FESeries::Legendre< dim, spacedim > Class Template Reference

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

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

Public Types
using	CoefficientType = double

Public Member Functions
	Legendre (const std::vector< unsigned int > &n_coefficients_per_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection)
	Legendre (const unsigned int n_coefficients_per_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 > &legendre_coefficients)
unsigned int	get_n_coefficients_per_direction (const unsigned int index) const
void	precalculate_all_transformation_matrices ()
template<class Archive >
void	save_transformation_matrices (Archive &ar, const unsigned int version)
template<class Archive >
void	load_transformation_matrices (Archive &ar, const unsigned int version)
bool	operator== (const Legendre< dim, spacedim > &legendre) const
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
const std::vector< unsigned int >	n_coefficients_per_direction
SmartPointer< const hp::FECollection< dim, spacedim > >	fe_collection
const hp::QCollection< dim >	q_collection
std::vector< FullMatrix< CoefficientType > >	legendre_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::Legendre< dim, spacedim >

A class to calculate expansion of a scalar FE field into series of Legendre functions on a reference element.

Legendre functions are solutions to Legendre's differential equation

\[ \frac{d}{dx}\left([1-x^2] \frac{d}{dx} P_n(x)\right) + n[n+1] P_n(x) = 0 \]

and can be expressed using Rodrigues' formula

\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[x^2-1]^n. \]

These polynomials are orthogonal with respect to the \( L^2 \) inner product on the interval \( [-1;1] \)

\[ \int_{-1}^1 P_m(x) P_n(x) = \frac{2}{2n + 1} \delta_{mn} \]

and are complete. A family of \( L^2 \)-orthogonal polynomials on \( [0;1] \) can be constructed via

\[ \widetilde P_m = \sqrt{2} P_m(2x-1). \]

An arbitrary scalar FE field on the reference element \( [0;1] \) can be expanded in the complete orthogonal basis as

\[ u(x) = \sum_{m} c_m \widetilde P_{m}(x). \]

From the orthogonality property of the basis, it follows that

\[ c_m = \frac{2m+1}{2} \int_0^1 u(x) \widetilde P_m(x) dx . \]

This class calculates coefficients \( c_{\bf k} \) using \( dim \)-dimensional Legendre polynomials constructed from \( \widetilde P_m(x) \) using tensor product rule.

Author

Denis Davydov, 2016.

Definition at line 259 of file fe_series.h.

Member Typedef Documentation

CoefficientType

template<int dim, int spacedim = dim>

using FESeries::Legendre< dim, spacedim >::CoefficientType = double

Definition at line 262 of file fe_series.h.

Constructor & Destructor Documentation

Legendre() [1/2]

template<int dim, int spacedim>

FESeries::Legendre< dim, spacedim >::Legendre	(	const std::vector< unsigned int > &	n_coefficients_per_direction,
		const hp::FECollection< dim, spacedim > &	fe_collection,
		const hp::QCollection< dim > &	q_collection
	)

Constructor that initializes all required data structures.

The n_coefficients_per_direction defines the number of coefficients 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 198 of file fe_series_legendre.cc.

Legendre() [2/2]

template<int dim, int spacedim>

FESeries::Legendre< dim, spacedim >::Legendre	(	const unsigned int	n_coefficients_per_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 coefficients 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 222 of file fe_series_legendre.cc.

Member Function Documentation

calculate()

template<int dim, int spacedim>

template<typename Number >

void FESeries::Legendre< dim, spacedim >::calculate	(	const ::Vector< Number > &	local_dof_values,
		const unsigned int	cell_active_fe_index,
		Table< dim, CoefficientType > &	legendre_coefficients
	)

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

Definition at line 281 of file fe_series_legendre.cc.

get_n_coefficients_per_direction()

template<int dim, int spacedim>

unsigned int FESeries::Legendre< dim, spacedim >::get_n_coefficients_per_direction

(

const unsigned int

index

)

const

Return the number of coefficients in each coordinate direction for the finite element associated with index in the provided hp::FECollection.

Definition at line 270 of file fe_series_legendre.cc.

precalculate_all_transformation_matrices()

template<int dim, int spacedim>

void FESeries::Legendre< dim, spacedim >::precalculate_all_transformation_matrices

Calculate all transformation matrices to transfer the finite element solution to the series expansion representation.

These matrices will be generated on demand by calling calculate() and stored for recurring purposes. Usually, this operation consumes a lot of workload. With this function, all matrices will be calculated in advance. This way, we can separate their costly generation from the actual application.

Definition at line 251 of file fe_series_legendre.cc.

save_transformation_matrices()

template<int dim, int spacedim = dim>

template<class Archive >

void FESeries::Legendre< dim, spacedim >::save_transformation_matrices	(	Archive &	ar,
		const unsigned int	version
	)

Write all transformation matrices of this object to a stream for the purpose of serialization.

Since any of its transformation matrices has to be generated only once for a given scenario, it is common practice to determine them in advance calling precalculate_all_transformation_matrices() and keep them via serialization.

load_transformation_matrices()

template<int dim, int spacedim = dim>

template<class Archive >

void FESeries::Legendre< dim, spacedim >::load_transformation_matrices	(	Archive &	ar,
		const unsigned int	version
	)

Read all transformation matrices from a stream and recover them for this object.

operator==()

template<int dim, int spacedim>

bool FESeries::Legendre< dim, spacedim >::operator== ( const Legendre< dim, spacedim > &  legendre ) const

inline

Test for equality of two series expansion objects.

Definition at line 238 of file fe_series_legendre.cc.

Member Data Documentation

n_coefficients_per_direction

template<int dim, int spacedim = dim>

const std::vector<unsigned int> FESeries::Legendre< dim, spacedim >::n_coefficients_per_direction

private

Number of coefficients in each direction for each finite element in the registered hp::FECollection.

Definition at line 352 of file fe_series.h.

fe_collection

template<int dim, int spacedim = dim>

SmartPointer<const hp::FECollection<dim, spacedim> > FESeries::Legendre< dim, spacedim >::fe_collection

private

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

Definition at line 357 of file fe_series.h.

q_collection

template<int dim, int spacedim = dim>

const hp::QCollection<dim> FESeries::Legendre< dim, spacedim >::q_collection

private

hp::QCollection used in calculation of transformation matrices.

Definition at line 362 of file fe_series.h.

legendre_transform_matrices

template<int dim, int spacedim = dim>

std::vector<FullMatrix<CoefficientType> > FESeries::Legendre< dim, spacedim >::legendre_transform_matrices

private

Transformation matrices for each FiniteElement.

Definition at line 367 of file fe_series.h.

unrolled_coefficients

template<int dim, int spacedim = dim>

std::vector<CoefficientType> FESeries::Legendre< dim, spacedim >::unrolled_coefficients

private

Auxiliary vector to store unrolled coefficients.

Definition at line 372 of file fe_series.h.

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

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