Matrix Equation Decomposition and Parallel Solution of Systems Resulting from Unstructured Finite Element Problems in Electromagnetics

Tom Cwik* and Daniel S. Katz**

*Jet Propulsion Laboratory, California Institute of Technology, Pasadena,
CA  91109

**Cray Research, 222 N. Sepulveda Blvd., Ste. 1406, El Segundo, CA  90245

        Finite element modeling has proven useful for accurately simulating
scattered or radiated electromagnetic fields from complex three-dimensional
objects whose geometry varies on the scale of a fraction of an electrical
wavelength.  An unstructured finite element model of realistic objects
leads to a large, sparse, system of equations that needs to be solved
efficiently with regard to machine memory and execution time.  Both
factorization and iterative solvers can be used to produce solutions to
these systems of equations.  Factorization leads to high memory
requirements that limit the electrical problem size of three-dimensional
objects that can be modeled.  An iterative solver can be used to
efficiently solve the system without excessive memory use and in a minimal
amount of time if the convergence rate is controlled.

        This paper will discuss a number of topics related to the parallel
creation and solution of matrices resulting from large unstructured
problems, in the context of an electromagnetic finite element code running
on the Cray T3D located at the Jet Propulsion Laboratory.  The JPL code,
named PHOEBUS, has been used to obtain solutions for systems with nearly
three-quarters of a million unknowns.

        One of these topics is mesh vs. matrix partitioning.  The code
running at JPL decomposes the finite element matrix in row slabs, as
compared with the usual strategy of decomposing the mesh.  Another topic is
the iterative solver, in this case a quasi minimum residual method.  An
examination of the computational kernel, a sparse matrix dense vector
multiply, is given, and the issue of solving for a single right hand side
vs. multiple right hand sides (possibly a block of right hand sides) is
discussed.  Another question related to the iterative solver is parallel
methods of preconditioning, such as using an incomplete Cholesky
factorization or a sparse approximate inverse.