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 >:

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. More...

	Basis_HGRAD_LINE_Hermite_FEM (int numPoints, const EPointType &pointType)
	Constructor. More...

void	getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
	Evaluation of a FEM basis on a reference Line cell. More...

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. More...

virtual int	getDegree () const
	Returns the degree of the basis. More...

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. More...

virtual EBasis	getBasisType () const
	Returns the basis type. More...

virtual ECoordinates	getCoordinateSystem () const
	Returns the type of coordinate system for which the basis is defined. More...

virtual int	getDofOrdinal (const int subcDim, const int subcOrd, const int subcDofOrd)
	DoF tag to ordinal lookup. More...

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. More...

virtual const std::vector < std::vector< int > > &	getAllDofTags ()
	Retrieves all DoF tags. More...

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 and at a particular point .

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 .

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. More...

std::vector< std::vector < std::vector< int > > >	tagToOrdinal_
	DoF tag to ordinal lookup table. More...

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 \form#185
   where \form#186

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

   The basis functions determined by the interpolation points \form#188
   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.

Constructor & Destructor Documentation

template<class Scalar , class ArrayScalar >

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

Constructor.

Parameters

int	pointType: [in] array containing interpolation points

Definition at line 80 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

template<class Scalar , class ArrayScalar >

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

Constructor.

Parameters

int	numPoints: [in] number of interpolation points
int	pointType: [in] type of points, either POINTTYPE_EQUISPACED or POINTTYPE_SPECTRAL

Definition at line 121 of file Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp.

Member Function Documentation

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.

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

http://docs.trilinos.org/r13.0/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Hermite_FEM.hpp
http://docs.trilinos.org/r13.0/packages/intrepid/src/Discretization/Basis/Intrepid_HGRAD_LINE_Hermite_FEMDef.hpp

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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