GetFEM  5.4.2
laplacian.cc
Go to the documentation of this file.
1 /*===========================================================================
2 
3  Copyright (C) 2002-2020 Yves Renard, Julien Pommier.
4 
5  This file is a part of GetFEM
6 
7  GetFEM is free software; you can redistribute it and/or modify it
8  under the terms of the GNU Lesser General Public License as published
9  by the Free Software Foundation; either version 3 of the License, or
10  (at your option) any later version along with the GCC Runtime Library
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13  WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
14  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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17  along with this program; if not, write to the Free Software Foundation,
18  Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
19 
20 ===========================================================================*/
21 
22 /**@file laplacian.cc
23  @brief Laplacian (Poisson) problem.
24 
25  The laplace equation is solved on a regular mesh of the unit
26  square, and is compared to the analytical solution.
27 
28  This program is used to check that getfem++ is working. This is
29  also a good example of use of GetFEM. This program does not use the
30  model bricks intentionally in order to serve as an example of solving
31  a pde directly with the assembly procedures.
32 */
33 
35 #include "getfem/getfem_export.h"
38 #include "getfem/getfem_superlu.h"
39 #include "gmm/gmm.h"
40 using std::endl; using std::cout; using std::cerr;
41 using std::ends; using std::cin;
42 
43 /* some GetFEM types that we will be using */
44 using bgeot::base_small_vector; /* special class for small (dim<16) vectors */
45 using bgeot::base_node; /* geometrical nodes (derived from base_small_vector)*/
46 using bgeot::scalar_type; /* = double */
47 using bgeot::size_type; /* = unsigned long */
48 
49 /* definition of some matrix/vector types. These ones are built
50  * using the predefined types in Gmm++
51  */
53 typedef gmm::row_matrix<sparse_vector_type> sparse_matrix_type;
54 typedef gmm::col_matrix<sparse_vector_type> col_sparse_matrix_type;
55 typedef std::vector<scalar_type> plain_vector;
56 
57 /* Definitions for the exact solution of the Laplacian problem,
58  * i.e. Delta(sol_u) + sol_f = 0
59  */
60 
61 base_small_vector sol_K; /* a coefficient */
62 /* exact solution */
63 scalar_type sol_u(const base_node &x) { return sin(gmm::vect_sp(sol_K, x)); }
64 /* righ hand side */
65 scalar_type sol_f(const base_node &x)
66 { return gmm::vect_sp(sol_K, sol_K) * sin(gmm::vect_sp(sol_K, x)); }
67 /* gradient of the exact solution */
68 base_small_vector sol_grad(const base_node &x)
69 { return sol_K * cos(gmm::vect_sp(sol_K, x)); }
70 
71 /*
72  structure for the Laplacian problem
73 */
74 struct laplacian_problem {
75 
76  enum { DIRICHLET_BOUNDARY_NUM = 0, NEUMANN_BOUNDARY_NUM = 1};
77  getfem::mesh mesh; /* the mesh */
78  getfem::mesh_im mim; /* the integration methods. */
79  getfem::mesh_fem mf_u; /* the main mesh_fem, for the Laplacian solution */
80  getfem::mesh_fem mf_rhs; /* the mesh_fem for the right hand side(f(x),..) */
81  getfem::mesh_fem mf_coef; /* the mesh_fem to represent pde coefficients */
82 
83  scalar_type residual; /* max residual for the iterative solvers */
84  size_type N;
85  bool gen_dirichlet;
86 
87  sparse_matrix_type SM; /* stiffness matrix. */
88  std::vector<scalar_type> U, B; /* main unknown, and right hand side */
89 
90  std::vector<scalar_type> Ud; /* reduced sol. for gen. Dirichlet condition. */
91  col_sparse_matrix_type NS; /* Dirichlet NullSpace
92  * (used if gen_dirichlet is true)
93  */
94  std::string datafilename;
95  bgeot::md_param PARAM;
96 
97  void assembly(void);
98  bool solve(void);
99  void init(void);
100  void compute_error();
101  laplacian_problem(void) : mim(mesh), mf_u(mesh), mf_rhs(mesh),
102  mf_coef(mesh) {}
103 };
104 
105 /* Read parameters from the .param file, build the mesh, set finite element
106  * and integration methods and selects the boundaries.
107  */
108 void laplacian_problem::init(void) {
109 
110  std::string MESH_TYPE = PARAM.string_value("MESH_TYPE","Mesh type ");
111  std::string FEM_TYPE = PARAM.string_value("FEM_TYPE","FEM name");
112  std::string INTEGRATION = PARAM.string_value("INTEGRATION",
113  "Name of integration method");
114 
115  cout << "MESH_TYPE=" << MESH_TYPE << "\n";
116  cout << "FEM_TYPE=" << FEM_TYPE << "\n";
117  cout << "INTEGRATION=" << INTEGRATION << "\n";
118 
119  /* First step : build the mesh */
122  N = pgt->dim();
123  std::vector<size_type> nsubdiv(N);
124  std::fill(nsubdiv.begin(),nsubdiv.end(),
125  PARAM.int_value("NX", "Nomber of space steps "));
126  getfem::regular_unit_mesh(mesh, nsubdiv, pgt,
127  PARAM.int_value("MESH_NOISED") != 0);
128 
129  bgeot::base_matrix M(N,N);
130  for (size_type i=0; i < N; ++i) {
131  static const char *t[] = {"LX","LY","LZ"};
132  M(i,i) = (i<3) ? PARAM.real_value(t[i],t[i]) : 1.0;
133  }
134  if (N>1) { M(0,1) = PARAM.real_value("INCLINE") * PARAM.real_value("LY"); }
135 
136  /* scale the unit mesh to [LX,LY,..] and incline it */
137  mesh.transformation(M);
138 
139  datafilename = PARAM.string_value("ROOTFILENAME","Base name of data files.");
140  scalar_type FT = PARAM.real_value("FT", "parameter for exact solution");
141  residual = PARAM.real_value("RESIDUAL");
142  if (residual == 0.) residual = 1e-10;
143  sol_K.resize(N);
144  for (size_type j = 0; j < N; j++)
145  sol_K[j] = ((j & 1) == 0) ? FT : -FT;
146 
147  /* set the finite element on the mf_u */
148  getfem::pfem pf_u = getfem::fem_descriptor(FEM_TYPE);
149  getfem::pintegration_method ppi = getfem::int_method_descriptor(INTEGRATION);
150 
151  mim.set_integration_method(mesh.convex_index(), ppi);
152  mf_u.set_finite_element(mesh.convex_index(), pf_u);
153 
154  /* set the finite element on mf_rhs (same as mf_u is DATA_FEM_TYPE is
155  not used in the .param file */
156  std::string data_fem_name = PARAM.string_value("DATA_FEM_TYPE");
157  if (data_fem_name.size() == 0) {
158  GMM_ASSERT1(pf_u->is_lagrange(), "You are using a non-lagrange FEM. "
159  << "In that case you need to set "
160  << "DATA_FEM_TYPE in the .param file");
161  mf_rhs.set_finite_element(mesh.convex_index(), pf_u);
162  } else {
163  mf_rhs.set_finite_element(mesh.convex_index(),
164  getfem::fem_descriptor(data_fem_name));
165  }
166 
167  /* set the finite element on mf_coef. Here we use a very simple element
168  * since the only function that need to be interpolated on the mesh_fem
169  * is f(x)=1 ... */
170  mf_coef.set_finite_element(mesh.convex_index(),
171  getfem::classical_fem(pgt,0));
172 
173  /* set boundary conditions
174  * (Neuman on the upper face, Dirichlet elsewhere) */
175  gen_dirichlet = PARAM.int_value("GENERIC_DIRICHLET");
176 
177  if (!pf_u->is_lagrange() && !gen_dirichlet)
178  GMM_WARNING2("With non lagrange fem prefer the generic "
179  "Dirichlet condition option");
180 
181  cout << "Selecting Neumann and Dirichlet boundaries\n";
182  getfem::mesh_region border_faces;
183  getfem::outer_faces_of_mesh(mesh, border_faces);
184  for (getfem::mr_visitor i(border_faces); !i.finished(); ++i) {
185  assert(i.is_face());
186  base_node un = mesh.normal_of_face_of_convex(i.cv(), i.f());
187  un /= gmm::vect_norm2(un);
188  if (gmm::abs(un[N-1] - 1.0) < 1.0E-7) { // new Neumann face
189  mesh.region(NEUMANN_BOUNDARY_NUM).add(i.cv(), i.f());
190  } else {
191  mesh.region(DIRICHLET_BOUNDARY_NUM).add(i.cv(), i.f());
192  }
193  }
194 }
195 
196 void laplacian_problem::assembly(void) {
197  size_type nb_dof = mf_u.nb_dof();
198  size_type nb_dof_rhs = mf_rhs.nb_dof();
199 
200  gmm::resize(B, nb_dof); gmm::clear(B);
201  gmm::resize(U, nb_dof); gmm::clear(U);
202  gmm::resize(SM, nb_dof, nb_dof); gmm::clear(SM);
203 
204  cout << "Number of dof : " << nb_dof << endl;
205  cout << "Assembling stiffness matrix" << endl;
206  getfem::asm_stiffness_matrix_for_laplacian(SM, mim, mf_u, mf_coef,
207  std::vector<scalar_type>(mf_coef.nb_dof(), 1.0));
208 
209  cout << "Assembling source term" << endl;
210  std::vector<scalar_type> F(nb_dof_rhs);
211  getfem::interpolation_function(mf_rhs, F, sol_f);
212  getfem::asm_source_term(B, mim, mf_u, mf_rhs, F);
213 
214  cout << "Assembling Neumann condition" << endl;
215  gmm::resize(F, nb_dof_rhs*N);
216  getfem::interpolation_function(mf_rhs, F, sol_grad);
217  getfem::asm_normal_source_term(B, mim, mf_u, mf_rhs, F,
218  NEUMANN_BOUNDARY_NUM);
219 
220  cout << "take Dirichlet condition into account" << endl;
221  if (!gen_dirichlet) {
222  std::vector<scalar_type> D(nb_dof);
223  getfem::interpolation_function(mf_u, D, sol_u);
224  getfem::assembling_Dirichlet_condition(SM, B, mf_u,
225  DIRICHLET_BOUNDARY_NUM, D);
226  } else {
227  gmm::resize(F, nb_dof_rhs);
228  getfem::interpolation_function(mf_rhs, F, sol_u);
229 
230  gmm::resize(Ud, nb_dof);
231  gmm::resize(NS, nb_dof, nb_dof);
232  col_sparse_matrix_type H(nb_dof_rhs, nb_dof);
233  std::vector<scalar_type> R(nb_dof_rhs);
234  std::vector<scalar_type> RHaux(nb_dof);
235 
236  /* build H and R such that U mush satisfy H*U = R */
237  getfem::asm_dirichlet_constraints(H, R, mim, mf_u, mf_rhs,
238  mf_rhs, F, DIRICHLET_BOUNDARY_NUM);
239 
240  gmm::clean(H, 1e-12);
241 // cout << "H = " << H << endl;
242 // cout << "R = " << R << endl;
243  int nbcols = int(getfem::Dirichlet_nullspace(H, NS, R, Ud));
244  // cout << "Number of irreductible unknowns : " << nbcols << endl;
245  gmm::resize(NS, gmm::mat_ncols(H),nbcols);
246 
247  gmm::mult(SM, Ud, gmm::scaled(B, -1.0), RHaux);
248  gmm::resize(B, nbcols);
249  gmm::resize(U, nbcols);
250  gmm::mult(gmm::transposed(NS), gmm::scaled(RHaux, -1.0), B);
251  sparse_matrix_type SMaux(nbcols, nb_dof);
252  gmm::mult(gmm::transposed(NS), SM, SMaux);
253  gmm::resize(SM, nbcols, nbcols);
254  /* NSaux = NS, but is stored by rows instead of by columns */
255  sparse_matrix_type NSaux(nb_dof, nbcols); gmm::copy(NS, NSaux);
256  gmm::mult(SMaux, NSaux, SM);
257  }
258 }
259 
260 
261 bool laplacian_problem::solve(void) {
262 
263  // see_schmidt(SM, U, B);
264 
265  cout << "Compute preconditionner\n";
266  gmm::iteration iter(residual, 1, 40000);
267  double time = gmm::uclock_sec();
268  if (1) {
269  // gmm::identity_matrix P;
270  // gmm::diagonal_precond<sparse_matrix_type> P(SM);
271  // gmm::mr_approx_inverse_precond<sparse_matrix_type> P(SM, 10, 10E-17);
272  // gmm::ildlt_precond<sparse_matrix_type> P(SM);
273  // gmm::ildltt_precond<sparse_matrix_type> P(SM, 20, 1E-6);
275  // gmm::ilutp_precond<sparse_matrix_type> P(SM, 20, 1E-6);
276  // gmm::ilu_precond<sparse_matrix_type> P(SM);
277  cout << "Time to compute preconditionner : "
278  << gmm::uclock_sec() - time << " seconds\n";
279 
280 
281  //gmm::HarwellBoeing_IO::write("SM", SM);
282 
283  // gmm::cg(SM, U, B, P, iter);
284  gmm::gmres(SM, U, B, P, 50, iter);
285  } else {
286  double rcond;
287  gmm::SuperLU_solve(SM, U, B, rcond);
288  cout << "cond = " << 1/rcond << "\n";
289  }
290 
291  cout << "Total time to solve : "
292  << gmm::uclock_sec() - time << " seconds\n";
293 
294  if (gen_dirichlet) {
295  std::vector<scalar_type> Uaux(mf_u.nb_dof());
296  gmm::mult(NS, U, Ud, Uaux);
297  gmm::resize(U, mf_u.nb_dof());
298  gmm::copy(Uaux, U);
299  }
300 
301  return (iter.converged());
302 }
303 
304 /* compute the error with respect to the exact solution */
305 void laplacian_problem::compute_error() {
306  std::vector<scalar_type> V(mf_rhs.nb_basic_dof());
307  getfem::interpolation(mf_u, mf_rhs, U, V);
308  for (size_type i = 0; i < mf_rhs.nb_basic_dof(); ++i)
309  V[i] -= sol_u(mf_rhs.point_of_basic_dof(i));
310  cout.precision(16);
311  cout << "L2 error = " << getfem::asm_L2_norm(mim, mf_rhs, V) << endl
312  << "H1 error = " << getfem::asm_H1_norm(mim, mf_rhs, V) << endl
313  << "Linfty error = " << gmm::vect_norminf(V) << endl;
314 }
315 
316 /**************************************************************************/
317 /* main program. */
318 /**************************************************************************/
319 
320 int main(int argc, char *argv[]) {
321 
322  GETFEM_MPI_INIT(argc, argv);
323  GMM_SET_EXCEPTION_DEBUG; // Exceptions make a memory fault, to debug.
324  FE_ENABLE_EXCEPT; // Enable floating point exception for Nan.
325 
326  try {
327  laplacian_problem p;
328  p.PARAM.read_command_line(argc, argv);
329  p.init();
330  p.mesh.write_to_file(p.datafilename + ".mesh");
331  p.assembly();
332  if (!p.solve()) GMM_ASSERT1(false, "Solve procedure has failed");
333  p.compute_error();
334  }
335  GMM_STANDARD_CATCH_ERROR;
336 
337  GETFEM_MPI_FINALIZE;
338 
339  return 0;
340 }
getfem_export.h
Export solutions to various formats.
getfem::asm_H1_norm
scalar_type asm_H1_norm(const mesh_im &mim, const mesh_fem &mf, const VEC &U, const mesh_region &rg=mesh_region::all_convexes())
compute the H1 norm of U.
Definition: getfem_assembling.h:302
gmm::resize
void resize(M &v, size_type m, size_type n)
*‍/
Definition: gmm_blas.h:231
getfem::asm_stiffness_matrix_for_laplacian
void asm_stiffness_matrix_for_laplacian(MAT &M, const mesh_im &mim, const mesh_fem &mf, const mesh_fem &mf_data, const VECT &A, const mesh_region &rg=mesh_region::all_convexes())
assembly of , where is scalar.
Definition: getfem_assembling.h:1152
bgeot::size_type
size_t size_type
used as the common size type in the library
Definition: bgeot_poly.h:49
getfem::int_method_descriptor
pintegration_method int_method_descriptor(std::string name, bool throw_if_not_found=true)
Get an integration method from its name .
Definition: getfem_integration.cc:1130
getfem_regular_meshes.h
Build regular meshes.
gmm::clear
void clear(L &l)
clear (fill with zeros) a vector or matrix.
Definition: gmm_blas.h:59
getfem::mesh_im
Describe an integration method linked to a mesh.
Definition: getfem_mesh_im.h:47
getfem::Dirichlet_nullspace
size_type Dirichlet_nullspace(const MAT1 &H, MAT2 &NS, const VECT1 &R, VECT2 &U0)
Build an orthogonal basis of the kernel of H in NS, gives the solution of minimal norm of H*U = R in ...
Definition: getfem_assembling.h:1656
bgeot::geometric_trans_descriptor
pgeometric_trans geometric_trans_descriptor(std::string name)
Get the geometric transformation from its string name.
Definition: bgeot_geometric_trans.cc:1163
getfem::classical_fem
pfem classical_fem(bgeot::pgeometric_trans pgt, short_type k, bool complete=false)
Give a pointer on the structures describing the classical polynomial fem of degree k on a given conve...
Definition: getfem_fem.cc:4141
getfem::mesh_fem
Describe a finite element method linked to a mesh.
Definition: getfem_mesh_fem.h:148
getfem::interpolation
void interpolation(const mesh_fem &mf_source, const mesh_fem &mf_target, const VECTU &U, VECTV &V, int extrapolation=0, double EPS=1E-10, mesh_region rg_source=mesh_region::all_convexes(), mesh_region rg_target=mesh_region::all_convexes())
interpolation/extrapolation of (mf_source, U) on mf_target.
Definition: getfem_interpolation.h:693
getfem_superlu.h
SuperLU interface for getfem.
gmm::iteration
The Iteration object calculates whether the solution has reached the desired accuracy,...
Definition: gmm_iter.h:53
getfem::asm_normal_source_term
void asm_normal_source_term(VECT1 &B, const mesh_im &mim, const mesh_fem &mf, const mesh_fem &mf_data, const VECT2 &F, const mesh_region &rg)
Normal source term (for boundary (Neumann) condition).
Definition: getfem_assembling.h:905
getfem::mesh_region::visitor
"iterator" class for regions.
Definition: getfem_mesh_region.h:237
getfem::asm_source_term
void asm_source_term(const VECT1 &B, const mesh_im &mim, const mesh_fem &mf, const mesh_fem &mf_data, const VECT2 &F, const mesh_region &rg=mesh_region::all_convexes())
source term (for both volumic sources and boundary (Neumann) sources).
Definition: getfem_assembling.h:877
gmm::vect_norm2
number_traits< typename linalg_traits< V >::value_type >::magnitude_type vect_norm2(const V &v)
Euclidean norm of a vector.
Definition: gmm_blas.h:557
getfem::asm_dirichlet_constraints
void asm_dirichlet_constraints(MAT &H, VECT1 &R, const mesh_im &mim, const mesh_fem &mf_u, const mesh_fem &mf_mult, const mesh_fem &mf_r, const VECT2 &r_data, const mesh_region &region, int version=ASMDIR_BUILDALL)
Assembly of Dirichlet constraints in a weak form.
Definition: getfem_assembling.h:1373
getfem::pfem
std::shared_ptr< const getfem::virtual_fem > pfem
type of pointer on a fem description
Definition: getfem_fem.h:244
bgeot::small_vector
container for small vectors of POD (Plain Old Data) types.
Definition: bgeot_small_vector.h:205
getfem_derivatives.h
Compute the gradient of a field on a getfem::mesh_fem.
getfem::asm_L2_norm
scalar_type asm_L2_norm(const mesh_im &mim, const mesh_fem &mf, const VEC &U, const mesh_region &rg=mesh_region::all_convexes())
compute , U might be real or complex
Definition: getfem_assembling.h:56
gmm::rsvector
sparse vector built upon std::vector.
Definition: gmm_def.h:488
getfem::mesh
Describe a mesh (collection of convexes (elements) and points).
Definition: getfem_mesh.h:95
bgeot::pgeometric_trans
std::shared_ptr< const bgeot::geometric_trans > pgeometric_trans
pointer type for a geometric transformation
Definition: bgeot_geometric_trans.h:186
gmm.h
Include common gmm files.
getfem::mesh_region
structure used to hold a set of convexes and/or convex faces.
Definition: getfem_mesh_region.h:55
getfem::regular_unit_mesh
void regular_unit_mesh(mesh &m, std::vector< size_type > nsubdiv, bgeot::pgeometric_trans pgt, bool noised=false)
Build a regular mesh of the unit square/cube/, etc.
Definition: getfem_regular_meshes.cc:238
getfem::outer_faces_of_mesh
void APIDECL outer_faces_of_mesh(const mesh &m, const dal::bit_vector &cvlst, convex_face_ct &flist)
returns a list of "exterior" faces of a mesh (i.e.
Definition: getfem_mesh.cc:822
gmm::vect_norminf
number_traits< typename linalg_traits< V >::value_type >::magnitude_type vect_norminf(const V &v)
Infinity norm of a vector.
Definition: gmm_blas.h:693
getfem::interpolation_function
void interpolation_function(mesh_fem &mf_target, const VECT &VV, F &f, mesh_region rg=mesh_region::all_convexes())
interpolation of a function f on mf_target.
Definition: getfem_interpolation.h:186
gmm::ilut_precond
Incomplete LU with threshold and K fill-in Preconditioner.
Definition: gmm_precond_ilut.h:102
getfem::fem_descriptor
pfem fem_descriptor(const std::string &name)
get a fem descriptor from its string name.
Definition: getfem_fem.cc:4232
getfem_assembling.h
Miscelleanous assembly routines for common terms. Use the low-level generic assembly....