Dune::MultiLinearGeometry< ct, mydim, cdim, Traits > Class Template Reference

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 user data More...
typedef FieldVector< ctype, coorddimension >	GlobalCoordinate
	type of global coordinates
typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianTransposed
	type of jacobian transposed
typedef JacobianInverseTransposed	Jacobian
	For backward-compatibility, export the type JacobianInverseTransposed as Jacobian. More...
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...
const JacobianTransposed &	jacobianTransposed (const LocalCoordinate &local) const
	obtain the transposed of the Jacobian More...
const 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.

As an additional feature, this class allows to attach arbitrary user data to each object. This is used in GeometryGrid to implement reference counting.

Member Typedef Documentation

Jacobian

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

typedef JacobianInverseTransposed Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::Jacobian

For backward-compatibility, export the type JacobianInverseTransposed as Jacobian.

Deprecated:

This typedef will be removed after the release of dune 2.3

LocalCoordinate

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

typedef FieldVector< ctype, mydimension > Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::LocalCoordinate

type of user data

For example, GeometryGrid uses this to implement reference counting. type of local coordinates

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::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 >

const 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 >>

const 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()

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()

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

References Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::integrationElement(), and Dune::ReferenceElement< ctype, dim >::volume().

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

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

• dune/geometry/multilineargeometry.hh