thresult.html

有限元学习研究用源代码(老外的),供科研人员参考

  <p>Let's consider now shortly the results of the previous
considerations.  The author has implemented in the concepts described
before in the 3D grid generation and geometry description package
IBG.

The idea of contravariant geometry description and the grid generation
algorithm was implemented in ANSI C in the "intersection-based grid
generation" package IBG.

  <p>The main differences between the concept of cogeometry described
here and the IBG package are:

 
 <ul>
 <li>IBG does not use flags. Instead of flags, only the intersection
points will be used in the functions f(k).
 
 <li>There are additional interface functions in IBG for the
description of topological neighbourhood relations.
 
 <li>It is not possible in IBG to transfer information about
orthogonal directions.
 
 <li>For an attribute, different boundary limits for different regions
cannot be managed.
 
 </ul>


But, even in this form, IBG shows the possibilities of the concept of
contravariant geometry description:

 
 <ul>
 <li>Fast prototyping strategy allows fast implementation.
 
 <li>Dimension-independence.
 
 <li>Many input possibilities.
 
 <li>Many possibilities to modify geometries.
 
 </ul>


 Here we show some examples of grids created by IBG.  The first
example shows a two-dimensional discretization of the region around
the island R&uuml;gen in Germany.

<p><b><a name="ruegen">picture:</a></b><a href="ruegen.gif">The island R&uuml;gen (Baltic coast of
Germany)</a>

 <p>The grid was created using a simple picture of the region with blue
colors for the water and brown colors for land.  The resulting grid
contains ~ 40 000 points. On a VAX 4000/90 workstation 24 sec. have
been used.
Segments with nontrivial topology may be simply created using a
characteristic function (&gt; 0 inside, &lt; 0 outside) of the region
--- the most trivial example of a cogeometry induced by a mapping.
The following grid was created using such a cogeometry:

<p><b><a name="hole">picture:</a></b><a href="hole.gif">Surface defined by a characteristic function</a>


This picture (as the following surface pictures) shows only the
internal surface between the different regions. It is only a part of
the 3D grid generated by IBG for all regions.

 To define the following geometry, partial intersection has been
used:

<p><b><a name="rohr">picture:</a></b><a href="rohr.gif">Partial intersection of two simple geometries </a>


The two parts have been described simply by their characteristic
functions f_1 = x^2 + (z-z_1)^2 - r_1^2 and f_2 = y^2 +
(z-z_2)^2 - r_2^2. To define the cogeometry it was not necessary to
compute explicitly the intersection line.

It is also easy to include external data. Using elevation data
obtained from the 
<a HREF="http://edcwww.cr.usgs.gov/glis/hyper/guide/1_dgr_dem"> 1-degree
USGS Digital Elevation Models</a> we have created the following
elevation profile:

<p><b><a name="canyon">picture:</a></b><a href="canyon.gif">Surface of the Grand Canyon (USA)</a>


The related 3D cogeometry has been simply defined by the
characteristic function f(x,y,z) = z - e(x,y) where e is the
elevation of the surface in the point (x,y).

  <p>In the 3D grid there have been around 200 000 nodes. The required time on an
alpha workstation was 500 s.