In the following, different algorithms for tetrahedral mesh adaptation are developed to fulfill state-of-the-art demands of TCAD simulations. First some definitions of commonly used technical terms have to be given, to provide the non-experienced reader with some convenient descriptions. Slightly different definitions can be found in literature, but the following ones are in the style of explanations given in [4].
For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex
is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron,
also known as pentatope (in each case with interior) in its adequate dimension.
A tetrahedron is a polyhedron composed of four triangular faces, three of which
meet at each vertex. A regular tetrahedron is one in which the four triangles
are regular, or "equilateral", and is one of the Platonic solids. A
tetrahedron is referred as a 3-simplex.
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(2.2) |
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(2.3) |
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(2.4) | |
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(2.5) |
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In
dimensions, the gift wrapping algorithm [33], which has
complexity
, where
is the
floor function, can be used. In two and three dimensions,
however, specialized algorithms exist with complexity
.
Let
be a (finite) set of points in
(
), the convex
hull of
, denoted as
, defines a domain
in
. Let
be a simplex, then the covering up
of
by means of such elements corresponds to the following:
Here is a natural definition: with respect to condition
, one can
see that
is the open set corresponding to the domain that means, in
particular, that
. Condition
is not strictly necessary to define a covering up, but it
is nevertheless practical with respect to the context and, thus, will be
assumed. Condition
means that element overlapping is proscribed [4].
Let
be a closed bounded domain in
(
). The
question is how to construct a confirming triangulation of this domain. Such a
triangulation will be referred to as a mesh of
and will be
denoted by
.
In contrast, Definition 5-
is no longer assumed, which means that the vertices are not in general, given
a priori and in Definition 7-
, the
s are not
necessarily simplices.
Most computational schemes using a mesh as a spatial support assume that this mesh is conformal (although, this property is not strictly necessary for some solution methods).
Elements are the basic components of a mesh. An element is defined by its geometric nature and a list of vertices.