1;
   for (int i = 0; i < mesh->GetNBE(); i++)
   {
      Element *el = mesh->GetBdrElement(i);
      int *v = el->GetVertices();
      int nv = el->GetNVertices();
      for (int j = 0; j < nv; j++)
      {
         v2v[v[j]] = 0;
      }
   }
   int nbvt = 0;
   for (int i = 0; i < v2v.Size(); i++)
   {
      if (v2v[i] == 0)
      {
         v2v[i] = nbvt++;
      }
   }

   // Create a new mesh for the boundary
   Mesh * bmesh = new Mesh(mesh->Dimension() - 1, nbvt, mesh->GetNBE(),
                           0, mesh->SpaceDimension());

   // Copy vertices to the boundary mesh
   nbvt = 0;
   for (int i = 0; i < v2v.Size(); i++)
   {
      if (v2v[i] >= 0)
      {
         double *c = mesh->GetVertex(i);
         bmesh->AddVertex(c);
         nbvt++;
      }
   }

   // Copy elements to the boundary mesh
   int bv[4];
   for (int i = 0; i < mesh->GetNBE(); i++)
   {
      Element *el = mesh->GetBdrElement(i);
      int *v = el->GetVertices();
      int nv = el->GetNVertices();

      for (int j = 0; j < nv; j++)
      {
         bv[j] = v2v[v[j]];
      }

      switch (el->GetGeometryType())
      {
         case Geometry::SEGMENT:
            bmesh->AddSegment(bv, el->GetAttribute());
            break;
         case Geometry::TRIANGLE:
            bmesh->AddTriangle(bv, el->GetAttribute());
            break;
         case Geometry::SQUARE:
            bmesh->AddQuad(bv, el->GetAttribute());
            break;
         default:
            break; // This should not happen
      }

   }
   bmesh->FinalizeTopology();

   // Copy GridFunction describing nodes if present
   if (mesh->GetNodes())
   {
      FiniteElementSpace *fes = mesh->GetNodes()->FESpace();
      const FiniteElementCollection *fec = fes->FEColl();
      if (dynamic_cast<const H1_FECollection*>(fec))
      {
         FiniteElementCollection *fec_copy =
            FiniteElementCollection::New(fec->Name());
         FiniteElementSpace *fes_copy =
            new FiniteElementSpace(*fes, bmesh, fec_copy);
         GridFunction *bdr_nodes = new GridFunction(fes_copy);
         bdr_nodes->MakeOwner(fec_copy);

         bmesh->NewNodes(*bdr_nodes, true);

         Array<int> vdofs;
         Array<int> bvdofs;
         Vector v;
         for (int i=0; i<mesh->GetNBE(); i++)
         {
            fes->GetBdrElementVDofs(i, vdofs);
            mesh->GetNodes()->GetSubVector(vdofs, v);

            fes_copy->GetElementVDofs(i, bvdofs);
            bdr_nodes->SetSubVector(bvdofs, v);
         }
      }
      else
      {
         cout << "\nDiscontinuous nodes not yet supported" << endl;
      }
   }

   return bmesh;
}

void recover_bdr_partitioning(const Mesh* mesh, const Array<int>& partitioning,
                              Array<int>& bdr_partitioning)
{
   bdr_partitioning.SetSize(mesh->GetNBE());
   int info, e;
   for (int be = 0; be < mesh->GetNBE(); be++)
   {
      mesh->GetBdrElementAdjacentElement(be, e, info);
      bdr_partitioning[be] = partitioning[e];
   }
}

int main (int argc, char *argv[])
{
   int np = 0;
   const char *mesh_file = "../../data/beam-hex.mesh";
   bool refine = true;

   OptionsParser args(argc, argv);
   args.AddOption(&mesh_file, "-m", "--mesh",
                  "Mesh file to visualize.");
   args.AddOption(&np, "-np", "--num-proc",
                  "Load mesh from multiple processors.");
   args.AddOption(&refine, "-ref", "--refinement", "-no-ref", "--no-refinement",
                  "Prepare the mesh for refinement or not.");
   args.Parse();
   if (!args.Good())
   {
      if (!args.Help())
      {
         args.PrintError(cout);
         cout << endl;
      }
      cout << "Visualize and manipulate a serial mesh:\n"
           << "   mesh-explorer -m <mesh_file>\n"
           << "Visualize and manipulate a parallel mesh:\n"
           << "   mesh-explorer -np <#proc> -m <mesh_prefix>\n" << endl
           << "All Options:\n";
      args.PrintHelp(cout);
      return 1;
   }
   args.PrintOptions(cout);

   Mesh *mesh;
   Mesh *bdr_mesh = NULL;

   // Helper to distinguish whether we use a parallel or serial mesh.
   const bool use_par_mesh = np > 0;

   // Helper for visualizing the partitioning.
   Array<int> partitioning;
   Array<int> bdr_partitioning;
   if (!use_par_mesh)
   {
      mesh = new Mesh(mesh_file, 1, refine);
      partitioning.SetSize(mesh->GetNE());
      partitioning = 0;
      bdr_partitioning.SetSize(mesh->GetNBE());
      bdr_partitioning = 0;
   }
   else
   {
      mesh = read_par_mesh(np, mesh_file, partitioning, bdr_partitioning);
      if (mesh == NULL)
      {
         return 3;
      }
   }
   int dim  = mesh->Dimension();
   int sdim = mesh->SpaceDimension();

   FiniteElementCollection *bdr_attr_fec = NULL;
   FiniteElementCollection *attr_fec;
   if (dim == 2)
   {
      attr_fec = new Const2DFECollection;
   }
   else
   {
      bdr_attr_fec = new Const2DFECollection;
      attr_fec = new Const3DFECollection;
   }

   int print_char = 1;
   while (1)
   {
      if (print_char)
      {
         cout << endl;
         mesh->PrintCharacteristics();
         cout << "boundary attribs   :";
         for (int i = 0; i < mesh->bdr_attributes.Size(); i++)
         {
            cout << ' ' << mesh->bdr_attributes[i];
         }
         cout << '\n' << "material attribs   :";
         for (int i = 0; i < mesh->attributes.Size(); i++)
         {
            cout << ' ' << mesh->attributes[i];
         }
         cout << endl;
         cout << "mesh curvature     : ";
         if (mesh->GetNodalFESpace() != NULL)
         {
            cout << mesh->GetNodalFESpace()->FEColl()->Name() << endl;
         }
         else
         {
            cout << "NONE" << endl;
         }
      }
      print_char = 0;
      cout << endl;
      cout << "What would you like to do?\n"
           "r) Refine\n"
           "c) Change curvature\n"
           "s) Scale\n"
           "t) Transform\n"
           "j) Jitter\n"
           "v) View mesh\n"
           "P) View partitioning\n"
           "m) View materials\n"
           "b) View boundary\n"
           "B) View boundary partitioning\n"
           "e) View elements\n"
           "h) View element sizes, h\n"
           "k) View element ratios, kappa\n"
           "J) View scaled Jacobian\n"
           "l) Plot a function\n"
           "x) Print sub-element stats\n"
           "f) Find physical point in reference space\n"
           "p) Generate a partitioning\n"
           "o) Reorder elements\n"
           "S) Save in MFEM format\n"
           "V) Save in VTK format (only linear and quadratic meshes)\n"
           "q) Quit\n"
#ifdef MFEM_USE_ZLIB
           "Z) Save in MFEM format with compression\n"
#endif
           "--> " << flush;
      char mk;
      cin >> mk;
      if (!cin) { break; }

      if (mk == 'q')
      {
         break;
      }

      if (mk == 'r')
      {
         cout <<
              "Choose type of refinement:\n"
              "s) standard refinement with Mesh::UniformRefinement()\n"
              "b) Mesh::UniformRefinement() (bisection for tet meshes)\n"
              "u) uniform refinement with a factor\n"
              "g) non-uniform refinement (Gauss-Lobatto) with a factor\n"
              "l) refine locally using the region() function\n"
              "--> " << flush;
         char sk;
         cin >> sk;
         switch (sk)
         {
            case 's':
               mesh->UniformRefinement();
               // Make sure tet-only meshes are marked for local refinement.
               mesh->Finalize(true);
               break;
            case 'b':
               mesh->UniformRefinement(1); // ref_algo = 1
               break;
            case 'u':
            case 'g':
            {
               cout << "enter refinement factor --> " << flush;
               int ref_factor;
               cin >> ref_factor;
               if (ref_factor <= 1 || ref_factor > 32) { break; }
               int ref_type = (sk == 'u') ? BasisType::ClosedUniform :
                              BasisType::GaussLobatto;
               *mesh = Mesh::MakeRefined(*mesh, ref_factor, ref_type);
               break;
            }
            case 'l':
            {
               Vector pt;
               Array<int> marked_elements;
               for (int i = 0; i < mesh->GetNE(); i++)
               {
                  // check all nodes of the element
                  IsoparametricTransformation T;
                  mesh->GetElementTransformation(i, &T);
                  for (int j = 0; j < T.GetPointMat().Width(); j++)
                  {
                     T.GetPointMat().GetColumnReference(j, pt);
                     if (region(pt) <= region_eps)
                     {
                        marked_elements.Append(i);
                        break;
                     }
                  }
               }
               mesh->GeneralRefinement(marked_elements);
               break;
            }
         }
         print_char = 1;
      }

      if (mk == 'c')
      {
         int p;
         cout << "enter new order for mesh curvature --> " << flush;
         cin >> p;
         mesh->SetCurvature(p > 0 ? p : -p, p <= 0);
         print_char = 1;
      }

      if (mk == 's')
      {
         double factor;
         cout << "scaling factor ---> " << flush;
         cin >> factor;

         GridFunction *nodes = mesh->GetNodes();
         if (nodes == NULL)
         {
            for (int i = 0; i < mesh->GetNV(); i++)
            {
               double *v = mesh->GetVertex(i);
               v[0] *= factor;
               v[1] *= factor;
               if (dim == 3)
               {
                  v[2] *= factor;
               }
            }
         }
         else
         {
            *nodes *= factor;
         }

         print_char = 1;
      }

      if (mk == 't')
      {
         mesh->Transform(transformation);
         print_char = 1;
      }

      if (mk == 'j')
      {
         double jitter;
         cout << "jitter factor ---> " << flush;
         cin >> jitter;

         GridFunction *nodes = mesh->GetNodes();

         if (nodes == NULL)
         {
            cerr << "The mesh should have nodes, introduce curvature first!\n";
         }
         else
         {
            FiniteElementSpace *fespace = nodes->FESpace();

            GridFunction rdm(fespace);
            rdm.Randomize();
            rdm -= 0.5; // shift to random values in [-0.5,0.5]
            rdm *= jitter;

            // compute minimal local mesh size
            Vector h0(fespace->GetNDofs());
            h0 = infinity();
            {
               Array<int> dofs;
               for (int i = 0; i < fespace->GetNE(); i++)
               {
                  fespace->GetElementDofs(i, dofs);
                  for (int j = 0; j < dofs.Size(); j++)
                  {
                     h0(dofs[j]) = std::min(h0(dofs[j]), mesh->GetElementSize(i));
                  }
               }
            }

            // scale the random values to be of order of the local mesh size
            for (int i = 0; i < fespace->GetNDofs(); i++)
            {
               for (int d = 0; d < dim; d++)
               {
                  rdm(fespace->DofToVDof(i,d)) *= h0(i);
               }
            }

            char move_bdr = 'n';
            cout << "move boundary nodes? [y/n] ---> " << flush;
            cin >> move_bdr;

            // don't perturb the boundary
            if (move_bdr == 'n')
            {
               Array<int> vdofs;
               for (int i = 0; i < fespace->GetNBE(); i++)
               {
                  fespace->GetBdrElementVDofs(i, vdofs);
                  for (int j = 0; j < vdofs.Size(); j++)
                  {
                     rdm(vdofs[j]) = 0.0;
                  }
               }
            }

            *nodes += rdm;
         }

         print_char = 1;
      }

      if (mk == 'x')
      {
         int sd, nz = 0;
         DenseMatrix J(dim);
         double min_det_J, max_det_J, min_det_J_z, max_det_J_z;
         double min_kappa, max_kappa, max_ratio_det_J_z;
         min_det_J = min_kappa = infinity();
         max_det_J = max_kappa = max_ratio_det_J_z = -infinity();
         cout << "subdivision factor ---> " << flush;
         cin >> sd;
         Array<int> bad_elems_by_geom(Geometry::NumGeom);
         bad_elems_by_geom = 0;
         // Only print so many to keep output compact
         const int max_to_print = 10;
         for (int i = 0; i < mesh->GetNE(); i++)
         {
            Geometry::Type geom = mesh->GetElementBaseGeometry(i);
            ElementTransformation *T = mesh->GetElementTransformation(i);

            RefinedGeometry *RefG = GlobGeometryRefiner.Refine(geom, sd, 1);
            IntegrationRule &ir = RefG->RefPts;

            min_det_J_z = infinity();
            max_det_J_z = -infinity();
            for (int j = 0; j < ir.GetNPoints(); j++)
            {
               T->SetIntPoint(&ir.IntPoint(j));
               Geometries.JacToPerfJac(geom, T->Jacobian(), J);

               double det_J = J.Det();
               double kappa =
                  J.CalcSingularvalue(0) / J.CalcSingularvalue(dim-1);

               min_det_J_z = fmin(min_det_J_z, det_J);
               max_det_J_z = fmax(max_det_J_z, det_J);

               min_kappa = fmin(min_kappa, kappa);
               max_kappa = fmax(max_kappa, kappa);
            }
            max_ratio_det_J_z =
               fmax(max_ratio_det_J_z, max_det_J_z/min_det_J_z);
            min_det_J = fmin(min_det_J, min_det_J_z);
            max_det_J = fmax(max_det_J, max_det_J_z);
            if (min_det_J_z <= 0.0)
            {
               if (nz < max_to_print)
               {
                  Vector center;
                  mesh->GetElementCenter(i, center);
                  cout << "det(J) < 0 = " << min_det_J_z << " in element "
                       << i << ", centered at: ";
                  center.Print();
               }
               nz++;
               bad_elems_by_geom[geom]++;
            }
         }
         if (nz >= max_to_print)
         {
            cout << "det(J) < 0 for " << nz - max_to_print << " more elements "
                 << "not printed.\n";
         }
         cout << "\nbad elements = " << nz;
         if (nz)
         {
            cout << "  --  ";
            Mesh::PrintElementsByGeometry(dim, bad_elems_by_geom, cout);
         }
         cout << "\nmin det(J)   = " << min_det_J
              << "\nmax det(J)   = " << max_det_J
              << "\nglobal ratio = " << max_det_J/min_det_J
              << "\nmax el ratio = " << max_ratio_det_J_z
              << "\nmin kappa    = " << min_kappa
              << "\nmax kappa    = " << max_kappa << endl;
      }

      if (mk == 'f')
      {
         DenseMatrix point_mat(sdim,1);
         cout << "\npoint in physical space ---> " << flush;
         for (int i = 0; i < sdim; i++)
         {
            cin >> point_mat(i,0);
         }
         Array<int> elem_ids;
         Array<IntegrationPoint> ips;

         // physical -> reference space
         mesh->FindPoints(point_mat, elem_ids, ips);

         cout << "point in reference space:";
         if (elem_ids[0] == -1)
         {
            cout << " NOT FOUND!\n";
         }
         else
         {
            cout << " element " << elem_ids[0] << ", ip =";
            cout << " " << ips[0].x;
            if (sdim > 1)
            {
               cout << " " << ips[0].y;
               if (sdim > 2)
               {
                  cout << " " << ips[0].z;
               }
            }
            cout << endl;
         }
      }

      if (mk == 'o')
      {
         cout << "What type of reordering?\n"
              "g) Gecko edge-product minimization\n"
              "h) Hilbert spatial sort\n"
              "--> " << flush;
         char rk;
         cin >> rk;

         Array<int> ordering, tentative;
         if (rk == 'h')
         {
            mesh->GetHilbertElementOrdering(ordering);
            mesh->ReorderElements(ordering);
         }
         else if (rk == 'g')
         {
            int outer, inner, window, period;
            cout << "Enter number of outer iterations (default 5): " << flush;
            cin >> outer;
            cout << "Enter number of inner iterations (default 4): " << flush;
            cin >> inner;
            cout << "Enter window size (default 4, beware of exponential cost): "
                 << flush;
            cin >> window;
            cout << "Enter period for window size increment (default 2): "
                 << flush;
            cin >> period;

            double best_cost = infinity();
            for (int i = 0; i < outer; i++)
            {
               int seed = i+1;
               double cost = mesh->GetGeckoElementOrdering(
                                tentative, inner, window, period, seed, true);

               if (cost < best_cost)
               {
                  ordering = tentative;
                  best_cost = cost;
               }
            }
            cout << "Final cost: " << best_cost << endl;

            mesh->ReorderElements(ordering);
         }
      }

      // These are most of the cases that open a new GLVis window
      if (mk == 'm' || mk == 'b' || mk == 'e' || mk == 'v' || mk == 'h' ||
          mk == 'k' || mk == 'J' || mk == 'p' || mk == 'B' || mk == 'P')
      {
         FiniteElementSpace *bdr_attr_fespace = NULL;
         FiniteElementSpace *attr_fespace =
            new FiniteElementSpace(mesh, attr_fec);
         GridFunction bdr_attr;
         GridFunction attr(attr_fespace);

         if (mk == 'm')
         {
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               attr(i) = mesh->GetAttribute(i);
            }
         }

         if (mk == 'P')
         {
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               attr(i) = partitioning[i] + 1;
            }
         }

         if (mk == 'b' || mk == 'B')
         {
            if (dim == 3)
            {
               delete bdr_mesh;
               bdr_mesh = skin_mesh(mesh);
               bdr_attr_fespace =
                  new FiniteElementSpace(bdr_mesh, bdr_attr_fec);
               bdr_attr.SetSpace(bdr_attr_fespace);
               if (mk == 'b')
               {
                  for (int i = 0; i < bdr_mesh->GetNE(); i++)
                  {
                     bdr_attr(i) = bdr_mesh->GetAttribute(i);
                  }
               }
               else if (mk == 'B')
               {
                  for (int i = 0; i < bdr_mesh->GetNE(); i++)
                  {
                     bdr_attr(i) = bdr_partitioning[i] + 1;
                  }
               }
               else
               {
                  MFEM_WARNING("Unimplemented case.");
               }
            }
            else
            {
               MFEM_WARNING("Unsupported mesh dimension.");
               attr = 1.0;
            }
         }

         if (mk == 'v')
         {
            attr = 1.0;
         }

         if (mk == 'e')
         {
            Array<int> coloring;
            srand(time(0));
            double a = double(rand()) / (double(RAND_MAX) + 1.);
            int el0 = (int)floor(a * mesh->GetNE());
            cout << "Generating coloring starting with element " << el0+1
                 << " / " << mesh->GetNE() << endl;
            mesh->GetElementColoring(coloring, el0);
            for (int i = 0; i < coloring.Size(); i++)
            {
               attr(i) = coloring[i];
            }
            cout << "Number of colors: " << attr.Max() + 1 << endl;
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               attr(i) = i; // coloring by element number
            }
         }

         if (mk == 'h')
         {
            DenseMatrix J(dim);
            double h_min, h_max;
            h_min = infinity();
            h_max = -h_min;
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               Geometry::Type geom = mesh->GetElementBaseGeometry(i);
               ElementTransformation *T = mesh->GetElementTransformation(i);
               T->SetIntPoint(&Geometries.GetCenter(geom));
               Geometries.JacToPerfJac(geom, T->Jacobian(), J);

               attr(i) = J.Det();
               if (attr(i) < 0.0)
               {
                  attr(i) = -pow(-attr(i), 1.0/double(dim));
               }
               else
               {
                  attr(i) = pow(attr(i), 1.0/double(dim));
               }
               h_min = min(h_min, attr(i));
               h_max = max(h_max, attr(i));
            }
            cout << "h_min = " << h_min << ", h_max = " << h_max << endl;
         }

         if (mk == 'k')
         {
            DenseMatrix J(dim);
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               Geometry::Type geom = mesh->GetElementBaseGeometry(i);
               ElementTransformation *T = mesh->GetElementTransformation(i);
               T->SetIntPoint(&Geometries.GetCenter(geom));
               Geometries.JacToPerfJac(geom, T->Jacobian(), J);
               attr(i) = J.CalcSingularvalue(0) / J.CalcSingularvalue(dim-1);
            }
         }

         if (mk == 'J')
         {
            // The "scaled Jacobian" is the determinant of the Jacobian scaled
            // by the l2 norms of its columns. It can be used to identify badly
            // skewed elements, since it takes values between 0 and 1, with 0
            // corresponding to a flat element, and 1 to orthogonal columns.
            DenseMatrix J(dim);
            int sd;
            cout << "subdivision factor ---> " << flush;
            cin >> sd;
            for (int i = 0; i < mesh->GetNE(); i++)
            {
               Geometry::Type geom = mesh->GetElementBaseGeometry(i);
               ElementTransformation *T = mesh->GetElementTransformation(i);

               RefinedGeometry *RefG = GlobGeometryRefiner.Refine(geom, sd, 1);
               IntegrationRule &ir = RefG->RefPts;

               // For each element, find the minimal scaled Jacobian in a
               // lattice of points with the given subdivision factor.
               attr(i) = infinity();
               for (int j = 0; j < ir.GetNPoints(); j++)
               {
                  T->SetIntPoint(&ir.IntPoint(j));
                  Geometries.JacToPerfJac(geom, T->Jacobian(), J);

                  // Jacobian determinant
                  double sJ = J.Det();

                  for (int k = 0; k < J.Width(); k++)
                  {
                     Vector col;
                     J.GetColumnReference(k,col);
                     // Scale by column norms
                     sJ /= col.Norml2();
                  }

                  attr(i) = fmin(sJ, attr(i));
               }
            }
         }

         if (mk == 'p')
         {
            cout << "What type of partitioning?\n"
                 "c) Cartesian\n"
                 "s) Simple 1D split of the element sequence\n"
                 "0) METIS_PartGraphRecursive (sorted neighbor lists)\n"
                 "1) METIS_PartGraphKway      (sorted neighbor lists)"
                 " (default)\n"
                 "2) METIS_PartGraphVKway     (sorted neighbor lists)\n"
                 "3) METIS_PartGraphRecursive\n"
                 "4) METIS_PartGraphKway\n"
                 "5) METIS_PartGraphVKway\n"
                 "--> " << flush;
            char pk;
            cin >> pk;
            if (pk == 'c')
            {
               int nxyz[3];
               cout << "Enter nx: " << flush;
               cin >> nxyz[0]; np = nxyz[0];
               if (mesh->Dimension() > 1)
               {
                  cout << "Enter ny: " << flush;
                  cin >> nxyz[1]; np *= nxyz[1];
                  if (mesh->Dimension() > 2)
                  {
                     cout << "Enter nz: " << flush;
                     cin >> nxyz[2]; np *= nxyz[2];
                  }
               }
               partitioning = Array<int>(mesh->CartesianPartitioning(nxyz), mesh->GetNE());
               recover_bdr_partitioning(mesh, partitioning, bdr_partitioning);
            }
            else if (pk == 's')
            {
               cout << "Enter number of processors: " << flush;
               cin >> np;

               partitioning.SetSize(mesh->GetNE());
               for (int i = 0; i < mesh->GetNE(); i++)
               {
                  partitioning[i] = i * np / mesh->GetNE();
               }
               recover_bdr_partitioning(mesh, partitioning, bdr_partitioning);
            }
            else
            {
               int part_method = pk - '0';
               if (part_method < 0 || part_method > 5)
               {
                  continue;
               }
               cout << "Enter number of processors: " << flush;
               cin >> np;
               partitioning = Array<int>(mesh->GeneratePartitioning(np, part_method),
                                         mesh->GetNE());
               recover_bdr_partitioning(mesh, partitioning, bdr_partitioning);
            }
            if (partitioning)
            {
               const char part_file[] = "partitioning.txt";
               ofstream opart(part_file);
               opart << "number_of_elements " << mesh->GetNE() << '\n'
                     << "number_of_processors " << np << '\n';
               for (int i = 0; i < mesh->GetNE(); i++)
               {
                  opart << partitioning[i] << '\n';
               }
               cout << "Partitioning file: " << part_file << endl;

               Array<int> proc_el(np);
               proc_el = 0;
               for (int i = 0; i < mesh->GetNE(); i++)
               {
                  proc_el[partitioning[i]]++;
               }
               int min_el = proc_el[0], max_el = proc_el[0];
               for (int i = 1; i < np; i++)
               {
                  if (min_el > proc_el[i])
                  {
                     min_el = proc_el[i];
                  }
                  if (max_el < proc_el[i])
                  {
                     max_el = proc_el[i];
                  }
               }
               cout << "Partitioning stats:\n"
                    << "           "
                    << setw(12) << "minimum"
                    << setw(12) << "average"
                    << setw(12) << "maximum"
                    << setw(12) << "total" << '\n';
               cout << " elements  "
                    << setw(12) << min_el
                    << setw(12) << double(mesh->GetNE())/np
                    << setw(12) << max_el
                    << setw(12) << mesh->GetNE() << endl;
            }
            else
            {
               continue;
            }

            for (int i = 0; i < mesh->GetNE(); i++)
            {
               attr(i) = partitioning[i] + 1;
            }
         }

         char vishost[] = "localhost";
         int  visport   = 19916;
         socketstream sol_sock(vishost, visport);
         if (sol_sock.is_open())
         {
            sol_sock.precision(14);
            if (sdim == 2)
            {
               sol_sock << "fem2d_gf_data_keys\n";
               if (mk != 'p')
               {
                  mesh->Print(sol_sock);
               }
               else
               {
                  // NURBS meshes do not support PrintWithPartitioning
                  if (mesh->NURBSext)
                  {
                     mesh->Print(sol_sock);
                     for (int i = 0; i < mesh->GetNE(); i++)
                     {
                        attr(i) = partitioning[i];
                     }
                  }
                  else
                  {
                     mesh->PrintWithPartitioning(partitioning, sol_sock, 1);
                  }
               }
               attr.Save(sol_sock);
               sol_sock << "RjlmAb***********";
               if (mk == 'v')
               {
                  sol_sock << "e";
               }
               else
               {
                  sol_sock << "\n";
               }
            }
            else
            {
               sol_sock << "fem3d_gf_data_keys\n";
               if (mk == 'v' || mk == 'h' || mk == 'k' || mk == 'J')
               {
                  mesh->Print(sol_sock);
               }
               else if (mk == 'b' || mk == 'B')
               {
                  bdr_mesh->Print(sol_sock);
                  bdr_attr.Save(sol_sock);
                  sol_sock << "mcaaA";
                  // Switch to a discrete color scale
                  sol_sock << "pppppp" << "pppppp" << "pppppp";
               }
               else
               {
                  // NURBS meshes do not support PrintWithPartitioning
                  if (mesh->NURBSext)
                  {
                     mesh->Print(sol_sock);
                     for (int i = 0; i < mesh->GetNE(); i++)
                     {
                        attr(i) = partitioning[i];
                     }
                  }
                  else
                  {
                     mesh->PrintWithPartitioning(partitioning, sol_sock);
                  }
               }
               if (mk != 'b' && mk != 'B')
               {
                  attr.Save(sol_sock);
                  sol_sock << "maaA";
                  if (mk == 'v')
                  {
                     sol_sock << "aa";
                  }
                  else
                  {
                     sol_sock << "\n";
                  }
               }
            }
            sol_sock << flush;
         }
         else
         {
            cout << "Unable to connect to "
                 << vishost << ':' << visport << endl;
         }
         delete attr_fespace;
         delete bdr_attr_fespace;
      }

      if (mk == 'l')
      {
         // Project and plot the function 'f'
         int p;
         FiniteElementCollection *fec = NULL;
         cout << "Enter projection space order: " << flush;
         cin >> p;
         if (p >= 1)
         {
            fec = new H1_FECollection(p, mesh->Dimension(),
                                      BasisType::GaussLobatto);
         }
         else
         {
            fec = new DG_FECollection(-p, mesh->Dimension(),
                                      BasisType::GaussLegendre);
         }
         FiniteElementSpace fes(mesh, fec);
         GridFunction level(&fes);
         FunctionCoefficient coeff(f);
         level.ProjectCoefficient(coeff);
         char vishost[] = "localhost";
         int  visport   = 19916;
         socketstream sol_sock(vishost, visport);
         if (sol_sock.is_open())
         {
            sol_sock.precision(14);
            sol_sock << "solution\n" << *mesh << level << flush;
         }
         else
         {
            cout << "Unable to connect to "
                 << vishost << ':' << visport << endl;
         }
         delete fec;
      }

      if (mk == 'S')
      {
         const char mesh_file[] = "mesh-explorer.mesh";
         ofstream omesh(mesh_file);
         omesh.precision(14);
         mesh->Print(omesh);
         cout << "New mesh file: " << mesh_file << endl;
      }

      if (mk == 'V')
      {
         const char mesh_file[] = "mesh-explorer.vtk";
         ofstream omesh(mesh_file);
         omesh.precision(14);
         mesh->PrintVTK(omesh);
         cout << "New VTK mesh file: " << mesh_file << endl;
      }

#ifdef MFEM_USE_ZLIB
      if (mk == 'Z')
      {
         const char mesh_file[] = "mesh-explorer.mesh.gz";
         ofgzstream omesh(mesh_file, "zwb9");
         omesh.precision(14);
         mesh->Print(omesh);
         cout << "New mesh file: " << mesh_file << endl;
      }
#endif

   }

   delete bdr_attr_fec;
   delete attr_fec;
   delete bdr_mesh;
   delete mesh;
   return 0;
}