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// ************************************************************************
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// Intrepid Package
// Copyright (2007) Sandia Corporation
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// 3. Neither the name of the Corporation nor the names of the
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//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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// Questions? Contact Pavel Bochev (pbboche@sandia.gov)
// Denis Ridzal (dridzal@sandia.gov), or
// Kara Peterson (kjpeter@sandia.gov)
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/** \file Intrepid_Basis.hpp
\brief Header file for the abstract base class Intrepid::Basis.
\author Created by P. Bochev and D. Ridzal.
*/
#ifndef INTREPID_BASIS_HPP
#define INTREPID_BASIS_HPP
#include "Intrepid_ConfigDefs.hpp"
#include "Intrepid_Types.hpp"
#include "Intrepid_Utils.hpp"
#include "Shards_CellTopology.hpp"
namespace Intrepid {
/** \class Intrepid::Basis
\brief An abstract base class that defines interface for concrete basis implementations for
Finite Element (FEM) and Finite Volume/Finite Difference (FVD) discrete spaces.
A FEM basis spans a discrete space whose type can be either COMPLETE or INCOMPLETE.
FEM basis functions are always defined on a reference cell and are dual to a unisolvent
set of degrees-of-freedom (DoF). FEM basis requires cell topology with a reference cell.
An FVD basis spans a discrete space whose type is typically BROKEN. The basis functions
are defined directly on the physical cell and are dual to a set of DoFs on that cell.
As a result, FVD bases require the vertex coordinates of the physical cell but the cell
itself is not required to have a reference cell.
Every DoF and its corresponding basis function from a given FEM or FVD basis set is
assigned an ordinal number which which specifies its numerical position in the DoF set,
and a 4-field DoF tag whose first 3 fields establish association between the DoF and a
subcell of particular dimension, and the last field gives the total number of basis
functions associated with that subcell; see Section \ref basis_dof_tag_ord_sec for details.
\remark To limit memory use by factory-type objects (basis factories will be included in future
releases of Intrepid), tag data is not initialized by basis ctors,
instead, whenever a function that requires tag data is invoked for a first time, it calls
initializeTags() to fill <var>ordinalToTag_</var> and <var>tagToOrdinal_</var>. Because
tag data is basis specific, every concrete basis class requires its own implementation
of initializeTags().
\todo restore test for inclusion of reference points in their resective reference cells in
getValues_HGRAD_Args, getValues_CURL_Args, getValues_DIV_Args
*/
template<class Scalar, class ArrayScalar>
class Basis {
private:
/** \brief Initializes <var>tagToOrdinal_</var> and <var>ordinalToTag_</var> lookup arrays.
*/
virtual void initializeTags() = 0;
protected:
/** \brief Cardinality of the basis, i.e., the number of basis functions/degrees-of-freedom
*/
int basisCardinality_;
/** \brief Degree of the largest complete polynomial space that can be represented by the basis
*/
int basisDegree_;
/** \brief 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.
*/
shards::CellTopology basisCellTopology_;
/** \brief Type of the basis
*/
EBasis basisType_;
/** \brief The coordinate system for which the basis is defined
*/
ECoordinates basisCoordinates_;
/** \brief "true" if <var>tagToOrdinal_</var> and <var>ordinalToTag_</var> have been initialized
*/
bool basisTagsAreSet_;
/** \brief DoF ordinal to tag lookup table.
Rank-2 array with dimensions (basisCardinality_, 4) containing the DoF tags. This array
is left empty at instantiation and filled by initializeTags() only when tag data is
requested.
\li ordinalToTag_[DofOrd][0] = dim. of the subcell associated with the specified DoF
\li ordinalToTag_[DofOrd][1] = ordinal of the subcell defined in the cell topology
\li ordinalToTag_[DodOrd][2] = ordinal of the specified DoF relative to the subcell
\li ordinalToTag_[DofOrd][3] = total number of DoFs associated with the subcell
*/
std::vector<std::vector<int> > ordinalToTag_;
/** \brief DoF tag to ordinal lookup table.
Rank-3 array with dimensions (maxScDim + 1, maxScOrd + 1, maxDfOrd + 1), i.e., the
columnwise maximums of the 1st three columns in the DoF tag table for the basis plus 1.
For every triple (subscDim, subcOrd, subcDofOrd) that is valid DoF tag data this array
stores the corresponding DoF ordinal. If the triple does not correspond to tag data,
the array stores -1. This array is left empty at instantiation and filled by
initializeTags() only when tag data is requested.
\li tagToOrdinal_[subcDim][subcOrd][subcDofOrd] = Degree-of-freedom ordinal
*/
std::vector<std::vector<std::vector<int> > > tagToOrdinal_;
public:
/** \brief Destructor
*/
virtual ~Basis() {}
/** \brief Evaluation of a FEM basis on a <strong>reference cell</strong>.
Returns values of <var>operatorType</var> acting on FEM basis functions for a set of
points in the <strong>reference cell</strong> for which the basis is defined.
\param outputValues [out] - variable rank array with the basis values
\param inputPoints [in] - rank-2 array (P,D) with the evaluation points
\param operatorType [in] - the operator acting on the basis functions
\remark For rank and dimension specifications of the output array see Section
\ref basis_md_array_sec. Dimensions of <var>ArrayScalar</var> arguments are checked
at runtime if HAVE_INTREPID_DEBUG is defined.
\remark A FEM basis spans a COMPLETE or INCOMPLETE polynomial space on the reference cell
which is a smooth function space. Thus, all operator types that are meaningful for the
approximated function space are admissible. When the order of the operator exceeds the
degree of the basis, the output array is filled with the appropriate number of zeros.
*/
virtual void getValues(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType) const = 0;
/** \brief Evaluation of an FVD basis evaluation on a <strong>physical cell</strong>.
Returns values of <var>operatorType</var> acting on FVD basis functions for a set of
points in the <strong>physical cell</strong> for which the FVD basis is defined.
\param outputValues [out] - variable rank array with the basis values
\param inputPoints [in] - rank-2 array (P,D) with the evaluation points
\param cellVertices [in] - rank-2 array (V,D) with the vertices of the physical cell
\param operatorType [in] - the operator acting on the basis functions
\remark For rank and dimension specifications of the output array see Section
\ref basis_md_array_sec. Dimensions of <var>ArrayScalar</var> arguments are checked
at runtime if HAVE_INTREPID_DEBUG is defined.
\remarks A typical FVD basis spans a BROKEN discrete space which is only piecewise smooth. For
example, it could be a piecewise constant space defined with respect to a partition
of the cell into simplices. Because differential operators are not meaningful for such
spaces, the default operator type in this method is set to OPERATOR_VALUE.
*/
virtual void getValues(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const ArrayScalar & cellVertices,
const EOperator operatorType = OPERATOR_VALUE) const = 0;
/** \brief Returns cardinality of the basis
\return the number of basis functions in the basis
*/
virtual int getCardinality() const;
/** \brief Returns the degree of the basis.
\return max. degree of the complete polynomials that can be represented by the basis.
*/
virtual int getDegree() const;
/** \brief 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.
\return Base cell topology
*/
virtual const shards::CellTopology getBaseCellTopology() const;
/** \brief Returns the basis type.
\return Basis type
*/
virtual EBasis getBasisType() const;
/** \brief Returns the type of coordinate system for which the basis is defined
\return Type of the coordinate system (Cartesian, polar, R-Z, etc.).
*/
virtual ECoordinates getCoordinateSystem() const;
/** \brief DoF tag to ordinal lookup.
\param subcDim [in] - tag field 0: dimension of the subcell associated with the DoF
\param subcOrd [in] - tag field 1: ordinal of the subcell defined by cell topology
\param subcDofOrd [in] - tag field 2: ordinal of the DoF relative to the subcell.
\return the DoF ordinal corresponding to the specified DoF tag data.
*/
virtual int getDofOrdinal(const int subcDim,
const int subcOrd,
const int subcDofOrd);
/** \brief DoF tag to ordinal data structure */
virtual const std::vector<std::vector<std::vector<int> > > &getDofOrdinalData( );
/** \brief DoF ordinal to DoF tag lookup.
\param dofOrd [in] - ordinal of the DoF whose tag is being retrieved
\return reference to a vector with dimension (4) such that \n
\li element [0] = tag field 0 -> dim. of the subcell associated with the specified DoF
\li element [1] = tag field 1 -> ordinal of the subcell defined by cell topology
\li element [2] = tag field 2 -> ordinal of the specified DoF relative to the subcell
\li element [3] = tag field 3 -> total number of DoFs associated with the subcell
*/
virtual const std::vector<int>& getDofTag(const int dofOrd);
/** \brief Retrieves all DoF tags.
\return reference to a vector of vectors with dimensions (basisCardinality_, 4) such that \n
\li element [DofOrd][0] = tag field 0 for the DoF with the specified ordinal
\li element [DofOrd][1] = tag field 1 for the DoF with the specified ordinal
\li element [DofOrd][2] = tag field 2 for the DoF with the specified ordinal
\li element [DofOrd][3] = tag field 3 for the DoF with the specified ordinal
*/
virtual const std::vector<std::vector<int> >& getAllDofTags();
}; // class Basis
//--------------------------------------------------------------------------------------------//
// //
// Helper functions of the Basis class //
// //
//--------------------------------------------------------------------------------------------//
//--------------------------------------------------------------------------------------------//
// //
// Argument checks //
// //
//--------------------------------------------------------------------------------------------//
/** \brief Runtime check of the arguments for the getValues method in an HGRAD-conforming
FEM basis. Verifies that ranks and dimensions of <var>ArrayScalar</var> input and output
arrays are consistent with the specified <var>operatorType</var>.
\param outputValues [in] - array of variable rank for the output basis values
\param inputPoints [in] - rank-2 array with dimensions (P,D) containing the points
\param operatorType [in] - operator applied to basis functions
\param cellTopo [in] - base cell topology on which the basis is defined
\param basisCard [in] - cardinality of the basis
*/
template<class Scalar, class ArrayScalar>
void getValues_HGRAD_Args(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType,
const shards::CellTopology& cellTopo,
const int basisCard);
/** \brief Runtime check of the arguments for the getValues method in an HCURL-conforming
FEM basis. Verifies that ranks and dimensions of <var>ArrayScalar</var> input and output
arrays are consistent with the specified <var>operatorType</var>.
\param outputValues [in] - array of variable rank for the output basis values
\param inputPoints [in] - rank-2 array with dimensions (P,D) containing the points
\param operatorType [in] - operator applied to basis functions
\param cellTopo [in] - base cell topology on which the basis is defined
\param basisCard [in] - cardinality of the basis
*/
template<class Scalar, class ArrayScalar>
void getValues_HCURL_Args(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType,
const shards::CellTopology& cellTopo,
const int basisCard);
/** \brief Runtime check of the arguments for the getValues method in an HDIV-conforming
FEM basis. Verifies that ranks and dimensions of <var>ArrayScalar</var> input and output
arrays are consistent with the specified <var>operatorType</var>.
\param outputValues [in] - array of variable rank for the output basis values
\param inputPoints [in] - rank-2 array with dimensions (P,D) containing the points
\param operatorType [in] - operator applied to basis functions
\param cellTopo [in] - base cell topology on which the basis is defined
\param basisCard [in] - cardinality of the basis
*/
template<class Scalar, class ArrayScalar>
void getValues_HDIV_Args(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType,
const shards::CellTopology& cellTopo,
const int basisCard);
/** \brief This is an interface class for bases whose degrees of freedom
can be associated with spatial locations in a reference element
(typically interpolation points for interpolatory bases).
*/
template<class ArrayScalar>
class DofCoordsInterface {
public:
/** \brief Pure virtual destructor (gives warnings if not included).
* Following "Effective C++: 3rd Ed." item 7 the implementation
* is included in the definition file.
*/
virtual ~DofCoordsInterface() = 0;
/** \brief Returns spatial locations (coordinates) of degrees of freedom on a
<strong>reference cell</strong>; defined for interpolatory bases.
\param DofCoords [out] - array with the coordinates of degrees of freedom,
dimensioned (F,D)
*/
virtual void getDofCoords(ArrayScalar & DofCoords) const = 0;
};
// include templated definitions
#include <Intrepid_BasisDef.hpp>
}// namespace Intrepid
//--------------------------------------------------------------------------------------------//
// //
// D O C U M E N T A T I O N P A G E S //
// //
//--------------------------------------------------------------------------------------------//
/**
\page basis_page Intrepid basis class
\section basis_dof_tag_ord_sec Degree of freedom ordinals and tags
Regardless of the basis type, i.e., FEM or FVD, each DoF is assigned an ordinal number which specifies
its numerical position in the DoF set, and a 4-field DoF tag whose first 3 fields establish association
between the DoF and a subcell of particular dimension. The last field in the DoF tag is for convenience
and stores the total number of DoFs associated with the specified subcell. In summary, the tag contains
the following information about a DoF with a given ordinal:
\li field 0: dimension of the subcell associated with the specified DoF ordinal;
\li field 1: ordinal of the subcell relative to its parent cell;
\li field 2: ordinal of the DoF relative to the subcell;
\li field 3: cardinality of the DoF set associated with this subcell.
DoF definition, DoF ordinals and DoF tags are basis-dependent and are documemented in the concrete
basis implementation. A typical entry in a DoF tag table has the following format:
\verbatim
|-------------------------------------------------------------------------------------------------|
| | degree-of-freedom-tag table | |
| DoF |----------------------------------------------------------| DoF definition |
| ordinal | subc dim | subc ordinal | subc DoF ord |subc num DoF | |
|-------------------------------------------------------------------------------------------------|
|---------|--------------|--------------|--------------|-------------|----------------------------|
| k | 1 | 2 | 1 | 3 | L_k(u) = (definition) |
|---------|--------------|--------------|--------------|-------------|----------------------------|
|-------------------------------------------------------------------------------------------------|
\endverbatim
The tag in this example establishes an association between the DoF with ordinal <var>k</var> and the 3rd
edge of the parent cell on which the basis is defined. Furthermore, the tag specifies that relative
to that edge, this DoF has ordinal 1, i.e., it is the second DoF on that edge. The last field in the
tag indicates that there are a total of 3 DoFs associated with that subcell.
\section basis_md_array_sec MD array template arguments for basis methods
FEM and FVD basis evaluation methods use generic MD arrays (see \ref md_array_page for details) to
pass the evaluation points (and cell vertices for FVD evaluation) and to return the basis values.
The ranks and the dimensions of the MD array arguments for these methods are as follows.
\subsection basis_md_array_out_sec Rank and dimensions of the output MD array
Rank and dimensions of the output array depend on the field rank of the basis functions, which can be 0
(scalar fields), 1 (vector fields), or 2 (tensor fields), the space dimension, and the <var>operatorType</var>.
The following table summarizes all admissible combinations:
\verbatim
|-------------------------------------------------------------------------------------------------|
| Rank and multi-dimensions of the output MD array in getValues methods |
|--------------------|-------------------------|-------------------------|------------------------|
|operator/field rank | rank 0 | rank 1 2D/3D | rank 2 2D/3D |
|--------------------|-------------------------|-------------------------|------------------------|
| VALUE | (F,P) | (F,P,D) | (F,P,D,D) |
|--------------------|-------------------------|-------------------------|------------------------|
| GRAD, D1 | (F,P,D) | (F,P,D,D) | (F,P,D,D,D) |
|--------------------|-------------------------|-------------------------|------------------------|
| CURL | (F,P,D) (undef. in 3D) | (F,P)/(F,P,D) | (F,P,D)/(F,P,D,D) |
|--------------------|-------------------------|-------------------------|------------------------|
| DIV | (F,P,D) (only in 1D) | (F,P) | (F,P,D) |
|--------------------|-------------------------|-------------------------|------------------------|
| D1,D2,..,D10 | (F,P,K) | (F,P,D,K) | (F,P,D,D,K) |
|-------------------------------------------------------------------------------------------------|
\endverbatim
\remarks
\li The totality of all derivatives whose order equals k (OPERATOR_Dk in Intrepid) forms a multiset;
see http://mathworld.wolfram.com/Multiset.html In Intrepid this multiset is enumerated using the
lexicographical order of the partial derivatives; see getDkEnumeration() for details.
\li The last dimension of the output array for D1,...,D10 is the cardinality of the Dk multiset
(computed by DkCardinality). The array is filled with zeroes whenever the order of the derivative
Dk exceeed the polynomial degree.
\subsection basis_md_array_in_sec Rank and dimensions of the input MD arrays
The FEM evaluation method has one MD array input argument which is used to pass the coordinates of
P evaluation points in the reference cell for which the concrete basis is defined. The FVD method
has two MD array input arguments. The first one passes the coordinates of P evaluation points in the
physical cell for which the concrete basis is defined. The second MD array passes the vertices of the
physical cell. Ranks and dimensions of these arrays are summarized in the following table:
\verbatim
|-------------------------------------------------------------------------------------------------|
| Rank and multi-dimensions of the input MD arrays in getValues methods |
|--------------------|------------------------|---------------------------------------------------|
| MD array | rank | multi-dimension | Description |
|--------------------|------------------------|---------------------------------------------------|
| evaluation points | 2 | (P,D) | Coordinates of P points in D-dimensions |
|--------------------|------------------------|---------------------------------------------------|
| cell vertices | 2 | (V,D) | Coordinates of V vertices of D-dimensional cell |
|-------------------------------------------------------------------------------------------------|
\endverbatim
*/
#endif
|