Publications

Most algorithms used in preconditioned iterative methods are generally applicable to complex valued linear systems, with real valued linear systems simply being a special case. However, most iterative solver packages available today focus exclusively on real valued systems, or deal with complex valued systems as an afterthought. One obvious approach to addressing this problem is to recast the complex problem into one of a several equivalent real forms and then use a real valued solver to solve the related system. However, well-known theoretical results showing unfavorable spectral properties for the equivalent real forms have diminished enthusiasm for this approach. At the same time, experience has shown that there are situations where using an equivalent real form can be very effective. In this paper, the authors explore this approach, giving both theoretical and experimental evidence that an equivalent real form can be useful for a number of practical situations. Furthermore, they show that by making good use of some of the advance features of modem solver packages, they can easily generate equivalent real form preconditioners that are computationally efficient and mathematically identical to their complex counterparts. Using their techniques, they are able to solve very ill-conditioned complex valued linear systems for a variety of large scale applications. However, more importantly, they shed more light on the effectiveness of equivalent real forms and more clearly delineate how and when they should be used.

As computational needs for structural finite element analysis increase, a robust implicit structural dynamics code is needed which can handle millions of degrees of freedom in the model and produce results with quick turn around time. A parallel code is needed to avoid limitations of serial platforms. Salinas is an implicit structural dynamics code specifically designed for massively parallel platforms. It computes the structural response of very large complex structures and provides solutions faster than any existing serial machine. This paper gives a current status of Salinas and uses demonstration problems to show Salinas' performance.

A fast precondition technique has been developed which accelerates the finite difference solutions of the 3D Maxwell's equations for geophysical modeling. The technique splits the electric field into its curl free and divergence free projections, and allows for the construction of an inverse operator. Test examples show an order of magnitude speed up compared with a simple Jacobi preconditioner. Using this preconditioner a low frequency Neumann series expansion is developed and used to compute responses at multiple frequencies very efficiently. Simulations requiring responses at multiple frequencies, show that the Neumann series is faster than the preconditioned solution, which must compute solutions at each discrete frequency. A Neumann series expansion has also been developed in the high frequency limit along with spectral Lanczos methods in both the high and low frequency cases for simulating multiple frequency responses with maximum efficiency. The research described in this report was to have been carried out over a two-year period. Because of communication difficulties, the project was funded for first year only. Thus the contents of this report are incomplete with respect to the original project objectives.

We report on the application of the one-level FETI method to the solution of a class of structural problems associated with the Department of Energy's Accelerated Strategic Computing Initiative (ASCI). We focus on numerical and parallel scalability issues,and discuss the treatment by FETI of severe structural heterogeneities. We also report on preliminary performance results obtained on the ASCI Option Red supercomputer configured with as many as one thousand processors, for problems with as many as 5 million degrees of freedom.

In this work the basic Finite Element Tearing and Interconnecting (FETI) linear system solver and the PARPACK eigensolver are combined to compute the smallest modes of symmetric generalized eigenvalue problems that arise from structures modeled primarily by solid finite elements. Problems with over one million unknowns are solved. A comprehensive and relatively self-contained description of the FETI method is presented.