EXTENDING SUBSTRUCTURE BASED ITERATIVE SOLVERS TO MULTIPLE LOAD AND REPEATED ANALYSES

Direct solvers currently dominate commercial finite element structural software, but do not scale well in the fine granularity regime target ed by emerging parallel processors. Substructure based iterative solve rs-often called also domain decomposition algorithms-lend themselves b etter to parallel processing, but must overcome several obstacles befo re earning their place in general purpose structural analysis programs . One such obstacle is the solution of systems with many or repeated r ight hand sides. Such systems arise, for example, in multiple load sta tic analyses and in implicit linear dynamics computations. Direct solv ers are well-suited for these problems because after the system matrix has been factored, the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions. On the other hand, iterative solvers in general are ill-suited for these problems because they often must restart from scratch for every diffe rent right hand side. In this paper, we present a methodology for exte nding the range of applications of domain decomposition methods to pro blems with multiple or repeated right hand sides. Basically, we formul ate the overall problem as a series of minimization problems over K-or thogonal and supplementary subspaces, and tailor the preconditioned co njugate gradient algorithm to solve them efficiently. The resulting so lution method is scalable, whereas direct factorization schemes and fo rward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structura l problems, and highlight its potential to outperform forward and back ward substitutions on parallel computers. As an example, we show that for a linear structural dynamics problem with 11640 degrees of freedom , every time-step beyond time-step 15 is solved in a single iteration and consumes 1.0 second on a 32 processor iPSC-860 system; for the sam e problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.