Francois Roux

Optimal convergence properties of the FETI domain decomposition method

Abstract The Finite Element Tearing and Interconnecting (FETI) method is a practical and efficient domain decomposition (DD) algorithm for the solution of self-adjoint elliptic partial differential equations. For large-scale structural problems discretized with shell and beam elements, this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers, and to compare favorably with leading DD methods. In this paper, we discuss some numerical properties of the FETI method that were not addressed before. In particular, we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the FETI method are equivalent to the construction and solution of a coarse problem that propagates the error globally, accelerates convergence, and ensures a performance that is independent of the number of subdomains. We also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios, the performance of the FETI method is also independent of the mesh size. However, we also argue that the FETI and other leading DD methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom, or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios. Finally, we report that for realistic problems, optimal preconditioners are not necessarily computationally efficient and can be outperformed by non-optimal ones.

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 targeted by emerging parallel processors. Substructure based iterative solvers—often called also domain decomposition algorithms—lend themselves better to parallel processing, but must overcome several obstacles before earning their place in general purpose structural analysis programs. One such obstacle is the solution of systems with many or repeated right hand sides. Such systems arise, for example, in multiple load static analyses and in implicit linear dynamics computations. Direct solvers 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 different right hand side. In this paper, we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides. Basically, we formulate the overall problem as a series of minimization problems over K-orthogonal and supplementary subspaces, and tailor the preconditioned conjugate gradient algorithm to solve them efficiently. The resulting solution method is scalable, whereas direct factorization schemes and forward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structural problems, and highlight its potential to outperform forward and backward 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 same problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.

The two-level FETI method. Part II: Extension to shell problems, parallel implementation and performance results

The two-level FETI method presented in Part I of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems. In this paper, we extend the two-level FETI method to shell problems, describe a variant approach, revisit the preconditioning problem, address the efficient implementation of the corresponding iterative solvers on massively parallel processors, highlight the computational price of mathematical optimality and discuss its consequences, and report some impressive performance results on two Paragon XP/S and IBM SP2 massively parallel processors for several realistic plate and shell problems.