The generation of
locally adapted conforming tetrahedral meshes is an important component
of many modern algorithms in the finite element solution of partial
differential equations. Typically, such meshes are produced by starting
with a coarse initial tetrahedral mesh followed by mesh adaption on
demand over space and time. During the calculation of a time step a
combination of error estimation and refinement mechanism is necessary
to deliver higher accuracy, if needed, by increasing the spatial
resolution. Features for refinement based on different kinds of error
estimations and refinement methods applied to an initial mesh have been
added to the three-dimensional Finite Element Diffusion and Oxidation
Simulator FEDOS.
Using strict isotropic meshes for three-dimensional process simulation
is not practicable. The need for calculation time and the limitation of
memory tend to result in anisotropic adapted meshes which are more
manageable. The idea is to create tetrahedral elements with special
geometric qualities by manipulation of an initial coarse mesh. The
basic manipulation step in our work is tetrahedral bisection. When
bisecting a tetrahedron, a particular edge -- called the refinement
edge -- is selected for the new vertex. As new tetrahedra are
constructed by refinement, their refinement edges must be selected
carefully to take an anisotropic shape into account and not to produce
degenerately shaped elements. Different kinds of refinement methods
have been implemented, investigated, and added to FEDOS.
In the numerical solution of practical problems of physics and
engineering, such as semiconductor device and process simulation, one
often encounters the difficulty that the overall accuracy of the
numerical approximation is deteriorated by local singularities. An
obvious remedy is to refine the discretization in the critical regions.
The question then is how to identify these regions and to obtain a
good balance between the refined and unrefined regions such that the
overall accuracy is optimal. These considerations clearly show the need
for error estimators which can be extracted a posteriori from the
computed numerical solution and the given data of the problem. The
error should be local and should yield reliable upper and lower bounds.
The global upper bounds are sufficient to obtain a numerical solution
with an accuracy below a prescribed tolerance. Local lower bounds are
necessary to ensure that the grid is correctly refined to obtain a
numerical solution with a prescribed tolerance using a minimal
number of grid points. Several error estimators have been implemented
in FEDOS. An interface which supports an error-driven
refinement has been added in order to provide powerful and flexible
coupling between error estimation and mesh refinement.
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