introduction_ex5.C File Reference

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Functions
void	assemble_poisson (EquationSystems &es, const std::string &system_name)
Real	exact_solution (const Real x, const Real y, const Real z=0.)
	This is the exact solution that we are trying to obtain. More...
void	exact_solution_wrapper (DenseVector< Number > &output, const Point &p, const Real)
int	main (int argc, char **argv)
void	assemble_poisson (EquationSystems &es, const std::string &libmesh_dbg_var(system_name))

Variables
QuadratureType	quad_type =INVALID_Q_RULE

Function Documentation

assemble_poisson() [1/2]

void assemble_poisson	(	EquationSystems &	es,
		const std::string &	system_name
	)

Definition at line 261 of file miscellaneous_ex16.C.

Referenced by main().

{
  // 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 the LinearImplicitSystem we are solving
  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>(system_name);

  // Get a pointer to the StaticCondensation class if it exists
  StaticCondensation * sc = nullptr;
  if (system.has_static_condensation())
    sc = &system.get_static_condensation();

  // 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.  We will talk more about the  DofMap
  // in future examples.
  const DofMap & dof_map = system.get_dof_map();

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

  // 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.  Introduction Example 4
  // describes some advantages of  std::unique_ptr's in the context of
  // quadrature rules.
  std::unique_ptr<FEBase> fe(FEBase::build(dim, fe_type));

  // A 5th order Gauss quadrature rule for numerical integration.
  QGauss qrule(dim, FIFTH);

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

  // Declare a special finite element object for
  // boundary integration.
  std::unique_ptr<FEBase> fe_face(FEBase::build(dim, fe_type));

  // Boundary integration requires one quadrature rule,
  // with dimensionality one less than the dimensionality
  // of the element.
  QGauss qface(dim - 1, FIFTH);

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

  // Here we define some references to cell-specific data that
  // will be used to assemble the linear system.
  //
  // The element Jacobian * quadrature weight at each integration point.
  const std::vector<Real> & JxW = fe->get_JxW();

  // The physical XY locations of the quadrature points on the element.
  // These might be useful for evaluating spatially varying material
  // properties at the quadrature points.
  const std::vector<Point> & q_point = fe->get_xyz();

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

  // The element shape function gradients evaluated at the quadrature
  // points.
  const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();

  // Define data structures to contain the element matrix
  // and right-hand-side vector contribution.  Following
  // basic finite element terminology we will denote these
  // "Ke" and "Fe".  These datatypes are templated on
  //  Number, which allows the same code to work for real
  // or complex numbers.
  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;

  // 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;

  // The global system matrix
  SparseMatrix<Number> & matrix = system.get_system_matrix();

  // Now we will loop over all the elements in the mesh.
  // We will compute the element matrix and right-hand-side
  // contribution.
  //
  // Element ranges are a nice way to iterate through all the
  // elements, or all the elements that have some property.  The
  // range will iterate from the first to the last element on
  // the local processor.
  // It is smart to make this one const so that we don't accidentally
  // mess it up!  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_local_element_ptr_range.
  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);

    // Cache the number of degrees of freedom on this element, for
    // use as a loop bound later.  We use cast_int to explicitly
    // convert from size() (which may be 64-bit) to unsigned int
    // (which may be 32-bit but which is definitely enough to count
    // *local* degrees of freedom.
    const unsigned int n_dofs = cast_int<unsigned int>(dof_indices.size());

    // 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);

    // With one variable, we should have the same number of degrees
    // of freedom as shape functions.
    libmesh_assert_equal_to(n_dofs, phi.size());

    // Zero the element matrix and right-hand side 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).

    // The  DenseMatrix::resize() and the  DenseVector::resize()
    // members will automatically zero out the matrix  and vector.
    Ke.resize(n_dofs, n_dofs);

    Fe.resize(n_dofs);

    // Now loop over the quadrature points.  This handles
    // the numeric integration.
    for (unsigned int qp = 0; qp < qrule.n_points(); qp++)
    {

      // Now 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 i = 0; i != n_dofs; i++)
        for (unsigned int j = 0; j != n_dofs; j++)
        {
          Ke(i, j) += JxW[qp] * (dphi[i][qp] * dphi[j][qp]);
        }

      // This is the end of the matrix summation loop
      // Now we build the element right-hand-side contribution.
      // This involves a single loop in which we integrate the
      // "forcing function" in the PDE against the test functions.
      {
        const Real x = q_point[qp](0);
        const Real y = q_point[qp](1);
        const Real eps = 1.e-3;

        // "fxy" is the forcing function for the Poisson equation.
        // In this case we set fxy to be a finite difference
        // Laplacian approximation to the (known) exact solution.
        //
        // We will use the second-order accurate FD Laplacian
        // approximation, which in 2D is
        //
        // u_xx + u_yy = (u(i,j-1) + u(i,j+1) +
        //                u(i-1,j) + u(i+1,j) +
        //                -4*u(i,j))/h^2
        //
        // Since the value of the forcing function depends only
        // on the location of the quadrature point (q_point[qp])
        // we will compute it here, outside of the i-loop
        const Real fxy =
            -(exact_solution(x, y - eps) + exact_solution(x, y + eps) + exact_solution(x - eps, y) +
              exact_solution(x + eps, y) - 4. * exact_solution(x, y)) /
            eps / eps;

        for (unsigned int i = 0; i != n_dofs; i++)
          Fe(i) += JxW[qp] * fxy * phi[i][qp];
      }
    }

    // We have now reached the end of the RHS summation,
    // and the end of quadrature point loop, so
    // the interior element integration has
    // been completed.  However, we have not yet addressed
    // boundary conditions.  For this example we will only
    // consider simple Dirichlet boundary conditions.
    //
    // There are several ways Dirichlet boundary conditions
    // can be imposed.  A simple approach, which works for
    // interpolary bases like the standard Lagrange polynomials,
    // is to assign function values to the
    // degrees of freedom living on the domain boundary. This
    // works well for interpolary bases, but is more difficult
    // when non-interpolary (e.g Legendre or Hierarchic) bases
    // are used.
    //
    // Dirichlet boundary conditions can also be imposed with a
    // "penalty" method.  In this case essentially the L2 projection
    // of the boundary values are added to the matrix. The
    // projection is multiplied by some large factor so that, in
    // floating point arithmetic, the existing (smaller) entries
    // in the matrix and right-hand-side are effectively ignored.
    //
    // This amounts to adding a term of the form (in latex notation)
    //
    // \frac{1}{\epsilon} \int_{\delta \Omega} \phi_i \phi_j = \frac{1}{\epsilon} \int_{\delta
    // \Omega} u \phi_i
    //
    // where
    //
    // \frac{1}{\epsilon} is the penalty parameter, defined such that \epsilon << 1
    {

      // The following loop is over the sides of the element.
      // If the element has no neighbor on a side then that
      // side MUST live on a boundary of the domain.
      for (auto side : elem->side_index_range())
        if (elem->neighbor_ptr(side) == nullptr)
        {
          // The value of the shape functions at the quadrature
          // points.
          const std::vector<std::vector<Real>> & phi_face = fe_face->get_phi();

          // The Jacobian * Quadrature Weight at the quadrature
          // points on the face.
          const std::vector<Real> & JxW_face = fe_face->get_JxW();

          // The XYZ locations (in physical space) of the
          // quadrature points on the face.  This is where
          // we will interpolate the boundary value function.
          const std::vector<Point> & qface_point = fe_face->get_xyz();

          // Compute the shape function values on the element
          // face.
          fe_face->reinit(elem, side);

          // Some shape functions will be 0 on the face, but for
          // ease of indexing and generality of code we loop over
          // them anyway
          libmesh_assert_equal_to(n_dofs, phi_face.size());

          // Loop over the face quadrature points for integration.
          for (unsigned int qp = 0; qp < qface.n_points(); qp++)
          {
            // The location on the boundary of the current
            // face quadrature point.
            const Real xf = qface_point[qp](0);
            const Real yf = qface_point[qp](1);

            // The penalty value.  \frac{1}{\epsilon}
            // in the discussion above.
            const Real penalty = 1.e10;

            // The boundary value.
            const Real value = exact_solution(xf, yf);

            // Matrix contribution of the L2 projection.
            for (unsigned int i = 0; i != n_dofs; i++)
              for (unsigned int j = 0; j != n_dofs; j++)
                Ke(i, j) += JxW_face[qp] * penalty * phi_face[i][qp] * phi_face[j][qp];

            // Right-hand-side contribution of the L2
            // projection.
            for (unsigned int i = 0; i != n_dofs; i++)
              Fe(i) += JxW_face[qp] * penalty * value * phi_face[i][qp];
          }
        }
    }

    // We have now finished the quadrature point loop,
    // and have therefore applied all the boundary conditions.

    // If this assembly program were to be used on an adaptive mesh,
    // we would have to apply any hanging node constraint equations
    dof_map.constrain_element_matrix_and_vector(Ke, Fe, dof_indices);

    if (sc)
      sc->set_current_elem(*elem);

    // The element matrix and right-hand-side are now built
    // for this element.  Add them to the global matrix and
    // right-hand-side vector.  The  SparseMatrix::add_matrix()
    // and  NumericVector::add_vector() members do this for us.
    matrix.add_matrix(Ke, dof_indices);
    system.rhs->add_vector(Fe, dof_indices);
  }

  matrix.close();
}

assemble_poisson() [2/2]

void assemble_poisson	(	EquationSystems &	es,
		const std::string &	libmesh_dbg_varsystem_name
	)

Definition at line 212 of file introduction_ex5.C.

References libMesh::SparseMatrix< T >::add_matrix(), libMesh::NumericVector< T >::add_vector(), libMesh::QBase::build(), libMesh::FEGenericBase< OutputType >::build(), dim, exact_solution(), libMesh::System::get_dof_map(), libMesh::EquationSystems::get_mesh(), libMesh::EquationSystems::get_system(), libMesh::ImplicitSystem::get_system_matrix(), mesh, libMesh::MeshBase::mesh_dimension(), quad_type, libMesh::Real, libMesh::DenseVector< T >::resize(), libMesh::DenseMatrix< T >::resize(), libMesh::ExplicitSystem::rhs, and libMesh::THIRD.

{
  libmesh_assert_equal_to (system_name, "Poisson");

  const MeshBase & mesh = es.get_mesh();

  const unsigned int dim = mesh.mesh_dimension();

  LinearImplicitSystem & system = es.get_system<LinearImplicitSystem>("Poisson");

  const DofMap & dof_map = system.get_dof_map();

  FEType fe_type = dof_map.variable_type(0);

  // 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>.  Below, the
  // functionality of std::unique_ptr's is described more detailed in
  // the context of building quadrature rules.
  std::unique_ptr<FEBase> fe (FEBase::build(dim, fe_type));

  // Now this deviates from example 4.  we create a
  // 5th order quadrature rule of user-specified type
  // for numerical integration.  Note that not all
  // quadrature rules support this order.
  std::unique_ptr<QBase> qrule(QBase::build(quad_type, dim, THIRD));

  // Tell the finite element object to use our
  // quadrature rule.  Note that a std::unique_ptr<QBase> returns
  // a QBase* pointer to the object it handles with get().
  // However, using get(), the std::unique_ptr<QBase> qrule is
  // still in charge of this pointer. I.e., when qrule goes
  // out of scope, it will safely delete the QBase object it
  // points to.  This behavior may be overridden using
  // std::unique_ptr<Xyz>::release(), but is currently not
  // recommended.
  fe->attach_quadrature_rule (qrule.get());

  // Declare a special finite element object for
  // boundary integration.
  std::unique_ptr<FEBase> fe_face (FEBase::build(dim, fe_type));

  // As already seen in example 3, boundary integration
  // requires a quadrature rule.  Here, however,
  // we use the more convenient way of building this
  // rule at run-time using quad_type.  Note that one
  // could also have initialized the face quadrature rules
  // with the type directly determined from qrule, namely
  // through:
  // \verbatim
  // std::unique_ptr<QBase>  qface (QBase::build(qrule->type(),
  // dim-1,
  // THIRD));
  // \endverbatim
  // And again: using the std::unique_ptr<QBase> relaxes
  // the need to delete the object afterward,
  // they clean up themselves.
  std::unique_ptr<QBase>  qface (QBase::build(quad_type,
                                              dim-1,
                                              THIRD));

  // Tell the finite element object to use our
  // quadrature rule.  Note that a std::unique_ptr<QBase> returns
  // a QBase* pointer to the object it handles with get().
  // However, using get(), the std::unique_ptr<QBase> qface is
  // still in charge of this pointer. I.e., when qface goes
  // out of scope, it will safely delete the QBase object it
  // points to.  This behavior may be overridden using
  // std::unique_ptr<Xyz>::release(), but is not recommended.
  fe_face->attach_quadrature_rule (qface.get());

  // This is again identical to example 4, and not commented.
  const std::vector<Real> & JxW = fe->get_JxW();

  const std::vector<Point> & q_point = fe->get_xyz();

  const std::vector<std::vector<Real>> & phi = fe->get_phi();

  const std::vector<std::vector<RealGradient>> & dphi = fe->get_dphi();

  DenseMatrix<Number> Ke;
  DenseVector<Number> Fe;
  std::vector<dof_id_type> dof_indices;

  // The global system matrix
  SparseMatrix<Number> & matrix = system.get_system_matrix();

  // Now we will loop over all the elements in the mesh.
  // See example 3 for details.
  for (const auto & elem : mesh.active_local_element_ptr_range())
    {
      dof_map.dof_indices (elem, dof_indices);

      const unsigned int n_dofs =
        cast_int<unsigned int>(dof_indices.size());

      fe->reinit (elem);

      libmesh_assert_equal_to (n_dofs, phi.size());

      Ke.resize (n_dofs, n_dofs);

      Fe.resize (n_dofs);

      // Now loop over the quadrature points.  This handles
      // the numeric integration.  Note the slightly different
      // access to the QBase members!
      for (unsigned int qp=0; qp<qrule->n_points(); qp++)
        {
          // Add the matrix contribution
          for (unsigned int i=0; i != n_dofs; i++)
            for (unsigned int j=0; j != n_dofs; j++)
              Ke(i,j) += JxW[qp]*(dphi[i][qp]*dphi[j][qp]);

          // fxy is the forcing function for the Poisson equation.
          // In this case we set fxy to be a finite difference
          // Laplacian approximation to the (known) exact solution.
          //
          // We will use the second-order accurate FD Laplacian
          // approximation, which in 2D on a structured grid is
          //
          // u_xx + u_yy = (u(i-1,j) + u(i+1,j) +
          //                u(i,j-1) + u(i,j+1) +
          //                -4*u(i,j))/h^2
          //
          // Since the value of the forcing function depends only
          // on the location of the quadrature point (q_point[qp])
          // we will compute it here, outside of the i-loop
          const Real x = q_point[qp](0);
          const Real y = q_point[qp](1);
          const Real z = q_point[qp](2);
          const Real eps = 1.e-3;

          const Real uxx = (exact_solution(x-eps, y, z) +
                            exact_solution(x+eps, y, z) +
                            -2.*exact_solution(x, y, z))/eps/eps;

          const Real uyy = (exact_solution(x, y-eps, z) +
                            exact_solution(x, y+eps, z) +
                            -2.*exact_solution(x, y, z))/eps/eps;

          const Real uzz = (exact_solution(x, y, z-eps) +
                            exact_solution(x, y, z+eps) +
                            -2.*exact_solution(x, y, z))/eps/eps;

          const Real fxy = - (uxx + uyy + ((dim==2) ? 0. : uzz));


          // Add the RHS contribution
          for (unsigned int i=0; i != n_dofs; i++)
            Fe(i) += JxW[qp]*fxy*phi[i][qp];
        }

      // If this assembly program were to be used on an adaptive mesh,
      // we would have to apply any hanging node constraint equations
      // Call heterogenously_constrain_element_matrix_and_vector to impose
      // non-homogeneous Dirichlet BCs
      dof_map.heterogenously_constrain_element_matrix_and_vector (Ke, Fe, dof_indices);

      // The element matrix and right-hand-side are now built
      // for this element.  Add them to the global matrix and
      // right-hand-side vector.  The SparseMatrix::add_matrix()
      // and NumericVector::add_vector() members do this for us.
      matrix.add_matrix (Ke, dof_indices);
      system.rhs->add_vector    (Fe, dof_indices);

    } // end of element loop
}

exact_solution()

Real exact_solution	(	const Real	x,
		const Real	y,
		const Real	t
	)

This is the exact solution that we are trying to obtain.

We will solve

• (u_xx + u_yy) = f

and take a finite difference approximation using this function to get f. This is the well-known "method of manufactured solutions".

Definition at line 43 of file exact_solution.C.

Referenced by assemble_poisson(), and exact_solution_wrapper().

{
  static const Real pi = acos(-1.);

  return cos(.5*pi*x)*sin(.5*pi*y)*cos(.5*pi*z);
}

exact_solution_wrapper()

void exact_solution_wrapper	(	DenseVector< Number > &	output,
		const Point &	p,
		const Real
	)

Definition at line 89 of file introduction_ex5.C.

References exact_solution().

Referenced by main().

{
  output(0) = exact_solution(p(0), p(1), p(2));
}

main()

int main	(	int	argc,
		char **	argv
	)

Definition at line 103 of file introduction_ex5.C.

References libMesh::EquationSystems::add_system(), assemble_poisson(), libMesh::MeshTools::Generation::build_cube(), libMesh::command_line_next(), libMesh::default_solver_package(), exact_solution_wrapper(), libMesh::FIRST, libMesh::EquationSystems::get_system(), libMesh::HEX8, libMesh::TriangleWrapper::init(), libMesh::EquationSystems::init(), libMesh::INVALID_SOLVER_PACKAGE, mesh, libMesh::out, libMesh::EquationSystems::print_info(), libMesh::MeshBase::print_info(), quad_type, and libMesh::ExodusII_IO::write_equation_systems().

{
  // Initialize libMesh and any dependent libraries, like in example 2.
  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");

  // Check for proper usage.  The quadrature rule
  // must be given at run time.
  libmesh_error_msg_if(argc < 3,
                       "Usage: " << argv[0] << " -q <rule>\n"
                       "  where <rule> is one of QGAUSS, QSIMPSON, or QTRAP.");

  // 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;

  // Set the quadrature rule type that the user wants
  quad_type = Utility::string_to_enum<QuadratureType>
    (libMesh::command_line_next("-q", std::string("QGAUSS")));

  // Skip this 3D example if libMesh was compiled as 1D-only.
  libmesh_example_requires(3 <= LIBMESH_DIM, "3D support");

  // We use Dirichlet boundary conditions here
#ifndef LIBMESH_ENABLE_DIRICHLET
  libmesh_example_requires(false, "--enable-dirichlet");
#endif

  // The following is identical to example 4, and therefore
  // not commented.  Differences are mentioned when present.
  Mesh mesh(init.comm());

  // We will use a linear approximation space in this example,
  // hence 8-noded hexahedral elements are sufficient.  This
  // is different than example 4 where we used 27-noded
  // hexahedral elements to support a second-order approximation
  // space.
  MeshTools::Generation::build_cube (mesh,
                                     16, 16, 16,
                                     -1., 1.,
                                     -1., 1.,
                                     -1., 1.,
                                     HEX8);

  mesh.print_info();

  EquationSystems equation_systems (mesh);

  equation_systems.add_system<LinearImplicitSystem> ("Poisson");

  unsigned int u_var = equation_systems.get_system("Poisson").add_variable("u", FIRST);

  equation_systems.get_system("Poisson").attach_assemble_function (assemble_poisson);

  // Construct a Dirichlet boundary condition object

  // Indicate which boundary IDs we impose the BC on
  // We either build a line, a square or a cube, and
  // here we indicate boundaries covering each case
  std::set<boundary_id_type> boundary_ids {0,1,2,3,4,5};

  // Create an AnalyticFunction object that we use to project the BC
  // This function just calls the function exact_solution via exact_solution_wrapper
  AnalyticFunction<> exact_solution_object(exact_solution_wrapper);

#ifdef LIBMESH_ENABLE_DIRICHLET
  // In general, when reusing a system-indexed exact solution, we want
  // to use the default system-ordering constructor for
  // DirichletBoundary, so we demonstrate that here.  In this case,
  // though, we have only one variable, so system- and local-
  // orderings are the same.
  DirichletBoundary dirichlet_bc
    (boundary_ids, {u_var}, exact_solution_object);

  // We must add the Dirichlet boundary condition _before_
  // we call equation_systems.init()
  equation_systems.get_system("Poisson").get_dof_map().add_dirichlet_boundary(dirichlet_bc);
#endif

  equation_systems.init();

  equation_systems.print_info();

  equation_systems.get_system("Poisson").solve();

  // "Personalize" the output, with the
  // number of the quadrature rule appended.
  std::ostringstream f_name;
  f_name << "out_" << quad_type << ".e";

#ifdef LIBMESH_HAVE_EXODUS_API
  ExodusII_IO(mesh).write_equation_systems (f_name.str(),
                                            equation_systems);
#endif // #ifdef LIBMESH_HAVE_EXODUS_API

  // All done.
  return 0;
}

Variable Documentation

quad_type

QuadratureType quad_type =INVALID_Q_RULE

Definition at line 98 of file introduction_ex5.C.

Referenced by assemble_poisson(), and main().