Mesh generation and mesh adaptation especially in the three-dimensional case
for unstructured meshes is still an unsolved problem. The demands forced by
discretization schemes are not covered easily. Nevertheless, different
developments over the last decade have shown that three-dimensional TCAD
simulations become more and more accepted in the semiconductor industry.
From the mesh adaptation view the task of removing points without violating
structural constraints is still an unresolved problem. One of the most
promising technique is called edge
collapsing where the length of a particular edge is set to zero by moving
one point identically over another mesh point. However, this strategy
is hard to implement, because a lot of additional constraints have to be
considered in the three-dimensional case of unstructured tetrahedral
meshes. For example, care has to be taken at an interface between
two segments, where any geometric change of the border is prohibited. Such
constraints can cause a complete crash of this strategy.
Furthermore, the impact of the mesh on the convergence of the numerical
solutions is still under exploration. Up to now there is no simple geometric
criterion for the mesh element, which guarantees a good discretization. It seems
that obtuse angles destroy the condition of the system matrix and should
therefore be avoided, but again it is not
clear how an element should look like related to a given set of equations
combined with a particular discretization scheme. The goal to achieve an
adequately accurate solution by a minimum number of mesh points and therefore,
also within a minimum on memory consumption and execution time for the
simulation, seems far away.
However, up to now reliable three-dimensional simulations of semiconductor manufacturing processes are possible, whereby the used numerical methods still need some improvements. Further research activities should be carried out especially in the area of anisotropic mesh generation and adaptation and their direct impact on the underlying discretization scheme. As demonstrated, anisotropy is an adequate instrument to keep the total amount of discretization elements as small as possible.