lf::geometry::Geometry Class Reference

Interface class for shape information on a mesh cell in the spirit of parametric finite element methods More...

#include <lf/geometry/geometry_interface.h>

Inheritance diagram for lf::geometry::Geometry:

Public Types
using	dim_t = base::RefEl::dim_t

Public Member Functions
virtual dim_t	DimLocal () const =0
	Dimension of the domain of this mapping.
virtual dim_t	DimGlobal () const =0
	Dimension of the image of this mapping.
virtual base::RefEl	RefEl () const =0
	The Reference element that defines the domain of this mapping.
virtual Eigen::MatrixXd	Global (const Eigen::MatrixXd &local) const =0
	Map a number of points in local coordinates into the global coordinate system.
virtual Eigen::MatrixXd	Jacobian (const Eigen::MatrixXd &local) const =0
	Evaluate the jacobian of the mapping simultaneously at numPoints points.
virtual Eigen::MatrixXd	JacobianInverseGramian (const Eigen::MatrixXd &local) const =0
	Evaluate the Jacobian * Inverse Gramian ( \( J (J^T J)^{-1}\)) simultaneously at numPoints.
virtual Eigen::VectorXd	IntegrationElement (const Eigen::MatrixXd &local) const =0
	The integration element (factor appearing in integral transformation formula, see below) at number of evaluation points (specified in local coordinates).
virtual std::unique_ptr< Geometry >	SubGeometry (dim_t codim, dim_t i) const =0
	Construct a new Geometry() object that describes the geometry of the i-th sub-entity with codimension=codim
virtual std::vector< std::unique_ptr< Geometry > >	ChildGeometry (const RefinementPattern &ref_pat, lf::base::dim_t codim) const =0
	Generate geometry objects for child entities created in the course of refinement.
virtual bool	isAffine () const
	element shape by affine mapping from reference element
virtual	~Geometry ()=default
	Virtual destructor.

Protected Member Functions
	Geometry ()=default
	Geometry (const Geometry &)=default
	Geometry (Geometry &&)=default
Geometry &	operator= (const Geometry &)=default
Geometry &	operator= (Geometry &&)=default

Related Symbols
(Note that these are not member symbols.)
void	PrintInfo (std::ostream &o, const Geometry &geom, int output_ctrl=0)
	Diagnostic output operator for Geometry.
std::ostream &	operator<< (std::ostream &stream, const Geometry &geom)
	Operator overload to print a Geometry to a stream, such as std::cout

Detailed Description

Interface class for shape information on a mesh cell in the spirit of parametric finite element methods

This abstract base class is supposed to provide complete information about the mapping \(\Phi:\widehat{K}\to K\) that takes the reference cell to a particular cell. The reference cell is defined through the topological type of a cell, see lf::base::RefEl.

Definition at line 21 of file geometry_interface.h.

Member Typedef Documentation

dim_t

using lf::geometry::Geometry::dim_t = base::RefEl::dim_t

Definition at line 30 of file geometry_interface.h.

Constructor & Destructor Documentation

Geometry() [1/3]

lf::geometry::Geometry::Geometry ( )

protecteddefault

Referenced by lf::mesh::Mesh::PrintInfo(), lf::io::writeMatlab(), lf::io::writeMatplotlib(), and lf::io::writeTikZ().

Geometry() [2/3]

lf::geometry::Geometry::Geometry ( const Geometry & )

protecteddefault

Geometry() [3/3]

lf::geometry::Geometry::Geometry ( Geometry && )

protecteddefault

~Geometry()

virtual lf::geometry::Geometry::~Geometry ( )

virtualdefault

Virtual destructor.

Member Function Documentation

ChildGeometry()

virtual std::vector< std::unique_ptr< Geometry > > lf::geometry::Geometry::ChildGeometry ( const RefinementPattern & ref_pat, lf::base::dim_t codim ) const

pure virtual

Generate geometry objects for child entities created in the course of refinement.

Parameters

ref_pat	An object encoding the details of refinement
codim	relative codimension of child entities whose shape is requested

Returns

an array of unique pointers pointers to geometry objects for child entities of the specified co-dimension and the given refinement pattern. The numbering of child entities is a convention.

Implementation of local mesh refinement entails splitting of elements into parts ("children"), whose shape will depend on the refinement pattern. This method creates the geometry objects describing the shape of children. The details of subdivisions corresponding to particular refinement patterns are fixed by the method RefinementPattern::ChildPolygons() and should be documented there.

See also

class lf::geometry::RefinementPattern

Implemented in lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::TriaO1, lf::geometry::TriaO2, and lf::geometry::Point.

Referenced by lf::geometry::test_utils::checkChildGeometry(), lf::geometry::test_utils::checkChildGeometryVolume(), and lf::refinement::MeshHierarchy::PerformRefinement().

DimGlobal()

virtual dim_t lf::geometry::Geometry::DimGlobal ( ) const

pure virtual

Dimension of the image of this mapping.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkChildGeometry(), lf::geometry::test_utils::checkIntegrationElement(), lf::geometry::test_utils::checkJacobian(), lf::geometry::test_utils::checkJacobianInverseGramian(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval(), lf::fe::DiffusionElementMatrixProvider< SCALAR, DIFF_COEFF >::Eval(), lf::fe::MassElementMatrixProvider< SCALAR, REACTION_COEFF >::Eval(), lf::uscalfe::LinearFELaplaceElementMatrix::Eval(), lf::uscalfe::LinearFELocalLoadVector< SCALAR, FUNCTOR >::Eval(), and PrintInfo().

DimLocal()

virtual dim_t lf::geometry::Geometry::DimLocal ( ) const

pure virtual

Dimension of the domain of this mapping.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkChildGeometry(), lf::geometry::test_utils::checkIntegrationElement(), lf::geometry::test_utils::checkJacobian(), lf::geometry::test_utils::checkJacobianInverseGramian(), lf::geometry::test_utils::checkSubGeometry(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval(), lf::uscalfe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::fe::DiffusionElementMatrixProvider< SCALAR, DIFF_COEFF >::Eval(), lf::fe::MassElementMatrixProvider< SCALAR, REACTION_COEFF >::Eval(), lf::fe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::uscalfe::LinearFELaplaceElementMatrix::Eval(), lf::uscalfe::LinearFELocalLoadVector< SCALAR, FUNCTOR >::Eval(), lf::fe::ScalarLoadEdgeVectorProvider< SCALAR, FUNCTOR, EDGESELECTOR >::Eval(), PrintInfo(), lf::mesh::Mesh::PrintInfo(), and lf::geometry::Volume().

Global()

virtual Eigen::MatrixXd lf::geometry::Geometry::Global ( const Eigen::MatrixXd & local ) const

pure virtual

Map a number of points in local coordinates into the global coordinate system.

Parameters

local

A Matrix of size DimLocal() x numPoints that contains in its columns the coordinates of the points at which the mapping function should be evaluated.

Returns

A Matrix of size DimGlobal() x numPoints that contains the mapped points as column vectors. Here numPoints is the number of columns of the matrix passed in the local argument.

\[ \mathtt{Global}\left(\left[\widehat{x}^1,\ldots,\widehat{x}^k\right]\right) = \left[ \mathbf{\Phi}_K(\widehat{x}^1),\ldots,\mathbf{\Phi}_K(\widehat{x}^k)\right]\;,\quad \widehat{x}^{\ell}\in\widehat{K}\;, \]

where \(\mathbf{\Phi}\) is the mapping taking the reference element to the current element \(K\).

This method provides a complete description of the shape of an entity through a parameterization over the corresponding reference element = parameter domain. The method takes as arguments a number of coordinate vectors of points in the reference element. For the sake of efficiency, these coordinate vectors are passed as the columns of a dynamic matrix type as supplied by Eigen.

For instance, this method is used in lf::geometry::Corners()

inline Eigen::MatrixXd Corners(const Geometry& geo) {
return geo.Global(geo.RefEl().NodeCoords()); }

Additional explanations in Lecture Document Paragraph 2.7.5.17.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkChildGeometry(), lf::mesh::test_utils::checkGeometryOrientation(), lf::geometry::test_utils::checkJacobian(), lf::geometry::test_utils::checkSubGeometry(), lf::geometry::Corners(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval(), lf::uscalfe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::fe::DiffusionElementMatrixProvider< SCALAR, DIFF_COEFF >::Eval(), lf::fe::MassElementMatrixProvider< SCALAR, REACTION_COEFF >::Eval(), lf::fe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::uscalfe::LinearFELaplaceElementMatrix::Eval(), lf::uscalfe::LinearFELocalLoadVector< SCALAR, FUNCTOR >::Eval(), lf::uscalfe::ScalarLoadEdgeVectorProvider< SCALAR, FUNCTOR, EDGESELECTOR >::Eval(), lf::refinement::EntityCenterPositionSelector< POSPRED >::operator()(), lf::mesh::utils::MeshFunctionGlobal< F >::operator()(), PrintInfo(), lf::mesh::Mesh::PrintInfo(), lf::mesh::Entity::PrintInfo(), lf::io::writeMatlab(), lf::io::writeMatplotlib(), and lf::io::writeTikZ().

IntegrationElement()

virtual Eigen::VectorXd lf::geometry::Geometry::IntegrationElement ( const Eigen::MatrixXd & local ) const

pure virtual

The integration element (factor appearing in integral transformation formula, see below) at number of evaluation points (specified in local coordinates).

Parameters

local

A Matrix of size DimLocal() x numPoints that contains the evaluation points (in local = reference coordinates) as column vectors.

Returns

A Vector of size numPoints x 1 that contains the integration elements at every evaluation point.

For a transformation \( \Phi : K \mapsto R^{\text{DimGlobal}}\) with Jacobian \( D\Phi : K \mapsto R^{\text{DimGlobal} \times \text{DimLocal}} \) the integration element \( g \) at point \( \xi \in K \) is defined by

\[ g(\xi) := \sqrt{\mathrm{det}\left|D\Phi^T(\xi) D\Phi(\xi) \right|} \]

More information also related to the use of lovcal quadrature rules is given in Lecture Document Paragraph 2.7.5.24.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkChildGeometryVolume(), lf::geometry::test_utils::checkIntegrationElement(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval(), lf::uscalfe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::fe::DiffusionElementMatrixProvider< SCALAR, DIFF_COEFF >::Eval(), lf::fe::MassElementMatrixProvider< SCALAR, REACTION_COEFF >::Eval(), lf::fe::ScalarLoadElementVectorProvider< SCALAR, MESH_FUNCTION >::Eval(), lf::uscalfe::LinearFELaplaceElementMatrix::Eval(), lf::uscalfe::MassEdgeMatrixProvider< SCALAR, COEFF, EDGESELECTOR >::Eval(), lf::uscalfe::ScalarLoadEdgeVectorProvider< SCALAR, FUNCTOR, EDGESELECTOR >::Eval(), lf::fe::MassEdgeMatrixProvider< SCALAR, COEFF, EDGESELECTOR >::Eval(), lf::fe::ScalarLoadEdgeVectorProvider< SCALAR, FUNCTOR, EDGESELECTOR >::Eval(), and lf::geometry::Volume().

isAffine()

virtual bool lf::geometry::Geometry::isAffine ( ) const

inlinevirtual

element shape by affine mapping from reference element

Returns

true, if the element is the affine image of a reference element

An affine map is a linear mapping plus a translation. For a 2D mesh an entity has an affine geometry, if

• a segement is a straight line
• a triangle is flat
• a quadrilateral is a flat parallelogram.

Note

The Jacobian/Gramian for an affine element are constant.

Reimplemented in lf::geometry::QuadO1, and lf::geometry::Parallelogram.

Definition at line 220 of file geometry_interface.h.

Jacobian()

virtual Eigen::MatrixXd lf::geometry::Geometry::Jacobian ( const Eigen::MatrixXd & local ) const

pure virtual

Evaluate the jacobian of the mapping simultaneously at numPoints points.

Parameters

local

A Matrix of size DimLocal x numPoints that contains the evaluation points as column vectors

Returns

A Matrix of size DimGlobal() x (DimLocal() * numPoints) that contains the jacobians at the evaluation points.

This method allows access to the derivative of the parametrization mapping in a number of points, passed as the columns of a dynamic matrix. The derivative of the parametrization in a point is a Jacobian matrix of size ‘DimGlobal() x DimLocal()’. For the sake of efficiency, these matrices are stacked horizontally and returned as one big dynamic matrix. Use Eigen's ‘block()’ method of Eigen::MatrixXd to extract the individual Jacobians from the returned matrix.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkIntegrationElement(), lf::geometry::test_utils::checkJacobian(), and lf::geometry::test_utils::checkJacobianInverseGramian().

JacobianInverseGramian()

virtual Eigen::MatrixXd lf::geometry::Geometry::JacobianInverseGramian ( const Eigen::MatrixXd & local ) const

pure virtual

Evaluate the Jacobian * Inverse Gramian ( \( J (J^T J)^{-1}\)) simultaneously at numPoints.

Parameters

local

A Matrix of size DimLocal() x numPoints that contains the evaluation points as column vectors.

Returns

A Matrix of size DimGlobal() x (DimLocal() * numPoints) that contains the Jacobian multiplied with the Inverse Gramian ( \( J (J^T J)^{-1}\)) at every evaluation point.

Note

If DimLocal() == DimGlobal() then \( J (J^T J)^{-1} = J^{-T} \), i.e. this method returns the inverse of the transposed jacobian.

Example for recovering a single transposed inverse Jacobian.

If both dimensions agree and have the value D, then the method returns the transposed of the inverse Jacobians of the transformation at the passed points. These are square DxD matrices.

To retrieve the j-th inverse transposed Jacobian from the returned matrix, use the block methdod of Eigen (case (D = DimLocal()) == DimGlobal())

JacobianInverseGramian(local).block(0,i*D,D,D)

More explanations in Paragraph 2.8.3.14.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkJacobianInverseGramian(), lf::uscalfe::ReactionDiffusionElementMatrixProvider< SCALAR, DIFF_COEFF, REACTION_COEFF >::Eval(), lf::fe::DiffusionElementMatrixProvider< SCALAR, DIFF_COEFF >::Eval(), lf::uscalfe::LinearFELaplaceElementMatrix::Eval(), and lf::fe::MeshFunctionGradFE< SCALAR_FE, SCALAR_COEFF >::operator()().

operator=() [1/2]

Geometry & lf::geometry::Geometry::operator= ( const Geometry & )

protecteddefault

operator=() [2/2]

Geometry & lf::geometry::Geometry::operator= ( Geometry && )

protecteddefault

RefEl()

virtual base::RefEl lf::geometry::Geometry::RefEl ( ) const

pure virtual

The Reference element that defines the domain of this mapping.

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkChildGeometry(), lf::geometry::test_utils::checkChildGeometryVolume(), lf::geometry::test_utils::checkSubGeometry(), lf::geometry::Corners(), operator<<(), PrintInfo(), lf::mesh::Mesh::PrintInfo(), and lf::geometry::Volume().

SubGeometry()

virtual std::unique_ptr< Geometry > lf::geometry::Geometry::SubGeometry ( dim_t codim, dim_t i ) const

pure virtual

Construct a new Geometry() object that describes the geometry of the i-th sub-entity with codimension=codim

Parameters

codim	The codimension of the sub-entity (w.r.t. DimLocal())
i	The zero-based index of the sub-entity.

Returns

A new Geometry object that describes the geometry of the specified sub-entity.

Let \( \mathbf{\Phi} : K \mapsto \mathbb{R}^\text{DimGlobal} \) be the mapping of this Geometry object and let \( \mathbf{\xi} : \mathbb{R}^{\text{DimLocal}-codim} \mapsto K\) be the first-order mapping that maps the reference element RefEl().SubType(codim,i) to the i-th sub-entity of RefEl(). I.e. for every node \( \mathbf{n_j} \) of RefEl().SubType(codim,i) it holds that \( \mathbf{\xi}(\mathbf{n_j}) = \) RefEl().NodeCoords(RefEl().SubSubEntity2SubEntity(codim, i, DimLocal()-codim, j)).

Then the geometry element returned by this method describes exactly the mapping \( \mathbf{\Phi} \circ \mathbf{\xi} \)

Implemented in lf::geometry::Point, lf::geometry::QuadO1, lf::geometry::Parallelogram, lf::geometry::QuadO2, lf::geometry::SegmentO1, lf::geometry::SegmentO2, lf::geometry::TriaO1, and lf::geometry::TriaO2.

Referenced by lf::geometry::test_utils::checkSubGeometry().

Friends And Related Symbol Documentation

operator<<()

std::ostream & operator<< ( std::ostream & stream, const Geometry & geom )

related

Operator overload to print a Geometry to a stream, such as std::cout

Parameters

stream	The stream to which this function should output
geom	The geometry to write to stream.

Returns

The stream itself.

Note

At the moment this will just write the type of the reference element to the stream.

Definition at line 26 of file print_info.cc.

References RefEl().

PrintInfo()

void PrintInfo ( std::ostream & o, const Geometry & geom, int output_ctrl = 0 )

related

Diagnostic output operator for Geometry.

Parameters

geom	The geometry to print info about
o	The stream to which this function should output
output_ctrl	Controls the level of detail of the written output (see below)

Output levels

• output_ctrl == 0: Reference element type of geometry is printed
• output_ctrl > 10: The above and global and local dimension and derived type of reference element is printed.
• output_ctrl > 90: The above and coordinates og the geometry is printed.

Definition at line 5 of file print_info.cc.

References DimGlobal(), DimLocal(), Global(), lf::base::RefEl::NodeCoords(), and RefEl().

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

• /__w/lehrfempp/lehrfempp/lib/lf/geometry/geometry_interface.h
• /__w/lehrfempp/lehrfempp/lib/lf/geometry/print_info.h