Reference documentation for deal.II version Git 3ecc854 2017-02-27 14:34:24 -0500
Public Member Functions | Private Member Functions | Private Attributes | List of all members
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 &&)
virtual ~Subscriptor ()
Subscriptoroperator= (const Subscriptor &)
Subscriptoroperator= (Subscriptor &&)
void subscribe (const char *identifier=0) const
void unsubscribe (const char *identifier=0) 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, char *arg2, std::string &arg3)
static::ExceptionBase & ExcNoSubscriber (char *arg1, char *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 \( \bf k \) .

Denis Davydov, 2016.

Definition at line 87 of file fe_series.h.

Constructor & Destructor Documentation

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

Member Function Documentation

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

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

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

Member Data Documentation

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

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

Definition at line 114 of file fe_series.h.

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

hp::QCollection used in calculation of transformation matrices.

Definition at line 119 of file fe_series.h.

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

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

Definition at line 130 of file fe_series.h.

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

Transformation matrices for each FiniteElement.

Definition at line 135 of file fe_series.h.

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

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: