MFEM  v4.2.0 Finite element discretization library
pminimal-surface.cpp
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1 // Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
3 // LICENSE and NOTICE for details. LLNL-CODE-806117.
4 //
5 // This file is part of the MFEM library. For more information and source code
6 // availability visit https://mfem.org.
7 //
8 // MFEM is free software; you can redistribute it and/or modify it under the
9 // terms of the BSD-3 license. We welcome feedback and contributions, see file
10 // CONTRIBUTING.md for details.
11 //
12 // --------------------------------------------------------------------
13 // Minimal Surface Miniapp: Compute minimal surfaces - Parallel Version
14 // --------------------------------------------------------------------
15 //
16 // This miniapp solves Plateau's problem: the Dirichlet problem for the minimal
17 // surface equation.
18 //
19 // Two problems can be run:
20 //
21 // - Problem 0 solves the minimal surface equation of parametric surfaces.
22 // The surface (-s) option allow the selection of different
23 // parametrization:
24 // s=0: Uses the given mesh from command line options
25 // s=1: Catenoid
26 // s=2: Helicoid
27 // s=3: Enneper
28 // s=4: Hold
29 // s=5: Costa
30 // s=6: Shell
31 // s=7: Scherk
32 // s=8: FullPeach
33 // s=9: QuarterPeach
34 // s=10: SlottedSphere
35 //
36 // - Problem 1 solves the minimal surface equation of the form z=f(x,y),
37 // for the Dirichlet problem, using Picard iterations:
38 // -div( q grad u) = 0, with q(u) = (1 + |∇u|²)^{-1/2}
39 //
40 // Compile with: make pminimal-surface
41 //
42 // Sample runs: mpirun -np 4 pminimal-surface
43 // mpirun -np 4 pminimal-surface -a
44 // mpirun -np 4 pminimal-surface -c
45 // mpirun -np 4 pminimal-surface -c -a
46 // mpirun -np 4 pminimal-surface -no-pa
47 // mpirun -np 4 pminimal-surface -no-pa -a
48 // mpirun -np 4 pminimal-surface -no-pa -a -c
49 // mpirun -np 4 pminimal-surface -p 1
50 //
51 // Device sample runs:
52 // mpirun -np 4 pminimal-surface -d debug
53 // mpirun -np 4 pminimal-surface -d debug -a
54 // mpirun -np 4 pminimal-surface -d debug -c
55 // mpirun -np 4 pminimal-surface -d debug -c -a
56 // mpirun -np 4 pminimal-surface -d cuda
57 // mpirun -np 4 pminimal-surface -d cuda -a
58 // mpirun -np 4 pminimal-surface -d cuda -c
59 // mpirun -np 4 pminimal-surface -d cuda -c -a
60 // mpirun -np 4 pminimal-surface -d cuda -no-pa
61 // mpirun -np 4 pminimal-surface -d cuda -no-pa -a
62 // mpirun -np 4 pminimal-surface -d cuda -no-pa -c
63 // mpirun -np 4 pminimal-surface -d cuda -no-pa -c -a
64
65 #include "mfem.hpp"
66 #include "../../general/forall.hpp"
67
68 using namespace mfem;
69
70 // Constant variables
71 constexpr int DIM = 2;
72 constexpr int SDIM = 3;
73 constexpr double PI = M_PI;
74 constexpr double NRM = 1.e-4;
75 constexpr double EPS = 1.e-14;
77 constexpr double NL_DMAX = std::numeric_limits<double>::max();
78
79 // Static variables for GLVis
80 static socketstream glvis;
81 constexpr int GLVIZ_W = 1024;
82 constexpr int GLVIZ_H = 1024;
83 constexpr int visport = 19916;
84 constexpr char vishost[] = "localhost";
85
86 // Context/Options for the solver
87 struct Opt
88 {
89  int sz, id;
90  int pb = 0;
91  int nx = 6;
92  int ny = 6;
93  int order = 3;
94  int refine = 2;
95  int niters = 8;
96  int surface = 5;
97  bool pa = true;
98  bool vis = true;
99  bool amr = false;
100  bool wait = false;
101  bool print = false;
103  bool by_vdim = false;
104  bool snapshot = false;
105  // bool vis_mesh = false; // Not supported by GLVis
106  double tau = 1.0;
107  double lambda = 0.1;
108  double amr_threshold = 0.6;
109  const char *keys = "gAm";
110  const char *device_config = "cpu";
111  const char *mesh_file = "../../data/mobius-strip.mesh";
112  void (*Tptr)(const Vector&, Vector&) = nullptr;
113 };
114
115 class Surface: public Mesh
116 {
117 protected:
118  Opt &opt;
119  ParMesh *mesh;
120  Array<int> bc;
121  H1_FECollection *fec;
123 public:
124  // Reading from mesh file
125  Surface(Opt &opt, const char *file): Mesh(file, true), opt(opt) { }
126
127  // Generate 2D empty surface mesh
128  Surface(Opt &opt, bool): Mesh(), opt(opt) { }
129
130  // Generate 2D quad surface mesh
131  Surface(Opt &opt): Mesh(opt.nx, opt.ny, QUAD, true), opt(opt) { }
132
133  // Generate 2D generic surface mesh
134  Surface(Opt &opt, int nv, int ne, int nbe):
135  Mesh(DIM, nv, ne, nbe, SDIM), opt(opt) { }
136
137  void Create()
138  {
139  if (opt.surface > 0)
140  {
141  Prefix();
142  Transform();
143  }
144  Postfix();
145  Refine();
146  Snap();
147  fec = new H1_FECollection(opt.order, DIM);
148  if (opt.amr) { EnsureNCMesh(); }
149  mesh = new ParMesh(MPI_COMM_WORLD, *this);
150  fes = new ParFiniteElementSpace(mesh, fec, opt.by_vdim ? 1 : SDIM);
151  BoundaryConditions();
152  }
153
154  int Solve()
155  {
156  // Initialize GLVis server if 'visualization' is set
157  if (opt.vis) { opt.vis = glvis.open(vishost, visport) == 0; }
158  // Send to GLVis the first mesh
159  if (opt.vis) { Visualize(opt, mesh, GLVIZ_W, GLVIZ_H); }
160  // Create and launch the surface solver
161  if (opt.by_vdim)
162  {
163  ByVDim(*this, opt).Solve();
164  }
165  else
166  {
167  ByNodes(*this, opt).Solve();
168  }
169  if (opt.vis && opt.snapshot)
170  {
171  opt.keys = "Sq";
172  Visualize(opt, mesh, mesh->GetNodes());
173  }
174  return 0;
175  }
176
177  ~Surface()
178  {
179  if (opt.vis) { glvis.close(); }
180  delete mesh; delete fec; delete fes;
181  }
182
183  virtual void Prefix()
184  {
185  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
186  }
187
188  virtual void Transform() { if (opt.Tptr) { Mesh::Transform(opt.Tptr); } }
189
190  virtual void Postfix()
191  {
192  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
193  }
194
195  virtual void Refine()
196  {
197  for (int l = 0; l < opt.refine; l++)
198  {
199  UniformRefinement();
200  }
201  }
202
203  virtual void Snap()
204  {
205  GridFunction &nodes = *GetNodes();
206  for (int i = 0; i < nodes.Size(); i++)
207  {
208  if (std::abs(nodes(i)) < EPS)
209  {
210  nodes(i) = 0.0;
211  }
212  }
213  }
214
215  void SnapNodesToUnitSphere()
216  {
217  Vector node(SDIM);
218  GridFunction &nodes = *GetNodes();
219  for (int i = 0; i < nodes.FESpace()->GetNDofs(); i++)
220  {
221  for (int d = 0; d < SDIM; d++)
222  {
223  node(d) = nodes(nodes.FESpace()->DofToVDof(i, d));
224  }
225  node /= node.Norml2();
226  for (int d = 0; d < SDIM; d++)
227  {
228  nodes(nodes.FESpace()->DofToVDof(i, d)) = node(d);
229  }
230  }
231  }
232
233  virtual void BoundaryConditions()
234  {
235  if (bdr_attributes.Size())
236  {
237  Array<int> ess_bdr(bdr_attributes.Max());
238  ess_bdr = 1;
240  fes->GetEssentialTrueDofs(ess_bdr, bc);
241  }
242  }
243
244  // Initialize visualization of some given mesh
245  static void Visualize(Opt &opt, const Mesh *mesh,
246  const int w, const int h,
247  const GridFunction *sol = nullptr)
248  {
249  const GridFunction &solution = sol ? *sol : *mesh->GetNodes();
250  glvis << "parallel " << opt.sz << " " << opt.id << "\n";
251  glvis << "solution\n" << *mesh << solution;
252  glvis.precision(8);
253  glvis << "window_size " << w << " " << h << "\n";
254  glvis << "keys " << opt.keys << "\n";
255  opt.keys = nullptr;
256  if (opt.wait) { glvis << "pause\n"; }
257  glvis << std::flush;
258  }
259
260  // Visualize some solution on the given mesh
261  static void Visualize(const Opt &opt, const Mesh *mesh,
262  const GridFunction *sol = nullptr)
263  {
264  glvis << "parallel " << opt.sz << " " << opt.id << "\n";
265  const GridFunction &solution = sol ? *sol : *mesh->GetNodes();
266  glvis << "solution\n" << *mesh << solution;
267  if (opt.wait) { glvis << "pause\n"; }
268  if (opt.snapshot && opt.keys) { glvis << "keys " << opt.keys << "\n"; }
269  glvis << std::flush;
270  }
271
272  using Mesh::Print;
273  static void Print(const Opt &opt, ParMesh *mesh, const GridFunction *sol)
274  {
275  const char *mesh_file = "surface.mesh";
276  const char *sol_file = "sol.gf";
277  if (!opt.id)
278  {
279  mfem::out << "Printing " << mesh_file << ", " << sol_file << std::endl;
280  }
281
282  std::ostringstream mesh_name;
283  mesh_name << mesh_file << "." << std::setfill('0') << std::setw(6) << opt.id;
284  std::ofstream mesh_ofs(mesh_name.str().c_str());
285  mesh_ofs.precision(8);
286  mesh->Print(mesh_ofs);
287  mesh_ofs.close();
288
289  std::ostringstream sol_name;
290  sol_name << sol_file << "." << std::setfill('0') << std::setw(6) << opt.id;
291  std::ofstream sol_ofs(sol_name.str().c_str());
292  sol_ofs.precision(8);
293  sol->Save(sol_ofs);
294  sol_ofs.close();
295  }
296
297  // Surface Solver class
298  class Solver
299  {
300  protected:
301  Opt &opt;
302  Surface &S;
303  CGSolver cg;
304  OperatorPtr A;
306  ParGridFunction x, x0, b;
308  mfem::Solver *M = nullptr;
309  const int print_iter = -1, max_num_iter = 2000;
310  const double RTOLERANCE = EPS, ATOLERANCE = EPS*EPS;
311  public:
312  Solver(Surface &S, Opt &opt): opt(opt), S(S), cg(MPI_COMM_WORLD),
313  a(S.fes), x(S.fes), x0(S.fes), b(S.fes), one(1.0)
314  {
315  cg.SetRelTol(RTOLERANCE);
316  cg.SetAbsTol(ATOLERANCE);
317  cg.SetMaxIter(max_num_iter);
318  cg.SetPrintLevel(print_iter);
319  }
320
321  ~Solver() { delete M; }
322
323  void Solve()
324  {
325  constexpr bool converged = true;
326  if (opt.pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL);}
327  for (int i=0; i < opt.niters; ++i)
328  {
329  if (opt.amr) { Amr(); }
330  if (opt.vis) { Surface::Visualize(opt, S.mesh); }
331  if (!opt.id) { mfem::out << "Iteration " << i << ": "; }
332  S.mesh->DeleteGeometricFactors();
333  a.Update();
334  a.Assemble();
335  if (Step() == converged) { break; }
336  }
337  if (opt.print) { Surface::Print(opt, S.mesh, S.mesh->GetNodes()); }
338  }
339
340  virtual bool Step() = 0;
341
342  protected:
343  bool Converged(const double rnorm) { return rnorm < NRM; }
344
345  bool ParAXeqB()
346  {
347  b = 0.0;
348  Vector X, B;
349  a.FormLinearSystem(S.bc, x, b, A, X, B);
350  if (!opt.pa) { M = new HypreBoomerAMG; }
351  if (M) { cg.SetPreconditioner(*M); }
352  cg.SetOperator(*A);
353  cg.Mult(B, X);
354  a.RecoverFEMSolution(X, b, x);
355  const bool by_vdim = opt.by_vdim;
356  GridFunction *nodes = by_vdim ? &x0 : S.fes->GetMesh()->GetNodes();
359  double rnorm = nodes->DistanceTo(x) / nodes->Norml2();
360  double glob_norm;
361  MPI_Allreduce(&rnorm, &glob_norm, 1, MPI_DOUBLE, MPI_MAX, MPI_COMM_WORLD);
362  rnorm = glob_norm;
363  if (!opt.id) { mfem::out << "rnorm = " << rnorm << std::endl; }
364  const double lambda = opt.lambda;
365  if (by_vdim)
366  {
368  return Converged(rnorm);
369  }
371  {
372  GridFunction delta(S.fes);
373  subtract(x, *nodes, delta); // delta = x - nodes
374  // position and Δ vectors at point i
375  Vector ni(SDIM), di(SDIM);
376  for (int i = 0; i < delta.Size()/SDIM; i++)
377  {
378  // extract local vectors
379  const int ndof = S.fes->GetNDofs();
380  for (int d = 0; d < SDIM; d++)
381  {
382  ni(d) = (*nodes)(d*ndof + i);
383  di(d) = delta(d*ndof + i);
384  }
385  // project the delta vector in radial direction
386  const double ndotd = (ni*di) / (ni*ni);
387  di.Set(ndotd,ni);
388  // set global vectors
389  for (int d = 0; d < SDIM; d++) { delta(d*ndof + i) = di(d); }
390  }
392  }
393  // x = lambda*nodes + (1-lambda)*x
394  add(lambda, *nodes, (1.0-lambda), x, x);
395  return Converged(rnorm);
396  }
397
398  void Amr()
399  {
400  MFEM_VERIFY(opt.amr_threshold >= 0.0 && opt.amr_threshold <= 1.0, "");
401  Mesh *mesh = S.mesh;
402  Array<Refinement> amr;
403  const int NE = mesh->GetNE();
405  for (int e = 0; e < NE; e++)
406  {
407  double minW = +NL_DMAX;
408  double maxW = -NL_DMAX;
410  const Geometry::Type &type = mesh->GetElement(e)->GetGeometryType();
411  const IntegrationRule *ir = &IntRules.Get(type, opt.order);
412  const int NQ = ir->GetNPoints();
413  for (int q = 0; q < NQ; q++)
414  {
415  eTr->SetIntPoint(&ir->IntPoint(q));
416  const DenseMatrix &J = eTr->Jacobian();
420  const double w = Jadjt.Weight();
421  minW = std::fmin(minW, w);
422  maxW = std::fmax(maxW, w);
423  }
424  if (std::fabs(maxW) != 0.0)
425  {
426  const double rho = minW / maxW;
427  MFEM_VERIFY(rho <= 1.0, "");
428  if (rho < opt.amr_threshold) { amr.Append(Refinement(e)); }
429  }
430  }
431  if (amr.Size()>0)
432  {
434  mesh->GeneralRefinement(amr);
435  S.fes->Update();
437  x.Update();
438  a.Update();
440  b.Update();
441  S.BoundaryConditions();
442  }
443  }
444  };
445
446  // Surface solver 'by vector'
447  class ByNodes: public Solver
448  {
449  public:
450  ByNodes(Surface &S, Opt &opt): Solver(S, opt)
452
453  bool Step()
454  {
455  x = *S.fes->GetMesh()->GetNodes();
456  bool converge = ParAXeqB();
457  S.mesh->SetNodes(x);
458  return converge ? true : false;
459  }
460  };
461
462  // Surface solver 'by ByVDim'
463  class ByVDim: public Solver
464  {
465  public:
466  void SetNodes(const GridFunction &Xi, const int c)
467  {
469  auto d_nodes = S.fes->GetMesh()->GetNodes()->Write();
470  const int ndof = S.fes->GetNDofs();
471  MFEM_FORALL(i, ndof, d_nodes[c*ndof + i] = d_Xi[i]; );
472  }
473
474  void GetNodes(GridFunction &Xi, const int c)
475  {
476  auto d_Xi = Xi.Write();
477  const int ndof = S.fes->GetNDofs();
479  MFEM_FORALL(i, ndof, d_Xi[i] = d_nodes[c*ndof + i]; );
480  }
481
482  ByVDim(Surface &S, Opt &opt): Solver(S, opt)
484
485  bool Step()
486  {
487  bool cvg[SDIM] {false};
488  for (int c=0; c < SDIM; ++c)
489  {
490  GetNodes(x, c);
491  x0 = x;
492  cvg[c] = ParAXeqB();
493  SetNodes(x, c);
494  }
495  const bool converged = cvg[0] && cvg[1] && cvg[2];
496  return converged ? true : false;
497  }
498  };
499 };
500
501 // #0: Default surface mesh file
502 struct MeshFromFile: public Surface
503 {
504  MeshFromFile(Opt &opt): Surface(opt, opt.mesh_file) { }
505 };
506
507 // #1: Catenoid surface
508 struct Catenoid: public Surface
509 {
510  Catenoid(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
511
512  void Prefix()
513  {
514  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
515  Array<int> v2v(GetNV());
516  for (int i = 0; i < v2v.Size(); i++) { v2v[i] = i; }
517  // identify vertices on vertical lines
518  for (int j = 0; j <= opt.ny; j++)
519  {
520  const int v_old = opt.nx + j * (opt.nx + 1);
521  const int v_new = j * (opt.nx + 1);
522  v2v[v_old] = v_new;
523  }
524  // renumber elements
525  for (int i = 0; i < GetNE(); i++)
526  {
527  Element *el = GetElement(i);
528  int *v = el->GetVertices();
529  const int nv = el->GetNVertices();
530  for (int j = 0; j < nv; j++)
531  {
532  v[j] = v2v[v[j]];
533  }
534  }
535  // renumber boundary elements
536  for (int i = 0; i < GetNBE(); i++)
537  {
538  Element *el = GetBdrElement(i);
539  int *v = el->GetVertices();
540  const int nv = el->GetNVertices();
541  for (int j = 0; j < nv; j++)
542  {
543  v[j] = v2v[v[j]];
544  }
545  }
546  RemoveUnusedVertices();
547  RemoveInternalBoundaries();
548  }
549
550  static void Parametrization(const Vector &x, Vector &p)
551  {
552  p.SetSize(SDIM);
553  // u in [0,2π] and v in [-π/6,π/6]
554  const double u = 2.0*PI*x[0];
555  const double v = PI*(x[1]-0.5)/3.;
556  p[0] = cos(u);
557  p[1] = sin(u);
558  p[2] = v;
559  }
560 };
561
562 // #2: Helicoid surface
563 struct Helicoid: public Surface
564 {
565  Helicoid(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
566
567  static void Parametrization(const Vector &x, Vector &p)
568  {
569  p.SetSize(SDIM);
570  // u in [0,2π] and v in [-2π/3,2π/3]
571  const double u = 2.0*PI*x[0];
572  const double v = 2.0*PI*(2.0*x[1]-1.0)/3.0;
573  p(0) = sin(u)*v;
574  p(1) = cos(u)*v;
575  p(2) = u;
576  }
577 };
578
579 // #3: Enneper's surface
580 struct Enneper: public Surface
581 {
582  Enneper(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
583
584  static void Parametrization(const Vector &x, Vector &p)
585  {
586  p.SetSize(SDIM);
587  // (u,v) in [-2, +2]
588  const double u = 4.0*(x[0]-0.5);
589  const double v = 4.0*(x[1]-0.5);
590  p[0] = +u - u*u*u/3.0 + u*v*v;
591  p[1] = -v - u*u*v + v*v*v/3.0;
592  p[2] = u*u - v*v;
593  }
594 };
595
596 // #4: Hold surface
597 struct Hold: public Surface
598 {
599  Hold(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
600
601  void Prefix()
602  {
603  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
604  Array<int> v2v(GetNV());
605  for (int i = 0; i < v2v.Size(); i++) { v2v[i] = i; }
606  // identify vertices on vertical lines
607  for (int j = 0; j <= opt.ny; j++)
608  {
609  const int v_old = opt.nx + j * (opt.nx + 1);
610  const int v_new = j * (opt.nx + 1);
611  v2v[v_old] = v_new;
612  }
613  // renumber elements
614  for (int i = 0; i < GetNE(); i++)
615  {
616  Element *el = GetElement(i);
617  int *v = el->GetVertices();
618  const int nv = el->GetNVertices();
619  for (int j = 0; j < nv; j++)
620  {
621  v[j] = v2v[v[j]];
622  }
623  }
624  // renumber boundary elements
625  for (int i = 0; i < GetNBE(); i++)
626  {
627  Element *el = GetBdrElement(i);
628  int *v = el->GetVertices();
629  const int nv = el->GetNVertices();
630  for (int j = 0; j < nv; j++)
631  {
632  v[j] = v2v[v[j]];
633  }
634  }
635  RemoveUnusedVertices();
636  RemoveInternalBoundaries();
637  }
638
639  static void Parametrization(const Vector &x, Vector &p)
640  {
641  p.SetSize(SDIM);
642  // u in [0,2π] and v in [0,1]
643  const double u = 2.0*PI*x[0];
644  const double v = x[1];
645  p[0] = cos(u)*(1.0 + 0.3*sin(3.*u + PI*v));
646  p[1] = sin(u)*(1.0 + 0.3*sin(3.*u + PI*v));
647  p[2] = v;
648  }
649 };
650
651 // #5: Costa minimal surface
652 #include <complex>
653 using cdouble = std::complex<double>;
654 #define I cdouble(0.0, 1.0)
655
656 // https://dlmf.nist.gov/20.2
657 cdouble EllipticTheta(const int a, const cdouble u, const cdouble q)
658 {
659  cdouble J = 0.0;
660  double delta = std::numeric_limits<double>::max();
661  switch (a)
662  {
663  case 1:
664  for (int n=0; delta > EPS; n+=1)
665  {
666  const cdouble j = pow(-1,n)*pow(q,n*(n+1.0))*sin((2.0*n+1.0)*u);
667  delta = abs(j);
668  J += j;
669  }
670  return 2.0*pow(q,0.25)*J;
671
672  case 2:
673  for (int n=0; delta > EPS; n+=1)
674  {
675  const cdouble j = pow(q,n*(n+1))*cos((2.0*n+1.0)*u);
676  delta = abs(j);
677  J += j;
678  }
679  return 2.0*pow(q,0.25)*J;
680  case 3:
681  for (int n=1; delta > EPS; n+=1)
682  {
683  const cdouble j = pow(q,n*n)*cos(2.0*n*u);
684  delta = abs(j);
685  J += j;
686  }
687  return 1.0 + 2.0*J;
688  case 4:
689  for (int n=1; delta > EPS; n+=1)
690  {
691  const cdouble j =pow(-1,n)*pow(q,n*n)*cos(2.0*n*u);
692  delta = abs(j);
693  J += j;
694  }
695  return 1.0 + 2.0*J;
696  }
697  return J;
698 }
699
700 // https://dlmf.nist.gov/23.6#E5
702  const cdouble w1 = 0.5,
703  const cdouble w3 = 0.5*I)
704 {
705  const cdouble tau = w3/w1;
706  const cdouble q = exp(I*M_PI*tau);
707  const cdouble e1 = M_PI*M_PI/(12.0*w1*w1)*
708  (1.0*pow(EllipticTheta(2,0,q),4) +
709  2.0*pow(EllipticTheta(4,0,q),4));
710  const cdouble u = M_PI*z / (2.0*w1);
711  const cdouble P = M_PI * EllipticTheta(3,0,q)*EllipticTheta(4,0,q) *
712  EllipticTheta(2,u,q)/(2.0*w1*EllipticTheta(1,u,q));
713  return P*P + e1;
714 }
715
716 cdouble EllipticTheta1Prime(const int k, const cdouble u, const cdouble q)
717 {
718  cdouble J = 0.0;
719  double delta = std::numeric_limits<double>::max();
720  for (int n=0; delta > EPS; n+=1)
721  {
722  const double alpha = 2.0*n+1.0;
723  const cdouble Dcosine = pow(alpha,k)*sin(k*M_PI/2.0 + alpha*u);
724  const cdouble j = pow(-1,n)*pow(q,n*(n+1.0))*Dcosine;
725  delta = abs(j);
726  J += j;
727  }
728  return 2.0*pow(q,0.25)*J;
729 }
730
731 // Logarithmic Derivative of Theta Function 1
733 {
734  cdouble J = 0.0;
735  double delta = std::numeric_limits<double>::max();
736  for (int n=1; delta > EPS; n+=1)
737  {
738  cdouble q2n = pow(q, 2*n);
739  if (abs(q2n) < EPS) { q2n = 0.0; }
740  const cdouble j = q2n/(1.0-q2n)*sin(2.0*n*u);
741  delta = abs(j);
742  J += j;
743  }
744  return 1.0/tan(u) + 4.0*J;
745 }
746
747 // https://dlmf.nist.gov/23.6#E13
749  const cdouble w1 = 0.5,
750  const cdouble w3 = 0.5*I)
751 {
752  const cdouble tau = w3/w1;
753  const cdouble q = exp(I*M_PI*tau);
754  const cdouble n1 = -M_PI*M_PI/(12.0*w1) *
755  (EllipticTheta1Prime(3,0,q)/
756  EllipticTheta1Prime(1,0,q));
757  const cdouble u = M_PI*z / (2.0*w1);
758  return z*n1/w1 + M_PI/(2.0*w1)*LogEllipticTheta1Prime(u,q);
759 }
760
761 // https://www.mathcurve.com/surfaces.gb/costa/costa.shtml
762 static double ALPHA[4] {0.0};
763 struct Costa: public Surface
764 {
765  Costa(Opt &opt): Surface((opt.Tptr = Parametrization, opt), false) { }
766
767  void Prefix()
768  {
769  ALPHA[3] = opt.tau;
770  const int nx = opt.nx, ny = opt.ny;
771  MFEM_VERIFY(nx>2 && ny>2, "");
772  const int nXhalf = (nx%2)==0 ? 4 : 2;
773  const int nYhalf = (ny%2)==0 ? 4 : 2;
774  const int nxh = nXhalf + nYhalf;
775  const int NVert = (nx+1) * (ny+1);
776  const int NElem = nx*ny - 4 - nxh;
777  const int NBdrElem = 0;
778  InitMesh(DIM, SDIM, NVert, NElem, NBdrElem);
779  // Sets vertices and the corresponding coordinates
780  for (int j = 0; j <= ny; j++)
781  {
782  const double cy = ((double) j / ny) ;
783  for (int i = 0; i <= nx; i++)
784  {
785  const double cx = ((double) i / nx);
786  const double coords[SDIM] = {cx, cy, 0.0};
788  }
789  }
790  // Sets elements and the corresponding indices of vertices
791  for (int j = 0; j < ny; j++)
792  {
793  for (int i = 0; i < nx; i++)
794  {
795  if (i == 0 && j == 0) { continue; }
796  if (i+1 == nx && j == 0) { continue; }
797  if (i == 0 && j+1 == ny) { continue; }
798  if (i+1 == nx && j+1 == ny) { continue; }
799  if ((j == 0 || j+1 == ny) && (abs(nx-(i<<1)-1)<=1)) { continue; }
800  if ((i == 0 || i+1 == nx) && (abs(ny-(j<<1)-1)<=1)) { continue; }
801  const int i0 = i + j*(nx+1);
802  const int i1 = i+1 + j*(nx+1);
803  const int i2 = i+1 + (j+1)*(nx+1);
804  const int i3 = i + (j+1)*(nx+1);
805  const int ind[4] = {i0, i1, i2, i3};
807  }
808  }
809  RemoveUnusedVertices();
811  FinalizeTopology();
812  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
813  }
814
815  static void Parametrization(const Vector &X, Vector &p)
816  {
817  const double tau = ALPHA[3];
818  Vector x = X;
819  x -= +0.5;
820  x *= tau;
821  x -= -0.5;
822
823  p.SetSize(3);
824  const bool y_top = x[1] > 0.5;
825  const bool x_top = x[0] > 0.5;
826  double u = x[0];
827  double v = x[1];
828  if (y_top) { v = 1.0 - x[1]; }
829  if (x_top) { u = 1.0 - x[0]; }
830  const cdouble w = u + I*v;
831  const cdouble w3 = I/2.;
832  const cdouble w1 = 1./2.;
833  const cdouble pw = WeierstrassP(w);
834  const cdouble e1 = WeierstrassP(0.5);
835  const cdouble zw = WeierstrassZeta(w);
836  const cdouble dw = WeierstrassZeta(w-w1) - WeierstrassZeta(w-w3);
837  p[0] = real(PI*(u+PI/(4.*e1))- zw +PI/(2.*e1)*(dw));
838  p[1] = real(PI*(v+PI/(4.*e1))-I*zw-PI*I/(2.*e1)*(dw));
839  p[2] = sqrt(PI/2.)*log(abs((pw-e1)/(pw+e1)));
840  if (y_top) { p[1] *= -1.0; }
841  if (x_top) { p[0] *= -1.0; }
842  const bool nan = std::isnan(p[0]) || std::isnan(p[1]) || std::isnan(p[2]);
843  MFEM_VERIFY(!nan, "nan");
844  ALPHA[0] = std::fmax(p[0], ALPHA[0]);
845  ALPHA[1] = std::fmax(p[1], ALPHA[1]);
846  ALPHA[2] = std::fmax(p[2], ALPHA[2]);
847  }
848
849  void Snap()
850  {
851  Vector node(SDIM);
852  MFEM_VERIFY(ALPHA[0] > 0.0,"");
853  MFEM_VERIFY(ALPHA[1] > 0.0,"");
854  MFEM_VERIFY(ALPHA[2] > 0.0,"");
855  GridFunction &nodes = *GetNodes();
856  const double phi = (1.0 + sqrt(5.0)) / 2.0;
857  for (int i = 0; i < nodes.FESpace()->GetNDofs(); i++)
858  {
859  for (int d = 0; d < SDIM; d++)
860  {
861  const double alpha = d==2 ? phi : 1.0;
862  const int vdof = nodes.FESpace()->DofToVDof(i, d);
863  nodes(vdof) /= alpha * ALPHA[d];
864  }
865  }
866  }
867 };
868
869 // #6: Shell surface model
870 struct Shell: public Surface
871 {
872  Shell(Opt &opt): Surface((opt.niters = 1, opt.Tptr = Parametrization, opt)) { }
873
874  static void Parametrization(const Vector &x, Vector &p)
875  {
876  p.SetSize(3);
877  // u in [0,2π] and v in [-15, 6]
878  const double u = 2.0*PI*x[0];
879  const double v = 21.0*x[1]-15.0;
880  p[0] = +1.0*pow(1.16,v)*cos(v)*(1.0+cos(u));
881  p[1] = -1.0*pow(1.16,v)*sin(v)*(1.0+cos(u));
882  p[2] = -2.0*pow(1.16,v)*(1.0+sin(u));
883  }
884 };
885
886 // #7: Scherk's doubly periodic surface
887 struct Scherk: public Surface
888 {
889  static void Parametrization(const Vector &x, Vector &p)
890  {
891  p.SetSize(SDIM);
892  const double alpha = 0.49;
893  // (u,v) in [-απ, +απ]
894  const double u = alpha*PI*(2.0*x[0]-1.0);
895  const double v = alpha*PI*(2.0*x[1]-1.0);
896  p[0] = u;
897  p[1] = v;
898  p[2] = log(cos(v)/cos(u));
899  }
900
901  Scherk(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
902 };
903
904 // #8: Full Peach street model
905 struct FullPeach: public Surface
906 {
907  static constexpr int NV = 8;
908  static constexpr int NE = 6;
909
910  FullPeach(Opt &opt):
911  Surface((opt.niters = std::min(4, opt.niters), opt), NV, NE, 0) { }
912
913  void Prefix()
914  {
916  {
917  {-1, -1, -1}, {+1, -1, -1}, {+1, +1, -1}, {-1, +1, -1},
918  {-1, -1, +1}, {+1, -1, +1}, {+1, +1, +1}, {-1, +1, +1}
919  };
921  {
922  {3, 2, 1, 0}, {0, 1, 5, 4}, {1, 2, 6, 5},
923  {2, 3, 7, 6}, {3, 0, 4, 7}, {4, 5, 6, 7}
924
925  };
926  for (int j = 0; j < NV; j++) { AddVertex(quad_v[j]); }
928
930  FinalizeTopology(false);
931  UniformRefinement();
932  SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
933  }
934
935  void Snap() { SnapNodesToUnitSphere(); }
936
937  void BoundaryConditions()
938  {
939  Vector X(SDIM);
940  Array<int> dofs;
941  Array<int> ess_vdofs, ess_tdofs;
942  ess_vdofs.SetSize(fes->GetVSize());
943  MFEM_VERIFY(fes->GetVSize() >= fes->GetTrueVSize(),"");
944  ess_vdofs = 0;
945  DenseMatrix PointMat;
947  for (int e = 0; e < fes->GetNE(); e++)
948  {
949  fes->GetElementDofs(e, dofs);
950  const IntegrationRule &ir = fes->GetFE(e)->GetNodes();
952  eTr->Transform(ir, PointMat);
953  Vector one(dofs.Size());
954  for (int dof = 0; dof < dofs.Size(); dof++)
955  {
956  one = 0.0;
957  one[dof] = 1.0;
958  const int k = dofs[dof];
959  MFEM_ASSERT(k >= 0, "");
960  PointMat.Mult(one, X);
961  const bool halfX = std::fabs(X[0]) < EPS && X[1] <= 0.0;
962  const bool halfY = std::fabs(X[2]) < EPS && X[1] >= 0.0;
963  const bool is_on_bc = halfX || halfY;
964  for (int c = 0; c < SDIM; c++)
965  { ess_vdofs[fes->DofToVDof(k, c)] = is_on_bc; }
966  }
967  }
968  const SparseMatrix *R = fes->GetRestrictionMatrix();
969  if (!R)
970  {
971  ess_tdofs.MakeRef(ess_vdofs);
972  }
973  else
974  {
975  R->BooleanMult(ess_vdofs, ess_tdofs);
976  }
979  }
980 };
981
982 // #9: 1/4th Peach street model
983 struct QuarterPeach: public Surface
984 {
985  QuarterPeach(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
986
987  static void Parametrization(const Vector &X, Vector &p)
988  {
989  p = X;
990  const double x = 2.0*X[0]-1.0;
991  const double y = X[1];
992  const double r = sqrt(x*x + y*y);
993  const double t = (x==0.0) ? PI/2.0 :
994  (y==0.0 && x>0.0) ? 0. :
995  (y==0.0 && x<0.0) ? PI : acos(x/r);
996  const double sqrtx = sqrt(1.0 + x*x);
997  const double sqrty = sqrt(1.0 + y*y);
998  const bool yaxis = PI/4.0<t && t < 3.0*PI/4.0;
999  const double R = yaxis?sqrtx:sqrty;
1000  const double gamma = r/R;
1001  p[0] = gamma * cos(t);
1002  p[1] = gamma * sin(t);
1003  p[2] = 1.0 - gamma;
1004  }
1005
1006  void Postfix()
1007  {
1008  for (int i = 0; i < GetNBE(); i++)
1009  {
1010  Element *el = GetBdrElement(i);
1011  const int fn = GetBdrElementEdgeIndex(i);
1012  MFEM_VERIFY(!FaceIsTrueInterior(fn),"");
1013  Array<int> vertices;
1014  GetFaceVertices(fn, vertices);
1015  const GridFunction *nodes = GetNodes();
1016  Vector nval;
1017  double R[2], X[2][SDIM];
1018  for (int v = 0; v < 2; v++)
1019  {
1020  R[v] = 0.0;
1021  const int iv = vertices[v];
1022  for (int d = 0; d < SDIM; d++)
1023  {
1024  nodes->GetNodalValues(nval, d+1);
1025  const double x = X[v][d] = nval[iv];
1026  if (d < 2) { R[v] += x*x; }
1027  }
1028  }
1029  if (std::fabs(X[0][1])<=EPS && std::fabs(X[1][1])<=EPS &&
1030  (R[0]>0.1 || R[1]>0.1))
1031  { el->SetAttribute(1); }
1032  else { el->SetAttribute(2); }
1033  }
1034  }
1035 };
1036
1037 // #10: Slotted sphere mesh
1038 struct SlottedSphere: public Surface
1039 {
1040  SlottedSphere(Opt &opt): Surface((opt.niters = 4, opt), 64, 40, 0) { }
1041
1042  void Prefix()
1043  {
1044  constexpr double delta = 0.15;
1045  constexpr int NV1D = 4;
1046  constexpr int NV = NV1D*NV1D*NV1D;
1047  constexpr int NEPF = (NV1D-1)*(NV1D-1);
1048  constexpr int NE = NEPF*6;
1049  const double V1D[NV1D] = {-1.0, -delta, delta, 1.0};
1050  double QV[NV][3];
1051  for (int iv=0; iv<NV; ++iv)
1052  {
1053  int ix = iv % NV1D;
1054  int iy = (iv / NV1D) % NV1D;
1055  int iz = (iv / NV1D) / NV1D;
1056
1057  QV[iv][0] = V1D[ix];
1058  QV[iv][1] = V1D[iy];
1059  QV[iv][2] = V1D[iz];
1060  }
1061  int QE[NE][4];
1062  for (int ix=0; ix<NV1D-1; ++ix)
1063  {
1064  for (int iy=0; iy<NV1D-1; ++iy)
1065  {
1066  int el_offset = ix + iy*(NV1D-1);
1067  // x = 0
1068  QE[0*NEPF + el_offset][0] = NV1D*ix + NV1D*NV1D*iy;
1069  QE[0*NEPF + el_offset][1] = NV1D*(ix+1) + NV1D*NV1D*iy;
1070  QE[0*NEPF + el_offset][2] = NV1D*(ix+1) + NV1D*NV1D*(iy+1);
1071  QE[0*NEPF + el_offset][3] = NV1D*ix + NV1D*NV1D*(iy+1);
1072  // x = 1
1073  int x_off = NV1D-1;
1074  QE[1*NEPF + el_offset][3] = x_off + NV1D*ix + NV1D*NV1D*iy;
1075  QE[1*NEPF + el_offset][2] = x_off + NV1D*(ix+1) + NV1D*NV1D*iy;
1076  QE[1*NEPF + el_offset][1] = x_off + NV1D*(ix+1) + NV1D*NV1D*(iy+1);
1077  QE[1*NEPF + el_offset][0] = x_off + NV1D*ix + NV1D*NV1D*(iy+1);
1078  // y = 0
1079  QE[2*NEPF + el_offset][0] = NV1D*NV1D*iy + ix;
1080  QE[2*NEPF + el_offset][1] = NV1D*NV1D*iy + ix + 1;
1081  QE[2*NEPF + el_offset][2] = NV1D*NV1D*(iy+1) + ix + 1;
1082  QE[2*NEPF + el_offset][3] = NV1D*NV1D*(iy+1) + ix;
1083  // y = 1
1084  int y_off = NV1D*(NV1D-1);
1085  QE[3*NEPF + el_offset][0] = y_off + NV1D*NV1D*iy + ix;
1086  QE[3*NEPF + el_offset][1] = y_off + NV1D*NV1D*iy + ix + 1;
1087  QE[3*NEPF + el_offset][2] = y_off + NV1D*NV1D*(iy+1) + ix + 1;
1088  QE[3*NEPF + el_offset][3] = y_off + NV1D*NV1D*(iy+1) + ix;
1089  // z = 0
1090  QE[4*NEPF + el_offset][0] = NV1D*iy + ix;
1091  QE[4*NEPF + el_offset][1] = NV1D*iy + ix + 1;
1092  QE[4*NEPF + el_offset][2] = NV1D*(iy+1) + ix + 1;
1093  QE[4*NEPF + el_offset][3] = NV1D*(iy+1) + ix;
1094  // z = 1
1095  int z_off = NV1D*NV1D*(NV1D-1);
1096  QE[5*NEPF + el_offset][0] = z_off + NV1D*iy + ix;
1097  QE[5*NEPF + el_offset][1] = z_off + NV1D*iy + ix + 1;
1098  QE[5*NEPF + el_offset][2] = z_off + NV1D*(iy+1) + ix + 1;
1099  QE[5*NEPF + el_offset][3] = z_off + NV1D*(iy+1) + ix;
1100  }
1101  }
1102  // Delete on x = 0 face
1103  QE[0*NEPF + 1 + 2*(NV1D-1)][0] = -1;
1104  QE[0*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1105  // Delete on x = 1 face
1106  QE[1*NEPF + 1 + 2*(NV1D-1)][0] = -1;
1107  QE[1*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1108  // Delete on y = 1 face
1109  QE[3*NEPF + 1 + 0*(NV1D-1)][0] = -1;
1110  QE[3*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1111  // Delete on z = 1 face
1112  QE[5*NEPF + 0 + 1*(NV1D-1)][0] = -1;
1113  QE[5*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1114  QE[5*NEPF + 2 + 1*(NV1D-1)][0] = -1;
1115  // Delete on z = 0 face
1116  QE[4*NEPF + 1 + 0*(NV1D-1)][0] = -1;
1117  QE[4*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1118  QE[4*NEPF + 1 + 2*(NV1D-1)][0] = -1;
1119  // Delete on y = 0 face
1120  QE[2*NEPF + 1 + 0*(NV1D-1)][0] = -1;
1121  QE[2*NEPF + 1 + 1*(NV1D-1)][0] = -1;
1122  for (int j = 0; j < NV; j++) { AddVertex(QV[j]); }
1123  for (int j = 0; j < NE; j++)
1124  {
1125  if (QE[j][0] < 0) { continue; }
1127  }
1128  RemoveUnusedVertices();
1130  EnsureNodes();
1131  FinalizeTopology();
1132  }
1133
1134  void Snap() { SnapNodesToUnitSphere(); }
1135 };
1136
1137 static int Problem0(Opt &opt)
1138 {
1139  // Create our surface mesh from command line options
1140  Surface *S = nullptr;
1141  switch (opt.surface)
1142  {
1143  case 0: S = new MeshFromFile(opt); break;
1144  case 1: S = new Catenoid(opt); break;
1145  case 2: S = new Helicoid(opt); break;
1146  case 3: S = new Enneper(opt); break;
1147  case 4: S = new Hold(opt); break;
1148  case 5: S = new Costa(opt); break;
1149  case 6: S = new Shell(opt); break;
1150  case 7: S = new Scherk(opt); break;
1151  case 8: S = new FullPeach(opt); break;
1152  case 9: S = new QuarterPeach(opt); break;
1153  case 10: S = new SlottedSphere(opt); break;
1154  default: MFEM_ABORT("Unknown surface (surface <= 10)!");
1155  }
1156  S->Create();
1157  S->Solve();
1158  delete S;
1159  return 0;
1160 }
1161
1162 // Problem 1: solve the Dirichlet problem for the minimal surface equation
1163 // of the form z=f(x,y), using Picard iterations.
1164 static double u0(const Vector &x) { return sin(3.0 * PI * (x[1] + x[0])); }
1165
1166 enum {NORM, AREA};
1167
1168 static double qf(const int order, const int ker, Mesh &m,
1170 {
1171  const Geometry::Type type = m.GetElementBaseGeometry(0);
1172  const IntegrationRule &ir(IntRules.Get(type, order));
1174
1175  const int NE(m.GetNE());
1176  const int ND(fes.GetFE(0)->GetDof());
1177  const int NQ(ir.GetNPoints());
1179  const GeometricFactors *geom = m.GetGeometricFactors(ir, flags);
1180
1181  const int D1D = fes.GetFE(0)->GetOrder() + 1;
1182  const int Q1D = IntRules.Get(Geometry::SEGMENT, ir.GetOrder()).GetNPoints();
1183  MFEM_VERIFY(ND == D1D*D1D, "");
1184  MFEM_VERIFY(NQ == Q1D*Q1D, "");
1185
1186  Vector Eu(ND*NE), grad_u(DIM*NQ*NE), sum(NE*NQ), one(NE*NQ);
1187  qi->SetOutputLayout(QVectorLayout::byVDIM);
1189  const Operator *G(fes.GetElementRestriction(e_ordering));
1190  G->Mult(u, Eu);
1192
1193  auto W = Reshape(ir.GetWeights().Read(), Q1D, Q1D);
1194  auto J = Reshape(geom->J.Read(), Q1D, Q1D, DIM, DIM, NE);
1195  auto detJ = Reshape(geom->detJ.Read(), Q1D, Q1D, NE);
1197  auto S = Reshape(sum.Write(), Q1D, Q1D, NE);
1198
1199  MFEM_FORALL_2D(e, NE, Q1D, Q1D, 1,
1200  {
1202  {
1204  {
1205  const double w = W(qx, qy);
1206  const double J11 = J(qx, qy, 0, 0, e);
1207  const double J12 = J(qx, qy, 1, 0, e);
1208  const double J21 = J(qx, qy, 0, 1, e);
1209  const double J22 = J(qx, qy, 1, 1, e);
1210  const double det = detJ(qx, qy, e);
1211  const double area = w * det;
1212  const double gu0 = grdU(0, qx, qy, e);
1213  const double gu1 = grdU(1, qx, qy, e);
1214  const double tgu0 = (J22*gu0 - J12*gu1)/det;
1215  const double tgu1 = (J11*gu1 - J21*gu0)/det;
1216  const double ngu = tgu0*tgu0 + tgu1*tgu1;
1217  const double s = (ker == AREA) ? sqrt(1.0 + ngu) :
1218  (ker == NORM) ? ngu : 0.0;
1219  S(qx, qy, e) = area * s;
1220  }
1221  }
1222  });
1223  one = 1.0;
1224  return sum * one;
1225 }
1226
1227 static int Problem1(Opt &opt)
1228 {
1229  const int order = opt.order;
1231  smesh.SetCurvature(opt.order, false, DIM, Ordering::byNODES);
1232  for (int l = 0; l < opt.refine; l++) { smesh.UniformRefinement(); }
1233  ParMesh mesh(MPI_COMM_WORLD, smesh);
1234  const H1_FECollection fec(order, DIM);
1235  ParFiniteElementSpace fes(&mesh, &fec);
1236  Array<int> ess_tdof_list;
1237  if (mesh.bdr_attributes.Size())
1238  {
1239  Array<int> ess_bdr(mesh.bdr_attributes.Max());
1240  ess_bdr = 1;
1241  fes.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
1242  }
1243  ParGridFunction uold(&fes), u(&fes), b(&fes);
1244  FunctionCoefficient u0_fc(u0);
1245  u.ProjectCoefficient(u0_fc);
1246  if (opt.vis) { opt.vis = glvis.open(vishost, visport) == 0; }
1247  if (opt.vis) { Surface::Visualize(opt, &mesh, GLVIZ_W, GLVIZ_H, &u); }
1248  Vector B, X;
1249  OperatorPtr A;
1250  CGSolver cg(MPI_COMM_WORLD);
1251  cg.SetRelTol(EPS);
1252  cg.SetAbsTol(EPS*EPS);
1253  cg.SetMaxIter(400);
1254  cg.SetPrintLevel(0);
1255  ParGridFunction eps(&fes);
1256  for (int i = 0; i < opt.niters; i++)
1257  {
1258  b = 0.0;
1259  uold = u;
1260  ParBilinearForm a(&fes);
1261  if (opt.pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
1262  const double q_uold = qf(order, AREA, mesh, fes, uold);
1263  MFEM_VERIFY(std::fabs(q_uold) > EPS,"");
1264  ConstantCoefficient q_uold_cc(1.0/sqrt(q_uold));
1266  a.Assemble();
1267  a.FormLinearSystem(ess_tdof_list, u, b, A, X, B);
1268  cg.SetOperator(*A);
1269  cg.Mult(B, X);
1270  a.RecoverFEMSolution(X, b, u);
1271  subtract(u, uold, eps);
1272  const double norm = sqrt(std::fabs(qf(order, NORM, mesh, fes, eps)));
1273  const double area = qf(order, AREA, mesh, fes, u);
1274  if (!opt.id)
1275  {
1276  mfem::out << "Iteration " << i << ", norm: " << norm
1277  << ", area: " << area << std::endl;
1278  }
1279  if (opt.vis) { Surface::Visualize(opt, &mesh, &u); }
1280  if (opt.print) { Surface::Print(opt, &mesh, &u); }
1281  if (norm < NRM) { break; }
1282  }
1283  return 0;
1284 }
1285
1286 int main(int argc, char *argv[])
1287 {
1288  Opt opt;
1289  MPI_Init(&argc, &argv);
1290  MPI_Comm_rank(MPI_COMM_WORLD, &opt.id);
1291  MPI_Comm_size(MPI_COMM_WORLD, &opt.sz);
1292  // Parse command-line options.
1293  OptionsParser args(argc, argv);
1294  args.AddOption(&opt.pb, "-p", "--problem", "Problem to solve.");
1295  args.AddOption(&opt.mesh_file, "-m", "--mesh", "Mesh file to use.");
1296  args.AddOption(&opt.wait, "-w", "--wait", "-no-w", "--no-wait",
1297  "Enable or disable a GLVis pause.");
1299  "Enable or disable radial constraints in solver.");
1301  "Number of elements in x-direction.");
1303  "Number of elements in y-direction.");
1304  args.AddOption(&opt.order, "-o", "--order", "Finite element order.");
1306  args.AddOption(&opt.niters, "-n", "--niter-max", "Max number of iterations");
1307  args.AddOption(&opt.surface, "-s", "--surface", "Choice of the surface.");
1309  "--no-partial-assembly", "Enable Partial Assembly.");
1310  args.AddOption(&opt.tau, "-t", "--tau", "Costa scale factor.");
1311  args.AddOption(&opt.lambda, "-l", "--lambda", "Lambda step toward solution.");
1312  args.AddOption(&opt.amr, "-a", "--amr", "-no-a", "--no-amr", "Enable AMR.");
1313  args.AddOption(&opt.amr_threshold, "-at", "--amr-threshold", "AMR threshold.");
1315  "Device configuration string, see Device::Configure().");
1316  args.AddOption(&opt.keys, "-k", "--keys", "GLVis configuration keys.");
1318  "--no-visualization", "Enable or disable visualization.");
1320  "-no-c", "--solve-bynodes",
1321  "Enable or disable the 'ByVdim' solver");
1322  args.AddOption(&opt.print, "-print", "--print", "-no-print", "--no-print",
1323  "Enable or disable result output (files in mfem format).");
1324  args.AddOption(&opt.snapshot, "-ss", "--snapshot", "-no-ss", "--no-snapshot",
1325  "Enable or disable GLVis snapshot.");
1326  args.Parse();
1327  if (!args.Good()) { args.PrintUsage(mfem::out); MPI_Finalize(); return 1; }
1328  MFEM_VERIFY(opt.lambda >= 0.0 && opt.lambda <= 1.0,"");
1329  if (!opt.id) { args.PrintOptions(mfem::out); }
1330
1331  // Initialize hardware devices
1332  Device device(opt.device_config);
1333  if (!opt.id) { device.Print(); }
1334
1335  if (opt.pb == 0) { Problem0(opt); }
1336
1337  if (opt.pb == 1) { Problem1(opt); }
1338
1339  MPI_Finalize();
1340  return 0;
1341 }
int GetNPoints() const
Returns the number of the points in the integration rule.
Definition: intrules.hpp:245
Geometry::Type GetGeometryType() const
Definition: element.hpp:52
Definition: array.hpp:310
int Size() const
Return the logical size of the array.
Definition: array.hpp:124
virtual void Print(std::ostream &out=mfem::out) const
Definition: mesh.hpp:1234
Definition: solvers.hpp:258
constexpr double PI
int GetNDofs() const
Returns number of degrees of freedom.
Definition: fespace.hpp:397
Class for an integration rule - an Array of IntegrationPoint.
Definition: intrules.hpp:90
Class for grid function - Vector with associated FE space.
Definition: gridfunc.hpp:30
int DofToVDof(int dof, int vd, int ndofs=-1) const
Definition: fespace.cpp:144
cdouble LogEllipticTheta1Prime(const cdouble u, const cdouble q)
const IntegrationRule & Get(int GeomType, int Order)
Returns an integration rule for given GeomType and Order.
Definition: intrules.cpp:915
A coefficient that is constant across space and time.
Definition: coefficient.hpp:78
virtual void GetVertices(Array< int > &v) const =0
Returns element&#39;s vertices.
cdouble WeierstrassZeta(const cdouble z, const cdouble w1=0.5, const cdouble w3=0.5 *I)
void SetSize(int s)
Resize the vector to size s.
Definition: vector.hpp:459
constexpr double NRM
const Geometry::Type geom
Definition: ex1.cpp:40
void BooleanMult(const Array< int > &x, Array< int > &y) const
y = A * x, treating all entries as booleans (zero=false, nonzero=true).
Definition: sparsemat.cpp:831
double Norml2() const
Returns the l2 norm of the vector.
Definition: vector.cpp:744
Pointer to an Operator of a specified type.
Definition: handle.hpp:33
std::complex< double > cdouble
constexpr int GLVIZ_W
void SetIntPoint(const IntegrationPoint *ip)
Set the integration point ip that weights and Jacobians will be evaluated at.
Definition: eltrans.hpp:85
Definition: densemat.cpp:2091
int GetOrder() const
Returns the order of the finite element. In the case of anisotropic orders, returns the maximum order...
Definition: fe.hpp:319
Data type dense matrix using column-major storage.
Definition: densemat.hpp:23
int Size() const
Returns the size of the vector.
Definition: vector.hpp:160
void Transform(void(*f)(const Vector &, Vector &))
Definition: mesh.cpp:10292
virtual void GetEssentialTrueDofs(const Array< int > &bdr_attr_is_ess, Array< int > &ess_tdof_list, int component=-1)
Get a list of essential true dofs, ess_tdof_list, corresponding to the boundary attributes marked in ...
Definition: fespace.cpp:415
int GetNE() const
Returns number of elements.
Definition: mesh.hpp:737
virtual void Mult(const Vector &x, Vector &y) const =0
Operator application: y=A(x).
void Print(std::ostream &out=mfem::out)
Print the configuration of the MFEM virtual device object.
Definition: device.cpp:261
const Array< double > & GetWeights() const
Return the quadrature weights in a contiguous array.
Definition: intrules.cpp:85
Abstract parallel finite element space.
Definition: pfespace.hpp:28
constexpr int DIM
struct _p_EPS * EPS
Definition: slepc.hpp:23
Structure for storing mesh geometric factors: coordinates, Jacobians, and determinants of the Jacobia...
Definition: mesh.hpp:1394
double * Write(bool on_dev=true)
Shortcut for mfem::Write(vec.GetMemory(), vec.Size(), on_dev).
Definition: vector.hpp:380
int main(int argc, char *argv[])
Definition: ex1.cpp:66
void GetNodalValues(int i, Array< double > &nval, int vdim=1) const
Returns the values in the vertices of i&#39;th element for dimension vdim.
Definition: gridfunc.cpp:350
Geometry::Type GetElementBaseGeometry(int i) const
Definition: mesh.hpp:834
void add(const Vector &v1, const Vector &v2, Vector &v)
Definition: vector.cpp:261
static void MarkerToList(const Array< int > &marker, Array< int > &list)
Convert a Boolean marker array to a list containing all marked indices.
Definition: fespace.cpp:434
int close()
Close the socketstream.
const Operator * GetElementRestriction(ElementDofOrdering e_ordering) const
Return an Operator that converts L-vectors to E-vectors.
Definition: fespace.cpp:894
DeviceTensor< sizeof...(Dims), T > Reshape(T *ptr, Dims...dims)
Wrap a pointer as a DeviceTensor with automatically deduced template parameters.
Definition: dtensor.hpp:134
virtual void Save(std::ostream &out) const
Save the GridFunction to an output stream.
Definition: gridfunc.cpp:3417
double Weight() const
Definition: densemat.cpp:508
Definition: vector.hpp:376
The BoomerAMG solver in hypre.
Definition: hypre.hpp:1079
IntegrationPoint & IntPoint(int i)
Returns a reference to the i-th integration point.
Definition: intrules.hpp:248
Vector J
Jacobians of the element transformations at all quadrature points.
Definition: mesh.hpp:1425
const GeometricFactors * GetGeometricFactors(const IntegrationRule &ir, const int flags)
Return the mesh geometric factors corresponding to the given integration rule.
Definition: mesh.cpp:768
void Parse()
Parse the command-line options. Note that this function expects all the options provided through the ...
Definition: optparser.cpp:150
cdouble WeierstrassP(const cdouble z, const cdouble w1=0.5, const cdouble w3=0.5 *I)
Data type sparse matrix.
Definition: sparsemat.hpp:46
int Append(const T &el)
Append element &#39;el&#39; to array, resize if necessary.
Definition: array.hpp:726
constexpr char vishost[]
Vector detJ
Determinants of the Jacobians at all quadrature points.
Definition: mesh.hpp:1431
A class that performs interpolation from an E-vector to quadrature point values and/or derivatives (Q...
double b
Definition: lissajous.cpp:42
constexpr int visport
const DenseMatrix & Jacobian()
Return the Jacobian matrix of the transformation at the currently set IntegrationPoint, using the method SetIntPoint().
Definition: eltrans.hpp:111
double delta
Definition: lissajous.cpp:43
Type
Constants for the classes derived from Element.
Definition: element.hpp:41
T Max() const
Find the maximal element in the array, using the comparison operator &lt; for class T.
Definition: array.cpp:68
cdouble EllipticTheta(const int a, const cdouble u, const cdouble q)
virtual void Print(std::ostream &out=mfem::out) const
Definition: pmesh.cpp:4169
const T * Read(bool on_dev=true) const
Definition: array.hpp:290
const Element * GetElement(int i) const
Definition: mesh.hpp:819
FiniteElementSpace * FESpace()
Definition: gridfunc.hpp:595
void PrintUsage(std::ostream &out) const
Print the usage message.
Definition: optparser.cpp:434
constexpr double NL_DMAX
double DistanceTo(const double *p) const
Compute the Euclidean distance to another vector.
Definition: vector.hpp:594
Array< int > bdr_attributes
A list of all unique boundary attributes used by the Mesh.
Definition: mesh.hpp:201
void Transpose()
(*this) = (*this)^t
Definition: densemat.cpp:1378
Class FiniteElementSpace - responsible for providing FEM view of the mesh, mainly managing the set of...
Definition: fespace.hpp:87
int GetDof() const
Returns the number of degrees of freedom in the finite element.
Definition: fe.hpp:315
void subtract(const Vector &x, const Vector &y, Vector &z)
Definition: vector.cpp:408
int GetOrder() const
Returns the order of the integration rule.
Definition: intrules.hpp:238
void AddOption(bool *var, const char *enable_short_name, const char *enable_long_name, const char *disable_short_name, const char *disable_long_name, const char *description, bool required=false)
Add a boolean option and set &#39;var&#39; to receive the value. Enable/disable tags are used to set the bool...
Definition: optparser.hpp:82
void SetSize(int nsize)
Change the logical size of the array, keep existing entries.
Definition: array.hpp:654
double a
Definition: lissajous.cpp:41
const double * Read(bool on_dev=true) const
Definition: vector.hpp:372
void GetElementTransformation(int i, IsoparametricTransformation *ElTr)
Definition: mesh.cpp:336
virtual void SetOperator(const Operator &op)
Set/update the solver for the given operator.
Definition: hypre.cpp:3441
Return a QuadratureInterpolator that interpolates E-vectors to quadrature point values and/or derivat...
Definition: fespace.cpp:950
ElementDofOrdering
Constants describing the possible orderings of the DOFs in one element.
Definition: fespace.hpp:65
const FiniteElement * GetFE(int i) const
Returns pointer to the FiniteElement in the FiniteElementCollection associated with i&#39;th element in t...
Definition: fespace.cpp:1798
constexpr int GLVIZ_H
void PrintOptions(std::ostream &out) const
Print the options.
Definition: optparser.cpp:304
constexpr int SDIM
Lexicographic ordering for tensor-product FiniteElements.
virtual int GetNVertices() const =0
void SetAttribute(const int attr)
Set element&#39;s attribute.
Definition: element.hpp:58
virtual void ProjectCoefficient(Coefficient &coeff)
Definition: gridfunc.cpp:2252
Class for parallel bilinear form.
int open(const char hostname[], int port)
Open the socket stream on &#39;port&#39; at &#39;hostname&#39;.
void MakeRef(T *, int)
Make this Array a reference to a pointer.
Definition: array.hpp:839
const double alpha
Definition: ex15.cpp:336
A general function coefficient.
Vector data type.
Definition: vector.hpp:51
void Mult(const double *x, double *y) const
Matrix vector multiplication.
Definition: densemat.cpp:175
virtual void Transform(const IntegrationPoint &, Vector &)=0
Transform integration point from reference coordinates to physical coordinates and store them in the ...
void GetNodes(Vector &node_coord) const
Definition: mesh.cpp:6603
Arbitrary order H1-conforming (continuous) finite elements.
Definition: fe_coll.hpp:159
cdouble EllipticTheta1Prime(const int k, const cdouble u, const cdouble q)
Base class for solvers.
Definition: operator.hpp:634
Class for parallel grid function.
Definition: pgridfunc.hpp:32
OutStream out(std::cout)
Global stream used by the library for standard output. Initially it uses the same std::streambuf as s...
Definition: globals.hpp:66
The MFEM Device class abstracts hardware devices such as GPUs, as well as programming models such as ...
Definition: device.hpp:118
Abstract operator.
Definition: operator.hpp:24
void GeneralRefinement(const Array< Refinement > &refinements, int nonconforming=-1, int nc_limit=0)
Definition: mesh.cpp:7977
Class for parallel meshes.
Definition: pmesh.hpp:32