generic geometry implementation based on corner coordinates More...

#include <dune/geometry/multilineargeometry.hh>

Public Types
typedef ct	ctype
	coordinate type

typedef FieldVector< ctype, mydimension >	LocalCoordinate
	type of local coordinates

typedef FieldVector< ctype, coorddimension >	GlobalCoordinate
	type of global coordinates

typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianTransposed
	type of jacobian transposed

typedef Dune::ReferenceElement< ctype, mydimension >	ReferenceElement
	type of reference element

Public Member Functions
template<class Corners >
	MultiLinearGeometry (const ReferenceElement &refElement, const Corners &corners)
	constructor More...

template<class Corners >
	MultiLinearGeometry (Dune::GeometryType gt, const Corners &corners)
	constructor More...

bool	affine () const
	is this mapping affine?

Dune::GeometryType	type () const
	obtain the name of the reference element

int	corners () const
	obtain number of corners of the corresponding reference element

GlobalCoordinate	corner (int i) const
	obtain coordinates of the i-th corner

GlobalCoordinate	center () const
	obtain the centroid of the mapping's image

GlobalCoordinate	global (const LocalCoordinate &local) const
	evaluate the mapping More...

	LocalCoordinate (const GlobalCoordinate &global) const
	evaluate the inverse mapping More...

ctype	integrationElement (const LocalCoordinate &local) const
	obtain the integration element More...

ctype	volume () const
	obtain the volume of the mapping's image More...

JacobianTransposed	jacobianTransposed (const LocalCoordinate &local) const
	obtain the transposed of the Jacobian More...

JacobianInverseTransposed	jacobianInverseTransposed (const LocalCoordinate &local) const
	obtain the transposed of the Jacobian's inverse More...

Static Public Attributes
static const int	mydimension = mydim
	geometry dimension

static const int	coorddimension = cdim
	coordinate dimension

Detailed Description

template<class ct, int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>
class Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >

generic geometry implementation based on corner coordinates

Based on the recursive definition of the reference elements, the MultiLinearGeometry provides a generic implementation of a geometry given the corner coordinates.

The geometric mapping is multilinear in the classical sense only in the case of cubes; for simplices it is linear. The name is still justified, because the mapping satisfies the important property of begin linear along edges.

Template Parameters

ct	coordinate type
mydim	geometry dimension
cdim	coordinate dimension
Traits	traits allowing to tweak some implementation details (optional)

The requirements on the traits are documented along with their default, MultiLinearGeometryTraits.

Constructor & Destructor Documentation

◆ MultiLinearGeometry() [1/2]

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

template<class Corners >

Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::MultiLinearGeometry	(	const ReferenceElement &	refElement,
		const Corners &	corners
	)

inline

constructor

Parameters

[in]	refElement	reference element for the geometry
[in]	corners	corners to store internally

Note: The type of corners is actually a template argument. It is only required that the internal corner storage can be constructed from this object.

◆ MultiLinearGeometry() [2/2]

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

template<class Corners >

Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::MultiLinearGeometry	(	Dune::GeometryType	gt,
		const Corners &	corners
	)

inline

constructor

Parameters

[in]	gt	geometry type
[in]	corners	corners to store internally

Note: The type of corners is actually a template argument. It is only required that the internal corner storage can be constructed from this object.

Member Function Documentation

◆ global()

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

GlobalCoordinate Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::global ( const LocalCoordinate & local ) const

inline

evaluate the mapping

Parameters

[in] local local coordinate to map

Returns: corresponding global coordinate

Referenced by Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::center(), Dune::P1VTKFunction< GV, V >::evaluate(), Dune::CachedMultiLinearGeometry< ct, mydim, cdim, Traits >::global(), and Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::LocalCoordinate().

◆ integrationElement()

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

ctype Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::integrationElement ( const LocalCoordinate & local ) const

inline

obtain the integration element

If the Jacobian of the mapping is denoted by $J(x)$, the integration integration element $\mu(x)$ is given by

\[ \mu(x) = \sqrt{|\det (J^T(x) J(x))|}.\]

Parameters

[in] local local coordinate to evaluate the integration element in

Returns: the integration element $\mu(x)$.

Note: For affine mappings, it is more efficient to call jacobianInverseTransposed before integrationElement, if both are required.

References Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed().

Referenced by Dune::CachedMultiLinearGeometry< ct, mydim, cdim, Traits >::integrationElement(), and Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::volume().

◆ jacobianInverseTransposed()

template<class ct , int mydim, int cdim, class Traits >

MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverseTransposed Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianInverseTransposed ( const LocalCoordinate & local ) const

inline

obtain the transposed of the Jacobian's inverse

The Jacobian's inverse is defined as a pseudo-inverse. If we denote the Jacobian by $J(x)$, the following condition holds:

\[J^{-1}(x) J(x) = I.\]

Referenced by Dune::CachedMultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianInverseTransposed().

◆ jacobianTransposed()

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

JacobianTransposed Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed ( const LocalCoordinate & local ) const

inline

obtain the transposed of the Jacobian

Parameters

[in] local local coordinate to evaluate Jacobian in

Returns: a reference to the transposed of the Jacobian

Note: The returned reference is reused on the next call to JacobianTransposed, destroying the previous value.

Referenced by Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::integrationElement(), Dune::CachedMultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed(), and Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::LocalCoordinate().

◆ LocalCoordinate()

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::LocalCoordinate ( const GlobalCoordinate & global ) const

inline

evaluate the inverse mapping

Parameters

[in] global global coordinate to map

Returns: corresponding local coordinate

Note: For given global coordinate y the returned local coordinate x that minimizes the following function over the local coordinate space spanned by the reference element.
(global( x ) - y).two_norm()

Dune::MultiLinearGeometry::global
GlobalCoordinate global(const LocalCoordinate &local) const
evaluate the mapping
Definition: multilineargeometry.hh:248

References Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::global(), Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed(), Dune::ReferenceElement< ctype, dim >::position(), and Dune::DenseVector< V >::two_norm2().

◆ volume()

template<class ct , int mydim, int cdim, class Traits = MultiLinearGeometryTraits< ct >>

ctype Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::volume ( ) const

inline

obtain the volume of the mapping's image

Note: The current implementation just returns
integrationElement( refElement().position( 0, 0 ) ) * refElement().volume()

Dune::MultiLinearGeometry::integrationElement
ctype integrationElement(const LocalCoordinate &local) const
obtain the integration element
Definition: multilineargeometry.hh:297

Dune::ReferenceElement::volume
ctype volume() const
obtain the volume of the reference element
Definition: referenceelements.hh:210

which is wrong for n-linear surface maps and other nonlinear maps.