Intrepid
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Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar > Class Template Reference

Implements Hermite interpolant basis of degree n on the reference Line cell. The basis has cardinality 2n and spans a COMPLETE linear polynomial space. More...

#include <Intrepid_HGRAD_LINE_Hermite_FEM.hpp>

Inheritance diagram for Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >:
Intrepid::Basis< Scalar, ArrayScalar >

Public Member Functions

 Basis_HGRAD_LINE_Hermite_FEM ()
 Default Constructor assumes the two interpolation points are the cell vertices. Cubic Hermite Interpolation.
 
 Basis_HGRAD_LINE_Hermite_FEM (const ArrayScalar &pts)
 Constructor.
 
 Basis_HGRAD_LINE_Hermite_FEM (int numPoints, const EPointType &pointType)
 Constructor.
 
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
 Evaluation of a FEM basis on a reference Line cell.
 
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const ArrayScalar &cellVertices, const EOperator operatorType=OPERATOR_VALUE) const
 FVD basis evaluation: invocation of this method throws an exception.
 
virtual void getDofCoords (ArrayScalar &DofCoords) const
 implements the dofcoords interface
 
void printTags (std::ostream &os)
 
- Public Member Functions inherited from Intrepid::Basis< Scalar, ArrayScalar >
virtual ~Basis ()
 Destructor.
 
virtual int getCardinality () const
 Returns cardinality of the basis.
 
virtual int getDegree () const
 Returns the degree of the basis.
 
virtual const shards::CellTopology getBaseCellTopology () const
 Returns the base cell topology for which the basis is defined. See Shards documentation http://trilinos.sandia.gov/packages/shards for definition of base cell topology.
 
virtual EBasis getBasisType () const
 Returns the basis type.
 
virtual ECoordinates getCoordinateSystem () const
 Returns the type of coordinate system for which the basis is defined.
 
virtual int getDofOrdinal (const int subcDim, const int subcOrd, const int subcDofOrd)
 DoF tag to ordinal lookup.
 
virtual const std::vector< std::vector< std::vector< int > > > & getDofOrdinalData ()
 DoF tag to ordinal data structure.
 
virtual const std::vector< int > & getDofTag (const int dofOrd)
 DoF ordinal to DoF tag lookup.
 
virtual const std::vector< std::vector< int > > & getAllDofTags ()
 Retrieves all DoF tags.
 

Private Types

template<typename T >
using RCP = Teuchos::RCP<T>
 
using SerialDenseMatrix = Teuchos::SerialDenseMatrix<int,Scalar>
 
using TAGS = std::vector<int>
 

Private Member Functions

void initializeTags ()
 Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.
 
void recurrence (ArrayScalar &P, ArrayScalar &Px, const Scalar x) const
 Evaluates $P_j(x_i)$ and $P_j'(x_i)$ at a particular point $x_i$.
 
void legendre_d (ArrayScalar &Pm, ArrayScalar &Pm1, const int m, const Scalar pt) const
 Evaluates $P_j^{(m)}(x_i) $ and $P_j^{(m+1)}(x_i)  $ at a particular point $x_i$.
 
void setupVandermonde (bool factor=true)
 Form the Legendre/Derivative Vandermonde matrix at the given lattice points and have the linear solver factor with equilibration.
 

Private Attributes

FieldContainer< Scalar > latticePts_
 Holds the points defining the Hermite basis.
 
SerialDenseMatrix V_
 Contains the values of the Legendre polynomials and their derivatives.
 
Teuchos::SerialDenseSolver< int, Scalar > solver_
 
TAGS tags_
 
bool isFactored_
 

Additional Inherited Members

- Protected Attributes inherited from Intrepid::Basis< Scalar, ArrayScalar >
int basisCardinality_
 Cardinality of the basis, i.e., the number of basis functions/degrees-of-freedom.
 
int basisDegree_
 Degree of the largest complete polynomial space that can be represented by the basis.
 
shards::CellTopology basisCellTopology_
 Base topology of the cells for which the basis is defined. See the Shards package http://trilinos.sandia.gov/packages/shards for definition of base cell topology.
 
EBasis basisType_
 Type of the basis.
 
ECoordinates basisCoordinates_
 The coordinate system for which the basis is defined.
 
bool basisTagsAreSet_
 "true" if tagToOrdinal_ and ordinalToTag_ have been initialized
 
std::vector< std::vector< int > > ordinalToTag_
 DoF ordinal to tag lookup table.
 
std::vector< std::vector< std::vector< int > > > tagToOrdinal_
 DoF tag to ordinal lookup table.
 

Detailed Description

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >

Implements Hermite interpolant basis of degree n on the reference Line cell. The basis has cardinality 2n and spans a COMPLETE linear polynomial space.

The Hermite interpolant approximation space is $\mathcal{H}_n\subset H_0^2(\Omega)$ where $\Omega=[-1,1]$

\[\mathcal{H}_n = \{h_{2j}(x),h_{2j+1}\}_{j=0}^{n-1} \]

The basis functions determined by the interpolation points $\{x_i\}_{i=0}^{n-1}$ which satisfy the ordering property

\[ -1\leq x_0 \leq \cdots \leq x_i \leq x_{i+1} \leq \cdots \leq x_n \leq 1 \]

while the basis functions satisfy

\[ h_{2j}(x_i) = \delta_{ij},\quad h'_{2j}(x_i) = 0 \]

\[ h_{2j+1}(x_i) = 0,\quad h'_{2j+1}(x_i) = \delta_{ij} \]

Basis functions and their derivatives are evaluated from the Legendre polynomials and their derivatives evaluated on the interpolation points and the input points in the getValues() method. Legendre polynomials and their derivatives are computed using the relations

\[ P_{j+1}(x) = x P_j(x) + \frac{x^2-1}{j+1}P_j^{(1)}(x),\quad P_0(x) = 1 \]

\[ P_{j+1}^{(m)(x)} = (j+m)P_j^{(m-1)}(x) + x P_j^{(m)}(x),\quad P_j^{(m)} = 0,\;\forall j<m\]

Definition at line 93 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

Member Typedef Documentation

◆ RCP

template<class Scalar , class ArrayScalar >
template<typename T >
using Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::RCP = Teuchos::RCP<T>
private

Definition at line 95 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

◆ SerialDenseMatrix

template<class Scalar , class ArrayScalar >
using Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::SerialDenseMatrix = Teuchos::SerialDenseMatrix<int,Scalar>
private

Definition at line 96 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

◆ TAGS

template<class Scalar , class ArrayScalar >
using Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::TAGS = std::vector<int>
private

Definition at line 98 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

Constructor & Destructor Documentation

◆ Basis_HGRAD_LINE_Hermite_FEM() [1/3]

template<class Scalar , class ArrayScalar >
Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::Basis_HGRAD_LINE_Hermite_FEM ( )

◆ Basis_HGRAD_LINE_Hermite_FEM() [2/3]

template<class Scalar , class ArrayScalar >
Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::Basis_HGRAD_LINE_Hermite_FEM ( const ArrayScalar & pts)

◆ Basis_HGRAD_LINE_Hermite_FEM() [3/3]

template<class Scalar , class ArrayScalar >
Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::Basis_HGRAD_LINE_Hermite_FEM ( int numPoints,
const EPointType & pointType )

Member Function Documentation

◆ getDofCoords()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::getDofCoords ( ArrayScalar & DofCoords) const
virtual

implements the dofcoords interface

Definition at line 549 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

◆ getValues() [1/2]

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const ArrayScalar & cellVertices,
const EOperator operatorType = OPERATOR_VALUE ) const
virtual

FVD basis evaluation: invocation of this method throws an exception.

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 539 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

◆ getValues() [2/2]

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType ) const
virtual

Evaluation of a FEM basis on a reference Line cell.

    Returns values of <var>operatorType</var> acting on FEM basis functions for a set of
    points in the <strong>reference Line</strong> cell. For rank and dimensions of
    I/O array arguments see Section \ref basis_md_array_sec .
Parameters
outputValues[out] - variable rank array with the basis values
inputPoints[in] - rank-n array (P,D) with the evaluation points
operatorType[in] - the operator acting on the basis functions

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 252 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

References Intrepid::isValidOperator().

◆ initializeTags()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::initializeTags ( )
privatevirtual

Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 377 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

References Intrepid::setOrdinalTagData().

◆ legendre_d()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::legendre_d ( ArrayScalar & Pm,
ArrayScalar & Pm1,
const int m,
const Scalar pt ) const
private

Evaluates $P_j^{(m)}(x_i) $ and $P_j^{(m+1)}(x_i)  $ at a particular point $x_i$.

Definition at line 222 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

◆ printTags()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::printTags ( std::ostream & os)

Definition at line 490 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

◆ recurrence()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::recurrence ( ArrayScalar & P,
ArrayScalar & Px,
const Scalar x ) const
private

Evaluates $P_j(x_i)$ and $P_j'(x_i)$ at a particular point $x_i$.

Definition at line 201 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

◆ setupVandermonde()

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::setupVandermonde ( bool factor = true)
private

Member Data Documentation

◆ isFactored_

template<class Scalar , class ArrayScalar >
bool Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::isFactored_
mutableprivate

Definition at line 112 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

◆ latticePts_

template<class Scalar , class ArrayScalar >
FieldContainer<Scalar> Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::latticePts_
private

◆ solver_

template<class Scalar , class ArrayScalar >
Teuchos::SerialDenseSolver<int,Scalar> Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::solver_
mutableprivate

Definition at line 108 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

◆ tags_

template<class Scalar , class ArrayScalar >
TAGS Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::tags_
mutableprivate

Definition at line 110 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.

◆ V_

template<class Scalar , class ArrayScalar >
SerialDenseMatrix Intrepid::Basis_HGRAD_LINE_Hermite_FEM< Scalar, ArrayScalar >::V_
mutableprivate

Contains the values of the Legendre polynomials and their derivatives.

Definition at line 106 of file Intrepid_HGRAD_LINE_Hermite_FEM.hpp.


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