systems_of_equations_ex6.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 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.
 
// This library 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 for more details.
 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 
 
 
// <h1> Systems Example 6 - 3D Linear Elastic Cantilever </h1>
// \author David Knezevic
// \date 2012
//
// This is a 3D version of systems_of_equations_ex4. The weak form PDE for
// equilibrium elasticity is:
//
//     \int_\Omega Sigma_ij v_i,j = \int_\Omega f_i v_i + \int_\Gamma g_i v_i ds,
//
// for all admissible test functions v, where:
//  * Sigma is the stress tensor, which for linear elasticity is
//    given by Sigma_ij = C_ijkl u_k,l.
//  * f is a body load.
//  * g is a surface traction on the surface \Gamma.
 
 
// C++ include files that we need
#include <iostream>
#include <algorithm>
#include <math.h>
 
// libMesh includes
#include "libmesh/libmesh_config.h"
#include "libmesh/libmesh.h"
#include "libmesh/mesh.h"
#include "libmesh/mesh_generation.h"
#include "libmesh/exodusII_io.h"
#include "libmesh/gnuplot_io.h"
#include "libmesh/linear_implicit_system.h"
#include "libmesh/equation_systems.h"
#include "libmesh/fe.h"
#include "libmesh/quadrature_gauss.h"
#include "libmesh/dof_map.h"
#include "libmesh/sparse_matrix.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/dense_submatrix.h"
#include "libmesh/dense_vector.h"
#include "libmesh/dense_subvector.h"
#include "libmesh/perf_log.h"
#include "libmesh/elem.h"
#include "libmesh/boundary_info.h"
#include "libmesh/zero_function.h"
#include "libmesh/dirichlet_boundaries.h"
#include "libmesh/string_to_enum.h"
#include "libmesh/getpot.h"
#include "libmesh/solver_configuration.h"
#include "libmesh/petsc_linear_solver.h"
#include "libmesh/petsc_macro.h"
#include "libmesh/enum_solver_package.h"
#include "libmesh/tensor_value.h"
#include "libmesh/vector_value.h"
#include "libmesh/utility.h"
 
#define x_scaling 1.3
 
// boundary IDs
#define BOUNDARY_ID_MIN_Z 0
#define BOUNDARY_ID_MIN_Y 1
#define BOUNDARY_ID_MAX_X 2
#define BOUNDARY_ID_MAX_Y 3
#define BOUNDARY_ID_MIN_X 4
#define BOUNDARY_ID_MAX_Z 5
#define NODE_BOUNDARY_ID 10
#define EDGE_BOUNDARY_ID 20
 
// Bring in everything from the libMesh namespace
using namespace libMesh;
 
#ifdef LIBMESH_HAVE_PETSC
// This class allows us to set the solver and preconditioner
// to be appropriate for linear elasticity.
class PetscSolverConfiguration : public SolverConfiguration
{
public:
 
  PetscSolverConfiguration(PetscLinearSolver<Number> & petsc_linear_solver) :
    _petsc_linear_solver(petsc_linear_solver)
  {
  }
 
  virtual void configure_solver()
  {
    LibmeshPetscCall2(_petsc_linear_solver.comm(), KSPSetType(_petsc_linear_solver.ksp(), const_cast<KSPType>(KSPCG)));
    LibmeshPetscCall2(_petsc_linear_solver.comm(), PCSetType(_petsc_linear_solver.pc(), const_cast<PCType>(PCBJACOBI)));
  }
 
  // The linear solver object that we are configuring
  PetscLinearSolver<Number> & _petsc_linear_solver;
 
};
#endif
 
class LinearElasticity : public System::Assembly
{
private:
  EquationSystems & es;
 
public:
 
  LinearElasticity (EquationSystems & es_in) :
    es(es_in)
  {}
 
  Real kronecker_delta(unsigned int i,
                       unsigned int j)
  {
    return i == j ? 1. : 0.;
  }
 
  Real elasticity_tensor(unsigned int i,
                         unsigned int j,
                         unsigned int k,
                         unsigned int l)
  {
    // Hard code material parameters for the sake of simplicity
    const Real poisson_ratio = 0.3;
    const Real young_modulus = 1.;
 
    // Define the Lame constants
    const Real lambda_1 = (young_modulus*poisson_ratio)/((1.+poisson_ratio)*(1.-2.*poisson_ratio));
    const Real lambda_2 = young_modulus/(2.*(1.+poisson_ratio));
 
    return lambda_1 * kronecker_delta(i, j) * kronecker_delta(k, l) +
      lambda_2 * (kronecker_delta(i, k) * kronecker_delta(j, l) + kronecker_delta(i, l) * kronecker_delta(j, k));
  }
 
  void assemble()
  {
    const MeshBase & mesh = es.get_mesh();
 
    const unsigned int dim = mesh.mesh_dimension();
 
    LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Elasticity");
 
    const unsigned int u_var = system.variable_number ("u");
 
    const DofMap & dof_map = system.get_dof_map();
    FEType fe_type = dof_map.variable_type(u_var);
    std::unique_ptr<FEBase> fe (FEBase::build(dim, fe_type));
    QGauss qrule (dim, fe_type.default_quadrature_order());
    fe->attach_quadrature_rule (&qrule);
 
    std::unique_ptr<FEBase> fe_face (FEBase::build(dim, fe_type));
    QGauss qface(dim-1, fe_type.default_quadrature_order());
    fe_face->attach_quadrature_rule (&qface);
 
    const std::vector<Real> & JxW = fe->get_JxW();
    const std::vector<std::vector<Real>> & phi = fe->get_phi();
    const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();
 
    DenseMatrix<Number> Ke;
    DenseSubMatrix<Number> Ke_var[3][3] =
      {
        {DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke)},
        {DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke)},
        {DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke), DenseSubMatrix<Number>(Ke)}
      };
 
    DenseVector<Number> Fe;
 
    DenseSubVector<Number> Fe_var[3] =
      {DenseSubVector<Number>(Fe),
       DenseSubVector<Number>(Fe),
       DenseSubVector<Number>(Fe)};
 
    std::vector<dof_id_type> dof_indices;
    std::vector<std::vector<dof_id_type>> dof_indices_var(3);
 
    SparseMatrix<Number> & matrix = system.get_system_matrix();
 
    for (const auto & elem : mesh.active_local_element_ptr_range())
      {
        dof_map.dof_indices (elem, dof_indices);
        for (unsigned int var=0; var<3; var++)
          dof_map.dof_indices (elem, dof_indices_var[var], var);
 
        const unsigned int n_dofs   = dof_indices.size();
        const unsigned int n_var_dofs = dof_indices_var[0].size();
 
        fe->reinit (elem);
 
        Ke.resize (n_dofs, n_dofs);
        for (unsigned int var_i=0; var_i<3; var_i++)
          for (unsigned int var_j=0; var_j<3; var_j++)
            Ke_var[var_i][var_j].reposition (var_i*n_var_dofs, var_j*n_var_dofs, n_var_dofs, n_var_dofs);
 
        Fe.resize (n_dofs);
        for (unsigned int var=0; var<3; var++)
          Fe_var[var].reposition (var*n_var_dofs, n_var_dofs);
 
        for (unsigned int qp=0; qp<qrule.n_points(); qp++)
          {
            // assemble \int_Omega C_ijkl u_k,l v_i,j \dx
            for (unsigned int dof_i=0; dof_i<n_var_dofs; dof_i++)
              for (unsigned int dof_j=0; dof_j<n_var_dofs; dof_j++)
                for (unsigned int i=0; i<3; i++)
                  for (unsigned int j=0; j<3; j++)
                    for (unsigned int k=0; k<3; k++)
                      for (unsigned int l=0; l<3; l++)
                        Ke_var[i][k](dof_i,dof_j) +=
                          JxW[qp] * elasticity_tensor(i,j,k,l) * dphi[dof_j][qp](l) * dphi[dof_i][qp](j);
 
            // assemble \int_Omega f_i v_i \dx
            VectorValue<Number> f_vec(0., 0., -1.);
            for (unsigned int dof_i=0; dof_i<n_var_dofs; dof_i++)
              for (unsigned int i=0; i<3; i++)
                Fe_var[i](dof_i) += JxW[qp] * (f_vec(i) * phi[dof_i][qp]);
          }
 
        // assemble \int_\Gamma g_i v_i \ds
        VectorValue<Number> g_vec(0., 0., -1.);
        {
          for (auto side : elem->side_index_range())
            if (elem->neighbor_ptr(side) == nullptr)
              {
                const std::vector<std::vector<Real>> & phi_face = fe_face->get_phi();
                const std::vector<Real> & JxW_face = fe_face->get_JxW();
 
                fe_face->reinit(elem, side);
 
                // Apply a traction
                for (unsigned int qp=0; qp<qface.n_points(); qp++)
                  if (mesh.get_boundary_info().has_boundary_id(elem, side, BOUNDARY_ID_MAX_X))
                    for (unsigned int dof_i=0; dof_i<n_var_dofs; dof_i++)
                      for (unsigned int i=0; i<3; i++)
                        Fe_var[i](dof_i) += JxW_face[qp] * (g_vec(i) * phi_face[dof_i][qp]);
              }
        }
 
        dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
 
        matrix.add_matrix         (Ke, dof_indices);
        system.rhs->add_vector    (Fe, dof_indices);
      }
  }
 
  // Post-process the solution to compute stresses
  void compute_stresses()
  {
    const MeshBase & mesh = es.get_mesh();
    const unsigned int dim = mesh.mesh_dimension();
 
    LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Elasticity");
 
    unsigned int displacement_vars[3];
    displacement_vars[0] = system.variable_number ("u");
    displacement_vars[1] = system.variable_number ("v");
    displacement_vars[2] = system.variable_number ("w");
    const unsigned int u_var = system.variable_number ("u");
 
    const DofMap & dof_map = system.get_dof_map();
    FEType fe_type = dof_map.variable_type(u_var);
    std::unique_ptr<FEBase> fe (FEBase::build(dim, fe_type));
    QGauss qrule (dim, fe_type.default_quadrature_order());
    fe->attach_quadrature_rule (&qrule);
 
    const std::vector<Real> & JxW = fe->get_JxW();
    const std::vector<std::vector<Real>> & phi = fe->get_phi();
    const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();
 
    // Also, get a reference to the ExplicitSystem
    ExplicitSystem & stress_system = es.get_system<ExplicitSystem>("StressSystem");
    const DofMap & stress_dof_map = stress_system.get_dof_map();
    unsigned int sigma_vars[6];
    sigma_vars[0] = stress_system.variable_number ("sigma_00");
    sigma_vars[1] = stress_system.variable_number ("sigma_01");
    sigma_vars[2] = stress_system.variable_number ("sigma_02");
    sigma_vars[3] = stress_system.variable_number ("sigma_11");
    sigma_vars[4] = stress_system.variable_number ("sigma_12");
    sigma_vars[5] = stress_system.variable_number ("sigma_22");
    unsigned int vonMises_var = stress_system.variable_number ("vonMises");
 
    // Storage for the stress dof indices on each element
    std::vector<std::vector<dof_id_type>> dof_indices_var(system.n_vars());
    std::vector<dof_id_type> stress_dof_indices_var;
    std::vector<dof_id_type> vonmises_dof_indices_var;
 
    for (const auto & elem : mesh.active_local_element_ptr_range())
      {
        for (unsigned int var=0; var<3; var++)
          dof_map.dof_indices (elem, dof_indices_var[var], displacement_vars[var]);
 
        const unsigned int n_var_dofs = dof_indices_var[0].size();
 
        fe->reinit (elem);
 
        std::vector<TensorValue<Number>> stress_tensor_qp(qrule.n_points());
        for (unsigned int qp=0; qp<qrule.n_points(); qp++)
          {
            // Row is variable u1, u2, or u3, column is x, y, or z
            TensorValue<Number> grad_u;
            for (unsigned int var_i=0; var_i<3; var_i++)
              for (unsigned int var_j=0; var_j<3; var_j++)
                for (unsigned int j=0; j<n_var_dofs; j++)
                  grad_u(var_i,var_j) += dphi[j][qp](var_j) * system.current_solution(dof_indices_var[var_i][j]);
 
            for (unsigned int var_i=0; var_i<3; var_i++)
              for (unsigned int var_j=0; var_j<3; var_j++)
                for (unsigned int k=0; k<3; k++)
                  for (unsigned int l=0; l<3; l++)
                    stress_tensor_qp[qp](var_i,var_j) += elasticity_tensor(var_i,var_j,k,l) * grad_u(k,l);
          }
 
        stress_dof_map.dof_indices (elem, vonmises_dof_indices_var, vonMises_var);
        std::vector<TensorValue<Number>> elem_sigma_vec(vonmises_dof_indices_var.size());
 
        // Below we project each component of the stress tensor onto a L2_LAGRANGE discretization.
        // Note that this gives a discontinuous stress plot on element boundaries, which is
        // appropriate. We then also get the von Mises stress from the projected stress tensor.
        unsigned int stress_var_index = 0;
        for (unsigned int var_i=0; var_i<3; var_i++)
          for (unsigned int var_j=var_i; var_j<3; var_j++)
            {
              stress_dof_map.dof_indices (elem, stress_dof_indices_var, sigma_vars[stress_var_index]);
 
              const unsigned int n_proj_dofs = stress_dof_indices_var.size();
 
              DenseMatrix<Real> Me(n_proj_dofs, n_proj_dofs);
              for (unsigned int qp=0; qp<qrule.n_points(); qp++)
                {
                  for(unsigned int i=0; i<n_proj_dofs; i++)
                    for(unsigned int j=0; j<n_proj_dofs; j++)
                      {
                        Me(i,j) += JxW[qp]*(phi[i][qp]*phi[j][qp]);
                      }
                }
 
              DenseVector<Number> Fe(n_proj_dofs);
              for (unsigned int qp=0; qp<qrule.n_points(); qp++)
                for(unsigned int i=0; i<n_proj_dofs; i++)
                  {
                    Fe(i) += JxW[qp] * stress_tensor_qp[qp](var_i,var_j) * phi[i][qp];
                  }
 
              DenseVector<Number> projected_data;
              Me.cholesky_solve(Fe, projected_data);
 
              for(unsigned int index=0; index<n_proj_dofs; index++)
                {
                  dof_id_type dof_index = stress_dof_indices_var[index];
                  if ((stress_system.solution->first_local_index() <= dof_index) &&
                      (dof_index < stress_system.solution->last_local_index()))
                    stress_system.solution->set(dof_index, projected_data(index));
 
                  elem_sigma_vec[index](var_i,var_j) = projected_data(index);
                }
 
              stress_var_index++;
            }
 
        for (std::size_t index=0; index<elem_sigma_vec.size(); index++)
          {
            elem_sigma_vec[index](1,0) = elem_sigma_vec[index](0,1);
            elem_sigma_vec[index](2,0) = elem_sigma_vec[index](0,2);
            elem_sigma_vec[index](2,1) = elem_sigma_vec[index](1,2);
 
            // Get the von Mises stress from the projected stress tensor
            Number vonMises_value = std::sqrt(0.5*(Utility::pow<2>(elem_sigma_vec[index](0,0) - elem_sigma_vec[index](1,1)) +
                                                   Utility::pow<2>(elem_sigma_vec[index](1,1) - elem_sigma_vec[index](2,2)) +
                                                   Utility::pow<2>(elem_sigma_vec[index](2,2) - elem_sigma_vec[index](0,0)) +
                                                   6.*(Utility::pow<2>(elem_sigma_vec[index](0,1)) +
                                                       Utility::pow<2>(elem_sigma_vec[index](1,2)) +
                                                       Utility::pow<2>(elem_sigma_vec[index](2,0)))));
 
            dof_id_type dof_index = vonmises_dof_indices_var[index];
 
            if ((stress_system.solution->first_local_index() <= dof_index) &&
                (dof_index < stress_system.solution->last_local_index()))
              stress_system.solution->set(dof_index, vonMises_value);
          }
      }
 
    // Should call close and update when we set vector entries directly
    stress_system.solution->close();
    stress_system.update();
  }
};
 
 
// Begin the main program.
int main (int argc, char ** argv)
{
  // Initialize libMesh and any dependent libraries
  LibMeshInit init (argc, argv);
 
  // This example requires a linear solver package.
  libmesh_example_requires(libMesh::default_solver_package() != INVALID_SOLVER_PACKAGE,
                           "--enable-petsc, --enable-trilinos, or --enable-eigen");
 
  // Initialize the cantilever mesh
  const unsigned int dim = 3;
 
  // Make sure libMesh was compiled for 3D
  libmesh_example_requires(dim == LIBMESH_DIM, "3D support");
 
  // We use Dirichlet boundary conditions here
#ifndef LIBMESH_ENABLE_DIRICHLET
  libmesh_example_requires(false, "--enable-dirichlet");
#endif
 
  // Get the mesh size from the command line.
  const int nx = libMesh::command_line_next("-nx", 32),
            ny = libMesh::command_line_next("-ny", 8),
            nz = libMesh::command_line_next("-nz", 4);
 
  // Create a 3D mesh distributed across the default MPI communicator.
  Mesh mesh(init.comm(), dim);
  MeshTools::Generation::build_cube (mesh,
                                     nx,
                                     ny,
                                     nz,
                                     0., 1.*x_scaling,
                                     0., 0.3,
                                     0., 0.1,
                                     HEX8);
 
  // Print information about the mesh to the screen.
  mesh.print_info();
 
  // Let's add some node and edge boundary conditions
  // Each processor should know about each boundary condition it can
  // see, so we loop over all elements, not just local elements.
  for (const auto & elem : mesh.element_ptr_range())
    {
      unsigned int
        side_max_x = 0, side_min_y = 0,
        side_max_y = 0, side_max_z = 0;
 
      bool
        found_side_max_x = false, found_side_max_y = false,
        found_side_min_y = false, found_side_max_z = false;
 
      for (auto side : elem->side_index_range())
        {
          if (mesh.get_boundary_info().has_boundary_id(elem, side, BOUNDARY_ID_MAX_X))
            {
              side_max_x = side;
              found_side_max_x = true;
            }
 
          if (mesh.get_boundary_info().has_boundary_id(elem, side, BOUNDARY_ID_MIN_Y))
            {
              side_min_y = side;
              found_side_min_y = true;
            }
 
          if (mesh.get_boundary_info().has_boundary_id(elem, side, BOUNDARY_ID_MAX_Y))
            {
              side_max_y = side;
              found_side_max_y = true;
            }
 
          if (mesh.get_boundary_info().has_boundary_id(elem, side, BOUNDARY_ID_MAX_Z))
            {
              side_max_z = side;
              found_side_max_z = true;
            }
        }
 
      // If elem has sides on boundaries
      // BOUNDARY_ID_MAX_X, BOUNDARY_ID_MAX_Y, BOUNDARY_ID_MAX_Z
      // then let's set a node boundary condition
      if (found_side_max_x && found_side_max_y && found_side_max_z)
        for (auto n : elem->node_index_range())
          if (elem->is_node_on_side(n, side_max_x) &&
              elem->is_node_on_side(n, side_max_y) &&
              elem->is_node_on_side(n, side_max_z))
            mesh.get_boundary_info().add_node(elem->node_ptr(n), NODE_BOUNDARY_ID);
 
 
      // If elem has sides on boundaries
      // BOUNDARY_ID_MAX_X and BOUNDARY_ID_MIN_Y
      // then let's set an edge boundary condition
      if (found_side_max_x && found_side_min_y)
        for (auto e : elem->edge_index_range())
          if (elem->is_edge_on_side(e, side_max_x) &&
              elem->is_edge_on_side(e, side_min_y))
            mesh.get_boundary_info().add_edge(elem, e, EDGE_BOUNDARY_ID);
    }
 
  // We're all done adding to the boundary_info; let's make sure its
  // caches know about it.
  mesh.get_boundary_info().regenerate_id_sets();
 
  // Create an equation systems object.
  EquationSystems equation_systems (mesh);
 
  // Declare the system and its variables.
  // Create a system named "Elasticity"
  LinearImplicitSystem & system =
    equation_systems.add_system<LinearImplicitSystem> ("Elasticity");
 
#ifdef LIBMESH_HAVE_PETSC
  // Attach a SolverConfiguration object to system.linear_solver
  PetscLinearSolver<Number> * petsc_linear_solver =
    cast_ptr<PetscLinearSolver<Number>*>(system.get_linear_solver());
  libmesh_assert(petsc_linear_solver);
  PetscSolverConfiguration petsc_solver_config(*petsc_linear_solver);
  petsc_linear_solver->set_solver_configuration(petsc_solver_config);
#endif
 
  LinearElasticity le(equation_systems);
  system.attach_assemble_object(le);
 
#ifdef LIBMESH_ENABLE_DIRICHLET
  // Add three displacement variables, u and v, to the system
  unsigned int u_var = system.add_variable("u", FIRST, LAGRANGE);
  unsigned int v_var = system.add_variable("v", FIRST, LAGRANGE);
  unsigned int w_var = system.add_variable("w", FIRST, LAGRANGE);
 
  // Create a ZeroFunction to initialize dirichlet_bc
  ZeroFunction<> zf;
 
  // Most DirichletBoundary users will want to supply a "locally
  // indexed" functor
  DirichletBoundary dirichlet_bc({BOUNDARY_ID_MIN_X, NODE_BOUNDARY_ID,
                                 EDGE_BOUNDARY_ID},
                                 {u_var, v_var, w_var}, zf,
                                 LOCAL_VARIABLE_ORDER);
 
  // We must add the Dirichlet boundary condition _before_
  // we call equation_systems.init()
  system.get_dof_map().add_dirichlet_boundary(dirichlet_bc);
#endif // LIBMESH_ENABLE_DIRICHLET
 
  // Also, initialize an ExplicitSystem to store stresses
  ExplicitSystem & stress_system =
    equation_systems.add_system<ExplicitSystem> ("StressSystem");
 
  stress_system.add_variable("sigma_00", FIRST, L2_LAGRANGE);
  stress_system.add_variable("sigma_01", FIRST, L2_LAGRANGE);
  stress_system.add_variable("sigma_02", FIRST, L2_LAGRANGE);
  stress_system.add_variable("sigma_11", FIRST, L2_LAGRANGE);
  stress_system.add_variable("sigma_12", FIRST, L2_LAGRANGE);
  stress_system.add_variable("sigma_22", FIRST, L2_LAGRANGE);
  stress_system.add_variable("vonMises", FIRST, L2_LAGRANGE);
 
  // Initialize the data structures for the equation system.
  equation_systems.init();
 
  // Print information about the system to the screen.
  equation_systems.print_info();
 
  // Solve the system
  system.solve();
 
  // Post-process the solution to compute the stresses
  le.compute_stresses();
 
  // Plot the solution
#ifdef LIBMESH_HAVE_EXODUS_API
 
  // Use single precision in this case (reduces the size of the exodus file)
  ExodusII_IO exo_io(mesh, /*single_precision=*/true);
  exo_io.write_discontinuous_exodusII("displacement_and_stress.exo", equation_systems);
 
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
 
  // All done.
  return 0;
}