The term ``mesh adaptation'' summarizes four mesh modification techniques,
namely mesh refinement, in analogy mesh coarsement,
point repositioning, and as fourth the so-called swap
operators [4]. ``Mesh refinement'', in simple terms, refers to
increasing the resolution of an initial mesh by increasing sample
points. Diametrically opposed, ``mesh coarsement'' reduces spatial resolution. The
third form called point repositioning does not change the amount of sampling
points at all. The core of this mesh adaptation method is the relocation of
present points according to particular methods which are usually driven by
the used discretization scheme.
A general rule of thumb is, that if the density of a mesh is increased,
the calculation error can be reduced, implying that the exact answer could
hypothetically be computed as the number of sample points is increased to
infinity, which is the basis for most numerical methods.
But it is not only the number of sampling points, which defines the quality of
the computed solution. A good solution can also be obtained on a coarse
mesh with smartly placed sampling points and mesh elements which reflects the
``nature'' of the numerical computations. John Chawer, president of Pointwise,
Inc. noted on the pre-conference short course of the 7
National
Congress on Computational Mechanics held 2003 in Albuquerque, New
Mexico [5], (presentations online available [6]):
``Case study: Twenty minutes of mesh smoothing reduces run time by four hours.''This gives rise to the fourth group of mesh modification the so-called swap operator methods [7], which are used to improve geometric properties of mesh elements. For three-dimensional tetrahedral based meshes, one can distinguish between two swap operators, namely the face swap and the edge swap operator. These operators change the local topology of the mesh and keep the number and the positions of points untouched. These methods do not fit in the mesh refinement, coarsement, or point repositioning classes and is therefore counted as a separate mesh adaptation group.
Over the last decade a wide spectrum of different mesh adaptation schemes have
been developed based on the four basic groups described in this section. Also
hybrid methods are in use which are constructed as particular sequence
of basic mesh adaptation procedures [8]. The next section gives an
overview of related state-of-the-art developments of mesh generators and
mathematical modeling tools with the capability of mesh refinement.
The most prominent group of tools with the capability of mesh refinement are
mesh generators themselves. Almost any state-of-the-art mesh generator has
different mesh improvement methods implemented based on the four modification
techniques presented in Section 1.2. The following
covers only a few mesh generation and numerical analysis software products with
strong refinement and mesh improvement features for unstructured tetrahedral
and hexahedral meshes. The survey is definitely not complete, it should
give just an impression of the manifoldness of mesh generation and
modifications. A good overview of recent research activities is given
in [9,10], a more up-to-date online reference can be found at Steven
J. Owen's Meshing Research Corner [11] or at the list about mesh
generation software from Robert Schneiders [12].
VGRID is a stand-alone mesh generator developed primarily for computational
fluid dynamics (CFD) which is one branch of fluid mechanics where numerical
methods are used to analyze fluid flows. The generator uses an advancing front
for unstructured meshes and an advancing layers approach for more structured
meshes and thin objects. The generator offers also local remeshing, grid
movement, and adaptive refinement features. VGRID is part of the NASA
Langley's Tetrahedral Unstructured Software System (TetrUSS) including geometry
set-up, mesh generation, flow solution, and analysis which is available to
U.S. entities, citizens, and permanent residents at [13]. The
generator has also the capability of generating anisotropic stretched grids for
improved efficiency. The user has also control over grid distribution through
adjustment of source parameters such as spacing, and intensity.
TRUEGRID is a general purpose mesh generation program with sophisticated
relaxation and parameterization capabilities [14]. It has been
optimized to produce
high quality, structured, quadrilateral and hexahedral meshes. Triangular, and
tetrahedral elements as sparingly as possible are generated, only when the
geometry demands it. TRUEGRID is a commercial software which provides complete
output files for many of the most popular analysis packages like
ABAQUS [15] and ANSYS [16]. For mesh improvement both
interactive graphical development and batch file capabilities are provided, so
that the user can visually display bad elements and then modify the
mesh. Also different mesh diagnostic tools are provided, to give the user a
good feedback about the generated mesh.
DELINK is the in-house mesh generator of the Institute for Microlectronics, mostly developed by Peter
Fleischmann [17]. DELINK is a three-dimensional conforming Delaunay mesh
generator which produces tetrahedral elements. One of the main features of this
mesh generator is that it provides a functional interface (API) which enables
the use as a library. This allows a strong integration of the mesh generator
into TCAD tools and enables various mesh adaptation techniques including also a
total remeshing step. The underlying meshing technique is a modified
advancing front approach with fast octree point location which handles all
degenerate Delaunay cases like cospherical points or Schoenhardt prisms and
untetrahedralizable polyhedra. DELINK offers also an automatically repair feature
and patches small holes in the surface descriptions. The software is available
free of charge to registered users at [18].
NETGEN is an automatic two- and three-dimensional mesh generator which was
mainly developed by Joachim Schöberl at the Johannes Kepler University
Linz, Austria [19]. The generator produces triangular or
quadrilateral meshes in two-dimensional, and tetrahedral meshes in
three-dimensional space, respectively. NETGEN contains modules for mesh
optimization based on node movement, element swapping, and splitting. Elements
are generated by a fast Delaunay algorithm in combination with a back-tracking
rule based procedure if the Delaunay tessellation fails.
MESH is a mesh generator based on a modified octree approach in combination
with conformal delaunization for triangles, tetrahedra, pyramid, and wedge
(prism) shaped elements. MESH was developed by ISE [20] which has been
taken over by SYNOPSIS [21]. The engineering discipline of MESH is
the semiconductor device and process simulation. For the construction of
three-dimensional meshes also an advancing front approach is available. This
generator is a remarkable one,
because it allows some automatic mesh adaptation based on the data stored on the
mesh. This enables a looped computational analysis cycle as depicted in
Figure 1.2.
Since the area of applications related to numerical computations is very
multifarious, an additional, more mathematical approach is undertaken to provide
the engineer with more general computational analysis tools. Such mathematical
modeling tools are not related to a particular scientific discipline but rather
to the nature of physical phenomenons. The engineer can choose a complete
system of predefined, mostly partial differential equations from a catalog,
and customizes the chosen mathematical skeletal structure related to a given
problem. In the following two software packages are presented, which enable a
full chain from the CAD model to the numerical analysis.
COMSOL Multiphysics [22] (formerly FEMLAB) is a finite element
analysis and software package for various physics applications, especially
coupled phenomena. The package provides also the so-called CAD Import
Module which simplifies the transition from geometric designs that engineers
create with specialized CAD tools to mathematical modeling. There is also a
strong inter-linkage between SolidWorks [23], a CAD environment, and
COMSOL Multiphysics which allows real-time geometry updating. This enables a design loop
between the numerical analysis tool and the geometric modeling process. For the
engineer, it is also possible to influence the meshing and a following
refinement procedure, to obtain a good spatial discretization. This
process becomes more and more self-acting, so that the fine-tuning process by
the user is kept as small as possible.
ANSYS [16] offers a wide spectrum of coupled physics tools combining
structural, thermal, CFD, acoustic, and electromagnetic simulations. In addition
a so-called ANSYS DesignSpace package is offered, which gives designers
a tool to conceptualize, design, and validate ideas. For the discretization a
very powerful module, the so-called ANSYS ICEM CFD package, has been
developed. ANSYS ICEM CFD provides sophisticated geometry
acquisition, mesh generation, mesh editing, a wide variety of solver outputs
and post-processing. It also includes mesh generation tools that offer the
capability to parametrically create grids from geometry in multi-block
structured, unstructured hexahedral, tetrahedral, hybrid grids consisting of
hexahedral, tetrahedral, pyramidal, and prismatic cells. Also Cartesian grid
formats combined with boundary conditions are available. The primary focus is on
computational fluid dynamics mesh generation but this tool can be used
for quite general finite element analysis and electromagnetics. It also features curvature
and proximity-based refinement.
At the Institute for Microlectronics, in the last decade different software products have been
developed to handle various areas in the field of TCAD computing. To summarize
this diversity of tools, a more generic approach which follows new programming
paradigms has been carried out. The generic scientific simulation environment
(GSSE) [24] separates topological mesh traversals and data access, as
in the well known C++ standard template library (STL) [25]. Solid
modeling, mesh generation, and adaptation are integrated components as well as
functional equation specifications for different discretization schemes such as
finite elements, finite differences, and finite volumes.
To strike a balance between accuracy on the one hand side and computational
time and memory consumption on the other hand side the mesh should be
constructed with reasonable mesh densities. This means that not every region of
a spatial simulation domain is of particular importance for the solution of
the numerical problem. So the idea is to use a finer mesh in simulation domains
where a high resolution is necessary and simultaneously reduce the memory
consumption by applying a coarse mesh in regions of less importance according
to an assumed error. So the goal of mesh adaptation is to increase accuracy of
numerical calculations under consideration of computational costs and a
feasible error.