Scalar Lagrange shape functions on the unit cube. More...
#include <lagrangeshapefunctions.hh>
Detailed Description
class Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >
Scalar Lagrange shape functions on the unit cube.
These are polynomial shape functions of fixed given order associated to certain interpolation nodes. A shape function is 1 at its associated node and 0 at all the other nodes.
The nodes form a cartesian grid with (order+1) nodes on each axis. The node placement is isotropic (the same node placement on each axis). In the current implementation, quasi Chebyshev nodes are used (due to their better interpolation properties compared to equidistant nodes). Each node is referenced by its index: a dim-tuple of integers with values between 0 and order (including), such that the maximum entry is at most order.
- Template Parameters
-
ctype the scalar type for coordinates dimension the spatial dimension Scalar the value type (a real number type) O the polynomial order
Definition at line 613 of file lagrangeshapefunctions.hh.
Public Types | |
| typedef ctype | CoordType |
| typedef Scalar | ResultType |
Public Member Functions | |
| LagrangeCubeShapeFunction () | |
| LagrangeCubeShapeFunction (LagrangeCubeShapeFunction const &other) | |
| LagrangeCubeShapeFunction (Dune::FieldVector< int, dimension > const &xi_) | |
| Creates a shape function associated to the node with indices xi_. More... | |
| ResultType | evaluateFunction (int, Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the shape function at point x. More... | |
| virtual Dune::FieldVector< Scalar, 1 > | evaluateFunction (Dune::FieldVector< ctype, dim > const &x) const |
| Evaluates the shape function (all components at once). More... | |
| ResultType | evaluateDerivative (int, int dir, Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the partial derivative of the shape function in coordinate direction dir. More... | |
| virtual Dune::FieldMatrix< ResultType, 1, dim > | evaluateDerivative (Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the derivative of the shape function (all components and all directions at once). More... | |
| virtual Tensor< ResultType, 1, dim, dim > | evaluate2ndDerivative (Dune::FieldVector< CoordType, dim > const &x) const |
| Evaluates the second derivative of the shape function (all components and all directions at once). More... | |
| int | localindex (int) const |
| Returns the local index of the shape function. More... | |
| int | codim () const |
| Returns the codimension of the subentity on which the associated node is located. More... | |
| int | entity () const |
| Returns the subentity number of the smallest codimension subentity on which the associated node is located. More... | |
| template<class Cell > | |
| int | entityIndex (Cell const &) const |
| Returns the local index on the subentity. More... | |
| Dune::FieldVector< CoordType, dim > | position () const |
| Returns the element-local position of the node associated to 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... | |
Static Public Attributes | |
| static int const | dim = dimension |
| static int const | comps = 1 |
| static int const | order = O |
Member Typedef Documentation
◆ CoordType
| typedef ctype Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::CoordType |
Definition at line 620 of file lagrangeshapefunctions.hh.
◆ ResultType
| typedef Scalar Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::ResultType |
Definition at line 621 of file lagrangeshapefunctions.hh.
Constructor & Destructor Documentation
◆ LagrangeCubeShapeFunction() [1/3]
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inline |
Default constructor. Constructs the shape function located at the origin of the reference element. Not particularly useful, but it's often convenient to have a default constructible class.
Definition at line 628 of file lagrangeshapefunctions.hh.
◆ LagrangeCubeShapeFunction() [2/3]
|
inline |
Definition at line 632 of file lagrangeshapefunctions.hh.
◆ LagrangeCubeShapeFunction() [3/3]
|
explicit |
Creates a shape function associated to the node with indices xi_.
The entries in xi_ have to be nonnegative, and their maximum must not exceed order.
Member Function Documentation
◆ codim()
|
inline |
Returns the codimension of the subentity on which the associated node is located.
Definition at line 702 of file lagrangeshapefunctions.hh.
◆ entity()
|
inline |
Returns the subentity number of the smallest codimension subentity on which the associated node is located.
Definition at line 708 of file lagrangeshapefunctions.hh.
◆ entityIndex()
|
inline |
Returns the local index on the subentity.
For higher order shape functions, globally unique numberings on intersections have to be used, therefore this depends on the actual element and an index set that covers this element.
Definition at line 718 of file lagrangeshapefunctions.hh.
◆ evaluate2ndDerivative()
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inlinevirtual |
Evaluates the second derivative of the shape function (all components and all directions at once).
Reimplemented from Kaskade::ShapeFunction< ctype, dimension, Scalar >.
Definition at line 685 of file lagrangeshapefunctions.hh.
◆ evaluateDerivative() [1/2]
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inlinevirtual |
Evaluates the derivative of the shape function (all components and all directions at once).
Implements Kaskade::ShapeFunction< ctype, dimension, Scalar >.
Definition at line 676 of file lagrangeshapefunctions.hh.
◆ evaluateDerivative() [2/2]
|
inline |
Evaluates the partial derivative of the shape function in coordinate direction dir.
0 <= dir < dim has to hold. The point x is not restricted to be inside the unit cube, but the meaning of evaluating a shape function outside of the unit cube is questionable.
Definition at line 670 of file lagrangeshapefunctions.hh.
◆ evaluateFunction() [1/2]
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inlinevirtual |
Evaluates the shape function (all components at once).
Implements Kaskade::ShapeFunction< ctype, dimension, Scalar >.
Definition at line 657 of file lagrangeshapefunctions.hh.
◆ evaluateFunction() [2/2]
|
inline |
Evaluates the shape function at point x.
The point x is not restricted to be inside the unit cube, but the meaning of evaluating a shape function outside of the unit cube is questionable.
Definition at line 652 of file lagrangeshapefunctions.hh.
◆ localindex()
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inline |
Returns the local index of the shape function.
Definition at line 697 of file lagrangeshapefunctions.hh.
◆ location()
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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, Scalar >.
Definition at line 744 of file lagrangeshapefunctions.hh.
◆ position()
|
inline |
Returns the element-local position of the node associated to the shape function.
Definition at line 736 of file lagrangeshapefunctions.hh.
Member Data Documentation
◆ comps
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static |
Definition at line 617 of file lagrangeshapefunctions.hh.
◆ dim
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static |
Definition at line 616 of file lagrangeshapefunctions.hh.
Referenced by Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::evaluate2ndDerivative(), Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::evaluateDerivative(), and Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::position().
◆ order
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static |
Definition at line 618 of file lagrangeshapefunctions.hh.
Referenced by Kaskade::LagrangeCubeShapeFunction< ctype, dimension, Scalar, O >::location().
The documentation for this class was generated from the following file:
