eigenproblems_ex1.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2025 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner

// This library 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 2.1 of the License, or (at your option) any later version.

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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// Lesser General Public License for more details.

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// <h1>Eigenproblems Example 1 - Solving an Eigen Problem</h1>
// \author Steffen Petersen
// \date 2005
//
// This example introduces the EigenSystem and shows
// how libMesh can be used for eigenvalue analysis.
//
// For solving eigen problems, libMesh interfaces
// SLEPc (www.grycap.upv.es/slepc/) which again is based on PETSc.
// Hence, this example will only work if the library is compiled
// with SLEPc support enabled.
//
// In this example some eigenvalues for a standard symmetric eigenvalue
// problem A*x=lambda*x are computed, where the matrix A
// is assembled according to a mass matrix.


// libMesh include files.
#include "libmesh/libmesh.h"
#include "libmesh/mesh.h"
#include "libmesh/mesh_generation.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/eigen_system.h"
#include "libmesh/equation_systems.h"
#include "libmesh/fe.h"
#include "libmesh/quadrature_gauss.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/sparse_matrix.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/dof_map.h"
#include "libmesh/getpot.h"

// Bring in everything from the libMesh namespace
using namespace libMesh;

// Function prototype.  This is the function that will assemble
// the eigen system. Here, we will simply assemble a mass matrix.
void assemble_mass(EquationSystems & es,
                   const std::string & system_name);

int main (int argc, char ** argv)
{
  // Initialize libMesh and the dependent libraries.
  LibMeshInit init (argc, argv);

#ifdef LIBMESH_DEFAULT_SINGLE_PRECISION
  // SLEPc currently gives us a nasty crash with Real==float
  libmesh_example_requires(false, "--disable-singleprecision");
#endif

#ifndef LIBMESH_HAVE_SLEPC
  libmesh_example_requires(false, "--enable-slepc");
#else
  // Check for proper usage.
  libmesh_error_msg_if(argc < 3, "\nUsage: " << argv[0] << " -n <number of eigen values>");

  // Tell the user what we are doing.
  libMesh::out << "Running " << argv[0];

  for (int i=1; i<argc; i++)
    libMesh::out << " " << argv[i];

  libMesh::out << std::endl << std::endl;

  // Get the number of eigen values to be computed from "-n", and
  // possibly the mesh size from -nx and -ny

  const int nev = libMesh::command_line_next("-n", 5),
            nx = libMesh::command_line_next("-nx", 20),
            ny = libMesh::command_line_next("-ny", 20);

  // Skip this 2D example if libMesh was compiled as 1D-only.
  libmesh_example_requires(2 <= LIBMESH_DIM, "2D support");

  // Create a mesh, with dimension to be overridden later, on the
  // default MPI communicator.
  Mesh mesh(init.comm());

  // Use the internal mesh generator to create a uniform
  // 2D grid on a square.
  MeshTools::Generation::build_square (mesh,
                                       nx, ny,
                                       -1., 1.,
                                       -1., 1.,
                                       QUAD4);

  // Print information about the mesh to the screen.
  mesh.print_info();

  // Create an equation systems object.
  EquationSystems equation_systems (mesh);

  // Create a EigenSystem named "Eigensystem" and (for convenience)
  // use a reference to the system we create.
  EigenSystem & eigen_system =
    equation_systems.add_system<EigenSystem> ("Eigensystem");

  // Declare the system variables.
  // Adds the variable "p" to "Eigensystem".   "p"
  // will be approximated using second-order approximation.
  eigen_system.add_variable("p", FIRST);

  // Give the system a pointer to the matrix assembly
  // function defined below.
  eigen_system.attach_assemble_function (assemble_mass);

  // Set necessary parameters used in EigenSystem::solve(),
  // i.e. the number of requested eigenpairs nev and the number
  // of basis vectors ncv used in the solution algorithm. Note that
  // ncv >= nev must hold and ncv >= 2*nev is recommended.
  equation_systems.parameters.set<unsigned int>("eigenpairs")    = nev;
  equation_systems.parameters.set<unsigned int>("basis vectors") = nev*3;

  // You may optionally change the default eigensolver used by SLEPc.
  // The Krylov-Schur method is mathematically equivalent to implicitly
  // restarted Arnoldi, the method of Arpack, so there is currently no
  // point in using SLEPc with Arpack.
  // ARNOLDI     = default in SLEPc 2.3.1 and earlier
  // KRYLOVSCHUR default in SLEPc 2.3.2 and later
  // eigen_system.get_eigen_solver().set_eigensolver_type(KRYLOVSCHUR);

  // Set the solver tolerance and the maximum number of iterations.
  equation_systems.parameters.set<Real>
    ("linear solver tolerance") = pow(TOLERANCE, 5./3.);
  equation_systems.parameters.set<unsigned int>
    ("linear solver maximum iterations") = 1000;

  // Initialize the data structures for the equation system.
  equation_systems.init();

  // Prints information about the system to the screen.
  equation_systems.print_info();

  // Solve the system "Eigensystem".
  eigen_system.solve();

  // Get the number of converged eigen pairs.
  unsigned int nconv = eigen_system.get_n_converged();

  libMesh::out << "Number of converged eigenpairs: " << nconv
               << "\n" << std::endl;

  // Get the last converged eigenpair
  if (nconv != 0)
    {
      eigen_system.get_eigenpair(nconv-1);

#ifdef LIBMESH_HAVE_EXODUS_API
      // Write the eigen vector to file.
      ExodusII_IO (mesh).write_equation_systems ("out.e", equation_systems);
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
    }
  else
    {
      libMesh::out << "WARNING: Solver did not converge!\n" << nconv << std::endl;
    }

  // All done.
  return 0;
#endif // LIBMESH_HAVE_SLEPC
}



void assemble_mass(EquationSystems & es,
                   const std::string & libmesh_dbg_var(system_name))
{
  // It is a good idea to make sure we are assembling
  // the proper system.
  libmesh_assert_equal_to (system_name, "Eigensystem");

#ifdef LIBMESH_HAVE_SLEPC

  // Get a constant reference to the mesh object.
  const MeshBase & mesh = es.get_mesh();

  // The dimension that we are running.
  const unsigned int dim = mesh.mesh_dimension();

  // Get a reference to our system.
  EigenSystem & eigen_system = es.get_system<EigenSystem> ("Eigensystem");

  // Get a constant reference to the Finite Element type
  // for the first (and only) variable in the system.
  FEType fe_type = eigen_system.get_dof_map().variable_type(0);

  // A reference to the system matrix
  SparseMatrix<Number> & matrix_A = eigen_system.get_matrix_A();

  // Build a Finite Element object of the specified type.  Since the
  // FEBase::build() member dynamically creates memory we will
  // store the object as a std::unique_ptr<FEBase>.  This can be thought
  // of as a pointer that will clean up after itself.
  std::unique_ptr<FEBase> fe (FEBase::build(dim, fe_type));

  // A  Gauss quadrature rule for numerical integration.
  // Use the default quadrature order.
  QGauss qrule (dim, fe_type.default_quadrature_order());

  // Tell the finite element object to use our quadrature rule.
  fe->attach_quadrature_rule (&qrule);

  // The element Jacobian * quadrature weight at each integration point.
  const std::vector<Real> & JxW = fe->get_JxW();

  // The element shape functions evaluated at the quadrature points.
  const std::vector<std::vector<Real>> & phi = fe->get_phi();

  // A reference to the DofMap object for this system.  The DofMap
  // object handles the index translation from node and element numbers
  // to degree of freedom numbers.
  const DofMap & dof_map = eigen_system.get_dof_map();

  // The element mass matrix.
  DenseMatrix<Number> Me;

  // This vector will hold the degree of freedom indices for
  // the element.  These define where in the global system
  // the element degrees of freedom get mapped.
  std::vector<dof_id_type> dof_indices;

  // Now we will loop over all the elements in the mesh that
  // live on the local processor. We will compute the element
  // matrix and right-hand-side contribution.  In case users
  // later modify this program to include refinement, we will
  // be safe and will only consider the active elements;
  // hence we use a variant of the active_elem_iterator.
  for (const auto & elem : mesh.active_local_element_ptr_range())
    {
      // Get the degree of freedom indices for the
      // current element.  These define where in the global
      // matrix and right-hand-side this element will
      // contribute to.
      dof_map.dof_indices (elem, dof_indices);

      // Compute the element-specific data for the current
      // element.  This involves computing the location of the
      // quadrature points (q_point) and the shape functions
      // (phi, dphi) for the current element.
      fe->reinit (elem);

      // Zero the element matrices and rhs before
      // summing them.  We use the resize member here because
      // the number of degrees of freedom might have changed from
      // the last element.  Note that this will be the case if the
      // element type is different (i.e. the last element was a
      // triangle, now we are on a quadrilateral).
      const unsigned int n_dofs =
        cast_int<unsigned int>(dof_indices.size());
      Me.resize (n_dofs, n_dofs);

      // Now loop over the quadrature points.  This handles
      // the numeric integration.
      //
      // We will build the element matrix.  This involves
      // a double loop to integrate the test functions (i) against
      // the trial functions (j).
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        for (unsigned int i=0; i != n_dofs; i++)
          for (unsigned int j=0; j != n_dofs; j++)
            Me(i,j) += JxW[qp]*phi[i][qp]*phi[j][qp];

      // On an unrefined mesh, constrain_element_matrix does
      // nothing.  If this assembly function is ever repurposed to
      // run on a refined mesh, getting the hanging node constraints
      // right will be important.  Note that, even with
      // asymmetric_constraint_rows = false, the constrained dof
      // diagonals still exist in the matrix, with diagonal entries
      // that are there to ensure non-singular matrices for linear
      // solves but which would generate positive non-physical
      // eigenvalues for eigensolves.
      dof_map.constrain_element_matrix(Me, dof_indices, false);

      // Finally, simply add the element contribution to the
      // overall matrix.
      matrix_A.add_matrix (Me, dof_indices);
    } // end of element loop

#else
  // Avoid compiler warnings
  libmesh_ignore(es);
#endif // LIBMESH_HAVE_SLEPC
}