2.3 State of the Art

Most of the research in mesh generation is done in the fields of computational fluid dynamics and stress mechanics. Three-dimensional mesh generation for applications in TCAD and for semiconductor process and device simulation is a recent topic and leaves room for much innovation. The following is a summary of modern approaches respectively in the order of cartesian-based, octree-based, unstructured tetrahedral, advancing front, hybrid, and Delaunay methods.

Aftosmis et al. [2] developed cartesian meshes for applications in computational fluid dynamics. They incorporate complex boundaries with a rigorous method to intersect the cartesian cells with the boundary representation. A limited flexibility in refinement is experienced, due to the cartesian nature of the mesh and the isotropic refinement technique. Each further refinement level yields about 2-4 times more elements. With intial 1.6 million elements a further increase in accuracy by adding one more refinement level would result in 3.3-6.6 million elements according to one of the examples in [2]. Still, in some areas the achieved accuracy is not optimal and refinement would be appropriate. Such a high point propagation due to refinement and the rigorous intersection method which greatly increases the number of surface elements is less likely to be afforded on a typical computer available to a user for everyday simulation purposes.

The cartesian-based mesh generator *OMEGA* well known for
semiconductor simulation applications is described in [53]. This
intersection-based method originates from previous works using
bisection-based and octree decomposition techniques
[66,65,35]. Improvements
include the ability to tessellate the cartesian cells into Delaunay
tetrahedra and a more anisotropic refinement approach. Subdivisions
for higher refinement levels are performed by splitting the nodes at
intersections or geometry-specific vertices instead of splitting the
nodes always at bisections. However, if vertices of the geometry lie
closely together without conveying important geometrical information as
for instance in the case of staircase-like approximations of slopes,
too thin elements are created and unnecessarily propagated throughout
the mesh. The key issue with cartesian methods is to merge
the coordinate system aligned cells with the unstructured surface of
the geometry. It seems that without rigorously generating a cell-consistent surface
triangulation as in [2], difficulties are experienced dealing
with the boundaries and interfaces of a semiconductor device.
To attain optimal flexibility regarding both the mesh density
distribution as well as the fitting of the boundary it will be
desireable to achieve anisotropy in arbitrary directions (not just
along the three coordinate axises).

Johnson et al. [76] use an unstructured tetrahedral method for
three-dimensional flow problems. The mesh generator is capable to
efficiently deal with moving objects in fluids. It is linked to an
input modeler for non-uniform rational B-spline (*NURBS*)
surface patches which however makes it less suitable for TCAD applications.

Advancing front methods [19,94] are mostly used because
of their good point placement properties which result in better
boundary-fitted meshes as opposed to mere boundary consistent meshes.
Elements are not intersected with the boundary, instead they are grown
from the boundary and are therefore well aligned with it.
An interesting technique to utilize the efficient and well defined
Delaunay Triangulation as a background mesh^{2.1} for
advancing front style mesh generation has been developed by
P.L. George et al. at *INRIA*, France [50].
Generally, it can be of interest to construct *protection layers*
around physically crucial boundaries or interfaces to provide the required
orthogonal mesh resolution for a correct representation of a physical
magnitude of the solution.
It remains an open problem how to effectively apply such
methods to the complexity of a three-dimensional semiconductor device.
In two dimensions approaches to enforce a certain anisotropy
along inversion layers or otherwise highly non-linear interfaces have
been investigated by
[4,112,177,82,133].
Most methods have come to a stage where they more or less rely on a
triangulation engine, e.g. *TRIANGLE* [160], to
generate a valid tessellation of a set of supporting grid lines and/or
mesh points supplied together with the boundary.
Conformal mapping techniques to ensure orthogonal boundary-fitted meshes by
means of mathematical transformations of parametrized non-planar
boundaries pose computation difficulties in three dimensions. The related
partial differential equation (PDE) gridding method^{2.2} seems better extendable for
three-dimensional TCAD applications. So far the two-dimensional
implementation *CGG* [26] of the PDE method has proven
powerful for semiconductor device simulation.

Pure octree-based solutions combined with tetrahedral templates are
hardly used for todays TCAD applications.
Considering the variety of existing approaches in three dimensions one
can observe that recently most methods have evolved to become hybrid
methods and require some sort of general tetrahedralization
[90,77,178,81].
The software package *LAGRIT* formerly known as *X3D*
provides a universal mesh generation toolkit [102].
Another hybrid mesh generator is *MESH**ise*
[54] which is a further development of *OMEGA*.

For moving boundary situations level set methods have gained great importance [90,1,27,127]. They require a mesh to define the magnitude which describes the moving boundary. At some stage a boundary consistent mesh must be derived from the level set representation by means of intersection and eventually tetrahedralization.

The history of methodologies in two-dimensional process and device
simulation leads to the observation that an unrestricted and stable
triangulation engine is one of the most important tools for mesh
generation purposes.
The theory of Delaunay provides a mathematical foundation for
provably terminating triangulation algorithms [15]. While the two-dimensional Delaunay Triangulation poses less
difficulties and a vast amount of literature exists
[41,70,145,175,86], the integration
of boundaries into a three-dimensional Delaunay Triangulation remains
a very active research area.
An extremely thin oxide layer on top of a comparatively big silicon
block calls for very sophisticated algorithms to automatically
generate an unstructured Delaunay mesh consistent with the
boundaries and interfaces.
This is one of the reasons why most of the software for a
three-dimensional Delaunay Triangulation available to the public
domain is less suitable for semiconductor simulation purposes.
A Delaunay method using local transformations is included in
*GEOMPACK* by B. Joe [74,72].
*QHULL* is a code which constructs a Delaunay Triangulation by
computing the convex hull in higher dimensions [12].
Excellent work related to Delaunay refinement mesh generation can be
found in [164].
The *QMG* software package for quality mesh generation
based on octrees is described in [109].
A detailed and much more complete survey of the worldwide research
activities in mesh generation is maintained by R. Schneiders [147].
Regarding future aspects of TCAD and automation issues one may refer
to [34,83].

2000-01-20