thgdint.html

有限元学习研究用源代码(老外的),供科研人员参考

In this part, we consider questions connected with the geometry
description. We introduce a new concept for geometry description we
denote <b>contravariant geometry description</b>. It has a lot of
advantages especially for the description of higher-dimensional
geometries.

<h1><a NAME="thgdintr">Introduction</a></h1>


In the simulation of various physical phenomena we often have
different regions with different physical properties which are
subdivided by sharp boundaries. Often we have not only discontinuous
changes in material properties on the boundaries, but we have to
consider different sets of equations and variables in different
region. Thus, the description of this subdivision into different
regions is a necessary part of many applications.

  <p>Usually the boundary itself also consists of different parts which
require different boundary conditions. This scheme may be continued
--- the boundary of the boundary also may be subdivided into parts
and so on. Thus, a complete geometry description has to define the
subdivision of the domain of interest into segments and their
boundaries for every codimension.

  <p>The most typical application for a geometry description is the
three-dimensional space. Often, usually caused by the complexity of
the 3D problem and the restricted computer power, 2D and 1D
simplifications will be used. But there are also interesting higher
dimensional applications: 4D for space-time, 6D for the phase space,
7D for the phase space in time. That's why it makes sense to consider
the general, n-dimensional problem.  In the following we consider
geometry descriptions on an arbitrary smooth manifold X. For the
segments of codimension 0 and 1 we use the denotations <b>region</b> and
<b>face</b>, for segments of dimension 0 and 1 the denotations <b>vertex</b> and
<b>line</b>.

  <p>If we have to consider an object on a space X (like in our case a
geometry on X), one of the first questions we have to consider is the
functional behaviour of the object. That means, assume we have some
mapping f: X --> Y between two spaces, is it possible to define
in a natural way a related object on Y for a given object on X
resp. an object on X for a given object on Y? If there is such a
natural transformation, such objects will be called <b>covariant</b>
resp.  <b>contravariant</b>.

 These notions come from category theory, but outside this
theory the notion covariant will be often used as for
covariant, as for contravariant objects, simply to emphasize that
they have a well-defined transformation behaviour. For example, a
function on a manifold is contravariant, but in general relativity we
use the notion covariant to describe this transformation
behaviour.
 

In our case, we can find an answer even without a formal definition of
a geometry:

  If we have a smooth mapping f: X --> Y and a geometry on
Y, this defines in general a natural geometry on X.
 

As the segments of this geometry in X we simply use the pre-images of
the segments of the geometry in Y. If we exclude some degenerate cases
(e.g. that the pre-image of a boundary segment has codimension 0), this
rule defines a geometry on X which we call the geometry <b>induced
by the mapping f</b>.

  <p>The standard way to formalize the notion of a geometry is a cell
complex. The k-dimensional boundary segments will be subdivided into
elementary cells, and every cell will be described by a mapping from a
standard cell into the space.

  <p>This definition leads to a problem --- there is no natural
realization of the induced geometry. The reason is that the natural
transformation of a cell complex points in the other direction. For a
cell complex on X, we easily obtain a cell complex on Y. But this
natural transformation does not define a geometry on Y. Thus, to
define a geometry we have used an object with another transformation
behaviour.

  <p>We introduce here another definition of a geometry. This
definition leads to the correct transformation behaviour, so we have
named it <b>contravariant geometry description</b> or shortly
<b>cogeometry</b>.
 The notion cogeometry we have formed in analogy to the pair
cohomology --- homology. There is indeed an analogy between the
cogeometry and cohomology compared with standard geometry description
and homology. The cogeometry can be considered also as a variant of
the dual construction of a cell complex.
 

It consists of functions f(k) which find intersections of
k-dimensional simplices with segments of the related codimensions ---
one function for each dimension. The first function f(0) is simply a
function which defines the region containing a given point, the second
allows to find intersections of an edge with boundary faces and so on.

  <p>We consider different algorithms which may be used to define geometries.
In this way, we obtain general algorithms which allow to implement

 
 <ul>
 <li>the "default" for the functions f(k) if the related boundary
has no nontrivial subdivision into different segments.
 
 <li>the geometry induced by a mapping.
 
 <li>the intersection of geometries.
 
 <li>the geometry described in the standard form --- by a cell complex.
 
 </ul>


Thus, we obtain a powerful set of methods to create and modify
cogeometries.  For example, the first algorithm allows to define a
cogeometry by an arbitrary function which defines the region
containing a given point. The operations, once implemented, may be
applied to every cogeometry, independent of the way used to define it.
Using these methods, it is simple to create complicate geometries not
only in 3D, but even in higher dimensional spaces.

  <p>The geometry description is closely connected with the attribute
description --- data which describe the properties of the segments and
functions defined on these segments. We show that these attributes may
be easily and in a modular way handled by a contravariant geometry
description.

  <p>The cogeometry is a natural object also from point of view of
object-oriented programming. The functions f(k) are simply the
methods of the related class "cogeometry". We describe shortly the
concept of a future C++ implementation.

  <p>Currently, the concept of contravariant geometry description was
implemented in C in the geometry description part IBGD of the general
purpose 3D grid generator IBG.