lf::geometry::SegmentO1 Class Reference

A straight edge defined by the location of its two endpoints. More...

#include <lf/geometry/segment_o1.h>

Inheritance diagram for lf::geometry::SegmentO1:

Public Member Functions
	SegmentO1 (Eigen::Matrix< double, Eigen::Dynamic, 2 > coords)
dim_t	DimLocal () const override
	Dimension of the domain of this mapping.
dim_t	DimGlobal () const override
	Dimension of the image of this mapping.
base::RefEl	RefEl () const override
	The Reference element that defines the domain of this mapping.
Eigen::MatrixXd	Global (const Eigen::MatrixXd &local) const override
	Map a number of points in local coordinates into the global coordinate system.
Eigen::MatrixXd	Jacobian (const Eigen::MatrixXd &local) const override
	Evaluate the jacobian of the mapping simultaneously at numPoints points.
Eigen::MatrixXd	JacobianInverseGramian (const Eigen::MatrixXd &local) const override
	Evaluate the Jacobian * Inverse Gramian ( \( J (J^T J)^{-1}\)) simultaneously at numPoints.
Eigen::VectorXd	IntegrationElement (const Eigen::MatrixXd &local) const override
	The integration element (factor appearing in integral transformation formula, see below) at number of evaluation points (specified in local coordinates).
std::unique_ptr< Geometry >	SubGeometry (dim_t codim, dim_t i) const override
	Construct a new Geometry() object that describes the geometry of the i-th sub-entity with codimension=codim
std::vector< std::unique_ptr< Geometry > >	ChildGeometry (const RefinementPattern &ref_pat, base::dim_t codim) const override
	Generate geometry objects for child entities created in the course of refinement.
Public Member Functions inherited from lf::geometry::Geometry
virtual bool	isAffine () const
	element shape by affine mapping from reference element
virtual	~Geometry ()=default
	Virtual destructor.

Private Attributes
Eigen::Matrix< double, Eigen::Dynamic, 2 >	coords_

Additional Inherited Members
Public Types inherited from lf::geometry::Geometry
using	dim_t = base::RefEl::dim_t
Protected Member Functions inherited from lf::geometry::Geometry
	Geometry ()=default
	Geometry (const Geometry &)=default
	Geometry (Geometry &&)=default
Geometry &	operator= (const Geometry &)=default
Geometry &	operator= (Geometry &&)=default
Related Symbols inherited from lf::geometry::Geometry
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

A straight edge defined by the location of its two endpoints.

Definition at line 11 of file segment_o1.h.

Constructor & Destructor Documentation

SegmentO1()

lf::geometry::SegmentO1::SegmentO1 ( Eigen::Matrix< double, Eigen::Dynamic, 2 > coords )

inlineexplicit

Definition at line 13 of file segment_o1.h.

Member Function Documentation

ChildGeometry()

std::vector< std::unique_ptr< Geometry > > lf::geometry::SegmentO1::ChildGeometry ( const RefinementPattern & ref_pat, base::dim_t codim ) const

overridevirtual

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

Implements lf::geometry::Geometry.

Definition at line 45 of file segment_o1.cc.

References lf::geometry::RefinementPattern::ChildPolygons(), Global(), lf::base::RefEl::kSegment(), lf::geometry::RefinementPattern::LatticeConst(), lf::geometry::RefinementPattern::NumChildren(), lf::geometry::RefinementPattern::RefEl(), and lf::base::RefEl::ToString().

DimGlobal()

dim_t lf::geometry::SegmentO1::DimGlobal ( ) const

inlineoverridevirtual

Dimension of the image of this mapping.

Implements lf::geometry::Geometry.

Definition at line 17 of file segment_o1.h.

References coords_.

Referenced by JacobianInverseGramian().

DimLocal()

dim_t lf::geometry::SegmentO1::DimLocal ( ) const

inlineoverridevirtual

Dimension of the domain of this mapping.

Implements lf::geometry::Geometry.

Definition at line 16 of file segment_o1.h.

Global()

Eigen::MatrixXd lf::geometry::SegmentO1::Global ( const Eigen::MatrixXd & local ) const

overridevirtual

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.

Implements lf::geometry::Geometry.

Definition at line 7 of file segment_o1.cc.

References coords_.

Referenced by ChildGeometry().

IntegrationElement()

Eigen::VectorXd lf::geometry::SegmentO1::IntegrationElement ( const Eigen::MatrixXd & local ) const

overridevirtual

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.

Implements lf::geometry::Geometry.

Definition at line 27 of file segment_o1.cc.

References coords_.

Jacobian()

Eigen::MatrixXd lf::geometry::SegmentO1::Jacobian ( const Eigen::MatrixXd & local ) const

overridevirtual

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.

Implements lf::geometry::Geometry.

Definition at line 11 of file segment_o1.cc.

References coords_.

JacobianInverseGramian()

Eigen::MatrixXd lf::geometry::SegmentO1::JacobianInverseGramian ( const Eigen::MatrixXd & local ) const

overridevirtual

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.

Implements lf::geometry::Geometry.

Definition at line 15 of file segment_o1.cc.

References coords_, and DimGlobal().

RefEl()

base::RefEl lf::geometry::SegmentO1::RefEl ( ) const

inlineoverridevirtual

The Reference element that defines the domain of this mapping.

Implements lf::geometry::Geometry.

Definition at line 18 of file segment_o1.h.

References lf::base::RefEl::kSegment().

SubGeometry()

std::unique_ptr< Geometry > lf::geometry::SegmentO1::SubGeometry ( dim_t codim, dim_t i ) const

overridevirtual

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 \( \mathbf{\Phi} \circ \mathbf{\xi} \)

Implements lf::geometry::Geometry.

Definition at line 33 of file segment_o1.cc.

References coords_.

Member Data Documentation

coords_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::SegmentO1::coords_

private

Definition at line 35 of file segment_o1.h.

Referenced by DimGlobal(), Global(), IntegrationElement(), Jacobian(), JacobianInverseGramian(), and SubGeometry().

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

• /__w/lehrfempp/lehrfempp/lib/lf/geometry/segment_o1.h
• /__w/lehrfempp/lehrfempp/lib/lf/geometry/segment_o1.cc