Dune::SGeometry< mydim, cdim, GridImp > Class Template Reference

#include <dune/grid/sgrid.hh>

Public Types
typedef GridImp::ctype	ctype
	define type used for coordinates in grid module
enum
	Dimension of the cube element.
enum
	Dimension of the world space that the cube element is embedded in.
typedef FieldVector< ctype, dim >	LocalCoordinate
	Type used for a vector of element coordinates.
typedef FieldVector< ctype, coorddim >	GlobalCoordinate
	Type used for a vector of world coordinates.
typedef conditional< dim==coorddim, DiagonalMatrix< ctype, dim >, FieldMatrix< ctype, dim, coorddim > >::type	JacobianTransposed
	Return type of jacobianTransposed. More...
typedef conditional< dim==coorddim, DiagonalMatrix< ctype, dim >, FieldMatrix< ctype, coorddim, dim > >::type	JacobianInverseTransposed
	Return type of jacobianInverseTransposed. More...

Public Member Functions
void	make (const FieldVector< ctype, cdim > &lower, const FieldMatrix< ctype, mydim, cdim > &A)
	Set up the geometry. More...
	SGeometry ()
	constructor
GeometryType	type () const
	Type of the cube. Here: a hypercube of the correct dimension.
GlobalCoordinate	global (const LocalCoordinate &local) const
	Map a point in local (element) coordinates to world coordinates.
LocalCoordinate	local (const GlobalCoordinate &global) const
	Map a point in global (world) coordinates to element coordinates.
JacobianTransposed	jacobianTransposed (DUNE_UNUSED const LocalCoordinate &local) const
	Jacobian transposed of the transformation from local to global coordinates.
JacobianInverseTransposed	jacobianInverseTransposed (DUNE_UNUSED const LocalCoordinate &local) const
	Jacobian transposed of the transformation from local to global coordinates.
ctype	integrationElement (DUNE_UNUSED const LocalCoordinate &local) const
	Return the integration element, i.e., the determinant term in the integral transformation formula.
GlobalCoordinate	center () const
	Return center of mass of the element.
int	corners () const
	Return the number of corners of the element.
GlobalCoordinate	corner (int k) const
	Return world coordinates of the k-th corner of the element.
ctype	volume () const
	Return the element volume.
bool	affine () const
	Return if the element is affine. Here: yes.

Detailed Description

template<int mydim, int cdim, class GridImp>
class Dune::SGeometry< mydim, cdim, GridImp >

SGeometry realizes the concept of the geometric part of a mesh entity.

The geometric part of a mesh entity is a $d$-dimensional object in $\mathbf{R}^w$ where $d$ corresponds the template parameter dim and $w$ corresponds to the template parameter dimworld.

The $d$-dimensional object is a polyhedron given by a certain number of corners, which are vectors in $\mathbf{R}^w$.

The member function global provides a map from a topologically equivalent polyhedron ("reference element") in $\mathbf{R}^d$ to the given polyhedron. This map can be inverted by the member function local, where an appropriate projection is applied first, when $d\neq w$.

In the case of a structured mesh discretizing a generalized cube this map is linear and can be described as

$$g(l) = s + \sum\limits_{i=0}^{d-1} l_ir^i$$

where $s\in\mathbf{R}^w$ is a given position vector, the $r^i\in\mathbf{R}^w$ are given direction vectors and $l\in\mathbf{R}^d$ is a local coordinate within the reference polyhedron. The direction vectors are assumed to be orthogonal with respect to the standard Eucliden inner product.

The $d$-dimensional reference polyhedron is given by the points $\{ (x_0,\ldots,x_{d-1}) \ | \ x_i\in\{0,1\}\ \}$.

In order to invert the map for a point $p$, we have to find a local coordinate $l$ such that $g(l)=p$. Of course this is only possible if $d=w$. In the general case $d\leq w$ we determine $l$ such that

$$(s,r^k) + \sum\limits_{i=0}^{d-1} l_i (r^i,r^k) = (p,r^k) \ \ \ \forall k=0,\ldots,d-1.$$

The resulting system is diagonal since the direction vectors are required to be orthogonal.

Member Typedef Documentation

JacobianInverseTransposed

typedef conditional<dim==coorddim,DiagonalMatrix<ctype,dim>,FieldMatrix<ctype,coorddim,dim>>::type Dune::AxisAlignedCubeGeometry< GridImp::ctype , dim, coorddim >::JacobianInverseTransposed

inherited

Return type of jacobianInverseTransposed.

This is a fast DiagonalMatrix if dim==coorddim, and a FieldMatrix otherwise. The FieldMatrix will never contain more than one entry per column, hence it could be replaced by something more efficient.

JacobianTransposed

typedef conditional<dim==coorddim,DiagonalMatrix<ctype,dim>,FieldMatrix<ctype,dim,coorddim>>::type Dune::AxisAlignedCubeGeometry< GridImp::ctype , dim, coorddim >::JacobianTransposed

inherited

Return type of jacobianTransposed.

This is a fast DiagonalMatrix if dim==coorddim, and a FieldMatrix otherwise. The FieldMatrix will never contain more than one entry per row, hence it could be replaced by something more efficient.

Member Function Documentation

make()

template<int mydim, int cdim, class GridImp >

void Dune::SGeometry< mydim, cdim, GridImp >::make ( const FieldVector< ctype, cdim > &  lower, const FieldMatrix< ctype, mydim, cdim > &  A  )

inline

Set up the geometry.

Parameters

lower	The lower left corner
A	The direction vectors

Allows a consistent treatment of all dimensions, including 0 (the vertex).

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

• dune/grid/sgrid.hh