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// ************************************************************************
//
// Intrepid Package
// Copyright (2007) Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
// license for use of this work by or on behalf of the U.S. Government.
//
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// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
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// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// 3. Neither the name of the Corporation nor the names of the
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
// PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
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// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Questions? Contact Pavel Bochev (pbboche@sandia.gov)
// Denis Ridzal (dridzal@sandia.gov), or
// Kara Peterson (kjpeter@sandia.gov)
//
// ************************************************************************
// @HEADER
/** \file Intrepid_FunctionSpaceTools.hpp
\brief Header file for the Intrepid::FunctionSpaceTools class.
\author Created by P. Bochev and D. Ridzal.
*/
#ifndef INTREPID_FUNCTIONSPACETOOLS_HPP
#define INTREPID_FUNCTIONSPACETOOLS_HPP
#include "Intrepid_ConfigDefs.hpp"
#include "Intrepid_ArrayTools.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_CellTools.hpp"
#include <Intrepid_KokkosRank.hpp>
namespace Intrepid {
/** \class Intrepid::FunctionSpaceTools
\brief Defines expert-level interfaces for the evaluation of functions
and operators in physical space (supported for FE, FV, and FD methods)
and FE reference space; in addition, provides several function
transformation utilities.
*/
class FunctionSpaceTools {
public:
/** \brief Transformation of a (scalar) value field in the H-grad space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of \e HGRAD functions \f$\Phi^*(\widehat{u}_f) = \widehat{u}_f\circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the function set \f$\{\widehat{u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p)
= \widehat{u}_f\circ F^{-1}_{c}(x_{c,p})
= \widehat{u}_f(\widehat{x}_p) = inVals(f,p) \qquad 0\le c < C \,,
\f]
i.e., it simply replicates the values in the user-provided container to every cell.
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeIn>
static void HGRADtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeIn & inVals);
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeIn>
static void HGRADtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeIn & inVals);*/
/** \brief Transformation of a gradient field in the H-grad space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of gradients of \e HGRAD functions
\f$\Phi^*(\nabla\widehat{u}_f) = \left((DF_c)^{-{\sf T}}\cdot\nabla\widehat{u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the gradients of the function set \f$\{\widehat{u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \nabla\widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left((DF_c)^{-{\sf T}}\cdot\nabla\widehat{u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= (DF_c)^{-{\sf T}}(\widehat{x}_p)\cdot\nabla\widehat{u}_f(\widehat{x}_p) \qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HGRADtransformGRAD(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HGRADtransformGRADTemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
*/
/** \brief Transformation of a (vector) value field in the H-curl space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of \e HCURL functions
\f$\Phi^*(\widehat{\bf u}_f) = \left((DF_c)^{-{\sf T}}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left((DF_c)^{-{\sf T}}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= (DF_c)^{-{\sf T}}(\widehat{x}_p)\cdot\widehat{\bf u}_f(\widehat{x}_p) \qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of native basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HCURLtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeIn>
static void HCURLtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobianInverse,
const ArrayTypeIn & inVals,
const char transpose = 'T');*/
/** \brief Transformation of a curl field in the H-curl space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of curls of \e HCURL functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c} DF_{c}\cdot\nabla\times\widehat{\bf u}_f\right)\circ F^{-1}_{c}\f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the curls of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \nabla\times\widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c} DF_{c}\cdot\nabla\times\widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) DF_{c}(\widehat{x}_p)\cdot\nabla\times\widehat{\bf u}_f(\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HCURLtransformCURL(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HCURLtransformCURLTemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');*/
/** \brief Transformation of a (vector) value field in the H-div space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P,D), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P,D).
Computes pullback of \e HDIV functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c} DF_{c}\cdot\widehat{\bf u}_f\right)\circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p,*) = \widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c} DF_{c}\cdot \widehat{\bf u}_f\right)\circ F^{-1}_{c}(x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) DF_{c}(\widehat{x}_p)\cdot\widehat{\bf u}_f(\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
| D | space dim | 0 <= D < spatial dimension |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeJac, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformVALUETemp(ArrayTypeOut & outVals,
const ArrayTypeJac & jacobian,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals,
const char transpose = 'N');*/
/** \brief Transformation of a divergence field in the H-div space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of the divergence of \e HDIV functions
\f$\Phi^*(\widehat{\bf u}_f) = \left(J^{-1}_{c}\nabla\cdot\widehat{\bf u}_{f}\right) \circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the divergencies of the vector function set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \nabla\cdot\widehat{\bf u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c}\nabla\cdot\widehat{\bf u}_{f}\right) \circ F^{-1}_{c} (x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) \nabla\cdot\widehat{\bf u}_{f} (\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformDIV(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
/*
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HDIVtransformDIVTemp(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
*/
/** \brief Transformation of a (scalar) value field in the H-vol space, defined at points on a
reference cell, stored in the user-provided container <var><b>inVals</b></var>
and indexed by (F,P), into the output container <var><b>outVals</b></var>,
defined on cells in physical space and indexed by (C,F,P).
Computes pullback of \e HVOL functions
\f$\Phi^*(\widehat{u}_f) = \left(J^{-1}_{c}\widehat{u}_{f}\right) \circ F^{-1}_{c} \f$
for points in one or more physical cells that are images of a given set of points in the reference cell:
\f[
\{ x_{c,p} \}_{p=0}^P = \{ F_{c} (\widehat{x}_p) \}_{p=0}^{P}\qquad 0\le c < C \,.
\f]
In this case \f$ F^{-1}_{c}(x_{c,p}) = \widehat{x}_p \f$ and the user-provided container
should contain the values of the functions in the set \f$\{\widehat{\bf u}_f\}_{f=0}^{F}\f$ at the
reference points:
\f[
inVals(f,p) = \widehat{u}_f(\widehat{x}_p) \,.
\f]
The method returns
\f[
outVals(c,f,p,*)
= \left(J^{-1}_{c}\widehat{u}_{f}\right) \circ F^{-1}_{c} (x_{c,p})
= J^{-1}_{c}(\widehat{x}_p) \widehat{u}_{f} (\widehat{x}_p)
\qquad 0\le c < C \,.
\f]
See Section \ref sec_pullbacks for more details about pullbacks.
\code
|------|----------------------|--------------------------------------------------|
| | Index | Dimension |
|------|----------------------|--------------------------------------------------|
| C | cell | 0 <= C < num. integration domains |
| F | field | 0 <= F < dim. of the basis |
| P | point | 0 <= P < num. integration points |
|------|----------------------|--------------------------------------------------|
\endcode
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeDet, class ArrayTypeIn>
static void HVOLtransformVALUE(ArrayTypeOut & outVals,
const ArrayTypeDet & jacobianDet,
const ArrayTypeIn & inVals);
/** \brief Contracts \a <b>leftValues</b> and \a <b>rightValues</b> arrays on the
point and possibly space dimensions and stores the result in \a <b>outputValues</b>;
this is a generic, high-level integration routine that calls either
FunctionSpaceTools::operatorIntegral, or FunctionSpaceTools::functionalIntegral,
or FunctionSpaceTools::dataIntegral methods, depending on the rank of the
\a <b>outputValues</b> array.
\param outputValues [out] - Output array.
\param leftValues [in] - Left input array.
\param rightValues [in] - Right input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar>
static void integrate(Intrepid::FieldContainer<Scalar> & outputValues,
const Intrepid::FieldContainer<Scalar> & leftValues,
const Intrepid::FieldContainer<Scalar> & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight>
static void integrate(ArrayOut & outputValues,
const ArrayInLeft & leftValues,
const ArrayInRight & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight>
static void integrateTemp(ArrayOut & outputValues,
const ArrayInLeft & leftValues,
const ArrayInRight & rightValues,
const ECompEngine compEngine,
const bool sumInto = false);
*/
template<class Scalar, class ArrayOut, class ArrayInLeft, class ArrayInRight,int leftrank,int outrank>
struct integrateTempSpec;
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of
two rank-3, 4, or 5 containers with dimensions (C,L,P) and (C,R,P),
or (C,L,P,D1) and (C,R,P,D1), or (C,L,P,D1,D2) and (C,R,P,D1,D2),
and returns the result in a rank-3 container with dimensions (C,L,R).
For a fixed index "C", (C,L,R) represents a rectangular L X R matrix
where L and R may be different.
\code
C - num. integration domains dim0 in both input containers
L - num. "left" fields dim1 in "left" container
R - num. "right" fields dim1 in "right" container
P - num. integration points dim2 in both input containers
D1- vector (1st tensor) dimension dim3 in both input containers
D2- 2nd tensor dimension dim4 in both input containers
\endcode
\param outputFields [out] - Output array.
\param leftFields [in] - Left input array.
\param rightFields [in] - Right input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInFieldsLeft, class ArrayInFieldsRight>
static void operatorIntegral(ArrayOutFields & outputFields,
const ArrayInFieldsLeft & leftFields,
const ArrayInFieldsRight & rightFields,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutFields, class ArrayInFieldsLeft, class ArrayInFieldsRight>
static void operatorIntegralTemp(ArrayOutFields & outputFields,
const ArrayInFieldsLeft & leftFields,
const ArrayInFieldsRight & rightFields,
const ECompEngine compEngine,
const bool sumInto = false);*/
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of a
rank-3, 4, or 5 container and a rank-2, 3, or 4 container, respectively,
with dimensions (C,F,P) and (C,P), or (C,F,P,D1) and (C,P,D1),
or (C,F,P,D1,D2) and (C,P,D1,D2), respectively, and returns the result
in a rank-2 container with dimensions (C,F).
For a fixed index "C", (C,F) represents a (column) vector of length F.
\code
C - num. integration domains dim0 in both input containers
F - num. fields dim1 in fields input container
P - num. integration points dim2 in fields input container and dim1 in tensor data container
D1 - first spatial (tensor) dimension index dim3 in fields input container and dim2 in tensor data container
D2 - second spatial (tensor) dimension index dim4 in fields input container and dim3 in tensor data container
\endcode
\param outputFields [out] - Output fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void functionalIntegral(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void functionalIntegralTemp(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const ECompEngine compEngine,
const bool sumInto = false);
*/
/** \brief Contracts the point (and space) dimensions P (and D1 and D2) of two
rank-2, 3, or 4 containers with dimensions (C,P), or (C,P,D1), or
(C,P,D1,D2), respectively, and returns the result in a rank-1 container
with dimensions (C).
\code
C - num. integration domains dim0 in both input containers
P - num. integration points dim1 in both input containers
D1 - first spatial (tensor) dimension index dim2 in both input containers
D2 - second spatial (tensor) dimension index dim3 in both input containers
\endcode
\param outputData [out] - Output data array.
\param inputDataLeft [in] - Left data input array.
\param inputDataRight [in] - Right data input array.
\param compEngine [in] - Computational engine.
\param sumInto [in] - If TRUE, sum into given output array,
otherwise overwrite it. Default: FALSE.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dataIntegral(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const ECompEngine compEngine,
const bool sumInto = false);
/* template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dataIntegralTemp(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const ECompEngine compEngine,
const bool sumInto = false);
*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of cell integrals, by multiplying absolute values
of the user-provided cell Jacobian determinants \a <b>inDet</b> with dimensions (C,P)
with the user-provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) = |\mbox{det}(DF_{c}(\widehat{x}_p))|\omega_{p} \,,
\f]
where \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on a reference cell
(a set of integration points and their associated weights; see
Intrepid::Cubature::getCubature for getting cubature rules on reference cells).
\warning
The user is responsible for providing input arrays with consistent data: the determinants
in \a <b>inDet</b> should be evaluated at integration points on the <b>reference cell</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all containers
P - num. integration points dim1 in all containers
\endcode
\param outVals [out] - Output array with weighted cell measures.
\param inDet [in] - Input array containing determinants of cell Jacobians.
\param inWeights [in] - Input integration weights.
*/
template<class Scalar, class ArrayOut, class ArrayDet, class ArrayWeights>
static void computeCellMeasure(ArrayOut & outVals,
const ArrayDet & inDet,
const ArrayWeights & inWeights);
/*template<class Scalar, class ArrayOut, class ArrayDet, class ArrayWeights>
static void computeCellMeasureTemp(ArrayOut & outVals,
const ArrayDet & inDet,
const ArrayWeights & inWeights);*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of face integrals, based on the provided
cell Jacobian array \a <b>inJac</b> with dimensions (C,P,D,D) and the
provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) =
\left\|\frac{\partial\Phi_c(\widehat{x}_p)}{\partial u}\times
\frac{\partial\Phi_c(\widehat{x}_p)}{\partial v}\right\|\omega_{p} \,,
\f]
where:
\li \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on \b reference
\b face \f$\widehat{\mathcal{F}}\f$, with ordinal \e whichFace relative to the specified parent reference cell;
\li \f$ \Phi_c : R \mapsto \mathcal{F} \f$ is parameterization of the physical face
corresponding to \f$\widehat{\mathcal{F}}\f$; see Section \ref sec_cell_topology_subcell_map.
\warning
The user is responsible for providing input arrays with consistent data: the Jacobians
in \a <b>inJac</b> should be evaluated at integration points on the <b>reference face</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
Cubature rules on reference faces are defined by a two-step process:
\li A cubature rule is defined on the parametrization domain \e R of the face
(\e R is the standard 2-simplex {(0,0),(1,0),(0,1)} or the standard 2-cube [-1,1] X [-1,1]).
\li The points are mapped to a reference face using Intrepid::CellTools::mapToReferenceSubcell
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all input containers
P - num. integration points dim1 in all input containers
D - spatial dimension dim2 and dim3 in Jacobian container
\endcode
\param outVals [out] - Output array with weighted face measures.
\param inJac [in] - Input array containing cell Jacobians.
\param inWeights [in] - Input integration weights.
\param whichFace [in] - Index of the face subcell relative to the parent cell; defines the domain of integration.
\param parentCell [in] - Parent cell topology.
*/
template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeFaceMeasure(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichFace,
const shards::CellTopology & parentCell);
/* template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeFaceMeasureTemp(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichFace,
const shards::CellTopology & parentCell);*/
/** \brief Returns the weighted integration measures \a <b>outVals</b> with dimensions
(C,P) used for the computation of edge integrals, based on the provided
cell Jacobian array \a <b>inJac</b> with dimensions (C,P,D,D) and the
provided integration weights \a <b>inWeights</b> with dimensions (P).
Returns a rank-2 array (C, P) array such that
\f[
\mbox{outVals}(c,p) =
\left\|\frac{d \Phi_c(\widehat{x}_p)}{d s}\right\|\omega_{p} \,,
\f]
where:
\li \f$\{(\widehat{x}_p,\omega_p)\}\f$ is a cubature rule defined on \b reference
\b edge \f$\widehat{\mathcal{E}}\f$, with ordinal \e whichEdge relative to the specified parent reference cell;
\li \f$ \Phi_c : R \mapsto \mathcal{E} \f$ is parameterization of the physical edge
corresponding to \f$\widehat{\mathcal{E}}\f$; see Section \ref sec_cell_topology_subcell_map.
\warning
The user is responsible for providing input arrays with consistent data: the Jacobians
in \a <b>inJac</b> should be evaluated at integration points on the <b>reference edge</b>
corresponding to the weights in \a <b>inWeights</b>.
\remark
Cubature rules on reference edges are defined by a two-step process:
\li A cubature rule is defined on the parametrization domain \e R = [-1,1] of the edge.
\li The points are mapped to a reference edge using Intrepid::CellTools::mapToReferenceSubcell
\remark
See Intrepid::CellTools::setJacobian for computation of \e DF and
Intrepid::CellTools::setJacobianDet for computation of its determinant.
\code
C - num. integration domains dim0 in all input containers
P - num. integration points dim1 in all input containers
D - spatial dimension dim2 and dim3 in Jacobian container
\endcode
\param outVals [out] - Output array with weighted edge measures.
\param inJac [in] - Input array containing cell Jacobians.
\param inWeights [in] - Input integration weights.
\param whichEdge [in] - Index of the edge subcell relative to the parent cell; defines the domain of integration.
\param parentCell [in] - Parent cell topology.
*/
template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeEdgeMeasure(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichEdge,
const shards::CellTopology & parentCell);
/* template<class Scalar, class ArrayOut, class ArrayJac, class ArrayWeights>
static void computeEdgeMeasureTemp(ArrayOut & outVals,
const ArrayJac & inJac,
const ArrayWeights & inWeights,
const int whichEdge,
const shards::CellTopology & parentCell);*/
/** \brief Multiplies fields \a <b>inVals</b> by weighted measures \a <b>inMeasure</b> and
returns the field array \a <b>outVals</b>; this is a simple redirection to the call
FunctionSpaceTools::scalarMultiplyDataField.
\param outVals [out] - Output array with scaled field values.
\param inMeasure [in] - Input array containing weighted measures.
\param inVals [in] - Input fields.
*/
template<class Scalar, class ArrayTypeOut, class ArrayTypeMeasure, class ArrayTypeIn>
static void multiplyMeasure(ArrayTypeOut & outVals,
const ArrayTypeMeasure & inMeasure,
const ArrayTypeIn & inVals);
/* template<class Scalar, class ArrayTypeOut, class ArrayTypeMeasure, class ArrayTypeIn>
static void multiplyMeasureTemp(ArrayTypeOut & outVals,
const ArrayTypeMeasure & inMeasure,
const ArrayTypeIn & inVals);*/
/** \brief Scalar multiplication of data and fields; please read the description below.
There are two use cases:
\li
multiplies a rank-3, 4, or 5 container \a <b>inputFields</b> with dimensions (C,F,P),
(C,F,P,D1) or (C,F,P,D1,D2), representing the values of a set of scalar, vector
or tensor fields, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P),
representing the values of scalar data, OR
\li
multiplies a rank-2, 3, or 4 container \a <b>inputFields</b> with dimensions (F,P),
(F,P,D1) or (F,P,D1,D2), representing the values of a scalar, vector or a
tensor field, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P),
representing the values of scalar data;
the output value container \a <b>outputFields</b> is indexed by (C,F,P), (C,F,P,D1)
or (C,F,P,D1,D2), regardless of which of the two use cases is considered.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\note The argument <var><b>inputFields</b></var> can be changed!
This enables in-place multiplication.
\param outputFields [out] - Output (product) fields array.
\param inputData [in] - Data (multiplying) array.
\param inputFields [in] - Input (being multiplied) fields array.
\param reciprocal [in] - If TRUE, <b>divides</b> input fields by the data
(instead of multiplying). Default: FALSE.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void scalarMultiplyDataField(ArrayOutFields & outputFields,
ArrayInData & inputData,
ArrayInFields & inputFields,
const bool reciprocal = false);
/** \brief Scalar multiplication of data and data; please read the description below.
There are two use cases:
\li
multiplies a rank-2, 3, or 4 container \a <b>inputDataRight</b> with dimensions (C,P),
(C,P,D1) or (C,P,D1,D2), representing the values of a set of scalar, vector
or tensor data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P),
representing the values of scalar data, OR
\li
multiplies a rank-1, 2, or 3 container \a <b>inputDataRight</b> with dimensions (P),
(P,D1) or (P,D1,D2), representing the values of scalar, vector or
tensor data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P),
representing the values of scalar data;
the output value container \a <b>outputData</b> is indexed by (C,P), (C,P,D1) or (C,P,D1,D2),
regardless of which of the two use cases is considered.
\code
C - num. integration domains
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\note The arguments <var><b>inputDataLeft</b></var>, <var><b>inputDataRight</b></var> can be changed!
This enables in-place multiplication.
\param outputData [out] - Output data array.
\param inputDataLeft [in] - Left (multiplying) data array.
\param inputDataRight [in] - Right (being multiplied) data array.
\param reciprocal [in] - If TRUE, <b>divides</b> input fields by the data
(instead of multiplying). Default: FALSE.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void scalarMultiplyDataData(ArrayOutData & outputData,
ArrayInDataLeft & inputDataLeft,
ArrayInDataRight & inputDataRight,
const bool reciprocal = false);
/** \brief Dot product of data and fields; please read the description below.
There are two use cases:
\li
dot product of a rank-3, 4 or 5 container \a <b>inputFields</b> with dimensions (C,F,P)
(C,F,P,D1) or (C,F,P,D1,D2), representing the values of a set of scalar, vector
or tensor fields, by the values in a rank-2, 3 or 4 container \a <b>inputData</b> indexed by
(C,P), (C,P,D1), or (C,P,D1,D2) representing the values of scalar, vector or
tensor data, OR
\li
dot product of a rank-2, 3 or 4 container \a <b>inputFields</b> with dimensions (F,P),
(F,P,D1) or (F,P,D1,D2), representing the values of a scalar, vector or tensor
field, by the values in a rank-2 container \a <b>inputData</b> indexed by (C,P), (C,P,D1) or
(C,P,D1,D2), representing the values of scalar, vector or tensor data;
the output value container \a <b>outputFields</b> is indexed by (C,F,P),
regardless of which of the two use cases is considered.
For input fields containers without a dimension index, this operation reduces to
scalar multiplication.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\param outputFields [out] - Output (dot product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void dotMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields);
/** \brief Dot product of data and data; please read the description below.
There are two use cases:
\li
dot product of a rank-2, 3 or 4 container \a <b>inputDataRight</b> with dimensions (C,P)
(C,P,D1) or (C,P,D1,D2), representing the values of a scalar, vector or a
tensor set of data, by the values in a rank-2, 3 or 4 container \a <b>inputDataLeft</b> indexed by
(C,P), (C,P,D1), or (C,P,D1,D2) representing the values of scalar, vector or
tensor data, OR
\li
dot product of a rank-2, 3 or 4 container \a <b>inputDataRight</b> with dimensions (P),
(P,D1) or (P,D1,D2), representing the values of scalar, vector or tensor
data, by the values in a rank-2 container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D1) or
(C,P,D1,D2), representing the values of scalar, vector, or tensor data;
the output value container \a <b>outputData</b> is indexed by (C,P),
regardless of which of the two use cases is considered.
For input fields containers without a dimension index, this operation reduces to
scalar multiplication.
\code
C - num. integration domains
P - num. integration points
D1 - first spatial (tensor) dimension index
D2 - second spatial (tensor) dimension index
\endcode
\param outputData [out] - Output (dot product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void dotMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight);
/** \brief Cross or outer product of data and fields; please read the description below.
There are four use cases:
\li
cross product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-3
container \a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
cross product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-3 container
\a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-3
container \a <b>inputData</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-3 container
\a <b>inputData</b> indexed by (C,P,D), representing the values of vector data;
for cross products, the output value container \a <b>outputFields</b> is indexed by
(C,F,P,D) in 3D (vector output) and by (C,F,P) in 2D (scalar output);
for outer products, the output value container \a <b>outputFields</b> is indexed by (C,F,P,D,D).
\code
C - num. integration domains
F - num. fields
P - num. integration points
D - spatial dimension, must be 2 or 3
\endcode
\param outputFields [out] - Output (cross or outer product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void vectorMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields);
/** \brief Cross or outer product of data and data; please read the description below.
There are four use cases:
\li
cross product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-3
container \a <b>inputDataLeft</b> indexed by (C,P,D) representing the values of vector data, OR
\li
cross product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-3 container
\a <b>inputDataLeft</b> indexed by (C,P,D), representing the values of vector data, OR
\li
outer product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-3
container \a <b>inputDataLeft</b> indexed by (C,P,D) representing the values of vector data, OR
\li
outer product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-3 container
\a <b>inputDataLeft</b> indexed by (C,P,D), representing the values of vector data;
for cross products, the output value container \a <b>outputData</b> is indexed by
(C,P,D) in 3D (vector output) and by (C,P) in 2D (scalar output);
for outer products, the output value container \a <b>outputData</b> is indexed by (C,P,D,D).
\code
C - num. integration domains
P - num. integration points
D - spatial dimension, must be 2 or 3
\endcode
\param outputData [out] - Output (cross or outer product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void vectorMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight);
/** \brief Matrix-vector or matrix-matrix product of data and fields; please read the description below.
There are four use cases:
\li
matrix-vector product of a rank-4 container \a <b>inputFields</b> with dimensions (C,F,P,D),
representing the values of a set of vector fields, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-vector product of a rank-3 container \a <b>inputFields</b> with dimensions (F,P,D),
representing the values of a vector field, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-5 container \a <b>inputFields</b> with dimensions (C,F,P,D,D),
representing the values of a set of tensor fields, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-4 container \a <b>inputFields</b> with dimensions (F,P,D,D),
representing the values of a tensor field, on the left by the values in a rank-2, 3, or 4
container \a <b>inputData</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data;
for matrix-vector products, the output value container \a <b>outputFields</b> is
indexed by (C,F,P,D);
for matrix-matrix products the output value container \a <b>outputFields</b> is
indexed by (C,F,P,D,D).
\remarks
The rank of \a <b>inputData</b> implicitly defines the type of tensor data:
\li rank = 2 corresponds to a constant diagonal tensor \f$ diag(a,\ldots,a) \f$
\li rank = 3 corresponds to a nonconstant diagonal tensor \f$ diag(a_1,\ldots,a_d) \f$
\li rank = 4 corresponds to a full tensor \f$ \{a_{ij}\}\f$
\note It is assumed that all tensors are square!
\note The method is defined for spatial dimensions D = 1, 2, 3
\code
C - num. integration domains
F - num. fields
P - num. integration points
D - spatial dimension
\endcode
\param outputFields [out] - Output (matrix-vector or matrix-matrix product) fields array.
\param inputData [in] - Data array.
\param inputFields [in] - Input fields array.
\param transpose [in] - If 'T', use transposed left data tensor; if 'N', no transpose. Default: 'N'.
*/
template<class Scalar, class ArrayOutFields, class ArrayInData, class ArrayInFields>
static void tensorMultiplyDataField(ArrayOutFields & outputFields,
const ArrayInData & inputData,
const ArrayInFields & inputFields,
const char transpose = 'N');
/** \brief Matrix-vector or matrix-matrix product of data and data; please read the description below.
There are four use cases:
\li
matrix-vector product of a rank-3 container \a <b>inputDataRight</b> with dimensions (C,P,D),
representing the values of a set of vector data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-vector product of a rank-2 container \a <b>inputDataRight</b> with dimensions (P,D),
representing the values of vector data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-4 container \a <b>inputDataRight</b> with dimensions (C,P,D,D),
representing the values of a set of tensor data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data, OR
\li
matrix-matrix product of a rank-3 container \a <b>inputDataRight</b> with dimensions (P,D,D),
representing the values of tensor data, on the left by the values in a rank-2, 3, or 4
container \a <b>inputDataLeft</b> indexed by (C,P), (C,P,D) or (C,P,D,D), respectively,
representing the values of tensor data;
for matrix-vector products, the output value container \a <b>outputData</b>
is indexed by (C,P,D);
for matrix-matrix products, the output value container \a <b>outputData</b>
is indexed by (C,P,D1,D2).
\remarks
The rank of <b>inputDataLeft</b> implicitly defines the type of tensor data:
\li rank = 2 corresponds to a constant diagonal tensor \f$ diag(a,\ldots,a) \f$
\li rank = 3 corresponds to a nonconstant diagonal tensor \f$ diag(a_1,\ldots,a_d) \f$
\li rank = 4 corresponds to a full tensor \f$ \{a_{ij}\}\f$
\note It is assumed that all tensors are square!
\note The method is defined for spatial dimensions D = 1, 2, 3
\code
C - num. integration domains
P - num. integration points
D - spatial dimension
\endcode
\param outputData [out] - Output (matrix-vector product) data array.
\param inputDataLeft [in] - Left input data array.
\param inputDataRight [in] - Right input data array.
\param transpose [in] - If 'T', use transposed tensor; if 'N', no transpose. Default: 'N'.
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void tensorMultiplyDataData(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const char transpose = 'N');
/* template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight>
static void tensorMultiplyDataDataTemp(ArrayOutData & outputData,
const ArrayInDataLeft & inputDataLeft,
const ArrayInDataRight & inputDataRight,
const char transpose = 'N');
*/
template<class Scalar, class ArrayOutData, class ArrayInDataLeft, class ArrayInDataRight,int outvalRank>
struct tensorMultiplyDataDataTempSpec;
/** \brief Applies left (row) signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,L), to the operator
<var><b>inoutOperator</b></var> indexed by (C,L,R).
Mathematically, this method computes the matrix-matrix product
\f[
\mathbf{K}^{c} = \mbox{diag}(\sigma^c_0,\ldots,\sigma^c_{L-1}) \mathbf{K}^c
\f]
where \f$\mathbf{K}^{c} \in \mathbf{R}^{L\times R}\f$ is array of matrices indexed by
cell number \e c and stored in the rank-3 array \e inoutOperator, and
\f$\{\sigma^c_l\}_{l=0}^{L-1}\f$ is array of left field signs indexed by cell number \e c
and stored in the rank-2 container \e fieldSigns;
see Section \ref sec_pullbacks for discussion of field signs. This operation is
required for operators generated by \e HCURL and \e HDIV-conforming vector-valued
finite element basis functions; see Sections \ref sec_pullbacks and Section
\ref sec_ops for applications of this method.
\code
C - num. integration domains
L - num. left fields
R - num. right fields
\endcode
\param inoutOperator [in/out] - Input / output operator array.
\param fieldSigns [in] - Left field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyLeftFieldSigns(ArrayTypeInOut & inoutOperator,
const ArrayTypeSign & fieldSigns);
/** \brief Applies right (column) signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,R), to the operator
<var><b>inoutOperator</b></var> indexed by (C,L,R).
Mathematically, this method computes the matrix-matrix product
\f[
\mathbf{K}^{c} = \mathbf{K}^c \mbox{diag}(\sigma^c_0,\ldots,\sigma^c_{R-1})
\f]
where \f$\mathbf{K}^{c} \in \mathbf{R}^{L\times R}\f$ is array of matrices indexed by
cell number \e c and stored in the rank-3 container \e inoutOperator, and
\f$\{\sigma^c_r\}_{r=0}^{R-1}\f$ is array of right field signs indexed by cell number \e c
and stored in the rank-2 container \e fieldSigns;
see Section \ref sec_pullbacks for discussion of field signs. This operation is
required for operators generated by \e HCURL and \e HDIV-conforming vector-valued
finite element basis functions; see Sections \ref sec_pullbacks and Section
\ref sec_ops for applications of this method.
\code
C - num. integration domains
L - num. left fields
R - num. right fields
\endcode
\param inoutOperator [in/out] - Input / output operator array.
\param fieldSigns [in] - Right field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyRightFieldSigns(ArrayTypeInOut & inoutOperator,
const ArrayTypeSign & fieldSigns);
/** \brief Applies field signs, stored in the user-provided container
<var><b>fieldSigns</b></var> and indexed by (C,F), to the function
<var><b>inoutFunction</b></var> indexed by (C,F), (C,F,P),
(C,F,P,D1) or (C,F,P,D1,D2).
Returns
\f[
\mbox{inoutFunction}(c,f,*) = \mbox{fieldSigns}(c,f)*\mbox{inoutFunction}(c,f,*)
\f]
See Section \ref sec_pullbacks for discussion of field signs.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - spatial dimension
D2 - spatial dimension
\endcode
\param inoutFunction [in/out] - Input / output function array.
\param fieldSigns [in] - Right field signs.
*/
template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyFieldSigns(ArrayTypeInOut & inoutFunction,
const ArrayTypeSign & fieldSigns);
/* template<class Scalar, class ArrayTypeInOut, class ArrayTypeSign>
static void applyFieldSignsTemp(ArrayTypeInOut & inoutFunction,
const ArrayTypeSign & fieldSigns);
*/
/** \brief Computes point values \a <b>outPointVals</b> of a discrete function
specified by the basis \a <b>inFields</b> and coefficients
\a <b>inCoeffs</b>.
The array \a <b>inFields</b> with dimensions (C,F,P), (C,F,P,D1),
or (C,F,P,D1,D2) represents the signed, transformed field (basis) values at
points in REFERENCE frame; the \a <b>outPointVals</b> array with
dimensions (C,P), (C,P,D1), or (C,P,D1,D2), respectively, represents
values of a discrete function at points in PHYSICAL frame.
The array \a <b>inCoeffs</b> dimensioned (C,F) supplies the coefficients
for the field (basis) array.
Returns rank-2,3 or 4 array such that
\f[
outPointValues(c,p,*) = \sum_{f=0}^{F-1} \sigma_{c,f} u_{c,f}(x_p)
\f]
where \f$\{u_{c,f}\}_{f=0}^{F-1} \f$ is scalar, vector or tensor valued finite element
basis defined on physical cell \f$\mathcal{C}\f$ and \f$\{\sigma_{c,f}\}_{f=0}^{F-1} \f$
are the field signs of the basis functions; see Section \ref sec_pullbacks.
This method implements the last step in a four step process; please see Section
\ref sec_evaluate for details about the first three steps that prepare the
necessary data for this method.
\code
C - num. integration domains
F - num. fields
P - num. integration points
D1 - spatial dimension
D2 - spatial dimension
\endcode
\param outPointVals [out] - Output point values of a discrete function.
\param inCoeffs [in] - Coefficients associated with the fields (basis) array.
\param inFields [in] - Field (basis) values.
*/
template<class Scalar, class ArrayOutPointVals, class ArrayInCoeffs, class ArrayInFields>
static void evaluate(ArrayOutPointVals & outPointVals,
const ArrayInCoeffs & inCoeffs,
const ArrayInFields & inFields);
}; // end FunctionSpaceTools
} // end namespace Intrepid
// include templated definitions
#include <Intrepid_FunctionSpaceToolsDef.hpp>
#endif
/***************************************************************************************************
** **
** D O C U M E N T A T I O N P A G E S **
** **
**************************************************************************************************/
/**
\page function_space_tools_page Function space tools
<b>Table of contents </b>
\li \ref sec_fst_overview
\li \ref sec_pullbacks
\li \ref sec_measure
\li \ref sec_evaluate
\section sec_fst_overview Overview
Intrepid::FunctionSpaceTools is a stateless class of \e expert \e methods for operations on finite
element subspaces of \f$H(grad,\Omega)\f$, \f$H(curl,\Omega)\f$, \f$H(div,\Omega)\f$ and \f$L^2(\Omega)\f$.
In Intrepid these spaces are referred to as \e HGRAD, \e HCURL, \e HDIV and \e HVOL. There are four
basic groups of methods:
- Transformation methods provide implementation of pullbacks for \e HGRAD, \e HCURL, \e HDIV and \e HVOL
finite element functions. Thease are essentialy the "change of variables rules" needed to transform
values of basis functions and their derivatives defined on a reference element \f$\widehat{{\mathcal C}}\f$
to a physical element \f${\mathcal C}\f$. See Section \ref sec_pullbacks for details
- Measure computation methods implement the volume, surface and line measures required for computation
of integrals in the physical frame by changing variables to reference frame. See Section \ref sec_measure
for details.
- Integration methods implement the algebraic operations to compute ubiquitous integrals of finite element
functions: integrals arising in bilinear forms and linear functionals.
- Methods for algebraic and vector-algebraic operations on multi-dimensional arrays with finite element
function values. These methods are used to prepare multidimensional arrays with data and finite
element function values for the integration routines. They also include evaluation methods to compute
finite element function values at some given points in physical frame; see Section \ref sec_evaluate.
\section sec_pullbacks Pullbacks
Notation in this section follows the standard definition of a finite element space by Ciarlet; see
<var> The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, SIAM, 2002. </var>
Given a reference cell \f$\{\widehat{{\mathcal C}},\widehat{P},\widehat{\Lambda}\}\f$ with a basis
\f$\{\widehat{u}_i\}_{i=0}^n\f$, the basis \f$\{{u}_i\}_{i=0}^n\f$ of \f$\{{\mathcal C},P,\Lambda\}\f$ is defined
as follows:
\f[
u_i = \sigma_i \Phi^*(\widehat{u}_i), \qquad i=1,\ldots,n \,.
\f]
In this formula \f$\{\sigma_i\}_{i=0}^n\f$, where \f$\sigma_i = \pm 1\f$, are the \e field \e signs,
and \f$\Phi^*\f$ is the \e pullback ("change of variables") transformation. For scalar spaces
such as \e HGRAD and \e HVOL the field signs are always equal to 1 and can be disregarded. For vector
field spaces such as \e HCURL or \e HDIV, the field sign of a basis function can be +1 or -1,
depending on the orientation of the physical edge or face, associated with the basis function.
The actual form of the pullback depends on which one of the four function spaces \e HGRAD, \e HCURL,
\e HDIV and \e HVOL is being approximated and is computed as follows. Let \f$F_{\mathcal C}\f$
denote the reference-to-physical map (see Section \ref sec_cell_topology_ref_map);
\f$DF_{\mathcal C}\f$ is its Jacobian (see Section \ref sec_cell_topology_ref_map_DF) and
\f$J_{\mathcal C} = \det(DF_{\mathcal C})\f$. Then,
\f[
\begin{array}{ll}
\Phi^*_G : HGRAD(\widehat{{\mathcal C}}) \mapsto HGRAD({\mathcal C})&
\qquad \Phi^*_G(\widehat{u}) = \widehat{u}\circ F^{-1}_{\mathcal C} \\[2ex]
\Phi^*_C : HCURL(\widehat{{\mathcal C}}) \mapsto HCURL({\mathcal C})&
\qquad \Phi^*_C(\widehat{\bf u}) = \left((DF_{\mathcal C})^{-{\sf T}}\cdot\widehat{\bf u}\right)\circ F^{-1}_{\mathcal C} \\[2ex]
\Phi^*_D : HDIV(\widehat{{\mathcal C}}) \mapsto HDIV({\mathcal C})&
\qquad \Phi^*_D(\widehat{\bf u}) = \left(J^{-1}_{\mathcal C} DF_{\mathcal C}\cdot\widehat{\bf u}\right)\circ F^{-1}_{\mathcal C}
\\[2ex]
\Phi^*_S : HVOL(\widehat{{\mathcal C}}) \mapsto HVOL({\mathcal C})&
\qquad \Phi^*_S(\widehat{u}) = \left(J^{-1}_{\mathcal C} \widehat{u}\right) \circ F^{-1}_{\mathcal C} \,.
\end{array}
\f]
Intrepid supports pullbacks only for cell topologies that have reference cells; see
\ref cell_topology_ref_cells.
\section sec_measure Measure
In Intrepid integrals of finite element functions over cells, 2-subcells (faces) and 1-subcells (edges)
are computed by change of variables to reference frame and require three different kinds of measures.
-# The integral of a scalar function over a cell \f${\mathcal C}\f$
\f[
\int_{{\mathcal C}} f(x) dx = \int_{\widehat{{\mathcal C}}} f(F(\widehat{x})) |J | d\widehat{x}
\f]
requires the volume measure defined by the determinant of the Jacobian. This measure is computed
by Intrepid::FunctionSpaceTools::computeCellMeasure
-# The integral of a scalar function over 2-subcell \f$\mathcal{F}\f$
\f[
\int_{\mathcal{F}} f(x) dx = \int_{R} f(\Phi(u,v))
\left\|\frac{\partial\Phi}{\partial u}\times \frac{\partial\Phi}{\partial v}\right\| du\,dv
\f]
requires the surface measure defined by the norm of the vector product of the surface tangents. This
measure is computed by Intrepid::FunctionSpaceTools::computeFaceMeasure. In this formula \e R is the parametrization
domain for the 2-subcell; see Section \ref sec_cell_topology_subcell_map for details.
-# The integral of a scalar function over a 1-subcell \f$\mathcal{E}\f$
\f[
\int_{\mathcal{E}} f(x) dx = \int_{R} f(\Phi(s)) \|\Phi'\| ds
\f]
requires the arc measure defined by the norm of the arc tangent vector. This measure is computed
by Intrepid::FunctionSpaceTools::computeEdgeMeasure. In this formula \e R is the parametrization
domain for the 1-subcell; see Section \ref sec_cell_topology_subcell_map for details.
\section sec_evaluate Evaluation of finite element fields
To make this example more specific, assume curl-conforming finite element spaces.
Suppose that we have a physical cell \f$\{{\mathcal C},P,\Lambda\}\f$ with a basis
\f$\{{\bf u}_i\}_{i=0}^n\f$. A finite element function on this cell is defined by a set of \e n
coefficients \f$\{c_i\}_{i=0}^n\f$:
\f[
{\bf u}^h(x) = \sum_{i=0}^n c_i {\bf u}_i(x) \,.
\f]
From Section \ref sec_pullbacks it follows that
\f[
{\bf u}^h(x) = \sum_{i=0}^n c_i \sigma_i
\left((DF_{\mathcal C})^{-{\sf T}}\cdot\widehat{\bf u}_i\right)\circ
F^{-1}_{\mathcal C}(x)
= \sum_{i=0}^n c_i \sigma_i
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,,
\f]
where \f$ \widehat{x} = F^{-1}_{\mathcal C}(x) \in \widehat{\mathcal C} \f$ is the pre-image
of \e x in the reference cell.
Consequently, evaluation of finite element functions at a given set of points
\f$\{x_p\}_{p=0}^P \subset {\mathcal C}\f$ comprises of the following four steps:
-# Application of the inverse map \f$F^{-1}_{\mathcal C}\f$ to obtain the pre-images
\f$\{\widehat{x}_p\}_{p=0}^P\f$ of the evaluation points in the reference cell
\f$\widehat{\mathcal{C}}\f$; see Intrepid::CellTools::mapToReferenceFrame
-# Evaluation of the appropriate reference basis set \f$\{\widehat{\bf u}_i\}_{i=1}^n\f$
at the pre-image set \f$\{\widehat{x}_p\}_{p=0}^P\f$; see Intrepid::Basis::getValues
-# Application of the appropriate transformation and field signs. In our example the finite
element space is curl-conforming and the appropriate transformation is implemented in
Intrepid::FunctionSpaceTools::HCURLtransformVALUE. Application of the signs to the
transformed functions is done by Intrepid::FunctionSpaceTools::applyFieldSigns.
-# The final step is to compute the sum of the transformed and signed basis function values
multiplied by the coefficients of the finite element function using
Intrepid::FunctionSpaceTools::evaluate.
Evaluation of adimssible derivatives of finite element functions is completely analogous
and follows the same four steps. Evaluation of scalar finite element functions is simpler
because application of the signes can be skipped for these functions.
\section sec_ops Evaluation of finite element operators and functionals
Assume the same setting as in Section \ref sec_evaluate. A finite element operator defined
by the finite element basis on the physical cell \f$\mathcal{C}\f$ is a matrix
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} = \int_{\mathcal C} {\mathcal L}_L {\bf u}_i(x)\, {\mathcal L}_R {\bf u}_j(x) \, dx \,.
\f]
where \f${\mathcal L}_L\f$ and \f${\mathcal L}_R \f$ are \e left and \e right operators acting on the basis
functions. Typically, when the left and the right basis functions are from the same finite
element basis (as in this example), the left and right operators are the same. If they are set
to \e VALUE we get a mass matrix; if they are set to an admissible differential operator we get
a stiffnesss matrix. Assume again that the basis is curl-conforming and the operators are
set to \e VALUE. Using the basis definition from Section \ref sec_pullbacks we have that
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} = \int_{\widehat{\mathcal C}} \sigma_i \sigma_j
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\cdot
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
It follows that
\f[
\mathbf{K}^{\mathcal{C}}_{i,j} =
\mbox{diag}(\sigma_0,\ldots,\sigma_n)\widehat{\mathbf{K}}^{\mathcal{C}}\mbox{diag}(\sigma_0,\ldots,\sigma_n)
\f]
where
\f[
\widehat{\mathbf{K}}^{\mathcal{C}}_{i,j} = \int_{\widehat{\mathcal C}}
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\cdot
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
is the raw cell operator matrix. The methods Intrepid::FunctionSpaceTools::applyLeftFieldSigns and
Intrepid::FunctionSpaceTools::applyRightFieldSigns apply the left and right diagonal sign matrices to
the raw cell operator.
A finite element operator defined by the finite element basis on the physical cell is a vector
\f[
\mathbf{f}^{\mathcal{C}}_{i} = \int_{\mathcal C} f(x) {\mathcal L}_R u_i(x) \, dx \,.
\f]
Assuming again operator \e VALUE and using the same arguments as above, we see that
\f[
\mathbf{f}^{\mathcal{C}} =
\mbox{diag}(\sigma_0,\ldots,\sigma_n)\widehat{\mathbf{f}}^{\mathcal{C}}\,,
\f]
where
\f[
\widehat{\mathbf{f}}^{\mathcal{C}} = \int_{\widehat{\mathcal C}}
\mathbf{f}\circ F_{\mathcal C}(\widehat{x})
(DF_{\mathcal C}(\widehat{x}))^{-{\sf T}}\cdot\widehat{\bf u}_i(\widehat{x})\,d\widehat{x}
\f]
is the raw cell functional.
*/
|