A 3D element map for a tetrahedron. More...

#include <elementMaps3D.hh>

Inheritance diagram for concepts::MapTetrahedron3d:

Public Member Functions
MapTetrahedron3d *	clone () const
	Returns a copy of the map. More...

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

Real	jacobian () const
	Returns the jacobian. More...

	MapTetrahedron3d (char *map, Real scX, Real scY, Real scZ)
	Constructor. More...

	MapTetrahedron3d (const MapTetrahedron3d &map)
	Copy constructor. More...

	MapTetrahedron3d (Real3d vtx0, Real3d vtx1, Real3d vtx2, Real3d vtx3)
	Constructor. More...

Real3d	operator() (Real x, Real y, Real z) const
	Returns a point in 3D mapped from the unit cube [0,1]³ onto the element in the original mesh. More...

	~MapTetrahedron3d ()

Private Member Functions
void	operator= (const MapTetrahedron3d &)
	Private assignement operator. More...

Private Attributes
MapReal3d	B_
	Computed map, matrix part. More...

Real3d	b_
	Computed map, vector part. More...

uchar *	map_
	Parsed formula for the map. More...

Real	scx_
	Right border of the x parameter domain. More...

Real	scy_
	Right border of the y parameter domain. More...

Real	scz_
	Right border of the z parameter domain. More...

uint	sz_
	Length of the parsed formula. More...

Detailed Description

A 3D element map for a tetrahedron.

The reference element is the unit cube.

There are two ways to fix the map of the tetrahedron: give the mapping explicitly or give the four corners and the map is computed internally.

Giving the map explicitly needs a few precautions: The reference element is the unit cube. For easier construction of the element map, a rectangle $[0,a] \times [0,b] \times [0,c]$ can be given as the parameter domain. To map this onto a tetrahedron, two sides have to degenerate using a so called Duffy transformation.

With a arbitrary Duffy transformation, one can choose the side which sould collapse. Here, the top face of the cube and the back edge of the bottom face should degenerate to a point, ie. the unit cube maps onto the unit simplex. This can be done with three Duffy transformations:

Collapse the top face against the 2 direction.
Collapse the resulting egde against the 1 direction.
Collapse the back edge of the bottom face against the 1 direction.

When giving the four corners, the first three have to be in the normal orientation of the first triangle of the tetrahedron, see Tetrahedron for the explanation of "normal orientation".

The application operator maps a point from the unit square onto the physical domain taking into account the parameter domain given in the constructor.

Definition at line 66 of file elementMaps3D.hh.

Constructor & Destructor Documentation

◆ MapTetrahedron3d() [1/3]

concepts::MapTetrahedron3d::MapTetrahedron3d	(	char *	map,
		Real	scX,
		Real	scY,
		Real	scZ
	)

Constructor.

The values of scX, scY and scZ are only for the map, they are not needed by the user for mapping a point from the reference element onto the Real element.

Parameters

map	The element map for this tetrehdron as a string, x, y and z are the allowed variables, the component are separated by a comma.
scX	The range of x is [0, scX].
scY	The range of y is [0, scY].
scZ	The range of z is [0, scZ].

◆ MapTetrahedron3d() [2/3]

concepts::MapTetrahedron3d::MapTetrahedron3d	(	Real3d	vtx0,
		Real3d	vtx1,
		Real3d	vtx2,
		Real3d	vtx3
	)

Constructor.

Takes the four physical corners of the tetrahedron and computes the element map: $F_K : S \rightarrow K$ with $\vec x = F_K(\xi) = B \cdot \vec \xi + \vec b$ , where $\vec b = \mbox{vtx0}$ and $B = [ (\mbox{vtx1} - \vec b) (\mbox{vtx2} - \vec b) (\mbox{vtx3} - \vec b) ]$ .

◆ MapTetrahedron3d() [3/3]

concepts::MapTetrahedron3d::MapTetrahedron3d ( const MapTetrahedron3d & map )

Copy constructor.

◆ ~MapTetrahedron3d()

concepts::MapTetrahedron3d::~MapTetrahedron3d ( )

inline

Definition at line 91 of file elementMaps3D.hh.

Member Function Documentation

◆ clone()

MapTetrahedron3d* concepts::MapTetrahedron3d::clone ( ) const

inline

Returns a copy of the map.

Definition at line 129 of file elementMaps3D.hh.

◆ info()

std::ostream& concepts::Map3d::info ( std::ostream & os ) const

inlinevirtualinherited

Returns information in an output stream.

Reimplemented from concepts::OutputOperator.

Reimplemented in concepts::PartMappingHexahedron3d, concepts::BlendingHexahedron3d, concepts::MappingHexahedron3d, and concepts::MapHexahedron3d.

Definition at line 28 of file elementMaps3D.hh.

◆ jacobian()

Real concepts::MapTetrahedron3d::jacobian ( ) const

inline

Returns the jacobian.

Definition at line 123 of file elementMaps3D.hh.

◆ operator()()

Real3d concepts::MapTetrahedron3d::operator()	(	Real	x,
		Real	y,
		Real	z
	)		const

inline

Returns a point in 3D mapped from the unit cube [0,1]³ onto the element in the original mesh.

If the map was given as formula in the constructor, the parameters are directly inserted into the formula. If the map was given by the four corners of the physical tetrahedron, then the point is first mapped from the unit cube ( $\eta$ ) onto the unit simplex ( $\xi$ ) using $\xi_1 = \eta_1(1-\eta_3)(1-\eta_2(1-\eta_3))$ , $\xi_2 = \eta_2(1-\eta_3)$ and $\xi_3 = \eta_3$ .

Returns

Parameters

x	$\in [0, 1]$
y	$\in [0, 1]$
z	$\in [0, 1]$

Definition at line 109 of file elementMaps3D.hh.

◆ operator=()

void concepts::MapTetrahedron3d::operator= ( const MapTetrahedron3d & )

private

Private assignement operator.

Member Data Documentation

◆ B_

MapReal3d concepts::MapTetrahedron3d::B_

private

Computed map, matrix part.

Definition at line 150 of file elementMaps3D.hh.

◆ b_

Real3d concepts::MapTetrahedron3d::b_

private

Computed map, vector part.

Definition at line 153 of file elementMaps3D.hh.

◆ map_

uchar* concepts::MapTetrahedron3d::map_

private

Parsed formula for the map.

Definition at line 135 of file elementMaps3D.hh.

◆ scx_

Real concepts::MapTetrahedron3d::scx_

private

Right border of the x parameter domain.

Definition at line 141 of file elementMaps3D.hh.

◆ scy_

Real concepts::MapTetrahedron3d::scy_

private

Right border of the y parameter domain.

Definition at line 144 of file elementMaps3D.hh.

◆ scz_

Real concepts::MapTetrahedron3d::scz_

private

Right border of the z parameter domain.

Definition at line 147 of file elementMaps3D.hh.

◆ sz_

uint concepts::MapTetrahedron3d::sz_

private

Length of the parsed formula.

Definition at line 138 of file elementMaps3D.hh.

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

geometry/elementMaps3D.hh

concepts::MapTetrahedron3d Class Reference

Public Member Functions

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ MapTetrahedron3d() [1/3]

◆ MapTetrahedron3d() [2/3]

◆ MapTetrahedron3d() [3/3]

◆ ~MapTetrahedron3d()

Member Function Documentation

◆ clone()

◆ info()

◆ jacobian()

◆ operator()()

◆ operator=()

Member Data Documentation

◆ B_

◆ b_

◆ map_

◆ scx_

◆ scy_

◆ scz_

◆ sz_