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有限元学习研究用源代码(老外的),供科研人员参考

<h1><a NAME="cogeomdef">Definition of a Cogeometry</a></h1>


There are a lot of different technical realizations of the dual concept. So, a
simple dualization leads to a function which returns for a given simplex and a
segment of the related codimension their intersection index. Such a realization
may be easier to use in theoretical considerations, but not for implementation.
We try to find here a variant which allows easy implementation and usage.

 
<h2><a NAME="defcont">The Continuous Case</a></h2>


Now we want to introduce the definition of a cogeometry for exact real
arithmetics. But, at first, let's consider shortly how we use some
notions like <b>in general</b> and <b>natural</b>. They look very
informal, but have a well-defined formal sense.

  <p>The notion <b>in general</b> may be formalized using the concept of
transversality. In this concept we distinguish between the "generic
situation" of "transverse intersection" and "degenerated situations".
They may be characterized by two properties:

 
 <ul>
 <li>A small modification allows to obtain a generic, transverse
situation.
 
 <li>On the other hand, a small modification of a generic situation
does not lead to a degenerate situation.
 
 </ul>


That means the generic situation defines an open, dense subset in the
set of all possible situations. Usually it is very easy to find in a
concrete situation what is the generic situation and what is
degenerated. In the generic situation, a sub-manifold of dimension k
has an intersection with a sub-manifold of codimension l if k >=l. It is a smooth sub-manifold of dimension (k-l), and the intersection
is transverse. That means, the tangential space in the intersection
point will be generated by the tangential subspaces of the two
sub-manifolds. Pre-images of sub-manifolds will be sub-manifolds.

  <p>To obtain formal results it is necessary to fix the related spaces
of manifolds. It is always possible to use smooth manifolds and
functions.  To obtain the correct number of derivatives which has to
be defined is more complicate, especially it depends on the dimensions
of the involved manifolds. The basic result is the theorem of Sard <a HREF="thbib.html#Sard1942"> [Sard1942] </a>. Related results and techniques
which allow to establish such results you can find in <a HREF="thbib.html#Hirsch1976"> [Hirsch1976] </a>.

  <p>There will be different possibilities to handle the degenerate
cases. In a degenerate situation we have different possible results
which may be obtained as the limit from different directions. There
are different strategies:

 
 <ul>
 <li> To leave the result undefined.
 
 <li> To allow every of the possible results.
 
 <li> To use the complete set of the possible results.
 
 <li> To fix some order and to use the lowest possible result.
 
 </ul>


The strategy used here is not relevant for the problems of
implementation, because the problem of degenerate cases will be
covered by another serious problem --- the rounding errors. That's why
we simply use the first strategy and define the cogeometry only for
the generic case. Compared with the other strategies, we sometimes
have to add "in general" even if the other strategies may allow to
omit this.

  <p>The notion <b>natural</b> will be used to describe an important
property of a construction related with mappings. Assume we have a
construction which allows to create some object on Y for a given
object on X and a mapping f: X --> Y. This construction is
natural if a composition property will be fulfilled: If we consider
the composition of f with g: Y --> Z, the construction for the
composition of the mappings leads to the same object on Z as the
composition of the constructions for the two mappings.

 <p>In the following, we use the short, informal notions "in general"
and "natural" having in mind that they may be transformed into exact
results. We do not give these exact formulations because they will be
straightforward from point of view of theoretical mathematics, and the
main interest of the "theoretical part" is to show that the concept of
cogeometry is very natural from theoretical point of view.

  <p>Let's now introduce the basic object. The required properties of these
objects we define later.

 
 <dl>
 <dt>A k-segment S_k 
 <dd>is a closed submanifold with
boundary of codimension k.

 
 <dt>A k-simplex 
 <dd>is a continuous mapping from the standard
k-dimensional reference simplex into X. In the case of a smooth
manifold, it makes sense to consider only smooth mappings.

 
 <dt>A side of a k-simplex 
 <dd>is a (k-1)-simplex defined by the
mapping of the related side of the reference simplex.  To avoid
exceptions we define the side of a 0-simplex as the empty object.

 
 <dt>A k-flag 
 <dd> consists of a point p (called position), a sequence
of (k+1) segments (S_0,...,S_k) and k orthogonal directions
(d(1),...,d(k)).  To avoid exceptions we define the flag also for
k = -1 as the empty object.

 
 <dt>An intersection of a k-simplex 
 <dd>may be a k-flag with position
in the simplex --- an <b>inner intersection</b> --- or a (k-1)-flag on
a side of the simplex --- a <b>boundary intersection</b>.

 
 <dt>The intersection function f(k) 
 <dd>allows to find intersections
of k-simplices.  Input is a k-simplex and an inititial intersection of
the simplex. The result is another intersection of the same simplex
which we call the continuation of the first intersection through the
simplex.

 
 <dt>A cogeometry G(X) 
 <dd>is a sequence of functions f(k) for k
>= 0.

 
 </dl>


Before we define the properties required for these objects, let's
define them for the case of a n-dimensional geometry described by a
smooth cell complex. We consider only smooth simplices.

  <p>In this case, the position of a k-flag is inside the segmentS_k and is a boundary point for all S_i with i &lt; k, S_i
is part of the boundary of S_j for i &gt; j, the direction d(i)
is in p tangential to S_j for j &lt; i, orthogonal to S_j . It
points into S_i-1.

  <p>For an inner intersection, we require not only that the position
of the flag is inside the simplex, but also that the projections of
the flag directions into the plane of the simplex define a
non-degenerate volume. This allows to define an orientation of the
intersection.

  <p> Now let's define the intersection function f(k). The input flag
defines a point on some (k-1)-segment S_k-1. Consider the
intersection of the k-simplex with this segment, especially the
component containing the initial point. In the generic situation we
obtain a smooth 1-dimensional manifold, and the initial intersection
is one of the two ends of this curve. The position of the return value
is the second end of this curve. The related flag we obtain using the
continuation of the flag along the curve. For degenerate cases we do
not define the function.

  <p>This description may fail, if it is not possible to continue the
flag because of a change of the neighbourhood relations in some
intermediate point of the curve. To avoid this effect we have to
require that such intermediate points must be part of some boundary of
codimension k.  For an arbitrary geometry, this may be obtained by
further subdivision of the related boundaries into parts with
identical neighbourhood relations.

  <p>Let's define now the properties of a cogeometry. The strategy we
use to fix these properties is to find properties which are fulfilled
for our example.  Let's list at first the most obvious properties:

 
 <ul>
 <li>At first, we have a list of "transversality and orthogonality
conditions" --- the directions of a flag have to be orthogonal, their
projection on the tangential plane of the simplex not degenerated.
 
 <li>We have a symmetry in the definition of f(k). The output of
a first call may be used as the input with the same simplex. Then the
result has to be the input of the first call.
 
 <li>The (k-1)-part of the list of segments of the two flags is
identical.
 
 <li>Usually the positions of the two flags are different. Only in the
case of inner intersections, the position may be the same. But in this
case the directions must be different.
 
 <li>The result for a simplex may be derived from the results for
smaller simplices obtained by subdivision of the initial simplex.
 
 </ul>


  <p>Obviously these properties make sense for every geometry.
Especially the last allows to localize the problem: The geometry will
be completely defined by the results of f(k) for arbitrary small
simplices.

  <p>To complete the definition, we have to add a condition which
describes the local behaviour of the geometry. This may be a condition
of the following type:

 
 <ul>
 <li>For every point there is a small neighbourhood so that there is a
homeomorphism which is smooth enough and allows to transfer the local
situation into the linear reference situation.
 
 </ul>


or another set of local regularity conditions which allows to build
such a homeomorphism. Different variants of this condition allow to
define different classes of smoothness of a cogeometry.

 
<h2><a NAME="codim">The Codimension of a Cogeometry</a></h2>


The <b>codimension</b> of a cogeometry is the highest codimension of a
segment of the cogeometry. For a cogeometry of codimension k it is not
possible to define input values for f(l) with l &gt; k+1. So, such
a cogeometry will be completely defined by the sequence of the f(l)
between 0 and k+1. Because of this simplification it is useful to have
information about the codimension.  We have the following obvious
properties of the codimension:

 
 <ul>
 <li>The codimension of a cogeometry in a n-dimensional space is <= n.

 
 <li>The codimension of the induced cogeometry is the same as of the original.

 
 <li>The codimension of the intersection of two cogeometries is the sum of the
codimensions of these cogeometries.

 
 </ul>


Thus, for the most interesting operations we can compute the
codimension explicitly.

 
<h2><a NAME="gdcovmorse">Connection to Morse Theory</a></h2>


There is a natural connection between the cogeometry and Morse functions (see
<a HREF="thbib.html#Morse1934"> [Morse1934] </a>,
<a HREF="thbib.html#Milnor1963"> [Milnor1963] </a>):

 A Morse function on a space X defines a cogeometry.

 Each segment of this cogeometry will be related to a
singularity of the Morse function. The segment may be defined as the
set of points so that the limit of the gradient flow is the related
singularity. The codimension of the segment is obviously the index of
the singularity.  

  <p>This connection shows that a cogeometry is a very natural object
from mathematical point of view. It also shows that a cogeometry may
be defined also for spaces of infinite dimension. A space which allow
to define a Morse function on it, allows also to define a cogeometry.
This shows that the class of spaces which allow a cogeometry is
greater than the class of spaces which allow a standard geometry
description.

 
<h2><a NAME="defround">The Implementation in Finite Precision Arithmetics</a></h2>


Now let's consider some modifications of the concept for the
continuous case which will be necessary or useful for an
implementation of the concept.

 
<h3><a NAME="affinesimpl">Affine Simplices</a></h3>


At first, for simplices we consider instead of arbitrary smooth
mappings only affine mappings. At a first look, this seems to be a
restriction, because this requires an affine structure on the basic
space X we consider. But because the geometry will be defined by the
results for arbitrary small simplices, and for such small simplices we
have some transformation into some standard situation, the affine
structure will not have any influence. For a manifold without affine
structure we can simply use the affine structure of some local
coordinates. The usage of other coordinates does not influence the
limit of arbitrary small simplices.

  <p>Usually in applications we consider only the n-dimensional
Euclidean space, so we have some well-defined global affine structure.
That's why, to use only affine simplices makes the interface much more
simpler to use. Instead of the definition of a mapping we have to define
only the coordinates of the corners of the simplex.

 
<h3><a NAME="findist">Finite Distances Instead of Infinitesimal Directions</a></h3>


To define a flag, we have to define a sequence of segments and
infinitesimal directions. In finite precision arithmetics, we use
instead a sequence of points (k, k-1,...,0) so that:

 
 <ul>
 <li>Each point i is inside the segment S_i of the flag.
 
 <li>The direction d(i) is defined by the vector from i to i-1.
 
 <li>The distance between the points is small: |d(i)| &lt; epsilon