FESeries::Fourier< dim > Class Template Reference

#include <deal.II/fe/fe_series.h>

Inheritance diagram for FESeries::Fourier< dim >:

Public Member Functions
	Fourier (const unsigned int size_in_each_direction, const hp::FECollection< dim > &fe_collection, const hp::QCollection< dim > &q_collection)
void	calculate (const ::Vector< double > &local_dof_values, const unsigned int cell_active_fe_index, Table< dim, std::complex< double > > &fourier_coefficients)
Public Member Functions inherited from Subscriptor
	Subscriptor ()
	Subscriptor (const Subscriptor &)
	Subscriptor (Subscriptor &&) noexcept
virtual	~Subscriptor ()
Subscriptor &	operator= (const Subscriptor &)
Subscriptor &	operator= (Subscriptor &&) noexcept
void	subscribe (const char *identifier=nullptr) const
void	unsubscribe (const char *identifier=nullptr) const
unsigned int	n_subscriptions () const
void	list_subscribers () const
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Member Functions
void	ensure_existence (const unsigned int fe_index)

Private Attributes
SmartPointer< const hp::FECollection< dim > >	fe_collection
SmartPointer< const hp::QCollection< dim > >	q_collection
Table< dim, Tensor< 1, dim > >	k_vectors
std::vector< FullMatrix< std::complex< double > > >	fourier_transform_matrices
std::vector< std::complex< double > >	unrolled_coefficients

Additional Inherited Members
Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)
static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Detailed Description

template<int dim>
class FESeries::Fourier< dim >

A class to calculate expansion of a scalar FE field into Fourier series on a reference element. The exponential form of the Fourier series is based on completeness and Hermitian orthogonality of the set of exponential functions \( \phi_{\bf k}({\bf x}) = \exp(2 \pi i\, {\bf k} \cdot {\bf x})\). For example in 1D the L2-orthogonality condition reads

\[ \int_0^1 \phi_k(x) \phi_l^\ast(x) dx=\delta_{kl}. \]

Note that \( \phi_{\bf k} = \phi_{-\bf k}^\ast \).

The arbitrary scalar FE field on the reference element can be expanded in the complete orthogonal exponential basis as

\[ u({\bf x}) = \sum_{\bf k} c_{\bf k} \phi_{\bf k}({\bf x}). \]

From the orthogonality property of the basis, it follows that

\[ c_{\bf k} = \int_{[0,1]^d} u({\bf x}) \phi_{\bf k}^\ast ({\bf x}) d{\bf x}\,. \]

It is this complex-valued expansion coefficients, that are calculated by this class. Note that \( u({\bf x}) = \sum_i u_i N_i({\bf x})\), where \( N_i({\bf x}) \) are real-valued FiniteElement shape functions. Consequently \( c_{\bf k} \equiv c_{-\bf k}^\ast \) and we only need to compute \( c_{\bf k} \) for positive indices \( \bf k \) .

Author

Denis Davydov, 2016.

Definition at line 87 of file fe_series.h.

Constructor & Destructor Documentation

Fourier()

template<int dim>

FESeries::Fourier< dim >::Fourier	(	const unsigned int	size_in_each_direction,
		const hp::FECollection< dim > &	fe_collection,
		const hp::QCollection< dim > &	q_collection
	)

A non-default constructor. The size_in_each_direction defines the number of modes in each direction, fe_collection is the hp::FECollection for which expansion will be used and q_collection is the hp::QCollection used to integrate the expansion for each FiniteElement in fe_collection.

Definition at line 74 of file fe_series.cc.

Member Function Documentation

calculate()

template<int dim>

void FESeries::Fourier< dim >::calculate	(	const ::Vector< double > &	local_dof_values,
		const unsigned int	cell_active_fe_index,
		Table< dim, std::complex< double > > &	fourier_coefficients
	)

Calculate fourier_coefficients of the cell vector field given by local_dof_values corresponding to FiniteElement with cell_active_fe_index .

Definition at line 87 of file fe_series.cc.

ensure_existence()

template<int dim>

void FESeries::Fourier< dim >::ensure_existence ( const unsigned int  fe_index )

private

Ensure that the transformation matrix for FiniteElement index fe_index is calculated. If not, calculate it.

Member Data Documentation

fe_collection

template<int dim>

SmartPointer<const hp::FECollection<dim> > FESeries::Fourier< dim >::fe_collection

private

hp::FECollection for which transformation matrices will be calculated.

Definition at line 114 of file fe_series.h.

q_collection

template<int dim>

SmartPointer<const hp::QCollection<dim> > FESeries::Fourier< dim >::q_collection

private

hp::QCollection used in calculation of transformation matrices.

Definition at line 119 of file fe_series.h.

k_vectors

template<int dim>

Table<dim, Tensor<1,dim> > FESeries::Fourier< dim >::k_vectors

private

Angular frequencies \( 2 \pi {\bf k} \) .

Definition at line 130 of file fe_series.h.

fourier_transform_matrices

template<int dim>

std::vector<FullMatrix<std::complex<double> > > FESeries::Fourier< dim >::fourier_transform_matrices

private

Transformation matrices for each FiniteElement.

Definition at line 135 of file fe_series.h.

unrolled_coefficients

template<int dim>

std::vector<std::complex<double> > FESeries::Fourier< dim >::unrolled_coefficients

private

Auxiliary vector to store unrolled coefficients.

Definition at line 140 of file fe_series.h.

---

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

• deal.II/fe/fe_series.h
• source/fe/fe_series.cc