Geometry implementation based on local-basis function parametrization. More...

#include <dune/geometry/localfiniteelementgeometry.hh>

Public Types
using	ctype = typename LocalBasisTraits::DomainFieldType
	coordinate type

using	LocalCoordinate = FieldVector< ctype, mydimension >
	type of local coordinates

using	GlobalCoordinate = FieldVector< ctype, coorddimension >
	type of global coordinates

using	Volume = decltype(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

using	ReferenceElements = Dune::ReferenceElements< ctype, mydimension >
	type of reference element

Public Member Functions
	LocalFiniteElementGeometry ()=default
	Default constructed geometry results in an empty/invalid representation.

	LocalFiniteElementGeometry (const ReferenceElement &refElement, const LocalFiniteElement &localFE, std::vector< GlobalCoordinate > vertices)
	Constructor from a vector of coefficients of the LocalBasis parametrizing the geometry. More...

template<class Param , std::enable_if_t< std::is_invocable_r_v< GlobalCoordinate, Param, LocalCoordinate >, int > = 0>
	LocalFiniteElementGeometry (const ReferenceElement &refElement, const LocalFiniteElement &localFE, Param &&parametrization)
	Constructor from a local parametrization function, mapping local to (curved) global coordinates. More...

template<class... Args>
	LocalFiniteElementGeometry (GeometryType gt, Args &&... args)
	Constructor, forwarding to the other constructors that take a reference-element. More...

int	order () const
	Obtain the (highest) polynomial order of the parametrization.

bool	affine () const
	Is this mapping affine? Geometries of order 1 might be affine, but it needs to be checked on non-simplex geometries.

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 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...

const LocalFiniteElement &	finiteElement () const
	Obtain the local finite-element.

const std::vector< GlobalCoordinate > &	coefficients () const
	Obtain the coefficients of the parametrization.

const LocalBasis &	localBasis () const
	The local basis of the stored local finite-element.

Static Public Attributes
static const int	mydimension = LocalBasisTraits::dimDomain
	geometry dimension

static const int	coorddimension = cdim
	coordinate dimension

Detailed Description

template<class LFE, int cdim>
class Dune::LocalFiniteElementGeometry< LFE, cdim >

Geometry implementation based on local-basis function parametrization.

Parametrization of the geometry by any localfunction interpolated into a local finite-element space.

Template Parameters

LFE	Type of a local finite-element.
cdim	Coordinate dimension.

Constructor & Destructor Documentation

◆ LocalFiniteElementGeometry() [1/3]

template<class LFE , int cdim>

Dune::LocalFiniteElementGeometry< LFE, cdim >::LocalFiniteElementGeometry	(	const ReferenceElement &	refElement,
		const LocalFiniteElement &	localFE,
		std::vector< GlobalCoordinate >	vertices
	)

inline

Constructor from a vector of coefficients of the LocalBasis parametrizing the geometry.

Parameters

[in]	refElement	reference element for the geometry
[in]	localFE	Local finite-element to use for the parametrization
[in]	vertices	Coefficients of the local interpolation into the basis

The vertices are the coefficients of a local interpolation in the local finite-element. For Lagrange local bases, these correspond to vertices on the curved geometry in the local Lagrange nodes.

Note: The vertices are stored internally, so if possible move an external vertex storage to this constructor

◆ LocalFiniteElementGeometry() [2/3]

template<class LFE , int cdim>

template<class Param , std::enable_if_t< std::is_invocable_r_v< GlobalCoordinate, Param, LocalCoordinate >, int > = 0>

Dune::LocalFiniteElementGeometry< LFE, cdim >::LocalFiniteElementGeometry	(	const ReferenceElement &	refElement,
		const LocalFiniteElement &	localFE,
		Param &&	parametrization
	)

inline

Constructor from a local parametrization function, mapping local to (curved) global coordinates.

Parameters

[in]	refElement	reference element for the geometry
[in]	localFE	Local finite-element to use for the parametrization
[in]	parametrization	parametrization function with signature `GlobalCoordinate(LocalCoordinate)`

The parametrization function is not stored in the class, but interpolated into the local finite-element basis and the computed interpolation coefficients are stored.

◆ LocalFiniteElementGeometry() [3/3]

template<class LFE , int cdim>

template<class... Args>

Dune::LocalFiniteElementGeometry< LFE, cdim >::LocalFiniteElementGeometry	(	GeometryType	gt,
		Args &&...	args
	)

inlineexplicit

Constructor, forwarding to the other constructors that take a reference-element.

Parameters

[in]	gt	geometry type
[in]	args...	arguments passed to the other constructors

Member Function Documentation

◆ global()

template<class LFE , int cdim>

GlobalCoordinate Dune::LocalFiniteElementGeometry< LFE, cdim >::global ( const LocalCoordinate & local ) const

inline

Evaluate the coordinate mapping.

Implements a linear combination of local basis functions scaled by the vertices as coefficients.

\[ global = \sum_i v_i \psi_i(local) \]

Parameters

[in] local local coordinate to map

Returns: corresponding global coordinate

References Dune::DenseVector< V >::axpy(), and Dune::LocalFiniteElementGeometry< LFE, cdim >::localBasis().

Referenced by Dune::LocalFiniteElementGeometry< LFE, cdim >::center(), and Dune::LocalFiniteElementGeometry< LFE, cdim >::corner().

◆ integrationElement()

template<class LFE , int cdim>

ctype Dune::LocalFiniteElementGeometry< LFE, cdim >::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::LocalFiniteElementGeometry< LFE, cdim >::jacobianTransposed().

Referenced by Dune::LocalFiniteElementGeometry< LFE, cdim >::volume().

◆ jacobian()

template<class LFE , int cdim>

Jacobian Dune::LocalFiniteElementGeometry< LFE, cdim >::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

References Dune::LocalFiniteElementGeometry< LFE, cdim >::localBasis(), and Dune::FieldMatrix< ctype, coorddimension, mydimension >::rows.

Referenced by Dune::LocalFiniteElementGeometry< LFE, cdim >::jacobianInverse(), and Dune::LocalFiniteElementGeometry< LFE, cdim >::jacobianTransposed().

◆ jacobianInverse()

template<class LFE , int cdim>

JacobianInverse Dune::LocalFiniteElementGeometry< LFE, 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::LocalFiniteElementGeometry< LFE, cdim >::jacobian().

Referenced by Dune::LocalFiniteElementGeometry< LFE, cdim >::jacobianInverseTransposed().

◆ jacobianInverseTransposed()

template<class LFE , int cdim>

JacobianInverseTransposed Dune::LocalFiniteElementGeometry< LFE, 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:

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

References Dune::LocalFiniteElementGeometry< LFE, cdim >::jacobianInverse(), and Dune::FieldMatrix< K, ROWS, COLS >::transposed().

◆ jacobianTransposed()

template<class LFE , int cdim>

JacobianTransposed Dune::LocalFiniteElementGeometry< LFE, cdim >::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::LocalFiniteElementGeometry< LFE, cdim >::jacobian(), and Dune::FieldMatrix< K, ROWS, COLS >::transposed().

Referenced by Dune::LocalFiniteElementGeometry< LFE, cdim >::integrationElement().

◆ LocalCoordinate()

template<class LFE , int cdim>

Dune::LocalFiniteElementGeometry< LFE, cdim >::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 LFE , int cdim>

Volume Dune::LocalFiniteElementGeometry< LFE, cdim >::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/localfiniteelementgeometry.hh

DUNE Distributed and Unified Numerics Environment

Dune Core Modules (unstable)

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Constructor & Destructor Documentation

◆ LocalFiniteElementGeometry() [1/3]

◆ LocalFiniteElementGeometry() [2/3]

◆ LocalFiniteElementGeometry() [3/3]

Member Function Documentation

◆ global()

◆ integrationElement()

◆ jacobian()

◆ jacobianInverse()

◆ jacobianInverseTransposed()

◆ jacobianTransposed()

◆ LocalCoordinate()

◆ volume()