laplacian.cc

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/*===========================================================================
 
 Copyright (C) 2002-2020 Yves Renard, Julien Pommier.
 
 This file is a part of GetFEM
 
 GetFEM  is  free software;  you  can  redistribute  it  and/or modify it
 under  the  terms  of the  GNU  Lesser General Public License as published
 by  the  Free Software Foundation;  either version 3 of the License,  or
 (at your option) any later version along with the GCC Runtime Library
 Exception either version 3.1 or (at your option) any later version.
 This program  is  distributed  in  the  hope  that it will be useful,  but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or  FITNESS  FOR  A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License and GCC Runtime Library Exception for more details.
 You  should  have received a copy of the GNU Lesser General Public License
 along  with  this program;  if not, write to the Free Software Foundation,
 Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA.
 
===========================================================================*/
 
/**@file laplacian.cc
   @brief Laplacian (Poisson) problem.
 
   The laplace equation is solved on a regular mesh of the unit
   square, and is compared to the analytical solution.
 
   This program is used to check that getfem++ is working. This is
   also a good example of use of GetFEM. This program  does not use the
   model bricks intentionally in order to serve as an example of solving
   a pde directly with the assembly procedures.
*/
 
#include "getfem/getfem_assembling.h"
#include "getfem/getfem_export.h"
#include "getfem/getfem_regular_meshes.h"
#include "getfem/getfem_derivatives.h"
#include "gmm/gmm.h"
using std::endl; using std::cout; using std::cerr;
using std::ends; using std::cin;
 
/* some GetFEM types that we will be using */
using bgeot::base_small_vector; /* special class for small (dim<16) vectors */
using bgeot::base_node; /* geometrical nodes (derived from base_small_vector)*/
using bgeot::scalar_type; /* = double */
using bgeot::size_type;   /* = unsigned long */
 
/* definition of some matrix/vector types. These ones are built
 * using the predefined types in Gmm++
 */
typedef gmm::rsvector<scalar_type> sparse_vector_type;
typedef gmm::row_matrix<sparse_vector_type> sparse_matrix_type;
typedef gmm::col_matrix<sparse_vector_type> col_sparse_matrix_type;
typedef std::vector<scalar_type> plain_vector;
 
/* Definitions for the exact solution of the Laplacian problem,
 *  i.e. Delta(sol_u) + sol_f = 0
 */
 
base_small_vector sol_K; /* a coefficient */
/* exact solution */
scalar_type sol_u(const base_node &x) { return sin(gmm::vect_sp(sol_K, x)); }
/* righ hand side */
scalar_type sol_f(const base_node &x)
{ return gmm::vect_sp(sol_K, sol_K) * sin(gmm::vect_sp(sol_K, x)); }
/* gradient of the exact solution */
base_small_vector sol_grad(const base_node &x)
{ return sol_K * cos(gmm::vect_sp(sol_K, x)); }
 
/*
  structure for the Laplacian problem
*/
struct laplacian_problem {
 
  enum { DIRICHLET_BOUNDARY_NUM = 0, NEUMANN_BOUNDARY_NUM = 1};
  getfem::mesh mesh;        /* the mesh */
  getfem::mesh_im mim;      /* the integration methods. */
  getfem::mesh_fem mf_u;    /* the main mesh_fem, for the Laplacian solution */
  getfem::mesh_fem mf_rhs;  /* the mesh_fem for the right hand side(f(x),..) */
  getfem::mesh_fem mf_coef; /* the mesh_fem to represent pde coefficients    */
 
  scalar_type residual;        /* max residual for the iterative solvers */
  size_type N;
  bool gen_dirichlet;
 
  sparse_matrix_type SM;     /* stiffness matrix.                           */
  std::vector<scalar_type> U, B;      /* main unknown, and right hand side  */
 
  std::vector<scalar_type> Ud; /* reduced sol. for gen. Dirichlet condition. */
  col_sparse_matrix_type NS; /* Dirichlet NullSpace
                              * (used if gen_dirichlet is true)
                              */
  std::string datafilename;
  bgeot::md_param PARAM;
 
  void assembly(void);
  bool solve(void);
  void init(void);
  void compute_error();
  laplacian_problem(void) : mim(mesh), mf_u(mesh), mf_rhs(mesh),
                            mf_coef(mesh) {}
};
 
/* Read parameters from the .param file, build the mesh, set finite element
 * and integration methods and selects the boundaries.
 */
void laplacian_problem::init(void) {
 
  std::string MESH_TYPE = PARAM.string_value("MESH_TYPE","Mesh type ");
  std::string FEM_TYPE  = PARAM.string_value("FEM_TYPE","FEM name");
  std::string INTEGRATION = PARAM.string_value("INTEGRATION",
                                               "Name of integration method");
 
  cout << "MESH_TYPE=" << MESH_TYPE << "\n";
  cout << "FEM_TYPE="  << FEM_TYPE << "\n";
  cout << "INTEGRATION=" << INTEGRATION << "\n";
 
  /* First step : build the mesh */
  bgeot::pgeometric_trans pgt =
    bgeot::geometric_trans_descriptor(MESH_TYPE);
  N = pgt->dim();
  std::vector<size_type> nsubdiv(N);
  std::fill(nsubdiv.begin(),nsubdiv.end(),
            PARAM.int_value("NX", "Nomber of space steps "));
  getfem::regular_unit_mesh(mesh, nsubdiv, pgt,
                            PARAM.int_value("MESH_NOISED") != 0);
 
  bgeot::base_matrix M(N,N);
  for (size_type i=0; i < N; ++i) {
    static const char *t[] = {"LX","LY","LZ"};
    M(i,i) = (i<3) ? PARAM.real_value(t[i],t[i]) : 1.0;
  }
  if (N>1) { M(0,1) = PARAM.real_value("INCLINE") * PARAM.real_value("LY"); }
 
  /* scale the unit mesh to [LX,LY,..] and incline it */
  mesh.transformation(M);
 
  datafilename = PARAM.string_value("ROOTFILENAME","Base name of data files.");
  scalar_type FT = PARAM.real_value("FT", "parameter for exact solution");
  residual = PARAM.real_value("RESIDUAL");
  if (residual == 0.) residual = 1e-10;
  sol_K.resize(N);
  for (size_type j = 0; j < N; j++)
    sol_K[j] = ((j & 1) == 0) ? FT : -FT;
 
  /* set the finite element on the mf_u */
  getfem::pfem pf_u = getfem::fem_descriptor(FEM_TYPE);
  getfem::pintegration_method ppi = getfem::int_method_descriptor(INTEGRATION);
 
  mim.set_integration_method(mesh.convex_index(), ppi);
  mf_u.set_finite_element(mesh.convex_index(), pf_u);
 
  /* set the finite element on mf_rhs (same as mf_u is DATA_FEM_TYPE is
     not used in the .param file */
  std::string data_fem_name = PARAM.string_value("DATA_FEM_TYPE");
  if (data_fem_name.size() == 0) {
    GMM_ASSERT1(pf_u->is_lagrange(), "You are using a non-lagrange FEM. "
                << "In that case you need to set "
                << "DATA_FEM_TYPE in the .param file");
    mf_rhs.set_finite_element(mesh.convex_index(), pf_u);
  } else {
    mf_rhs.set_finite_element(mesh.convex_index(),
                              getfem::fem_descriptor(data_fem_name));
  }
 
  /* set the finite element on mf_coef. Here we use a very simple element
   *  since the only function that need to be interpolated on the mesh_fem
   * is f(x)=1 ... */
  mf_coef.set_finite_element(mesh.convex_index(),
                             getfem::classical_fem(pgt,0));
 
  /* set boundary conditions
   * (Neuman on the upper face, Dirichlet elsewhere) */
  gen_dirichlet = PARAM.int_value("GENERIC_DIRICHLET");
 
  if (!pf_u->is_lagrange() && !gen_dirichlet)
    GMM_WARNING2("With non lagrange fem prefer the generic "
                 "Dirichlet condition option");
 
  cout << "Selecting Neumann and Dirichlet boundaries\n";
  getfem::mesh_region border_faces;
  getfem::outer_faces_of_mesh(mesh, border_faces);
  for (getfem::mr_visitor i(border_faces); !i.finished(); ++i) {
    assert(i.is_face());
    base_node un = mesh.normal_of_face_of_convex(i.cv(), i.f());
    un /= gmm::vect_norm2(un);
    if (gmm::abs(un[N-1] - 1.0) < 1.0E-7) { // new Neumann face
      mesh.region(NEUMANN_BOUNDARY_NUM).add(i.cv(), i.f());
    } else {
      mesh.region(DIRICHLET_BOUNDARY_NUM).add(i.cv(), i.f());
    }
  }
}
 
void laplacian_problem::assembly(void) {
  size_type nb_dof = mf_u.nb_dof();
  size_type nb_dof_rhs = mf_rhs.nb_dof();
 
  gmm::resize(B, nb_dof); gmm::clear(B);
  gmm::resize(U, nb_dof); gmm::clear(U);
  gmm::resize(SM, nb_dof, nb_dof); gmm::clear(SM);
 
  cout << "Number of dof : " << nb_dof << endl;
  cout << "Assembling stiffness matrix" << endl;
  getfem::asm_stiffness_matrix_for_laplacian(SM, mim, mf_u, mf_coef,
     std::vector<scalar_type>(mf_coef.nb_dof(), 1.0));
 
  cout << "Assembling source term" << endl;
  std::vector<scalar_type> F(nb_dof_rhs);
  getfem::interpolation_function(mf_rhs, F, sol_f);
  getfem::asm_source_term(B, mim, mf_u, mf_rhs, F);
 
  cout << "Assembling Neumann condition" << endl;
  gmm::resize(F, nb_dof_rhs*N);
  getfem::interpolation_function(mf_rhs, F, sol_grad);
  getfem::asm_normal_source_term(B, mim, mf_u, mf_rhs, F,
                                 NEUMANN_BOUNDARY_NUM);
 
  cout << "take Dirichlet condition into account" << endl;
  if (!gen_dirichlet) {
    std::vector<scalar_type> D(nb_dof);
    getfem::interpolation_function(mf_u, D, sol_u);
    getfem::assembling_Dirichlet_condition(SM, B, mf_u,
                                           DIRICHLET_BOUNDARY_NUM, D);
  } else {
    gmm::resize(F, nb_dof_rhs);
    getfem::interpolation_function(mf_rhs, F, sol_u);
 
    gmm::resize(Ud, nb_dof);
    gmm::resize(NS, nb_dof, nb_dof);
    col_sparse_matrix_type H(nb_dof_rhs, nb_dof);
    std::vector<scalar_type> R(nb_dof_rhs);
    std::vector<scalar_type> RHaux(nb_dof);
 
    /* build H and R such that U mush satisfy H*U = R */
    getfem::asm_dirichlet_constraints(H, R, mim, mf_u, mf_rhs,
      mf_rhs, F, DIRICHLET_BOUNDARY_NUM);
 
    gmm::clean(H, 1e-12);
//     cout << "H = " << H << endl;
//     cout << "R = " << R << endl;
    int nbcols = int(getfem::Dirichlet_nullspace(H, NS, R, Ud));
    // cout << "Number of irreductible unknowns : " << nbcols << endl;
    gmm::resize(NS, gmm::mat_ncols(H),nbcols);
 
    gmm::mult(SM, Ud, gmm::scaled(B, -1.0), RHaux);
    gmm::resize(B, nbcols);
    gmm::resize(U, nbcols);
    gmm::mult(gmm::transposed(NS), gmm::scaled(RHaux, -1.0), B);
    sparse_matrix_type SMaux(nbcols, nb_dof);
    gmm::mult(gmm::transposed(NS), SM, SMaux);
    gmm::resize(SM, nbcols, nbcols);
    /* NSaux = NS, but is stored by rows instead of by columns */
    sparse_matrix_type NSaux(nb_dof, nbcols); gmm::copy(NS, NSaux);
    gmm::mult(SMaux, NSaux, SM);
  }
}
 
 
bool laplacian_problem::solve(void) {
 
  // see_schmidt(SM, U, B);
 
  cout << "Compute preconditionner\n";
  gmm::iteration iter(residual, 1, 40000);
  double time = gmm::uclock_sec();
  if (1) {
    // gmm::identity_matrix P;
    // gmm::diagonal_precond<sparse_matrix_type> P(SM);
    // gmm::mr_approx_inverse_precond<sparse_matrix_type> P(SM, 10, 10E-17);
    // gmm::ildlt_precond<sparse_matrix_type> P(SM);
    // gmm::ildltt_precond<sparse_matrix_type> P(SM, 20, 1E-6);
    gmm::ilut_precond<sparse_matrix_type> P(SM, 20, 1E-6);
    // gmm::ilutp_precond<sparse_matrix_type> P(SM, 20, 1E-6);
    // gmm::ilu_precond<sparse_matrix_type> P(SM);
    cout << "Time to compute preconditionner : "
         << gmm::uclock_sec() - time << " seconds\n";
 
 
    //gmm::HarwellBoeing_IO::write("SM", SM);
 
    // gmm::cg(SM, U, B, P, iter);
    gmm::gmres(SM, U, B, P, 50, iter);
  } else {
    double rcond;
#if defined(GMM_USES_SUPERLU)
    gmm::SuperLU_solve(SM, U, B, rcond);
#endif
    cout << "cond = " << 1/rcond << "\n";
  }
 
  cout << "Total time to solve : "
       << gmm::uclock_sec() - time << " seconds\n";
 
  if (gen_dirichlet) {
    std::vector<scalar_type> Uaux(mf_u.nb_dof());
    gmm::mult(NS, U, Ud, Uaux);
    gmm::resize(U, mf_u.nb_dof());
    gmm::copy(Uaux, U);
  }
 
  return (iter.converged());
}
 
/* compute the error with respect to the exact solution */
void laplacian_problem::compute_error() {
  std::vector<scalar_type> V(mf_rhs.nb_basic_dof());
  getfem::interpolation(mf_u, mf_rhs, U, V);
  for (size_type i = 0; i < mf_rhs.nb_basic_dof(); ++i)
    V[i] -= sol_u(mf_rhs.point_of_basic_dof(i));
  cout.precision(16);
  cout << "L2 error = " << getfem::asm_L2_norm(mim, mf_rhs, V) << endl
       << "H1 error = " << getfem::asm_H1_norm(mim, mf_rhs, V) << endl
       << "Linfty error = " << gmm::vect_norminf(V) << endl;
}
 
/**************************************************************************/
/*  main program.                                                         */
/**************************************************************************/
 
int main(int argc, char *argv[]) {
 
  GETFEM_MPI_INIT(argc, argv);
  GMM_SET_EXCEPTION_DEBUG; // Exceptions make a memory fault, to debug.
  FE_ENABLE_EXCEPT;        // Enable floating point exception for Nan.
 
  try {
    laplacian_problem p;
    p.PARAM.read_command_line(argc, argv);
    p.init();
    p.mesh.write_to_file(p.datafilename + ".mesh");
    p.assembly();
    if (!p.solve()) GMM_ASSERT1(false, "Solve procedure has failed");
    p.compute_error();
  }
  GMM_STANDARD_CATCH_ERROR;
 
  GETFEM_MPI_FINALIZE;
 
  return 0;
}