Intrepid
Public Member Functions | Private Member Functions
Intrepid::Basis_HCURL_HEX_I1_FEM< Scalar, ArrayScalar > Class Template Reference

Implementation of the default H(curl)-compatible FEM basis of degree 1 on Hexahedron cell. More...

#include <Intrepid_HCURL_HEX_I1_FEM.hpp>

Inheritance diagram for Intrepid::Basis_HCURL_HEX_I1_FEM< Scalar, ArrayScalar >:
Intrepid::Basis< Scalar, ArrayScalar > Intrepid::DofCoordsInterface< ArrayScalar >

List of all members.

Public Member Functions

 Basis_HCURL_HEX_I1_FEM ()
 Constructor.
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const EOperator operatorType) const
 Evaluation of a FEM basis on a reference Hexahedron cell.
void getValues (ArrayScalar &outputValues, const ArrayScalar &inputPoints, const ArrayScalar &cellVertices, const EOperator operatorType=OPERATOR_VALUE) const
 FVD basis evaluation: invocation of this method throws an exception.
void getDofCoords (ArrayScalar &DofCoords) const
 Returns spatial locations (coordinates) of degrees of freedom on a reference Quadrilateral.

Private Member Functions

void initializeTags ()
 Initializes tagToOrdinal_ and ordinalToTag_ lookup arrays.

Detailed Description

template<class Scalar, class ArrayScalar>
class Intrepid::Basis_HCURL_HEX_I1_FEM< Scalar, ArrayScalar >

Implementation of the default H(curl)-compatible FEM basis of degree 1 on Hexahedron cell.

Implements Nedelec basis of degree 1 on the reference Hexahedron cell. The basis has cardinality 12 and spans an INCOMPLETE tri-linear polynomial space. Basis functions are dual to a unisolvent set of degrees-of-freedom (DoF) defined and enumerated as follows:

  ===================================================================================================
  |         |           degree-of-freedom-tag table                    |                            |
  |   DoF   |----------------------------------------------------------|       DoF definition       |
  | ordinal |  subc dim    | subc ordinal | subc DoF ord |subc num DoF |                            |
  |=========|==============|==============|==============|=============|============================|
  |    0    |       1      |       0      |       0      |      1      |   L_0(u) = (u.t)(0,-1,-1)  |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    1    |       1      |       1      |       0      |      1      |   L_1(u) = (u.t)(1,0,-1)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    2    |       1      |       2      |       0      |      1      |   L_2(u) = (u.t)(0,1,-1)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    3    |       1      |       3      |       0      |      1      |   L_3(u) = (u.t)(-1,0,-1)  |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    4    |       1      |       4      |       0      |      1      |   L_4(u) = (u.t)(0,-1,1)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    5    |       1      |       5      |       0      |      1      |   L_5(u) = (u.t)(1,0,1)    |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    6    |       1      |       6      |       0      |      1      |   L_6(u) = (u.t)(0,1,1)    |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    7    |       1      |       7      |       0      |      1      |   L_7(u) = (u.t)(-1,0,1)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    8    |       1      |       8      |       0      |      1      |   L_8(u) = (u.t)(-1,-1,0)  |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |    9    |       1      |       9      |       0      |      1      |   L_9(u) = (u.t)(1,-1,0)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |   10    |       1      |      10      |       0      |      1      |   L_10(u) = (u.t)(1,1,0)   |
  |---------|--------------|--------------|--------------|-------------|----------------------------|
  |   11    |       1      |      11      |       0      |      1      |   L_11(u) = (u.t)(-1,1,0)  |
  |=========|==============|==============|==============|=============|============================|
  |   MAX   |  maxScDim=1  |  maxScOrd=11 |  maxDfOrd=0  |      -      |                            |
  |=========|==============|==============|==============|=============|============================|
Remarks:
  • In the DoF functional ${\bf t}$ is an edge tangent. Direction of edge tangents follows the vertex order of the edges in the cell topology and runs from edge vertex 0 to edge vertex 1, whereas their length is set equal to edge length. For example, edge 8 of all Hexahedron reference cells has vertex order {0,4}, i.e., its tangent runs from vertex 0 of the reference Hexahedron to vertex 4 of that cell. On the reference Hexahedron the coordinates of these vertices are (-1,-1,-1) and (-1,-1,1), respectively. Therefore, the tangent to edge 8 is (-1,-1, 1) - (-1,-1,-1) = (0, 0, 2). Because its length already equals edge length, no further rescaling of the edge tangent is needed.
  • The length of the edge tangent equals the edge length. As a result, the DoF functional is the value of the tangent component of a vector field at the edge midpoint times the edge length. The resulting basis is equivalent to a basis defined by using the edge circulation as a DoF functional. Note that all edges of reference Hexahedron<> cells have length 2.

Definition at line 115 of file Intrepid_HCURL_HEX_I1_FEM.hpp.

Member Function Documentation

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HCURL_HEX_I1_FEM< Scalar, ArrayScalar >::getDofCoords ( ArrayScalar &  DofCoords) const [virtual]

Returns spatial locations (coordinates) of degrees of freedom on a reference Quadrilateral.

Parameters:
DofCoords[out] - array with the coordinates of degrees of freedom, dimensioned (F,D)

Implements Intrepid::DofCoordsInterface< ArrayScalar >.

Definition at line 302 of file Intrepid_HCURL_HEX_I1_FEMDef.hpp.

template<class Scalar , class ArrayScalar >
void Intrepid::Basis_HCURL_HEX_I1_FEM< Scalar, ArrayScalar >::getValues ( ArrayScalar &  outputValues,
const ArrayScalar &  inputPoints,
const EOperator  operatorType 
) const [virtual]

Evaluation of a FEM basis on a reference Hexahedron cell.

Returns values of operatorType acting on FEM basis functions for a set of points in the reference Hexahedron cell. For rank and dimensions of I/O array arguments see Section MD array template arguments for basis methods.

Parameters:
outputValues[out] - rank-3 array with the computed basis values
inputPoints[in] - rank-2 array with dimensions (P,D) containing reference points
operatorType[in] - operator applied to basis functions

Implements Intrepid::Basis< Scalar, ArrayScalar >.

Definition at line 100 of file Intrepid_HCURL_HEX_I1_FEMDef.hpp.

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