Dune::AffineGeometry< ct, mydim, cdim > Class Template Reference

Implementation of the Geometry interface for affine geometries. More...

#include <dune/geometry/affinegeometry.hh>

Public Types
typedef ct	ctype
	Type used for coordinates.
typedef FieldVector< ctype, mydimension >	LocalCoordinate
	Type for local coordinate vector.
typedef FieldVector< ctype, coorddimension >	GlobalCoordinate
	Type for coordinate vector in world space.
typedef ctype	Volume
	Type used for volume.
typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianTransposed
	Type for the transposed Jacobian matrix.
typedef FieldMatrix< ctype, coorddimension, mydimension >	JacobianInverseTransposed
	Type for the transposed inverse Jacobian matrix.
typedef FieldMatrix< ctype, coorddimension, mydimension >	Jacobian
	Type for the Jacobian matrix.
typedef FieldMatrix< ctype, mydimension, coorddimension >	JacobianInverse
	Type for the inverse Jacobian matrix.

Public Member Functions
	AffineGeometry ()=default
	Constructs an empty geometry. More...
	AffineGeometry (const ReferenceElement &refElement, const GlobalCoordinate &origin, const JacobianTransposed &jt)
	Create affine geometry from reference element, one vertex, and the Jacobian matrix.
	AffineGeometry (Dune::GeometryType gt, const GlobalCoordinate &origin, const JacobianTransposed &jt)
	Create affine geometry from GeometryType, one vertex, and the Jacobian matrix.
template<class CoordVector >
	AffineGeometry (const ReferenceElement &refElement, const CoordVector &coordVector)
	Create affine geometry from reference element and a vector of vertex coordinates.
template<class CoordVector >
	AffineGeometry (Dune::GeometryType gt, const CoordVector &coordVector)
	Create affine geometry from GeometryType and a vector of vertex coordinates.
bool	affine () const
	Always true: this is an affine geometry.
Dune::GeometryType	type () const
	Obtain the type 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...
Volume	volume () const
	Obtain the volume of the element.
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...
Jacobian	jacobian (const LocalCoordinate &local) const
	Obtain the Jacobian. More...
JacobianInverse	jacobianInverse (const LocalCoordinate &local) const
	Obtain the Jacobian's inverse. More...

Static Public Attributes
static const int	mydimension = mydim
	Dimension of the geometry.
static const int	coorddimension = cdim
	Dimension of the world space.

Detailed Description

template<class ct, int mydim, int cdim>
class Dune::AffineGeometry< ct, mydim, cdim >

Implementation of the Geometry interface for affine geometries.

Template Parameters

ct	Type used for coordinates
mydim	Dimension of the geometry
cdim	Dimension of the world space

Constructor & Destructor Documentation

AffineGeometry()

template<class ct , int mydim, int cdim>

Dune::AffineGeometry< ct, mydim, cdim >::AffineGeometry ( )

default

Constructs an empty geometry.

This constructor creates an empty (invalid) affine geometry. It may not be used in any way except for assigning other affine geometries to it. After assigning a valid geometry, it may be used without restrictions.

Member Function Documentation

global()

template<class ct , int mydim, int cdim>

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

inline

Evaluate the mapping.

Parameters

[in]

local

local coordinate to map

Returns

corresponding global coordinate

References Dune::AffineGeometry< ct, mydim, cdim >::global(), and Dune::DenseMatrix< MAT >::umtv().

Referenced by Dune::AffineGeometry< ct, mydim, cdim >::center(), Dune::AffineGeometry< ct, mydim, cdim >::corner(), Dune::AffineGeometry< ct, mydim, cdim >::global(), and Dune::AffineGeometry< ct, mydim, cdim >::LocalCoordinate().

integrationElement()

template<class ct , int mydim, int cdim>

ctype Dune::AffineGeometry< ct, mydim, cdim >::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 \(\mu(x)\).

jacobian()

template<class ct , int mydim, int cdim>

Jacobian Dune::AffineGeometry< ct, mydim, cdim >::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

References Dune::FieldMatrix< K, ROWS, COLS >::transposed().

jacobianInverse()

template<class ct , int mydim, int cdim>

JacobianInverse Dune::AffineGeometry< ct, mydim, cdim >::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:

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

References Dune::FieldMatrix< K, ROWS, COLS >::transposed().

jacobianInverseTransposed()

template<class ct , int mydim, int cdim>

const JacobianInverseTransposed & Dune::AffineGeometry< ct, mydim, cdim >::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()

template<class ct , int mydim, int cdim>

const JacobianTransposed & Dune::AffineGeometry< ct, mydim, cdim >::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

LocalCoordinate()

template<class ct , int mydim, int cdim>

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

inline

Evaluate the inverse mapping.

Parameters

[in]

global

global coordinate to map

Returns

corresponding local coordinate

The returned local coordinate y minimizes

(global( y ) - x).two_norm()

on the entire affine hull of the reference element. This degenerates to the inverse map if the argument y is in the range of the map.

References Dune::AffineGeometry< ct, mydim, cdim >::global(), and Dune::DenseMatrix< MAT >::mtv().

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

• dune/geometry/affinegeometry.hh