LaplaceYoung Class Reference

A class which provides the residual and jacobian assembly functions for the Laplace-Young system of equations. More...

Inheritance diagram for LaplaceYoung:

Public Member Functions
	LaplaceYoung ()
virtual void	residual (const NumericVector< Number > &X, NumericVector< Number > &R, NonlinearImplicitSystem &S)
	Function which computes the residual.
virtual void	jacobian (const NumericVector< Number > &X, SparseMatrix< Number > &J, NonlinearImplicitSystem &S)
	Function which computes the jacobian.
virtual void	postcheck (const NumericVector< Number > &old_soln, NumericVector< Number > &search_direction, NumericVector< Number > &new_soln, bool &changed_search_direction, bool &changed_new_soln, NonlinearImplicitSystem &S)
	Function which performs a postcheck on the solution.

Private Attributes
Real	_kappa
Real	_sigma
Real	_gamma

Detailed Description

A class which provides the residual and jacobian assembly functions for the Laplace-Young system of equations.

Definition at line 78 of file miscellaneous_ex3.C.

Constructor & Destructor Documentation

LaplaceYoung()

LaplaceYoung::LaplaceYoung ( )

inline

Definition at line 84 of file miscellaneous_ex3.C.

                 :
    _kappa(1.),
    _sigma(0.2),
    _gamma(1.0)
  {}

Member Function Documentation

jacobian()

void LaplaceYoung::jacobian ( const NumericVector< Number > &  X, SparseMatrix< Number > &  J, NonlinearImplicitSystem &  S  )

virtual

Function which computes the jacobian.

Definition at line 456 of file miscellaneous_ex3.C.

{
  // Get a reference to the equation system.
  EquationSystems & es = sys.get_equation_systems();
 
  // 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 NonlinearImplicitSystem we are solving
  NonlinearImplicitSystem & system =
    es.get_system<NonlinearImplicitSystem>("Laplace-Young");
 
  // 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.
  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);
 
  // Here we define some references to cell-specific data that
  // will be used to assemble the linear system.
  // We begin with 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();
 
  // 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 Jacobian element matrix.
  // Following basic finite element terminology we will denote these
  // "Ke". More detail is in example 3.
  DenseMatrix<Number> Ke;
 
  // 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 active elements in the mesh which
  // are local to this processor.
  // We will compute the element Jacobian contribution.
  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 Jacobian 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());
      Ke.resize (n_dofs, n_dofs);
 
      // Now we will build the element Jacobian.  This involves
      // a double loop to integrate the test functions (i) against
      // the trial functions (j). Note that the Jacobian depends
      // on the current solution x, which we access using the soln
      // vector.
      //
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          Gradient grad_u;
 
          for (unsigned int i=0; i<n_dofs; i++)
            grad_u += dphi[i][qp]*soln(dof_indices[i]);
 
          const Number
            sa = 1. + grad_u*grad_u,
            K  = 1. / std::sqrt(sa),
            dK = -K / sa;
 
          for (unsigned int i=0; i<n_dofs; i++)
            for (unsigned int j=0; j<n_dofs; j++)
              Ke(i,j) += JxW[qp]*(
                                  K * (dphi[i][qp]*dphi[j][qp]) +
                                  dK * (grad_u*dphi[j][qp]) * (grad_u*dphi[i][qp]) +
                                  _kappa * phi[i][qp] * phi[j][qp]
                                  );
        }
 
      dof_map.constrain_element_matrix (Ke, dof_indices);
 
      // Add the element matrix to the system Jacobian.
      jacobian.add_matrix (Ke, dof_indices);
    }
 
  // That's it.
}

References _kappa, libMesh::FEGenericBase< OutputType >::build(), dim, libMesh::FIFTH, libMesh::System::get_dof_map(), libMesh::System::get_equation_systems(), libMesh::EquationSystems::get_mesh(), libMesh::EquationSystems::get_system(), jacobian(), mesh, libMesh::MeshBase::mesh_dimension(), libMesh::QBase::n_points(), and libMesh::DenseMatrix< T >::resize().

Referenced by jacobian().

postcheck()

void LaplaceYoung::postcheck ( const NumericVector< Number > &  old_soln, NumericVector< Number > &  search_direction, NumericVector< Number > &  new_soln, bool &  changed_search_direction, bool &  changed_new_soln, NonlinearImplicitSystem &  S  )

virtual

Function which performs a postcheck on the solution.

In this example, we take a "damped" Newton step defined by:

u_new = u_old + gamma*delta_u

where delta_u is the search direction, and 0 < gamma <= 1 is a damping parameter. gamma=1 corresponds to a full Newton step.

This is really for demonstration purposes only, as it just degrades the rate of nonlinear convergence in this particular example.

Definition at line 584 of file miscellaneous_ex3.C.

{
  // Back up along the search direction by some amount.  Since Newton
  // already works well for this problem, the only affect of this is
  // to degrade the rate of convergence.
  //
  // The minus sign is due to the sign of the "search_direction"
  // vector which comes in from the Newton solve. The RHS of the
  // nonlinear system, i.e. the residual, is generally not multiplied
  // by -1, so the solution vector, i.e. the search_direction, has a
  // factor -1.
  if (_gamma != 1.0)
    {
      new_soln = old_soln;
      new_soln.add(-_gamma, search_direction);
      changed_new_soln = true;
    }
  else
    changed_new_soln = false;
}

References _gamma, and libMesh::NumericVector< T >::add().

residual()

void LaplaceYoung::residual ( const NumericVector< Number > &  X, NumericVector< Number > &  R, NonlinearImplicitSystem &  S  )

virtual

Function which computes the residual.

Definition at line 286 of file miscellaneous_ex3.C.

{
  EquationSystems & es = sys.get_equation_systems();
 
  // 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();
  libmesh_assert_equal_to (dim, 2);
 
  // Get a reference to the NonlinearImplicitSystem we are solving
  NonlinearImplicitSystem & system =
    es.get_system<NonlinearImplicitSystem>("Laplace-Young");
 
  // 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.
  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.
  // We begin with 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();
 
  // 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 residual contributions
  DenseVector<Number> Re;
 
  // 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 active elements in the mesh which
  // are local to this processor.
  // We will compute the element residual.
  residual.zero();
 
  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);
 
      // 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());
      Re.resize (n_dofs);
 
      // Now we will build the residual. This involves
      // the construction of the matrix K and multiplication of it
      // with the current solution x. We rearrange this into two loops:
      // In the first, we calculate only the contribution of
      // K_ij*x_j which is independent of the row i. In the second loops,
      // we multiply with the row-dependent part and add it to the element
      // residual.
 
      for (unsigned int qp=0; qp<qrule.n_points(); qp++)
        {
          Number u = 0;
          Gradient grad_u;
 
          for (unsigned int j=0; j<n_dofs; j++)
            {
              u      += phi[j][qp]*soln(dof_indices[j]);
              grad_u += dphi[j][qp]*soln(dof_indices[j]);
            }
 
          const Number K = 1./std::sqrt(1. + grad_u*grad_u);
 
          for (unsigned int i=0; i<n_dofs; i++)
            Re(i) += JxW[qp]*(
                              K*(dphi[i][qp]*grad_u) +
                              _kappa*phi[i][qp]*u
                              );
        }
 
      // At this point the interior element integration has
      // been completed.  However, we have not yet addressed
      // boundary conditions.
 
      // The following loops 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();
 
            // Compute the shape function values on the element face.
            fe_face->reinit(elem, side);
 
            // Loop over the face quadrature points for integration.
            for (unsigned int qp=0; qp<qface.n_points(); qp++)
              {
                // This is the right-hand-side contribution (f),
                // which has to be subtracted from the current residual
                for (unsigned int i=0; i<n_dofs; i++)
                  Re(i) -= JxW_face[qp]*_sigma*phi_face[i][qp];
              }
          }
 
      dof_map.constrain_element_vector (Re, dof_indices);
      residual.add_vector (Re, dof_indices);
    }
 
  // That's it.
}

References _kappa, _sigma, libMesh::FEGenericBase< OutputType >::build(), dim, libMesh::FIFTH, libMesh::System::get_dof_map(), libMesh::System::get_equation_systems(), libMesh::EquationSystems::get_mesh(), libMesh::EquationSystems::get_system(), mesh, libMesh::MeshBase::mesh_dimension(), libMesh::QBase::n_points(), residual(), libMesh::DenseVector< T >::resize(), and libMesh::DenseVector< T >::zero().

Referenced by residual().

Member Data Documentation

_gamma

Real LaplaceYoung::_gamma

private

Definition at line 129 of file miscellaneous_ex3.C.

Referenced by postcheck().

_kappa

Real LaplaceYoung::_kappa

private

Definition at line 125 of file miscellaneous_ex3.C.

Referenced by jacobian(), and residual().

_sigma

Real LaplaceYoung::_sigma

private

Definition at line 126 of file miscellaneous_ex3.C.

Referenced by residual().

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The documentation for this class was generated from the following file:

• examples/miscellaneous/miscellaneous_ex3/miscellaneous_ex3.C