Reference documentation for deal.II version Git 350fd17bfc 2020-02-26 19:16:00 +0100
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Public Member Functions | Private Attributes | List of all members
FESeries::Legendre< dim, spacedim > Class Template Reference

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

Inheritance diagram for FESeries::Legendre< dim, spacedim >:
[legend]

Public Member Functions

 Legendre (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 > &legendre_coefficients)
 
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 ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (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 unsigned int N
 
SmartPointer< const hp::FECollection< dim, spacedim > > fe_collection
 
SmartPointer< 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 ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (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 229 of file fe_series.h.

Constructor & Destructor Documentation

◆ Legendre()

template<int dim, int spacedim>
FESeries::Legendre< dim, spacedim >::Legendre ( 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 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 191 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 236 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 219 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 207 of file fe_series_legendre.cc.

Member Data Documentation

◆ N

template<int dim, int spacedim = dim>
const unsigned int FESeries::Legendre< dim, spacedim >::N
private

Number of coefficients in each direction

Definition at line 300 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 305 of file fe_series.h.

◆ q_collection

template<int dim, int spacedim = dim>
SmartPointer<const hp::QCollection<dim> > FESeries::Legendre< dim, spacedim >::q_collection
private

hp::QCollection used in calculation of transformation matrices.

Definition at line 310 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 315 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 320 of file fe_series.h.


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