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

generic geometry implementation based on corner coordinates More...

#include <dune/geometry/multilineargeometry.hh>

Inheritance diagram for Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >:

Classes
class	JacobianInverseTransposed

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 ctype	Volume
	type of volume
typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianTransposed
	type of jacobian transposed
typedef FieldMatrix< ctype, coorddimension, mydimension >	Jacobian
	Type for the Jacobian matrix.
typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianInverse
	Type for the inverse Jacobian matrix.
typedef ReferenceElements::ReferenceElement	ReferenceElement
	type of reference element

Public Member Functions
template<class Corners >
	MultiLinearGeometry (const ReferenceElement &refElement, const Corners &corners)
	constructor
template<class Corners >
	MultiLinearGeometry (Dune::GeometryType gt, const Corners &corners)
	constructor
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
LocalCoordinate	local (const GlobalCoordinate &globalCoord) const
	evaluate the inverse mapping
Volume	integrationElement (const LocalCoordinate &local) const
	obtain the integration element
Volume	volume () const
	obtain the volume of the mapping's image
JacobianTransposed	jacobianTransposed (const LocalCoordinate &local) const
	obtain the transposed of the Jacobian
JacobianInverseTransposed	jacobianInverseTransposed (const LocalCoordinate &local) const
	obtain the transposed of the Jacobian's inverse
Jacobian	jacobian (const LocalCoordinate &local) const
	Obtain the Jacobian.
JacobianInverse	jacobianInverse (const LocalCoordinate &local) const
	Obtain the Jacobian's inverse.

Static Public Attributes
static constexpr int	mydimension = mydim
	geometry dimension
static constexpr int	coorddimension = cdim
	coordinate dimension

Protected Types
typedef Dune::ReferenceElements< ctype, mydimension >	ReferenceElements
typedef Traits::MatrixHelper	MatrixHelper
typedef std::conditional< hasSingleGeometryType, std::integral_constant< unsignedint, Traits::templatehasSingleGeometryType< mydimension >::topologyId >, unsignedint >::type	TopologyId

Protected Member Functions
ReferenceElement	refElement () const
TopologyId	topologyId () const
bool	affine (JacobianTransposed &jacobianT) const

Static Protected Member Functions
template<bool add, int dim, class CornerIterator >
static void	global (TopologyId topologyId, std::integral_constant< int, dim >, CornerIterator &cit, const ctype &df, const LocalCoordinate &x, const ctype &rf, GlobalCoordinate &y)
template<bool add, class CornerIterator >
static void	global (TopologyId topologyId, std::integral_constant< int, 0 >, CornerIterator &cit, const ctype &df, const LocalCoordinate &x, const ctype &rf, GlobalCoordinate &y)
template<bool add, int rows, int dim, class CornerIterator >
static void	jacobianTransposed (TopologyId topologyId, std::integral_constant< int, dim >, CornerIterator &cit, const ctype &df, const LocalCoordinate &x, const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt)
template<bool add, int rows, class CornerIterator >
static void	jacobianTransposed (TopologyId topologyId, std::integral_constant< int, 0 >, CornerIterator &cit, const ctype &df, const LocalCoordinate &x, const ctype &rf, FieldMatrix< ctype, rows, cdim > &jt)
template<int dim, class CornerIterator >
static bool	affine (TopologyId topologyId, std::integral_constant< int, dim >, CornerIterator &cit, JacobianTransposed &jt)
template<class CornerIterator >
static bool	affine (TopologyId topologyId, std::integral_constant< int, 0 >, CornerIterator &cit, JacobianTransposed &jt)

Friends
ReferenceElement	referenceElement (const MultiLinearGeometry &geometry)

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.

Member Typedef Documentation

ctype

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

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

coordinate type

GlobalCoordinate

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

typedef FieldVector< ctype, coorddimension > Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::GlobalCoordinate

type of global coordinates

Jacobian

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

typedef FieldMatrix< ctype, coorddimension, mydimension > Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::Jacobian

Type for the Jacobian matrix.

JacobianInverse

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

typedef FieldMatrix< ctype, mydimension, coorddimension > Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianInverse

Type for the inverse Jacobian matrix.

JacobianTransposed

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

typedef FieldMatrix< ctype, mydimension, coorddimension > Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::JacobianTransposed

type of jacobian transposed

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 local coordinates

MatrixHelper

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

typedef Traits::MatrixHelper Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::MatrixHelper

protected

ReferenceElement

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

typedef ReferenceElements::ReferenceElement Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::ReferenceElement

type of reference element

ReferenceElements

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

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

protected

TopologyId

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

typedef std::conditional<hasSingleGeometryType,std::integral_constant<unsignedint,Traits::templatehasSingleGeometryType<mydimension>::topologyId>,unsignedint>::type Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::TopologyId

protected

Volume

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

typedef ctype Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::Volume

type of volume

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

affine() [1/4]

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

bool Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::affine ( ) const

inline

is this mapping affine?

affine() [2/4]

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

bool Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::affine ( JacobianTransposed &  jacobianT ) const

inlineprotected

affine() [3/4]

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

template<class CornerIterator >

bool Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::affine ( TopologyId  topologyId, std::integral_constant< int, 0 >  , CornerIterator &  cit, JacobianTransposed &  jt  )

inlinestaticprotected

affine() [4/4]

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

template<int dim, class CornerIterator >

bool Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::affine ( TopologyId  topologyId, std::integral_constant< int, dim >  , CornerIterator &  cit, JacobianTransposed &  jt  )

inlinestaticprotected

center()

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

GlobalCoordinate Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::center ( ) const

inline

obtain the centroid of the mapping's image

corner()

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

GlobalCoordinate Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::corner ( int  i ) const

inline

obtain coordinates of the i-th corner

corners()

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

int Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::corners ( ) const

inline

obtain number of corners of the corresponding reference element

global() [1/3]

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

global() [2/3]

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

template<bool add, class CornerIterator >

void Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::global ( TopologyId  topologyId, std::integral_constant< int, 0 >  , CornerIterator &  cit, const ctype &  df, const LocalCoordinate &  x, const ctype &  rf, GlobalCoordinate &  y  )

inlinestaticprotected

global() [3/3]

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

template<bool add, int dim, class CornerIterator >

void Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::global ( TopologyId  topologyId, std::integral_constant< int, dim >  , CornerIterator &  cit, const ctype &  df, const LocalCoordinate &  x, const ctype &  rf, GlobalCoordinate &  y  )

inlinestaticprotected

integrationElement()

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

Volume 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.

jacobian()

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

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

inline

Obtain the Jacobian.

Parameters

[in]

local

local coordinate to evaluate Jacobian in

Returns

a copy of the transposed of the Jacobian

jacobianInverse()

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

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

inline

Obtain 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:

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

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:

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

jacobianTransposed() [1/3]

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.

jacobianTransposed() [2/3]

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

template<bool add, int rows, class CornerIterator >

void Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed ( TopologyId  topologyId, std::integral_constant< int, 0 >  , CornerIterator &  cit, const ctype &  df, const LocalCoordinate &  x, const ctype &  rf, FieldMatrix< ctype, rows, cdim > &  jt  )

inlinestaticprotected

jacobianTransposed() [3/3]

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

template<bool add, int rows, int dim, class CornerIterator >

void Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::jacobianTransposed ( TopologyId  topologyId, std::integral_constant< int, dim >  , CornerIterator &  cit, const ctype &  df, const LocalCoordinate &  x, const ctype &  rf, FieldMatrix< ctype, rows, cdim > &  jt  )

inlinestaticprotected

local()

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

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

inline

evaluate the inverse mapping

Parameters

[in]

globalCoord

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()

refElement()

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

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

inlineprotected

topologyId()

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

TopologyId Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::topologyId ( ) const

inlineprotected

type()

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

Dune::GeometryType Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::type ( ) const

inline

obtain the name of the reference element

volume()

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

Volume 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.

Friends And Related Symbol Documentation

referenceElement

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

ReferenceElement referenceElement ( const MultiLinearGeometry< ct, mydim, cdim, Traits > &  geometry )

friend

Member Data Documentation

coorddimension

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

constexpr int Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::coorddimension = cdim

staticconstexpr

coordinate dimension

mydimension

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

constexpr int Dune::MultiLinearGeometry< ct, mydim, cdim, Traits >::mydimension = mydim

staticconstexpr

geometry dimension

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

• multilineargeometry.hh