EL.SEVIER Comput. Methods Appl. Mech. Engrg. 155 (1998) 129-151 Computer methods in applied mechanics and engineering The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems Charbel Farhat”‘“, Jan Mandela-b “Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado at Boulder, Boulder, CO 80309-0429, USA ‘Center for Computational Mathematics, University of Colorado at Denver, Denver, CO 80217-3364, USA Received 24 March 1996 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ Abstract We present a Lagrange multiplier based substructuring method for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems. The proposed method is essentially an extension of the FETf domain decomposition algorithm to fourth-order problems. The main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations. This results in a two-level FETf substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures, and grows at most polylogatithmically with the number of elements per substructure. These theoretically proven optimal convergence properties of the new FETI method are numerically demonstrated for several finite element plate bending static and transient problems. The two-level iterative solver presented in this paper is applicable to a large family of biharmonic time-independent as well as time-dependent systems. It is also extendible to shell problems. 0 1998 Elsevier Science S.A. 1. Introduction Direct solution methods are popular in structural and solid mechanics and currently dominate commercial finite element structural software, essentially because: (a) they are robust and reliable even for ill-conditioned systems, (b) their performance depends only on the problem size and sparsity and not on the underlying physics, differential order, and discretization scheme, (c) their execution time can be predicted for any given problem and computational platform, and (d) they are well-suited for the solution of systems of equations with many or repeated right-hand sides. Such systems arise, for example, in static analysis with multiple load cases, in implicit linear dynamics computations, in the solution of nonlinear problems via a quasi-Newton scheme, and in various structural eigenvalue problems. However, direct solvers have also disadvantages. For example, they suffer from excessive storage requirements especially for large-scale three-dimensional solid problems, they can be expensive when an out-of-core solution strategy is required, and they do not scale well in the fine granularity regime targeted by emerging parallel processors (we note though that some progress has been recently achieved in this particular area [l]). On the other hand, classical iterative solvers such as Jacobi, Gauss-Seidel, successive over-relaxation, and conjugate gradient [2] are almost continuously avoided in commercial and production finite element structural software, essentially because: (a) they are unreliable in structural mechanics where the finite element matrices * Corresponding author. 0045-7825/98/$19.00 0 1998 Elsevier Science S.A. All rights reserved PII SOO45-7825(97)00146-l