Vectorial Nedelec shape functions on the unit simplex. More...
#include <nedelecshapefunctions.hh>
Detailed Description
class Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >
Vectorial Nedelec shape functions on the unit simplex.
These are linear shape functions associated to edges. In barycentric coordinates \( \lambda \), with \( \lambda_i = x_i \) for \( i=0,\dots, \mathrm{dim}-1 \) and \( \lambda_\mathrm{dim} = 1-\sum_{i=0}^{\mathrm{dim}-1} x_i \) on the unit simplex, the shape function associated to the edge \( e = (p_i,p_j) \) is
\[ \phi_e = \pm (\lambda_i \nabla \lambda_j - \lambda_j \nabla \lambda i). \]
Here, \( p_k \) is the vertex with \( \lambda_k = 1 \). The sign of the shape function depends on the numbering of vertices relative to the edge implemented by the grid manager.
- Template Parameters
-
ctype the coordinate type (a real number type) T the value type (a real number type) D the dimension of the unit simplex over which the shape functions are defined
Definition at line 45 of file nedelecshapefunctions.hh.
Public Types | |
| typedef ctype | CoordType |
| typedef T | ResultType |
Public Member Functions | |
| NedelecSimplexShapeFunction () | |
| NedelecSimplexShapeFunction (int e) | |
| virtual Dune::FieldVector< T, dim > | evaluateFunction (Dune::FieldVector< ctype, dim > const &x) const |
| Evaluates the shape function at point x. More... | |
| virtual Dune::FieldMatrix< ResultType, dim, dim > | evaluateDerivative (Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the derivative of the shape function. More... | |
| virtual Tensor< ResultType, 1, dim, dim > | evaluate2ndDerivative (Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the second derivative of the shape function. More... | |
| virtual std::tuple< int, int, int, int > | location () const |
| Returns a tuple (nominalOrder,codim,entity,index) giving detailed information about the location of the shape function. More... | |
| Dune::FieldVector< CoordType, dim > | position () const |
| Returns the element-local position of the node associated to the shape function. More... | |
| Dune::FieldVector< CoordType, dim > | edge () const |
| Returns the edge vector of the edge associated to this shape function. More... | |
Static Public Attributes | |
| static int const | dim = dimension |
| static int const | comps = dim |
| static int const | order = 1 |
Member Typedef Documentation
◆ CoordType
| typedef ctype Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >::CoordType |
Definition at line 52 of file nedelecshapefunctions.hh.
◆ ResultType
| typedef T Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >::ResultType |
Definition at line 53 of file nedelecshapefunctions.hh.
Constructor & Destructor Documentation
◆ NedelecSimplexShapeFunction() [1/2]
|
inline |
Default constructor. Constructs the shape function associated with the edge 0. Not particularly useful, but it's often convenient to have a default constructible class.
Definition at line 60 of file nedelecshapefunctions.hh.
◆ NedelecSimplexShapeFunction() [2/2]
|
inlineexplicit |
Creates a shape function associated to the edge e.
- Parameters
-
e subentity index of the edge (codim=dim-1 subentity index)
Definition at line 70 of file nedelecshapefunctions.hh.
Member Function Documentation
◆ edge()
|
inline |
Returns the edge vector of the edge associated to this shape function.
Definition at line 175 of file nedelecshapefunctions.hh.
◆ evaluate2ndDerivative()
|
inlinevirtual |
Evaluates the second derivative of the shape function.
The result is r[i][j][k] = \( \partial^2 \phi_i / (\partial x_j \partial x_k) = 0. \)
The point x is not restricted to be inside the unit simplex, but the meaning of evaluating a shape function outside of the unit simplex is questionable.
Reimplemented from Kaskade::ShapeFunction< ctype, dimension, T, dimension >.
Definition at line 156 of file nedelecshapefunctions.hh.
◆ evaluateDerivative()
|
inlinevirtual |
Evaluates the derivative of the shape function.
The point x is not restricted to be inside the unit simplex, but the meaning of evaluating a shape function outside of the unit simplex is questionable.
Implements Kaskade::ShapeFunction< ctype, dimension, T, dimension >.
Definition at line 115 of file nedelecshapefunctions.hh.
◆ evaluateFunction()
|
inlinevirtual |
Evaluates the shape function at point x.
The point x is not restricted to be inside the unit simplex, but the meaning of evaluating a shape function outside of the unit simplex is questionable.
Implements Kaskade::ShapeFunction< ctype, dimension, T, dimension >.
Definition at line 102 of file nedelecshapefunctions.hh.
◆ location()
|
inlinevirtual |
Returns a tuple (nominalOrder,codim,entity,index) giving detailed information about the location of the shape function.
Each shape function is associated to a certain subentity of the element.
nominalOrder is a nonnegative ordering parameter that is usually the polynomial order of the shape function, but need not coincide.
codim is the codimension of the subentity to which the shape function is associated, entity is the number of the subentity, and index is the number of the shape function among those that are associated to the same subentity.
Implements Kaskade::ShapeFunction< ctype, dimension, T, dimension >.
Definition at line 162 of file nedelecshapefunctions.hh.
◆ position()
|
inline |
Returns the element-local position of the node associated to the shape function.
Definition at line 170 of file nedelecshapefunctions.hh.
Member Data Documentation
◆ comps
|
static |
Definition at line 49 of file nedelecshapefunctions.hh.
◆ dim
|
static |
Definition at line 48 of file nedelecshapefunctions.hh.
Referenced by Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >::evaluateDerivative(), Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >::location(), and Kaskade::NedelecSimplexShapeFunction< ctype, dimension, T >::NedelecSimplexShapeFunction().
◆ order
|
static |
Definition at line 50 of file nedelecshapefunctions.hh.
The documentation for this class was generated from the following file:
