thmotiv.html

有限元学习研究用源代码(老外的),供科研人员参考

<h1><a NAME="introduction">Introduction</a></h1>


At first, let us describe the motivation for our considerations.

<h2><a NAME="rolesimulation">General Motivation</a></h2>


One of the most interesting applications of scientific computing is
simulation. For many problems in very different fields of physics and
chemistry (thermal analysis, mechanics of solids, electromagnetism,
fluid mechanics, cosmology) and their applications (from semiconductor
devices to weather prediction) computer simulation is a very powerful
instrument. It is usually much cheaper than a real experiment,
and often a real experiment is even not possible (f.e. in global
climate prediction, cosmology). Numerical simulation also allows to
obtain detailed information about the process. One can get the value
of every interesting function at every point and for every moment of
time. In a real experiment usually only some integral values or a low
number of values at measurement points can be measured.
   <p> The simulation of natural processes in our three-dimensional
world is usually based on the numerical solution of three-dimensional
time-dependent partial differential equations. The most popular
techniques of discretization of such PDEs --- Finite Element Methods
(FEM) and Finite Volume or Box Methods (FV) are based on grids.  So
grid generation is a necessary and important part of almost all
computer simulation programs.  Caused by the three dimensions of the
space we live in, for many applications 3D grids are required.
  <p> Another problem of many applications is the geometry description.
In the real world, the domain of interest is usually subdivided into
parts --- regions --- with well-defined, sharp boundaries between
them.  This subdivision is a necessary part of the input data. In 3D,
this geometry description is a nontrivial, very complicate object, and
to create and modify geometry descriptions is in general as complicate
as the grid generation itself.
   <p> In the past, caused by limited computer resources and high
costs, usually only 2D and even 1D simulation was possible, especially
for time-dependent problems. So, simulation was restricted to
situations with high symmetry in the geometry and the initial
conditions. Now the hardware necessary for 3D computations is
available. Today 3D computations seems to be possible even on a
PC. Thus, a lot of new applications for simulation will be
possible. All they need good grid generation programs. So, there seems
to be a market for general purpose grid generation and geometry
description packages.
<h2><a NAME="authorsMotivation">Author's Motivation </a></h2>


  <p> From 1987 I was concerned with DIOS, a program for 2D simulation
of semiconductor technology. I was responsible for geometry handling
and grid generation. During this time I have considered not only the
2D problems, but also possible algorithms for 3D technology
simulation, especially for geometry handling. As a result, some basic
ideas of the concept described here have been developed.
  <p> After the unification of Germany in 1990, caused by general
problems of industrial research in the further GDR and the termination
of the GDR academy of science, the simulation of semiconductor
technology with DIOS has not been continued in Berlin. The further
development of DIOS now is located in the ETH Zurich.
  <p> I am convinced that the concept for 3D grid generation and
geometry description I have developed for technology simulation will
be useful not only for technology simulation, but also for a great
class of other applications. So, my idea was to implement the concept
as a general purpose grid generation package, independent of a direct
application in semiconductor technology or device simulation. In 1992
I have started to implement the IBG package.
  <p> This thesis will be also used as the "theoretical part" of the
documentation of the IBG package.
<h2><a NAME="currentSituation">The Current Situation</a></h2>


At the current state, many simulation programs use their own grid
generator.  There are a lot of reasons for this situation:

 
 <ul>
 <li> The requirements of different applications are different.
 
 <li> Good grid generation software is often too expensive or only for
internal use.
 
 <li> The result of the simulation depends too much on the
grid. That's why the programmer needs a lot of control over the grid
generator. To write an own grid generator may be the easier way to
obtain this control instead of learning the internals of an external
grid generator.
 
 <li> A grid requires a complicate, nontrivial data structure. To build-in
an external grid generator requires nontrivial data exchange.
 
 <li>It is very easy to write a trivial grid generator (f.e. for a
regular, rectangular grid). Thus, fast prototyping often leads to an
own implementation for the grid generator.
 
 <li> Once introduced, it is often too difficult to replace the
implementation by an external grid generator because of bad modular
design. That's why it is also often not possible to extract the grid
generation part from the simulation program for external use.
 
 <li> A "hand-made" special grid for the concrete situation is often
better than the grid created by the general-purpose grid generator.
 
 </ul>


This situation leads to a lot of double work for the programmer and
less quality of the resulting grids. But this situation may change:

 
 <ul>
 <li> Simulation is no longer restricted to 1D/2D and/or
time-independent computations. So a lot of new applications can be
considered.
 
 <li> On the other hand, the number of different grid generation
concepts will not increase very fast. That means, the number of
different applications using the same algorithm will increase.
 
 <li> Increasing computer power allows to use also non-optimal grids
created in non-optimal time.
 
 <li> The supply of cheap and public domain software for grid
generation and their quality also will increase.
 
 <li> Modular design becomes more common.
 
 </ul>


We see, that cheap or public domain software for 3D grid generation
and geometry description of good quality, with a good modular
interface, is necessary to speed up this process.

  <p> But, the result of this process will not be one universal grid
generator for all applications. The reason is that different
equations, different discretizations of these equation, different
hardware and different application requirements lead to different
optimal grids:

 
 <ul>
 <li> Vectorization requires structured grids.
 
 <li> A simple, fixed geometry allows to use "hand-made" optimal grids.
 
 <li> Time-dependent or very complicate geometries require automatic
grid generation.
 
 <li> A complicate geometry requires unstructured grids.
 
 <li> If effects of interest are located in small parts, local refinement
is necessary.
 
 <li> Different discretization even of the same equation leads to different
optimal grids. We show this later for FEM and FV.
 
 </ul>



<h2><a NAME="exapplication">Example Application: Semiconductor Technology</a></h2>


An example for an application where simulation was already very
successful is process simulation - the simulation of different
technological steps in the process of fabrication of semiconductor
devices.  Because to perform a real test of a new technological
process is very expensive and needs a lot of time, it is much easier
and cheaper to make a simulation. The output of the technology
simulation may be used for the simulation of the behaviour of the
device (device simulation).  In the past, 1D and 2D simulators have
been used successfully for the simulation of such devices. But for the
problems of current device fabrication 3D simulation is necessary.
   <p> Let's consider now some of the steps of semiconductor technology
and the problems connected with their simulation especially in 3D:
<h3><a NAME="stgeom">Geometrical Properties</a></h3>


Let's consider at first the geometrical bounds of the problem. Usually
only a small part of the complete chip will be simulated (e.g. one
memory cell).  A typical domain for computation may be around 10 mum,
but domains up to 100 &mu;m are possible. The typical length for the
minimal geometrical object is much smaller, may be around 100 nm, may
be smaller. So, even to describe the geometry it is not possible to
use a regular coarse grid.
  <p> There are great parts on the device where "nothing happens". But
we also have small regions which require high accuracy. So, local
refinement will be necessary. Caused by the technology and the typical
design, we often have regions which have approximately 2D or even 1D
geometry and concentrations.  That's why, 1D and 2D simulation
programs have been very successful. For such regions anisotropic
refinement is necessary. A possibility for integrated 1D, 2D and 3D
computations is useful. For example, for the first steps 1D
computation may be enough, and only after the usage of a nontrivial
mask 2D or 3D is useful.
<h3><a NAME="stdevice">Device Description</a></h3>


At first let's consider the central data structure of the technology
simulation. This data structure is the input and the output of every
technology step. This description has to include the data which are
necessary for the device simulation:

 
 <ul>
 <li> The geometry of the device. The number and the topology of the
regions is not fixed. Regions may be not connected and not simply
connected.
 
 <li> The dopant concentration in silicon. We have to consider here
different sorts of dopants (P, B, As).
 
 </ul>


But this information is not enough for technology simulation. Often we need
more detailed information: