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// Intrepid Package
// Copyright (2007) Sandia Corporation
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// 1. Redistributions of source code must retain the above copyright
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// 2. Redistributions in binary form must reproduce the above copyright
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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
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// Denis Ridzal (dridzal@sandia.gov), or
// Kara Peterson (kjpeter@sandia.gov)
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/** \file Intrepid_HDIV_TET_I1_FEM.hpp
\brief Header file for the Intrepid::HDIV_TET_I1_FEM class.
\author Created by P. Bochev and D. Ridzal and K. Petrson.
*/
#ifndef INTREPID_HDIV_TET_I1_FEM_HPP
#define INTREPID_HDIV_TET_I1_FEM_HPP
#include "Intrepid_Basis.hpp"
namespace Intrepid {
/** \class Intrepid::Basis_HDIV_TET_I1_FEM
\brief Implementation of the default H(div)-compatible FEM basis of degree 1 on Tetrahedron cell
Implements Raviart-Thomas basis of degree 1 on the reference Tetrahedron cell. The basis has
cardinality 4 and spans an INCOMPLETE linear polynomial space. Basis functions are dual
to a unisolvent set of degrees-of-freedom (DoF) defined and enumerated as follows:
\verbatim
=========================================================================================================
| | degree-of-freedom-tag table | |
| DoF |----------------------------------------------------------| DoF definition |
| ordinal | subc dim | subc ordinal | subc DoF ord |subc num DoF | |
|=========|==============|==============|==============|=============|==================================|
| 0 | 2 | 0 | 0 | 1 | L_0(u) = (u.n)(1/3,0,1/3) |
|---------|--------------|--------------|--------------|-------------|----------------------------------|
| 1 | 2 | 1 | 0 | 1 | L_1(u) = (u.n)(1/3,1/3,1/3) |
|---------|--------------|--------------|--------------|-------------|----------------------------------|
| 2 | 2 | 2 | 0 | 1 | L_2(u) = (u.n)(0,1/3,1/3) |
|---------|--------------|--------------|--------------|-------------|----------------------------------|
| 3 | 2 | 3 | 0 | 1 | L_3(u) = (u.n)(1/3,1/3,0) |
|=========|==============|==============|==============|=============|==================================|
| MAX | maxScDim=2 | maxScOrd=3 | maxDfOrd=0 | - | |
|=========|==============|==============|==============|=============|==================================|
\endverbatim
\remarks
\li In the DoF functional \f${\bf n}\f$ is a face normal. Direction of face normals
is determined by the right-hand rule applied to faces oriented by their vertex order
in the cell topology, from face vertex 0 to last face vertex, whereas their length is
set equal to face area (see http://mathworld.wolfram.com/Right-HandRule.html for definition
of right-hand rule). For example, face 1 of all Tetrahedron cells has vertex order {1,2,3}
and its right-hand rule normal can be computed, e.g., by the vector product of edge
tangents to edges {1,2} and {2,3}. On the reference Tetrahedron the coordinates of
face 1 vertices are (1,0,0), (0,1,0), and (0,0,1), the edge tangents are (-1,1,0) and
(0,-1,1) and the face normal direction is (-1,1,0) X (0,-1,1) = (1,1,1). Length of this
raw face normal is twice the face area of face 1 and so the final face normal to face 1 is
obtained by scaling the raw normal by 1/2: (1/2,1/2,1/2).
\li The length of the face normal equals the face area. As a result, the DoF functional
is the value of the normal component of a vector field at the face center times the
face area. The resulting basis is equivalent to a basis defined by using the face
flux as a DoF functional. Note that faces 0, 2, and 3 of reference Tetrahedron<>
cells have area 1/2 and face 1 has area Sqrt(3)/2.
*/
template<class Scalar, class ArrayScalar>
class Basis_HDIV_TET_I1_FEM : public Basis<Scalar, ArrayScalar>, public DofCoordsInterface<ArrayScalar> {
private:
/** \brief Initializes <var>tagToOrdinal_</var> and <var>ordinalToTag_</var> lookup arrays.
*/
void initializeTags();
public:
/** \brief Constructor.
*/
Basis_HDIV_TET_I1_FEM();
/** \brief Evaluation of a FEM basis on a <strong>reference Tetrahedron</strong> cell.
Returns values of <var>operatorType</var> acting on FEM basis functions for a set of
points in the <strong>reference Tetrahedron</strong> cell. For rank and dimensions of
I/O array arguments see Section \ref basis_md_array_sec.
\param outputValues [out] - rank-3 or 4 array with the computed basis values
\param inputPoints [in] - rank-2 array with dimensions (P,D) containing reference points
\param operatorType [in] - operator applied to basis functions
*/
void getValues(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const EOperator operatorType) const;
/** \brief FVD basis evaluation: invocation of this method throws an exception.
*/
void getValues(ArrayScalar & outputValues,
const ArrayScalar & inputPoints,
const ArrayScalar & cellVertices,
const EOperator operatorType = OPERATOR_VALUE) const;
/** \brief Returns spatial locations (coordinates) of degrees of freedom on a
<strong>reference Tetrahedron</strong>.
\param DofCoords [out] - array with the coordinates of degrees of freedom,
dimensioned (F,D)
*/
void getDofCoords(ArrayScalar & DofCoords) const;
};
}// namespace Intrepid
#include "Intrepid_HDIV_TET_I1_FEMDef.hpp"
#endif
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