systems_of_equations_ex9.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 9 - Linear elasticity problem with periodic constraints </h1>
//
// In this example we illustrate periodic constraints with a linear elasticity example.
// We consider a sector of a circular domain, and hence we must impose an "azimuthal"
// periodic condition, which entails a dof-transformation between the two periodic
// boundaries to account for the rotation of the coordinate system.
//
// The baseline code is from systems_of_equations_ex6, and we edit to use a different mesh
// and to impose the periodic boundary condition.
 
 
// 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/mesh_refinement.h"
#include "libmesh/solver_configuration.h"
#include "libmesh/petsc_linear_solver.h"
#include "libmesh/petsc_macro.h"
#include "libmesh/periodic_boundaries.h"
#include "libmesh/periodic_boundary.h"
#include "libmesh/enum_solver_package.h"
#include "libmesh/tensor_value.h"
#include "libmesh/vector_value.h"
 
// Bring in everything from the libMesh namespace
using namespace libMesh;
 
#ifdef LIBMESH_ENABLE_PERIODIC
// Here we define the azimuthal periodic boundary condition class.
// This class assumes that (u,v,w) are variables 0, 1, and 2 in
// the System, but this could be made more general if needed.
class AzimuthalPeriodicBoundary : public PeriodicBoundaryBase
{
public:
  AzimuthalPeriodicBoundary(
    Point center,
    Point axis,
    Real angle)
  :
    PeriodicBoundaryBase(),
    _center(center),
    _axis(axis),
    _theta(angle)
  {
    set_variable(0);
    set_variable(1);
    set_variable(2);
    set_up_rotation_matrix();
  }
 
  AzimuthalPeriodicBoundary(const AzimuthalPeriodicBoundary & o, TransformationType t = FORWARD)
    :
    PeriodicBoundaryBase(o),
    _center(o._center),
    _axis(o._axis),
    _theta(o._theta)
  {
    if (t == INVERSE)
      {
        std::swap(myboundary, pairedboundary);
        _theta *= -1.0;
      }
 
    set_variable(0);
    set_variable(1);
    set_variable(2);
    set_up_rotation_matrix();
  }
 
  virtual ~AzimuthalPeriodicBoundary() = default;
 
  void set_up_rotation_matrix()
  {
    // Formula for rotation matrix about an axis is given on wikipedia:
    // en.wikipedia.org/wiki/Rotation_matrix
    // We rotate by angle theta about the axis defined by u, which is a
    // unit vector in the direction of _axis.
    // Note: this example already requires LIBMESH_DIM==3 so we don't
    // explicitly check for that here.
    Point u = _axis.unit();
    Real u_x = u(0);
    Real u_y = u(1);
    Real u_z = u(2);
    Real cost = std::cos(_theta);
    Real sint = std::sin(_theta);
    _R = RealTensorValue
      (cost + u_x*u_x*(1.0 - cost),     u_x*u_y*(1.0 - cost) - u_z*sint, u_x*u_z*(1.0 - cost) + u_y*sint,
       u_y*u_x*(1.0 - cost) + u_z*sint, cost + u_y*u_y*(1.0 - cost),     u_y*u_z*(1.0 - cost) - u_x*sint,
       u_z*u_x*(1.0 - cost) - u_y*sint, u_z*u_y*(1.0 - cost) + u_x*sint, cost + u_z*u_z*(1.0 - cost));
 
    // For generality, PeriodicBoundaryBase::set_transformation_matrix() takes an
    // (n_vars * n_vars) DenseMatrix which we construct on the fly.
    this->set_transformation_matrix
      (DenseMatrix<Real> (3, 3, {_R(0,0), _R(0,1), _R(0,2), _R(1,0), _R(1,1), _R(1,2), _R(2,0), _R(2,1), _R(2,2)}));
  }
 
  virtual Point get_corresponding_pos(const Point & pt) const override
  {
    // Note that since _theta defines the angle from "paired boundary" to
    // "my boundary", and we want the inverse of that here, we multiply
    // by the transpose of the transformation matrix below.
    return _R.left_multiply(pt - _center) + _center;
  }
 
  virtual std::unique_ptr<PeriodicBoundaryBase> clone(TransformationType t = FORWARD) const override
  {
    return std::make_unique<AzimuthalPeriodicBoundary>(*this, t);
  }
 
private:
 
  // Define the properties needed for the azimuthal periodic boundary.
  // Note that _theta specifies the angle from "paired boundary" to
  // "my boundary".
  Point _center;
  Point _axis;
  Real _theta;
  RealTensorValue _R;
};
 
#endif // LIBMESH_ENABLE_PERIODIC
 
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
            // The mesh for this example has two subdomains with ids 1
            // and 101, and the forcing function is different on each.
            auto f_vec = (elem->subdomain_id() == 101 ?
                          VectorValue<Number>(1., 1., 0.) :
                          VectorValue<Number>(0.36603, 1.36603, 0.));
 
            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]);
          }
 
        dof_map.constrain_element_matrix_and_vector (Ke, Fe, dof_indices);
 
        matrix.add_matrix         (Ke, dof_indices);
        system.rhs->add_vector    (Fe, dof_indices);
      }
  }
};
 
 
// 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");
 
#ifndef LIBMESH_HAVE_EXODUS_API
  // example requires ExodusII to load the mesh
  libmesh_example_requires(false, "--enable-exodus");
#endif
 
  // We use Dirichlet and Periodic boundary conditions here
#ifndef LIBMESH_ENABLE_DIRICHLET
  libmesh_example_requires(false, "--enable-dirichlet");
#endif
#ifndef LIBMESH_ENABLE_PERIODIC
  libmesh_example_requires(false, "--enable-periodic");
#endif
 
  // Initialize the cantilever mesh
  const unsigned int dim = 3;
 
  // Make sure libMesh was compiled for 3D
  libmesh_example_requires(dim == LIBMESH_DIM, "3D support");
 
  // Make sure libMesh has normal boundary id sizes
  libmesh_example_requires(sizeof(boundary_id_type) > 1, "boundary_id_size > 1");
 
#if LIBMESH_BOUNDARY_ID_BYTES > 1
  // Create a 3D mesh distributed across the default MPI communicator.
  Mesh mesh(init.comm());
 
  // 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_ENABLE_PERIODIC
  // Add two azimuthal periodic boundaries on two adjacent domains.
  // We do this to show that the periodic boundary condition that
  // we impose leads to a continuous solution across adjacent domains.
  //
  // We add the periodic boundaries *before* reading the Mesh, so
  // that periodic neighbors will be retained when a DistributedMesh
  // is distributed.
  //
  // The angle specified below defines the mapping
  // from "pairedboundary" to "myboundary".
  {
    Point center(0., 0., 0.);
    Point axis(0., 0., 1.);
    Real angle = 2*libMesh::pi/12.0;
    AzimuthalPeriodicBoundary periodic_bc(center, axis, angle);
    periodic_bc.myboundary = 301;
    periodic_bc.pairedboundary = 302;
    system.get_dof_map().add_periodic_boundary(periodic_bc);
  }
  {
    Point center(0., 0., 0.);
    Point axis(0., 0., 1.);
    Real angle = 2*libMesh::pi/12.0;
    AzimuthalPeriodicBoundary periodic_bc(center, axis, angle);
    periodic_bc.myboundary = 401;
    periodic_bc.pairedboundary = 402;
    system.get_dof_map().add_periodic_boundary(periodic_bc);
  }
#endif // LIBMESH_ENABLE_PERIODIC
 
  mesh.read("systems_of_equations_ex9.exo");
 
  GetPot input(argc, argv);
  const unsigned int n_refinements = input("n_refinements", 0);
  // Skip adaptive runs on a non-adaptive libMesh build
#ifndef LIBMESH_ENABLE_AMR
  libmesh_example_requires(n_refinements==0, "--enable-amr");
#else
  MeshRefinement mesh_refinement(mesh);
  mesh_refinement.uniformly_refine(n_refinements);
#endif
 
  // Print information about the mesh to the screen.
  mesh.print_info();
 
  // 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);
 
  LinearElasticity le(equation_systems);
  system.attach_assemble_object(le);
 
  ZeroFunction<> zf;
 
#ifdef LIBMESH_ENABLE_DIRICHLET
  DirichletBoundary clamped_bc({300,400}, {u_var,v_var,w_var}, zf);
  system.get_dof_map().add_dirichlet_boundary(clamped_bc);
#endif
 
  // 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();
 
  // Plot the solution
#ifdef LIBMESH_HAVE_EXODUS_API
 
  ExodusII_IO (mesh).write_equation_systems("solution.exo",
                                            equation_systems);
 
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
#endif // #if LIBMESH_BOUNDARY_ID_BYTES > 1
 
  // All done.
  return 0;
}