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//
// Copyright (C) 2005 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, 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.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__fe_poly_tensor_h
#define dealii__fe_poly_tensor_h
#include <deal.II/lac/full_matrix.h>
#include <deal.II/fe/fe.h>
#include <deal.II/base/derivative_form.h>
#include <deal.II/base/quadrature.h>
DEAL_II_NAMESPACE_OPEN
/**
* This class gives a unified framework for the implementation of
* FiniteElement classes based on Tensor valued polynomial spaces like
* PolynomialsBDM and PolynomialsRaviartThomas.
*
* Every class that implements following function can be used as template
* parameter PolynomialType.
*
* @code
* void compute (const Point<dim> &unit_point,
* std::vector<Tensor<1,dim> > &values,
* std::vector<Tensor<2,dim> > &grads,
* std::vector<Tensor<3,dim> > &grad_grads) const;
* @endcode
*
* In many cases, the node functionals depend on the shape of the mesh cell,
* since they evaluate normal or tangential components on the faces. In order
* to allow for a set of transformations, the variable #mapping_type has been
* introduced. It should also be set in the constructor of a derived class.
*
* This class is not a fully implemented FiniteElement class, but implements
* some common features of vector valued elements based on vector valued
* polynomial classes. What's missing here in particular is information on the
* topological location of the node values.
*
* For more information on the template parameter <tt>spacedim</tt>, see the
* documentation for the class Triangulation.
*
* <h3>Deriving classes</h3>
*
* Any derived class must decide on the polynomial space to use. This
* polynomial space should be implemented simply as a set of vector valued
* polynomials like PolynomialsBDM and PolynomialsRaviartThomas. In order to
* facilitate this implementation, the basis of this space may be arbitrary.
*
* <h4>Determining the correct basis</h4>
*
* In most cases, the set of desired node values $N_i$ and the basis functions
* $v_j$ will not fulfill the interpolation condition $N_i(v_j) =
* \delta_{ij}$.
*
* The use of the member data #inverse_node_matrix allows to compute the basis
* $v_j$ automatically, after the node values for each original basis function
* of the polynomial space have been computed.
*
* Therefore, the constructor of a derived class should have a structure like
* this (example for interpolation in support points):
*
* @code
* fill_support_points();
*
* const unsigned int n_dofs = this->dofs_per_cell;
* FullMatrix<double> N(n_dofs, n_dofs);
*
* for (unsigned int i=0;i<n_dofs;++i)
* {
* const Point<dim>& p = this->unit_support_point[i];
*
* for (unsigned int j=0;j<n_dofs;++j)
* for (unsigned int d=0;d<dim;++d)
* N(i,j) += node_vector[i][d]
* * this->shape_value_component(j, p, d);
* }
*
* this->inverse_node_matrix.reinit(n_dofs, n_dofs);
* this->inverse_node_matrix.invert(N);
* @endcode
*
* @note The matrix #inverse_node_matrix should have dimensions zero before
* this piece of code is executed. Only then, shape_value_component() will
* return the raw polynomial <i>j</i> as defined in the polynomial space
* PolynomialType.
*
* <h4>Setting the transformation</h4>
*
* In most cases, vector valued basis functions must be transformed when
* mapped from the reference cell to the actual grid cell. These
* transformations can be selected from the set MappingType and stored in
* #mapping_type. Therefore, each constructor should contain a line like:
* @code
* this->mapping_type = this->mapping_none;
* @endcode
*
* @see PolynomialsBDM, PolynomialsRaviartThomas
* @ingroup febase
* @author Guido Kanschat
* @date 2005
*/
template <class PolynomialType, int dim, int spacedim=dim>
class FE_PolyTensor : public FiniteElement<dim,spacedim>
{
public:
/**
* Constructor.
*
* @arg @c degree: constructor argument for poly. May be different from @p
* fe_data.degree.
*/
FE_PolyTensor (const unsigned int degree,
const FiniteElementData<dim> &fe_data,
const std::vector<bool> &restriction_is_additive_flags,
const std::vector<ComponentMask> &nonzero_components);
// for documentation, see the FiniteElement base class
virtual
UpdateFlags
requires_update_flags (const UpdateFlags update_flags) const;
/**
* Compute the (scalar) value of shape function @p i at the given quadrature
* point @p p. Since the elements represented by this class are vector
* valued, there is no such scalar value and the function therefore throws
* an exception.
*/
virtual double shape_value (const unsigned int i,
const Point<dim> &p) const;
// documentation inherited from the base class
virtual double shape_value_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Compute the gradient of (scalar) shape function @p i at the given
* quadrature point @p p. Since the elements represented by this class are
* vector valued, there is no such scalar value and the function therefore
* throws an exception.
*/
virtual Tensor<1,dim> shape_grad (const unsigned int i,
const Point<dim> &p) const;
// documentation inherited from the base class
virtual Tensor<1,dim> shape_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
/**
* Compute the Hessian of (scalar) shape function @p i at the given
* quadrature point @p p. Since the elements represented by this class are
* vector valued, there is no such scalar value and the function therefore
* throws an exception.
*/
virtual Tensor<2,dim> shape_grad_grad (const unsigned int i,
const Point<dim> &p) const;
// documentation inherited from the base class
virtual Tensor<2,dim> shape_grad_grad_component (const unsigned int i,
const Point<dim> &p,
const unsigned int component) const;
protected:
/**
* The mapping type to be used to map shape functions from the reference
* cell to the mesh cell.
*/
MappingType mapping_type;
/* NOTE: The following function has its definition inlined into the class declaration
because we otherwise run into a compiler error with MS Visual Studio. */
virtual
typename FiniteElement<dim,spacedim>::InternalDataBase *
get_data(const UpdateFlags update_flags,
const Mapping<dim,spacedim> &/*mapping*/,
const Quadrature<dim> &quadrature,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &/*output_data*/) const
{
// generate a new data object and
// initialize some fields
InternalData *data = new InternalData;
data->update_each = requires_update_flags(update_flags);
const unsigned int n_q_points = quadrature.size();
// some scratch arrays
std::vector<Tensor<1,dim> > values(0);
std::vector<Tensor<2,dim> > grads(0);
std::vector<Tensor<3,dim> > grad_grads(0);
std::vector<Tensor<4,dim> > third_derivatives(0);
std::vector<Tensor<5,dim> > fourth_derivatives(0);
if (update_flags & (update_values | update_gradients | update_hessians) )
data->sign_change.resize (this->dofs_per_cell);
// initialize fields only if really
// necessary. otherwise, don't
// allocate memory
if (update_flags & update_values)
{
values.resize (this->dofs_per_cell);
data->shape_values.reinit (this->dofs_per_cell, n_q_points);
if (mapping_type != mapping_none)
data->transformed_shape_values.resize (n_q_points);
}
if (update_flags & update_gradients)
{
grads.resize (this->dofs_per_cell);
data->shape_grads.reinit (this->dofs_per_cell, n_q_points);
data->transformed_shape_grads.resize (n_q_points);
if ( (mapping_type == mapping_raviart_thomas)
||
(mapping_type == mapping_piola)
||
(mapping_type == mapping_nedelec)
||
(mapping_type == mapping_contravariant))
data->untransformed_shape_grads.resize(n_q_points);
}
if (update_flags & update_hessians)
{
grad_grads.resize (this->dofs_per_cell);
data->shape_grad_grads.reinit (this->dofs_per_cell, n_q_points);
data->transformed_shape_hessians.resize (n_q_points);
if ( mapping_type != mapping_none )
data->untransformed_shape_hessian_tensors.resize(n_q_points);
}
// Compute shape function values
// and derivatives and hessians on
// the reference cell.
// Make sure, that for the
// node values N_i holds
// N_i(v_j)=\delta_ij for all basis
// functions v_j
if (update_flags & (update_values | update_gradients))
for (unsigned int k=0; k<n_q_points; ++k)
{
poly_space.compute(quadrature.point(k),
values, grads, grad_grads,
third_derivatives,
fourth_derivatives);
if (update_flags & update_values)
{
if (inverse_node_matrix.n_cols() == 0)
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
data->shape_values[i][k] = values[i];
else
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
{
Tensor<1,dim> add_values;
for (unsigned int j=0; j<this->dofs_per_cell; ++j)
add_values += inverse_node_matrix(j,i) * values[j];
data->shape_values[i][k] = add_values;
}
}
if (update_flags & update_gradients)
{
if (inverse_node_matrix.n_cols() == 0)
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
data->shape_grads[i][k] = grads[i];
else
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
{
Tensor<2,dim> add_grads;
for (unsigned int j=0; j<this->dofs_per_cell; ++j)
add_grads += inverse_node_matrix(j,i) * grads[j];
data->shape_grads[i][k] = add_grads;
}
}
if (update_flags & update_hessians)
{
if (inverse_node_matrix.n_cols() == 0)
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
data->shape_grad_grads[i][k] = grad_grads[i];
else
for (unsigned int i=0; i<this->dofs_per_cell; ++i)
{
Tensor<3,dim> add_grad_grads;
for (unsigned int j=0; j<this->dofs_per_cell; ++j)
add_grad_grads += inverse_node_matrix(j,i) * grad_grads[j];
data->shape_grad_grads[i][k] = add_grad_grads;
}
}
}
return data;
}
virtual
void
fill_fe_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const CellSimilarity::Similarity cell_similarity,
const Quadrature<dim> &quadrature,
const Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
virtual
void
fill_fe_face_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const Quadrature<dim-1> &quadrature,
const Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
virtual
void
fill_fe_subface_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
const unsigned int face_no,
const unsigned int sub_no,
const Quadrature<dim-1> &quadrature,
const Mapping<dim,spacedim> &mapping,
const typename Mapping<dim,spacedim>::InternalDataBase &mapping_internal,
const dealii::internal::FEValues::MappingRelatedData<dim, spacedim> &mapping_data,
const typename FiniteElement<dim,spacedim>::InternalDataBase &fe_internal,
dealii::internal::FEValues::FiniteElementRelatedData<dim, spacedim> &output_data) const;
/**
* Fields of cell-independent data for FE_PolyTensor. Stores the values of
* the shape functions and their derivatives on the reference cell for later
* use.
*
* All tables are organized in a way, that the value for shape function
* <i>i</i> at quadrature point <i>k</i> is accessed by indices
* <i>(i,k)</i>.
*/
class InternalData : public FiniteElement<dim,spacedim>::InternalDataBase
{
public:
/**
* Array with shape function values in quadrature points. There is one row
* for each shape function, containing values for each quadrature point.
*/
Table<2,Tensor<1,dim> > shape_values;
/**
* Array with shape function gradients in quadrature points. There is one
* row for each shape function, containing values for each quadrature
* point.
*/
Table<2,DerivativeForm<1, dim, spacedim> > shape_grads;
/**
* Array with shape function hessians in quadrature points. There is one
* row for each shape function, containing values for each quadrature
* point.
*/
Table<2,DerivativeForm<2, dim, spacedim> > shape_grad_grads;
/**
* Scratch arrays for intermediate computations
*/
mutable std::vector<double> sign_change;
mutable std::vector<Tensor<1, spacedim> > transformed_shape_values;
// for shape_gradient computations
mutable std::vector<Tensor<2, spacedim > > transformed_shape_grads;
mutable std::vector<Tensor<2, dim > > untransformed_shape_grads;
// for shape_hessian computations
mutable std::vector<Tensor<3, spacedim > > transformed_shape_hessians;
mutable std::vector<Tensor<3, dim > > untransformed_shape_hessian_tensors;
};
/**
* The polynomial space. Its type is given by the template parameter
* PolynomialType.
*/
PolynomialType poly_space;
/**
* The inverse of the matrix <i>a<sub>ij</sub></i> of node values
* <i>N<sub>i</sub></i> applied to polynomial <i>p<sub>j</sub></i>. This
* matrix is used to convert polynomials in the "raw" basis provided in
* #poly_space to the basis dual to the node functionals on the reference
* cell.
*
* This object is not filled by FE_PolyTensor, but is a chance for a derived
* class to allow for reorganization of the basis functions. If it is left
* empty, the basis in #poly_space is used.
*/
FullMatrix<double> inverse_node_matrix;
/**
* If a shape function is computed at a single point, we must compute all of
* them to apply #inverse_node_matrix. In order to avoid too much overhead,
* we cache the point and the function values for the next evaluation.
*/
mutable Point<dim> cached_point;
/**
* Cached shape function values after call to shape_value_component().
*/
mutable std::vector<Tensor<1,dim> > cached_values;
/**
* Cached shape function gradients after call to shape_grad_component().
*/
mutable std::vector<Tensor<2,dim> > cached_grads;
/**
* Cached second derivatives of shape functions after call to
* shape_grad_grad_component().
*/
mutable std::vector<Tensor<3,dim> > cached_grad_grads;
};
DEAL_II_NAMESPACE_CLOSE
#endif
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