"NUMERICAL SIMULATIONS BY SYMBOLIC COMPUTATIONS?"

Gustav Amberg

Department of Mechanics, KTH, 100 44 Stockholm, Sweden


Many problems in science and engineering are formulated as systems of
partial differential equations. Time dependent PDEs in two and three
dimensions can describe for instance different kinds of fluid flow
problems, heat transfer, deformable bodies, model equations for phase
transformations, etc. To analyze a problem then requires some means to
simulate the mathematical model. That requires some kind of coding, either
of a completely new program for the particular application, or adaptation
of existing commercial software. Both have drawbacks, in that many wheels
have to be reinvented when a specialized code is written, and that it is
hard to have the kind of control over the model and solution procedure that
is often necessary, when using a commercial software.

In our work on modelling of phase change, fluid flow and heat transfer in
materials processes we need to have full control of the equations, boundary
conditions etc. that make up our mathematical model. At the same time we
must make the simulations reasonably efficient in order to reach to
interesting applications. In answer to these demands, we have developed a
toolbox, ('femLego') in Maple which can be used to generate complete Finite
Element codes in 1, 2 or 3 dimensions, from a symbolic specification of the
mathematical problem.

The specification of the model, i.e. PDEs, boundary and initial conditions,
is entered into Maple, and from this all problem dependent parts of a
fortran FEM code are generated. The user may choose from different elements
in one to three dimensions, different numerical linear algebra routines
etc, to set up a solution algorithm. In its present state, the software is
geared towards timedependent problems and uses unstructured grids. Adaptive
remeshing can be added. The pre- and postprocessing, i.e. mesh generation
and plotting, is done by interfacing to other applications, such as AVS
(Express).

We have used this toolbox to create codes for the study of thermocapillary
convection, growth of dendritic crystals, flow of viscoelastic liquids and
much more, and we have found that it can reduce the time from idea to a
working simulation to a matter of hours, also for quite complicated
problems. One advantage of working inside Maple is that the Maple worksheet
automatically provides a readable and complete documentation. Also it
invites you to do the derivations of your problem in Maple, which can then
immediately be used as input to the code generation commands. For example,
when setting up an optimal control problem, the adjoint problem for the
control is derived symbolically from the original PDEs, and then directly
fed to the code generation commands. This reduces bugs and mistakes to an
absolute minimum and makes it possible to try out ideas quickly.