A second-order quadrilateral in the plane or in 3D space. More...

#include <lf/geometry/quad_o2.h>

Inheritance diagram for lf::geometry::QuadO2:

Public Member Functions
	QuadO2 (Eigen::Matrix< double, Eigen::Dynamic, 8 > coords)
	Constructor building quadrilateral from vertex/midpoint coordinates.

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, lf::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, 8 >	coords_
	Coordinates of the 8 vertices/midpoints, stored in matrix columns.

Eigen::Matrix< double, Eigen::Dynamic, 1 >	alpha_

Eigen::Matrix< double, Eigen::Dynamic, 2 >	beta_

Eigen::Matrix< double, Eigen::Dynamic, 2 >	gamma_

Eigen::Matrix< double, Eigen::Dynamic, 1 >	delta_

Eigen::Matrix< double, Eigen::Dynamic, 2 >	epsilon_

Eigen::Matrix< double, Eigen::Dynamic, 2 >	gamma_x_2_

Eigen::Matrix< double, Eigen::Dynamic, 2 >	epsilon_x_2_

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 second-order quadrilateral in the plane or in 3D space.

Coordinates \( coords = [A, B, C, D, E, F, G, H] \) are mapped to the reference element as follows:

(0.0, 1.0) - (0.5, 1.0) - (1.0, 1.0)                  D - G - C
     |                         |                      |       |
(0.0, 0.5)                (1.0, 0.5)        ->        H       F
     |                         |                      |       |
(0.0, 0.0) - (0.5, 0.0) - (1.0, 0.0)                  A - E - B

Definition at line 29 of file quad_o2.h.

Constructor & Destructor Documentation

◆ QuadO2()

lf::geometry::QuadO2::QuadO2 ( Eigen::Matrix< double, Eigen::Dynamic, 8 > coords )

explicit

Constructor building quadrilateral from vertex/midpoint coordinates.

Parameters

coords w x 8 matrix, w = world dimension, whose columns contain the world coordinates of the vertices/midpoints

Definition at line 19 of file quad_o2.cc.

References alpha_, beta_, coords_, delta_, epsilon_, epsilon_x_2_, gamma_, and gamma_x_2_.

Member Function Documentation

◆ ChildGeometry()

std::vector< std::unique_ptr< Geometry > > lf::geometry::QuadO2::ChildGeometry	(	const RefinementPattern &	ref_pat,
		lf::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

For a detailed description of the indexing of the vertices of child entities see Refinement.xoj

Implements lf::geometry::Geometry.

Definition at line 184 of file quad_o2.cc.

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

◆ DimGlobal()

dim_t lf::geometry::QuadO2::DimGlobal ( ) const

inlineoverridevirtual

Dimension of the image of this mapping.

Implements lf::geometry::Geometry.

Definition at line 39 of file quad_o2.h.

References coords_.

Referenced by IntegrationElement(), JacobianInverseGramian(), and SubGeometry().

◆ DimLocal()

dim_t lf::geometry::QuadO2::DimLocal ( ) const

inlineoverridevirtual

Dimension of the domain of this mapping.

Implements lf::geometry::Geometry.

Definition at line 38 of file quad_o2.h.

◆ Global()

Eigen::MatrixXd lf::geometry::QuadO2::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()); }

lf::base::RefEl::NodeCoords

const Eigen::MatrixXd & NodeCoords() const

Get the coordinates of the nodes of this reference element.

Definition ref_el.h:241

lf::geometry::Geometry

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

Definition geometry_interface.h:21

lf::geometry::Geometry::RefEl

virtual base::RefEl RefEl() const =0

The Reference element that defines the domain of this mapping.

lf::geometry::Geometry::Global

virtual Eigen::MatrixXd Global(const Eigen::MatrixXd &local) const =0

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

lf::geometry::Corners

Eigen::MatrixXd Corners(const Geometry &geo)

The corners of a shape with piecewise smooth boundary.

Definition geometry_interface.h:261

Additional explanations in Lecture Document Paragraph 2.7.5.17.

Implements lf::geometry::Geometry.

Definition at line 56 of file quad_o2.cc.

References alpha_, beta_, delta_, epsilon_, and gamma_.

Referenced by ChildGeometry().

◆ IntegrationElement()

Eigen::VectorXd lf::geometry::QuadO2::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 134 of file quad_o2.cc.

References DimGlobal(), and Jacobian().

◆ Jacobian()

Eigen::MatrixXd lf::geometry::QuadO2::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 69 of file quad_o2.cc.

References beta_, delta_, epsilon_, epsilon_x_2_, gamma_, and gamma_x_2_.

Referenced by IntegrationElement(), and JacobianInverseGramian().

◆ JacobianInverseGramian()

Eigen::MatrixXd lf::geometry::QuadO2::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)

lf::geometry::QuadO2::JacobianInverseGramian

Eigen::MatrixXd JacobianInverseGramian(const Eigen::MatrixXd &local) const override

Evaluate the Jacobian * Inverse Gramian ( ) simultaneously at numPoints.

Definition quad_o2.cc:100

More explanations in Paragraph 2.8.3.14.

Implements lf::geometry::Geometry.

Definition at line 100 of file quad_o2.cc.

References DimGlobal(), and Jacobian().

◆ RefEl()

base::RefEl lf::geometry::QuadO2::RefEl ( ) const

inlineoverridevirtual

The Reference element that defines the domain of this mapping.

Implements lf::geometry::Geometry.

Definition at line 40 of file quad_o2.h.

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

◆ SubGeometry()

std::unique_ptr< Geometry > lf::geometry::QuadO2::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 160 of file quad_o2.cc.

References coords_, and DimGlobal().

Member Data Documentation

◆ alpha_

Eigen::Matrix<double, Eigen::Dynamic, 1> lf::geometry::QuadO2::alpha_

private

Definition at line 77 of file quad_o2.h.

Referenced by Global(), and QuadO2().

◆ beta_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::QuadO2::beta_

private

Definition at line 78 of file quad_o2.h.

Referenced by Global(), Jacobian(), and QuadO2().

◆ coords_

Eigen::Matrix<double, Eigen::Dynamic, 8> lf::geometry::QuadO2::coords_

private

Coordinates of the 8 vertices/midpoints, stored in matrix columns.

Definition at line 70 of file quad_o2.h.

Referenced by DimGlobal(), QuadO2(), and SubGeometry().

◆ delta_

Eigen::Matrix<double, Eigen::Dynamic, 1> lf::geometry::QuadO2::delta_

private

Definition at line 80 of file quad_o2.h.

Referenced by Global(), Jacobian(), and QuadO2().

◆ epsilon_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::QuadO2::epsilon_

private

Definition at line 81 of file quad_o2.h.

Referenced by Global(), Jacobian(), and QuadO2().

◆ epsilon_x_2_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::QuadO2::epsilon_x_2_

private

Definition at line 85 of file quad_o2.h.

Referenced by Jacobian(), and QuadO2().

◆ gamma_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::QuadO2::gamma_

private

Definition at line 79 of file quad_o2.h.

Referenced by Global(), Jacobian(), and QuadO2().

◆ gamma_x_2_

Eigen::Matrix<double, Eigen::Dynamic, 2> lf::geometry::QuadO2::gamma_x_2_

private

Definition at line 84 of file quad_o2.h.

Referenced by Jacobian(), and QuadO2().

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

/__w/lehrfempp/lehrfempp/lib/lf/geometry/quad_o2.h
/__w/lehrfempp/lehrfempp/lib/lf/geometry/quad_o2.cc

Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ QuadO2()

Member Function Documentation

◆ ChildGeometry()

◆ DimGlobal()

◆ DimLocal()

◆ Global()

◆ IntegrationElement()

◆ Jacobian()

◆ JacobianInverseGramian()

Example for recovering a single transposed inverse Jacobian.

◆ RefEl()

◆ SubGeometry()

Member Data Documentation

◆ alpha_

◆ beta_

◆ coords_

◆ delta_

◆ epsilon_

◆ epsilon_x_2_

◆ gamma_

◆ gamma_x_2_