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

Inheritance diagram for FESeries::Fourier< dim, spacedim >:

Public Member Functions
	Fourier (const unsigned int size_in_each_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection)

template<typename Number >
void	calculate (const ::Vector< Number > &local_dof_values, const unsigned int cell_active_fe_index, Table< dim, CoefficientType > &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 (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Attributes
SmartPointer< const hp::FECollection< dim, spacedim > >	fe_collection

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

Table< dim, Tensor< 1, dim > >	k_vectors

std::vector< FullMatrix< CoefficientType > >	fourier_transform_matrices

std::vector< CoefficientType >	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, int spacedim = dim>
class FESeries::Fourier< dim, spacedim >

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 91 of file fe_series.h.

Constructor & Destructor Documentation

◆ Fourier()

template<int dim, int spacedim>

FESeries::Fourier< dim, spacedim >::Fourier	(	const unsigned int	size_in_each_direction,
		const hp::FECollection< dim, spacedim > &	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 170 of file fe_series_fourier.cc.