#include <elementMaps3D.hh>

Inheritance diagram for concepts::MapTetrahedron3d:

Public Member Functions
	MapTetrahedron3d (char *map, Real scX, Real scY, Real scZ)

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

	MapTetrahedron3d (const MapTetrahedron3d &map)
	Copy constructor.

Real3d	operator() (Real x, Real y, Real z) const

Real	jacobian () const
	Returns the jacobian.

MapTetrahedron3d *	clone () const
	Returns a copy of the map.

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

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() [1/2]

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() [2/2]

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()

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::MapHexahedron3d, concepts::MappingHexahedron3d, and concepts::PartMappingHexahedron3d.

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 $\xi_3 = \eta_3$ .