A 2D cell: quadrilateral. More...

#include <cell2D.hh>

Inheritance diagram for concepts::Quad2d:

Public Types
typedef uint	index_type

Public Member Functions
Real3d	center () const
	Returns the center of the cell. More...

Real2d	chi (Real xi, Real eta) const
	Evaluates the element map. More...

virtual Quad2d *	child (uint i)
	Returns a child. More...

virtual const Quad2d *	child (uint i) const
	Returns a child. More...

const Quad2d *	child (uint i, Real xi, Real eta, Real &xiC, Real &etaC) const
	Returns a child. More...

Quad2d *	clone (Quad &cntr, MappingQuad2d *map) const
	Returns a copy of itself. More...

Quad &	connector () const
	Returns the quadrilateral connector (topology) More...

virtual uint	dim () const
	Returns the dimension of the space. More...

MappingEdge2d *	edgeMap (uint edge) const
	Returns the mapping of an edge. More...

virtual Real3d	elemMap (const Real coord_local) const
	Element map from point local coordinates in 1D. More...

Real3d	elemMap (const Real2d &coord_local) const
	Element map from point local coordinates in 2D. More...

virtual Real3d	elemMap (const Real3d &coord_local) const
	Element map from point local coordinates in 3D. More...

const Quad2dSubdivision *	getStrategy () const
	Returns the subdivision strategy of this quad. More...

virtual Real	gramDeterminantRoot (const Real xi, const Real eta) const
	Returns the square root of the Gram determinant. More...

bool	hasChildren () const

MapReal2d	inverseLaplace (const Real xi, const Real eta) const
	Computes the second partial derivatives of the inverse Mapping. More...

MapReal2d	jacobian (const concepts::Real2d &p) const

MapReal2d	jacobian (const Real xi, const Real eta) const
	Computes the Jacobian for `xi`, `eta` $\in [0,1]$ . More...

Real	jacobianDeterminant (const Real xi, const Real eta) const
	Returns the determinant of the Jacobian for `xi`, `eta` $\in [0,1]$ . More...

MapReal2d	jacobianInverse (const Real xi, const Real eta) const
	Returns the inverse of the Jacobian for `xi`, `eta` $\in [0,1]$ . More...

Level< 2 >	level () const
	Returns the level of the cell. More...

virtual Real	lineElement (const Real xi, const uint edge) const
	Computes the differential element on an edge. More...

const MappingQuad2d *	map () const
	Returns the stored element map (same for all family related to one ancestor). More...

	Quad2d (Quad &cntr, const MappingQuad2d &map)
	Constructor. More...

virtual void	setStrategy (const Quad2dSubdivision *strategy=0) throw (StrategyChange)
	Sets the subdivision strategy of this quad. More...

Real3d	vertex (uint i) const
	Returns the coordinates of the ith vertex. More...

virtual	~Quad2d ()

Static Public Attributes
static uint	MAX_LEVEL

Protected Member Functions
Quad2d *	createChild_ (Quad &cntr, const Index &idx, bool flag) const
	Non-virtual form of the private createChild_() method. More...

virtual std::ostream &	info (std::ostream &os) const
	Returns information in an output stream. More...

	Quad2d (Quad &cntr, MappingQuad2d *map, const Index &idx)
	Private constructor. More...

Protected Attributes
Quad &	cntr_
	Reference to the quadrilateral connector (topology) More...

Index	idx_
	Index of this element. More...

MappingQuad2d *	map_
	Pointer to the element map. More...

Private Member Functions
virtual Quad2d *	createChild_ (Quad &cntr, const Index &idx) const
	Creates and returns a child. More...

Quad2d &	operator= (const Quad2d &other)

	Quad2d (const Quad2d &other)

Private Attributes
Quad2d *	chld_
	Pointer to the first child. More...

concepts::Array< Edge2d * >	edge_

Quad2d *	lnk_
	Pointer to a sibling. More...

const Quad2dSubdivision *	subdivStrategy_
	Subdivision strategy for the quadrilateral. More...

Friends
class	Quad2dSubdiv2H

class	Quad2dSubdiv2V

class	Quad2dSubdiv4

class	Quad2dSubdivision

Detailed Description

A 2D cell: quadrilateral.

If a quad is subdivided, two or four new quads are created. This happens automatically, if a child is requested, depending on the strategy that has been set. If two or four children are created depends on the chosen strategy. The default strategy is the create four children.

The directions of the edges in the quadrilateral on the coarsest level are externally defined, see Quad::rho(). The edges of quadrilaterals which are created by subdivision are always counter-clockwise.

See also: Cell for more information on cells in a mesh.

Test:

test::TestJacobian2D

test::TestMatrices2D

Examples: meshes.cc.

Definition at line 378 of file cell2D.hh.

Member Typedef Documentation

◆ index_type

typedef uint concepts::QuadNd::index_type

inherited

Definition at line 194 of file cell2D.hh.

Constructor & Destructor Documentation

◆ Quad2d() [1/3]

concepts::Quad2d::Quad2d	(	Quad &	cntr,
		const MappingQuad2d &	map
	)

Constructor.

Takes the connector cntr and the element map map and creates a cell.

Parameters

cntr	Topological information of the quadrilateral
map	Element map of the quadrilateral

◆ ~Quad2d()

virtual concepts::Quad2d::~Quad2d ( )

virtual

◆ Quad2d() [2/3]

concepts::Quad2d::Quad2d ( const Quad2d & other )

private

◆ Quad2d() [3/3]

concepts::Quad2d::Quad2d	(	Quad &	cntr,
		MappingQuad2d *	map,
		const Index &	idx
	)

protected

Private constructor.

Member Function Documentation

◆ center()

Real3d concepts::Quad2d::center ( ) const

inlinevirtual

Returns the center of the cell.

Implements concepts::Cell2.

Definition at line 516 of file cell2D.hh.

◆ chi()

Real2d concepts::Quad2d::chi	(	Real	xi,
		Real	eta
	)		const

Evaluates the element map.

Maps a point from the the reference coordinates in [0,1]² onto the physical coordinates.

Returns: Point in 2D in physical coordinates.

Parameters

xi	$\in [0,1]$
eta	$\in [0,1]$

◆ child() [1/3]

virtual Quad2d* concepts::Quad2d::child ( uint i )

virtual

Returns a child.

If no children exist, four new children are created. The quadrilateral is cut into four new quadrilaterals by joining the midpoints of its edges.

Parameters

i	Index of the child to be returned.

Implements concepts::Cell2.

◆ child() [2/3]

virtual const Quad2d* concepts::Quad2d::child ( uint i ) const

virtual

Returns a child.

If no children exist, none are created and 0 is returned.

Parameters

i	Index of the child to be returned.

Implements concepts::Cell2.

◆ child() [3/3]

const Quad2d* concepts::Quad2d::child	(	uint	i,
		Real	xi,
		Real	eta,
		Real &	xiC,
		Real &	etaC
	)		const

inline

Returns a child.

If not a valid children exist, none are created and 0 is returned. FIXME: xiC and etaC are useless parameters

Parameters

i	Index of the child to be returned.
xi,eta	local coordinates, where the child is located
xiC,etaC	local coordinates in the child

Definition at line 420 of file cell2D.hh.

◆ clone()

Quad2d* concepts::Quad2d::clone	(	Quad &	cntr,
		MappingQuad2d *	map
	)		const

inline

Returns a copy of itself.

Definition at line 536 of file cell2D.hh.

◆ connector()

Quad& concepts::QuadNd::connector ( ) const

inlinevirtualinherited

Returns the quadrilateral connector (topology)

Implements concepts::Cell2.

Definition at line 234 of file cell2D.hh.

◆ createChild_() [1/2]

virtual Quad2d* concepts::Quad2d::createChild_	(	Quad &	cntr,
		const Index &	idx
	)		const

privatevirtual

Creates and returns a child.

Can only called by the object itself or friend classes. Its virtual, so that inheritated classes with additional properties can create a child of the same kind.

◆ createChild_() [2/2]

Quad2d* concepts::Quad2d::createChild_	(	Quad &	cntr,
		const Index &	idx,
		bool	flag
	)		const

protected

Non-virtual form of the private createChild_() method.

Can be called by inheritated classes.

◆ dim()

virtual uint concepts::Quad2d::dim ( ) const

inlinevirtual

Returns the dimension of the space.

Implements concepts::QuadNd.

Definition at line 561 of file cell2D.hh.

◆ edgeMap()

MappingEdge2d* concepts::Quad2d::edgeMap ( uint edge ) const

Returns the mapping of an edge.

Construct a cell for an edge of quad with

Edge2d edge(*quad.connector().edge(0), *quad.edgeMap(0));

The edgeMap can be useful for egde integrals in a 2D quadrilateral mesh.

◆ elemMap() [1/3]

virtual Real3d concepts::Cell::elemMap ( const Real coord_local ) const

virtualinherited

Element map from point local coordinates in 1D.

Reimplemented in concepts::Edge2d, concepts::Edge1d, concepts::Sphere3d, and concepts::SphericalSurface3d.

◆ elemMap() [2/3]

Real3d concepts::Quad2d::elemMap ( const Real2d & coord_local ) const

inlinevirtual

Element map from point local coordinates in 2D.

Implements concepts::Cell2.

Definition at line 447 of file cell2D.hh.

◆ elemMap() [3/3]

virtual Real3d concepts::Cell2::elemMap ( const Real3d & coord_local ) const

inlinevirtualinherited

Element map from point local coordinates in 3D.

Reference element is 2D, third component is omitted.

Reimplemented from concepts::Cell.

Definition at line 104 of file cell.hh.

◆ getStrategy()

const Quad2dSubdivision* concepts::Quad2d::getStrategy ( ) const

inline

Returns the subdivision strategy of this quad.

If you want to find check against another strategy use

quad.getStrategy() == QuadSubdiv2V::instance()

Definition at line 559 of file cell2D.hh.

◆ gramDeterminantRoot()

virtual Real concepts::Quad2d::gramDeterminantRoot	(	const Real	xi,
		const Real	eta
	)		const

inlinevirtual

Returns the square root of the Gram determinant.

The Gram determinant is given by

$\det(A^\top A) \,,$

where A is the Jacobian of the mapping. Note that for a Quad2d this is the same as the determinant of the Jacobian.

Implements concepts::QuadNd.

Definition at line 491 of file cell2D.hh.

◆ hasChildren()

bool concepts::Quad2d::hasChildren ( ) const

inline

Definition at line 409 of file cell2D.hh.

◆ info()

virtual std::ostream& concepts::Quad2d::info ( std::ostream & os ) const

protectedvirtual

Returns information in an output stream.

Implements concepts::Cell.

◆ inverseLaplace()

MapReal2d concepts::Quad2d::inverseLaplace	(	const Real	xi,
		const Real	eta
	)		const

Computes the second partial derivatives of the inverse Mapping.

Those from a cell in the initial mesh are computed by the element map itsself. For a subdivided cell, with the use of the subdivision map , the values are computed by the element map at the Point $(a \cdot xi, b \cdot eta)$ . For notation check out the documentation of the method jacobian() of this class.

xi, eta $\in [0,1]$

◆ jacobian() [1/2]

MapReal2d concepts::Quad2d::jacobian ( const concepts::Real2d & p ) const

◆ jacobian() [2/2]

MapReal2d concepts::Quad2d::jacobian	(	const Real	xi,
		const Real	eta
	)		const

Computes the Jacobian for xi, eta $\in [0,1]$ .

The Jacobian of a cell in the initial mesh is computed by the element map. The Jacobian of a subdivided cell is computed as follows. Let $H : \hat K \rightarrow \hat K'$ be the subdivision map in the reference coordinates (0,1)² and $F_K : \hat K \rightarrow K$ , $F_{K'} : \hat K \rightarrow K'$ the element maps of K and K' respectively. Then,

$F_K : \hat K' \rightarrow K' \quad \text{and} \quad F_K \circ H = F_{K'}.$

The subdivision map H in reference coordinates is

$H : \vec\xi \mapsto \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \vec \xi$

where $a = 2^{-l_0}$ , $b = 2^{-l_1}$ . $\vec l = (l_0, l_1)$ is the level of the cell with respect to the cell in the initial mesh. The derivative of H is $dH = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ .

Then, the Jacobian of $F_{K'}$ is

$dF_{K'} = dF_K \circ H \cdot dH.$

The part $dF_K \circ H$ (without H) is computed by the element map and the part $\cdot dH$ is computed in jacobian().

◆ jacobianDeterminant()

Real concepts::Quad2d::jacobianDeterminant	(	const Real	xi,
		const Real	eta
	)		const

inline

Returns the determinant of the Jacobian for xi, eta $\in [0,1]$ .

Definition at line 487 of file cell2D.hh.

◆ jacobianInverse()

MapReal2d concepts::Quad2d::jacobianInverse	(	const Real	xi,
		const Real	eta
	)		const

inline

Returns the inverse of the Jacobian for xi, eta $\in [0,1]$ .

Definition at line 480 of file cell2D.hh.

◆ level()

Level<2> concepts::QuadNd::level ( ) const

inlineinherited

Returns the level of the cell.

The level describes the refinements in both directions with respect to the parent cell in the initial mesh. This level information is taken into account when evaluating the element map or the Jacobian.

Definition at line 257 of file cell2D.hh.

◆ lineElement()

virtual Real concepts::Quad2d::lineElement	(	const Real	xi,
		const uint	edge
	)		const

virtual

Computes the differential element on an edge.

xi stand for one of the variables

$\xi$

,

$\eta$

depending on the edge.

Implements concepts::QuadNd.

◆ map()

const MappingQuad2d* concepts::Quad2d::map ( ) const

inline

Returns the stored element map (same for all family related to one ancestor).

Definition at line 521 of file cell2D.hh.

◆ operator=()

Quad2d& concepts::Quad2d::operator= ( const Quad2d & other )

private

◆ setStrategy()

virtual void concepts::Quad2d::setStrategy	(	const Quad2dSubdivision *	strategy = `0`	)
throw	(	StrategyChange
	)

virtual

Sets the subdivision strategy of this quad.

If the parameter is set to 0 (or if the method is called without parameter) the strategy is set to the default (if not already set). The default subdivision strategy is subdivision into 4 children.

Parameters

strategy Pointer to an instance of a subdivision strategy.

Exceptions

StrategyChange if the change is not allowed (the change is not allowed if there are children present or the topological strategy is set).

◆ vertex()

Real3d concepts::Quad2d::vertex ( uint i ) const

virtual

Returns the coordinates of the ith vertex.

Implements concepts::Cell2.

Friends And Related Function Documentation

◆ Quad2dSubdiv2H

friend class Quad2dSubdiv2H

friend

Definition at line 381 of file cell2D.hh.

◆ Quad2dSubdiv2V

friend class Quad2dSubdiv2V

friend

Definition at line 382 of file cell2D.hh.

◆ Quad2dSubdiv4

friend class Quad2dSubdiv4

friend

Definition at line 380 of file cell2D.hh.

◆ Quad2dSubdivision

friend class Quad2dSubdivision

friend

Definition at line 379 of file cell2D.hh.

Member Data Documentation

◆ chld_

Quad2d* concepts::Quad2d::chld_

private

Pointer to the first child.

The children are stored in a linked list.

Definition at line 582 of file cell2D.hh.

◆ cntr_

Quad& concepts::QuadNd::cntr_

protectedinherited

Reference to the quadrilateral connector (topology)

Definition at line 261 of file cell2D.hh.

◆ edge_

concepts::Array<Edge2d*> concepts::Quad2d::edge_

mutableprivate

Definition at line 588 of file cell2D.hh.

◆ idx_

Index concepts::QuadNd::idx_

protectedinherited

Index of this element.

Definition at line 263 of file cell2D.hh.

◆ lnk_

Quad2d* concepts::Quad2d::lnk_

private

Pointer to a sibling.

Definition at line 577 of file cell2D.hh.

◆ map_

MappingQuad2d* concepts::Quad2d::map_

protected

Pointer to the element map.

Definition at line 566 of file cell2D.hh.

◆ MAX_LEVEL

uint concepts::QuadNd::MAX_LEVEL

staticinherited

Definition at line 198 of file cell2D.hh.

◆ subdivStrategy_

const Quad2dSubdivision* concepts::Quad2d::subdivStrategy_

private

Subdivision strategy for the quadrilateral.

Definition at line 585 of file cell2D.hh.

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

geometry/cell2D.hh

concepts::Quad2d Class ReferenceGeometry

Public Types

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ index_type

Constructor & Destructor Documentation

◆ Quad2d() [1/3]

◆ ~Quad2d()

◆ Quad2d() [2/3]

◆ Quad2d() [3/3]

Member Function Documentation

◆ center()

◆ chi()

◆ child() [1/3]

◆ child() [2/3]

◆ child() [3/3]

◆ clone()

◆ connector()

◆ createChild_() [1/2]

◆ createChild_() [2/2]

◆ dim()

◆ edgeMap()

◆ elemMap() [1/3]

◆ elemMap() [2/3]

◆ elemMap() [3/3]

◆ getStrategy()

◆ gramDeterminantRoot()

◆ hasChildren()

◆ info()

◆ inverseLaplace()

◆ jacobian() [1/2]

◆ jacobian() [2/2]

◆ jacobianDeterminant()

◆ jacobianInverse()

◆ level()

◆ lineElement()

◆ map()

◆ operator=()

◆ setStrategy()

◆ vertex()

Friends And Related Function Documentation

◆ Quad2dSubdiv2H

◆ Quad2dSubdiv2V

◆ Quad2dSubdiv4

◆ Quad2dSubdivision

Member Data Documentation

◆ chld_

◆ cntr_

◆ edge_

◆ idx_

◆ lnk_

◆ map_

◆ MAX_LEVEL

◆ subdivStrategy_

concepts::Quad2d Class Reference
Geometry