Geometry parametrized by a LocalFunction and a LocalGeometry. More...

#include <dune/geometry/mappedgeometry.hh>

Public Types
using	LocalCoordinate = typename Geo::LocalCoordinate
	type of local coordinates

using	GlobalCoordinate = std::remove_reference_t< decltype(std::declval< Map >()(std::declval< typename Geo::GlobalCoordinate >()))>
	type of global coordinates

using	ctype = typename Geo::ctype
	coordinate type

using	Volume = std::remove_reference_t< decltype(Dune::power(std::declval< ctype >(), mydimension))>
	type of volume

using	Jacobian = FieldMatrix< ctype, coorddimension, mydimension >
	type of jacobian

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

using	JacobianInverse = FieldMatrix< ctype, mydimension, coorddimension >
	type of jacobian inverse

using	JacobianInverseTransposed = FieldMatrix< ctype, coorddimension, mydimension >
	type of jacobian inverse transposed

Public Member Functions
template<class Geo_ , class Map_ , std::enable_if_t< Dune::IsInteroperable< Map, Map_ >::value, int > = 0, std::enable_if_t< Dune::IsInteroperable< Geo, Geo_ >::value, int > = 0>
	MappedGeometry (Map_ &&mapping, Geo_ &&geometry, bool affine=false)
	Constructor from mapping to parametrize the geometry. More...

bool	affine () const
	Is this mapping affine? Not in general, since we don't know anything about the mapping. The returned value can be given in the constructor of the class.

GeometryType	type () const
	Obtain the geometry type from 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
	Map the center of the wrapped geometry.

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

	LocalCoordinate (const GlobalCoordinate &y, Impl::GaussNewtonOptions< ctype > opts={}) const
	Evaluate the inverse coordinate mapping. More...

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

Volume	volume (Impl::ConvergenceOptions< ctype > opts={}) const
	Obtain the volume of the mapping's image. More...

Volume	volume (const QuadratureRule< ctype, mydimension > &quadRule) const
	Obtain the volume of the mapping's image by given quadrature rules.

Jacobian	jacobian (const LocalCoordinate &local) const
	Obtain the Jacobian. More...

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

JacobianInverse	jacobianInverse (const LocalCoordinate &local) const
	Obtain the Jacobian's inverse. More...

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

Static Public Attributes
static constexpr int	mydimension = LocalCoordinate::size()
	geometry dimension

static constexpr int	coorddimension = GlobalCoordinate::size()
	coordinate dimension

Detailed Description

template<class Map, class Geo>
class Dune::MappedGeometry< Map, Geo >

Geometry parametrized by a LocalFunction and a LocalGeometry.

This class represents a geometry that is parametrized by the chained mapping of a geometry g of type Geo and the given (differentiable) function f of type Map as f o g.

The geometry g: LG -> GG maps points from the local domain LG to its global domain GG. It must fulfill the dune-grid geometry-concept. Examples are the geometry of a grid element, the geometry of a `ReferenceElement` or any other geometry implementation in dune-geometry. The local domain LG represents the local coordinates of the MappedGeometry.

The function f: LF -> GF is a differentiable function in the sense of the dune-functions concept. It maps coordinates from the global domain of the geometry, LF=GG into the range type GF representing the global coordinates of the MappedGeometry.

Requirements: Let local be of type LG, g of type Geo and f of type Map.

g must be a model of Concept::Geometry
f must be a model of Concept::DifferentiableFunction<GF(LF)>

The following expressions must be valid:

GG gg = g.global(local): the "evaluation" of the geometry in its local domain.
GF gf = f(gg): the evaluation of the mapping f in its local domain that is the geometries' range.
auto df = derivative(f), df(gg): The derivative of f w.r.t. its global coordinate GF evaluated at its local coordinated LF=GG.

Template Parameters

Map	Differentiable mapping `f: LF -> GF` with `LF = GG` the range of the geometry `g`.
Geo	A geometry type `g: LG -> GG` fulfilling the concept `Concept::Geometry`.

Constructor & Destructor Documentation

◆ MappedGeometry()

template<class Map , class Geo >

template<class Geo_ , class Map_ , std::enable_if_t< Dune::IsInteroperable< Map, Map_ >::value, int > = 0, std::enable_if_t< Dune::IsInteroperable< Geo, Geo_ >::value, int > = 0>

Dune::MappedGeometry< Map, Geo >::MappedGeometry	(	Map_ &&	mapping,
		Geo_ &&	geometry,
		bool	affine = `false`
	)

inline

Constructor from mapping to parametrize the geometry.

Parameters

[in]	mapping	A differentiable function for the parametrization of the geometry
[in]	geometry	The geometry that is mapped
[in]	affine	Flag indicating whether the `MappedGeometry` represents an affine mapping

Member Function Documentation

◆ global()

template<class Map , class Geo >

GlobalCoordinate Dune::MappedGeometry< Map, Geo >::global ( const LocalCoordinate & local ) const

inline

Evaluate the coordinate mapping.

Chained evaluation of the geometry and mapping from local coordinates in the reference element associated to the wrapped geometry.

Parameters

[in] local local coordinates

Returns: corresponding global coordinate

◆ integrationElement()

template<class Map , class Geo >

ctype Dune::MappedGeometry< Map, Geo >::integrationElement ( const LocalCoordinate & local ) const

inline

Obtain the integration element.

If the Jacobian of the geometry is denoted by \(J(x)\), the 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 \(\mu(x)\).

References Dune::MappedGeometry< Map, Geo >::jacobianTransposed().

Referenced by Dune::MappedGeometry< Map, Geo >::volume().

◆ jacobian()

template<class Map , class Geo >

Jacobian Dune::MappedGeometry< Map, Geo >::jacobian ( const LocalCoordinate & local ) const

inline

Obtain the Jacobian.

Parameters

[in] local local coordinate to evaluate Jacobian in

Returns: the matrix corresponding to the Jacobian

Referenced by Dune::MappedGeometry< Map, Geo >::jacobianInverse(), and Dune::MappedGeometry< Map, Geo >::jacobianTransposed().

◆ jacobianInverse()

template<class Map , class Geo >

JacobianInverse Dune::MappedGeometry< Map, Geo >::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::MappedGeometry< Map, Geo >::jacobian().

Referenced by Dune::MappedGeometry< Map, Geo >::jacobianInverseTransposed().

◆ jacobianInverseTransposed()

template<class Map , class Geo >

JacobianInverseTransposed Dune::MappedGeometry< Map, Geo >::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:

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

References Dune::MappedGeometry< Map, Geo >::jacobianInverse(), and Dune::transpose().

◆ jacobianTransposed()

template<class Map , class Geo >

JacobianTransposed Dune::MappedGeometry< Map, Geo >::jacobianTransposed ( const LocalCoordinate & local ) const

inline

Obtain the transposed of the Jacobian.

Parameters

[in] local local coordinate to evaluate Jacobian in

Returns: the matrix corresponding to the transposed of the Jacobian

References Dune::MappedGeometry< Map, Geo >::jacobian(), and Dune::transpose().

Referenced by Dune::MappedGeometry< Map, Geo >::integrationElement().

◆ LocalCoordinate()

template<class Map , class Geo >

Dune::MappedGeometry< Map, Geo >::LocalCoordinate	(	const GlobalCoordinate &	y,
		Impl::GaussNewtonOptions< ctype >	opts = `{}`
	)		const

inline

Evaluate the inverse coordinate mapping.

Parameters

[in]	y	Global coordinate to map
[in]	opts	Parameters to control the behavior of the Gauss-Newton algorithm.

Returns: For given global coordinate y the local coordinate x that minimizes the function (global(x) - y).two_norm2() over the local coordinate space spanned by the reference element.

Exceptions

In	case of an error indicating that the local coordinate can not be obtained, a `Dune::Exception` is thrown, with an error code from `GaussNewtonErrorCode`.

Note: It is not guaranteed that the resulting local coordinate is inside the reference element domain.

◆ volume()

template<class Map , class Geo >

Volume Dune::MappedGeometry< Map, Geo >::volume ( Impl::ConvergenceOptions< ctype > opts = {} ) const

inline

Obtain the volume of the mapping's image.

Calculates the volume of the entity by numerical integration. Since the polynomial order of the volume element is not known, iteratively compute numerical integrals with increasing order of the quadrature rules, until tolerance is reached.

Parameters

opts	An optional control over the convergence, providing a break tolerance and a maximal iteration count.

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

dune/geometry/mappedgeometry.hh

DUNE Distributed and Unified Numerics Environment

DUNE-FEM (unstable)

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Constructor & Destructor Documentation

◆ MappedGeometry()

Member Function Documentation

◆ global()

◆ integrationElement()

◆ jacobian()

◆ jacobianInverse()

◆ jacobianInverseTransposed()

◆ jacobianTransposed()

◆ LocalCoordinate()

◆ volume()