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

#include <sgrid.hh>

Inheritance diagram for Dune::SGeometry< mydim, cdim, GridImp >:

Dune::GeometryDefaultImplementation< mydim, cdim, GridImp, GeometryImp >

List of all members.


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.

Public Types

typedef sgrid_ctype ctype
 define type used for coordinates in grid module

Public Member Functions

GeometryType type () const
 return the element type identifier
int corners () const
 return the number of corners of this element. Corners are numbered 0...n-1
const FieldVector< sgrid_ctype,
cdim > & 
operator[] (int i) const
 access to coordinates of corners. Index is the number of the corner
FieldVector< sgrid_ctype, cdim > global (const FieldVector< sgrid_ctype, mydim > &local) const
 maps a local coordinate within reference element to global coordinate in element
FieldVector< sgrid_ctype, mydim > local (const FieldVector< sgrid_ctype, cdim > &global) const
 maps a global coordinate within the element to a local coordinate in its reference element
bool checkInside (const FieldVector< sgrid_ctype, mydim > &local) const
 returns true if the point in local coordinates is located within the refelem
sgrid_ctype integrationElement (const FieldVector< sgrid_ctype, mydim > &local) const
const FieldMatrix< sgrid_ctype,
cdim, mydim > & 
jacobianInverseTransposed (const FieldVector< sgrid_ctype, mydim > &local) const
 can only be called for dim=dimworld!
void print (std::ostream &ss, int indent) const
 print internal data
void make (FieldMatrix< sgrid_ctype, mydim+1, cdim > &__As)
 SGeometry ()
 constructor
ctype volume () const
 return volume of the geometry


Member Function Documentation

template<int mydim, int cdim, class GridImp>
sgrid_ctype Dune::SGeometry< mydim, cdim, GridImp >::integrationElement ( const FieldVector< sgrid_ctype, mydim > &  local  )  const

Integration over a general element is done by integrating over the reference element and using the transformation from the reference element to the global element as follows:

\[\int\limits_{\Omega_e} f(x) dx = \int\limits_{\Omega_{ref}} f(g(l)) A(l) dl \]

where $g$ is the local to global mapping and $A(l)$ is the integration element.

For a general map $g(l)$ involves partial derivatives of the map (surface element of the first kind if $d=2,w=3$, determinant of the Jacobian of the transformation for $d=w$, $\|dg/dl\|$ for $d=1$).

For linear elements, the derivatives of the map with respect to local coordinates do not depend on the local coordinates and are the same over the whole element.

For a structured mesh where all edges are parallel to the coordinate axes, the computation is the length, area or volume of the element is very simple to compute.

Each grid module implements the integration element with optimal efficieny. This will directly translate in substantial savings in the computation of finite element stiffness matrices.

template<int mydim, int cdim, class GridImp>
void Dune::SGeometry< mydim, cdim, GridImp >::make ( FieldMatrix< sgrid_ctype, mydim+1, cdim > &  __As  ) 

The first dim columns of As contain the dim direction vectors. Column dim is the position vector. This format allows a consistent treatement of all dimensions, including 0 (the vertex).


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

Generated on Thu Apr 2 10:40:48 2009 for dune-grid by  doxygen 1.5.6