Wrapper class for geometries. More...

#include <dune/grid/common/geometry.hh>

Public Types
typedef GeometryImp< mydim, cdim, GridImp >	Implementation
	type of underlying implementation More...

typedef GridImp::ctype	ctype
	define type used for coordinates in grid module

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

typedef FieldVector< ctype, cdim >	GlobalCoordinate
	type of the global coordinates

typedef decltype(std::declval< Implementation >().volume())	Volume
	Number type used for the geometry volume.

typedef Implementation::JacobianInverseTransposed	JacobianInverseTransposed
	type of jacobian inverse transposed More...

typedef Implementation::JacobianTransposed	JacobianTransposed
	type of jacobian transposed More...

using	JacobianInverse = Std::detected_or_t< JacobianInverseDefault, JacobianInverseOfImplementation, Implementation >
	type of jacobian inverse More...

using	Jacobian = Std::detected_or_t< JacobianDefault, JacobianOfImplementation, Implementation >
	type of jacobian More...

Public Member Functions
Implementation &	impl ()
	access to the underlying implementation More...

const Implementation &	impl () const
	access to the underlying implementation More...

GeometryType	type () const
	Return the type of the reference element. The type can be used to access the Dune::ReferenceElement.

bool	affine () const
	Return true if the geometry mapping is affine and false otherwise.

int	corners () const
	Return the number of corners of the reference element. More...

GlobalCoordinate	corner (int i) const
	Obtain a corner of the geometry. More...

GlobalCoordinate	global (const LocalCoordinate &local) const
	Evaluate the map \( g\). More...

	LocalCoordinate (const GlobalCoordinate &global) const
	Evaluate the inverse map \( g^{-1}\). More...

Volume	integrationElement (const LocalCoordinate &local) const
	Return the factor appearing in the integral transformation formula. More...

Volume	volume () const
	return volume of geometry

GlobalCoordinate	center () const
	return center of geometry More...

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

JacobianInverseTransposed	jacobianInverseTransposed (const LocalCoordinate &local) const
	Return inverse of transposed of Jacobian. More...

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

JacobianInverse	jacobianInverse (const LocalCoordinate &local) const
	Return inverse of Jacobian. More...

Static Public Attributes
static constexpr int	mydimension = mydim
	geometry dimension

static constexpr int	coorddimension = cdim
	dimension of embedding coordinate system

Related Functions
(Note that these are not member functions.)
template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp, typename Impl >
auto	referenceElement (const Geometry< mydim, cdim, GridImp, GeometryImp > &geo, const Impl &) -> decltype(referenceElement< typename GridImp::ctype, mydim >(geo.type()))
	Second-level dispatch to select the correct reference element for a grid geometry. More...

Interface for grid implementers
Implementation	realGeometry

	Geometry (const Implementation &impl)
	copy constructor from implementation

Detailed Description

template<int mydim, int cdim, class GridImp, template< int, int, class > class GeometryImp>
class Dune::Geometry< mydim, cdim, GridImp, GeometryImp >

Wrapper class for geometries.

Template Parameters

mydim	Dimension of the domain
cdim	Dimension of the range
GridImp	Type that is a model of Dune::Grid
GeometryImp	Class template that is a model of Dune::Geometry

Maps

A Geometry defines a map

\[ g : D \to W\]

where \(D\subseteq\mathbf{R}^\textrm{mydim}\) and \(W\subseteq\mathbf{R}^\textrm{cdim}\). The domain \(D\) is one of a set of predefined convex polytopes, the so-called reference elements (see Dune::ReferenceElement). The dimensionality of \(D\) is mydim. In general \(\textrm{mydim}\leq\textrm{cdim}\), i.e. the convex polytope may be mapped to a manifold. Moreover, we require that \( g\in \left( C^1(D) \right)^\textrm{cdim}\) and one-to-one.

Engine Concept

The Geometry class template wraps an object of type GeometryImp and forwards all member function calls to corresponding members of this class. In that sense Geometry defines the interface and GeometryImp supplies the implementation.

Member Typedef Documentation

◆ Implementation

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

typedef GeometryImp< mydim, cdim, GridImp > Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::Implementation

type of underlying implementation

Warning: Implementation details may change without prior notification.

◆ Jacobian

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

using Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::Jacobian = Std::detected_or_t<JacobianDefault, JacobianOfImplementation, Implementation>

type of jacobian

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

You can multiply it from the right to a suitable FieldMatrix
It is copy constructible and copy assignable.

◆ JacobianInverse

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

using Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::JacobianInverse = Std::detected_or_t<JacobianInverseDefault, JacobianInverseOfImplementation, Implementation>

type of jacobian inverse

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

You can multiply it from the right to a suitable FieldMatrix
It is copy constructible and copy assignable.

◆ JacobianInverseTransposed

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

typedef Implementation::JacobianInverseTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::JacobianInverseTransposed

type of jacobian inverse transposed

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

It satisfies the ConstMatrix interface.
It is copy constructible and copy assignable.

◆ JacobianTransposed

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

typedef Implementation::JacobianTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::JacobianTransposed

type of jacobian transposed

The exact type is implementation-dependent. However, it is guaranteed to have the following properties:

It satisfies the ConstMatrix interface.
It is copy constructible and copy assignable.

Member Function Documentation

◆ center()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::center ( ) const

inline

return center of geometry

Note that this method is still subject to a change of name and semantics. At the moment, the center is not required to be the centroid of the geometry, or even the centroid of its corners. This makes the current default implementation acceptable, which maps the centroid of the reference element to the geometry. We may change the name (and semantic) of the method to centroid() if we find reasonably efficient ways to implement it properly.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ corner()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::corner ( int i ) const

inline

Obtain a corner of the geometry.

This method is for convenient access to the corners of the geometry. The same result could be achieved by calling

global( referenceElement.position( i, mydimension ) )

Dune::Geometry::global

GlobalCoordinate global(const LocalCoordinate &local) const

Evaluate the map .

Definition: geometry.hh:228

Dune::Geometry::referenceElement

auto referenceElement(const Geometry< mydim, cdim, GridImp, GeometryImp > &geo, const Impl &) -> decltype(referenceElement< typename GridImp::ctype, mydim >(geo.type()))

Second-level dispatch to select the correct reference element for a grid geometry.

Definition: geometry.hh:581

Dune::Geometry::mydimension

static constexpr int mydimension

geometry dimension

Definition: geometry.hh:94

Parameters

[in] i number of the corner (with respect to the reference element)

Returns: position of the i-th corner

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::EdgeS0_5Basis< Geometry, RF >::EdgeS0_5Basis(), and Dune::EdgeS0_5Interpolation< Geometry, Traits_ >::EdgeS0_5Interpolation().

◆ corners()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

int Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::corners ( ) const

inline

Return the number of corners of the reference element.

Since a geometry is a convex polytope the number of corners is a well-defined concept. The method is redundant because this information is also available via the reference element. It is here for efficiency and ease of use.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ global()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

GlobalCoordinate Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::global ( const LocalCoordinate & local ) const

inline

Evaluate the map \( g\).

Parameters

[in] local Position in the reference element \(D\)

Returns: Position in \(W\)

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::HierarchicSearch< Grid, IS >::findEntity(), and Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::LocalCoordinate().

◆ impl() [1/2]

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

Implementation & Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl ( )

inline

access to the underlying implementation

Warning: Implementation details may change without prior notification.

◆ impl() [2/2]

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

const Implementation & Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl ( ) const

inline

access to the underlying implementation

Warning: Implementation details may change without prior notification.

◆ integrationElement()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

Volume Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::integrationElement ( const LocalCoordinate & local ) const

inline

Return the factor appearing in the integral transformation formula.

Let \( g : D \to W\) denote the transformation described by the Geometry. Then the jacobian of the transformation is defined as the \(\textrm{cdim}\times\textrm{mydim}\) matrix

\[ J_g(x) = \left( \begin{array}{ccc} \frac{\partial g_0}{\partial x_0} & \cdots & \frac{\partial g_0}{\partial x_{n-1}} \\ \vdots & \ddots & \vdots \\ \frac{\partial g_{m-1}}{\partial x_0} & \cdots & \frac{\partial g_{m-1}}{\partial x_{n-1}} \end{array} \right).\]

Here we abbreviated \(m=\textrm{cdim}\) and \(n=\textrm{mydim}\) for ease of readability.

The integration element \(\mu(x)\) for any \(x\in D\) is then defined as

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

Parameters

[in] local Position \(x\in D\)

Returns: integration element \(\mu(x)\)

Note: Each implementation computes the integration element with optimal efficiency. For example in an equidistant structured mesh it may be as simple as \(h^\textrm{mydim}\).

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::PDELab::integrateGridFunction().

◆ jacobian()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

Jacobian Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobian ( const LocalCoordinate & local ) const

inline

Return the Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters

[in] local position \(x\in D\)

Returns: \(J_g(x)\)

Note: The exact return type is implementation defined.

◆ jacobianInverse()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

JacobianInverse Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobianInverse ( const LocalCoordinate & local ) const

inline

Return inverse of Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters

[in] local position \(x\in D\)

Returns: \(J_g^{-1}(x)\)

The use of this function is to compute the jacobians of some function \(f : W \to \textbf{R}\) at some position \(y=g(x)\), where \(x\in D\) and \(g\) the transformation of the Geometry. When we set \(\hat{f}(x) = f(g(x))\) and apply the chain rule we obtain

\[\nabla f(g(x)) = J_{\hat{f}}(x) J_g^{-1}(x).\]

Note: In the non-quadratic case \(\textrm{cdim} \neq \textrm{mydim}\), the pseudoinverse of \(J_g(x)\) is returned. This means that its transposed is inverse for all tangential vectors in \(g(x)\) while mapping all normal vectors to zero.; The exact return type is implementation defined.

◆ jacobianInverseTransposed()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

JacobianInverseTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobianInverseTransposed ( const LocalCoordinate & local ) const

inline

Return inverse of transposed of Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters

[in] local position \(x\in D\)

Returns: \(J_g^{-T}(x)\)

The use of this function is to compute the gradient of some function \(f : W \to \textbf{R}\) at some position \(y=g(x)\), where \(x\in D\) and \(g\) the transformation of the Geometry. When we set \(\hat{f}(x) = f(g(x))\) and apply the chain rule we obtain

\[\nabla f(g(x)) = J_g^{-T}(x) \nabla \hat{f}(x).\]

Note: In the non-quadratic case \(\textrm{cdim} \neq \textrm{mydim}\), the pseudoinverse of \(J_g^T(x)\) is returned. This means that it is inverse for all tangential vectors in \(g(x)\) while mapping all normal vectors to zero.; The exact return type is implementation defined.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Referenced by Dune::PDELab::VectorBasisInterfaceSwitch< Basis, typename std::enable_if< Dune::AlwaysTrue< std::integral_constant< std::size_t, Basis::Traits::dimDomain > >::value >::type >::jacobian().

◆ jacobianTransposed()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

JacobianTransposed Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::jacobianTransposed ( const LocalCoordinate & local ) const

inline

Return the transposed of the Jacobian.

The Jacobian is defined in the documentation of integrationElement.

Parameters

[in] local position \(x\in D\)

Returns: \(J_g^T(x)\)

Note: The exact return type is implementation defined.

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

◆ LocalCoordinate()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp>

Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::LocalCoordinate ( const GlobalCoordinate & global ) const

inline

Evaluate the inverse map \( g^{-1}\).

Parameters

[in] global Position in \(W\)

Returns: Position in \(D\) that maps to global

References Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::global(), and Dune::Geometry< mydim, cdim, GridImp, GeometryImp >::impl().

Friends And Related Function Documentation

◆ referenceElement()

template<int mydim, int cdim, class GridImp , template< int, int, class > class GeometryImp, typename Impl >

auto referenceElement	(	const Geometry< mydim, cdim, GridImp, GeometryImp > &	geo,
		const Impl &
	)		-> decltype(referenceElement<typename GridImp::ctype,mydim>(geo.type()))

This function is the default implementation of the second-level reference element dispatch performed by Geometry.

Note: This function is only important for grid implementors with geometries that do not have a standard reference element.

When referenceElement() is called with a Geometry, it will forward the call to referenceElement(const Geometry&,const GeometryImplementation&). This default implementation will do the right thing as long as your geometry is based on a standard Dune ReferenceElement. If it is not and you want to supply your own reference element implementation, provide an override of this function for your specific geometry implementation.

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

dune/grid/common/geometry.hh

DUNE Distributed and Unified Numerics Environment

DUNE PDELab (unstable)

Public Types

Public Member Functions

Static Public Attributes

Related Functions

Interface for grid implementers

Detailed Description

Maps

Engine Concept

Member Typedef Documentation

◆ Implementation

◆ Jacobian

◆ JacobianInverse

◆ JacobianInverseTransposed

◆ JacobianTransposed

Member Function Documentation

◆ center()

◆ corner()

◆ corners()

◆ global()

◆ impl() [1/2]

◆ impl() [2/2]

◆ integrationElement()

◆ jacobian()

◆ jacobianInverse()

◆ jacobianInverseTransposed()

◆ jacobianTransposed()

◆ LocalCoordinate()

Friends And Related Function Documentation

◆ referenceElement()