INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 44, 489–516 (1999) A SIMPLE AND EFFICIENT EXTENSION OF A CLASS OF SUBSTRUCTURE BASED PRECONDITIONERS TO HETEROGENEOUS STRUCTURAL MECHANICS PROBLEMS DANIEL J. RIXEN AND CHARBEL FARHAT * Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado at Boulder, Boulder, CO 80309-0429, U.S.A. SUMMARY Several domain decomposition methods with Lagrange multipliers have been recently designed for solving iteratively large-scale systems of nite element equations. While these methods dier typically by implemen- tational details, they share in most cases the same substructure based preconditioners that were originally developed for the FETI method. The success of these preconditioners is due to the fact that, for homoge- neous structural mechanics problems, they ensure a computational performance that scales with the problem size. In this paper, we address the suboptimal behaviour of these preconditioners in the presence of material and/or discretization heterogeneities. We propose a simple and virtually no-cost extension of these precon- ditioners that exhibits scalability even for highly heterogeneous systems of equations. We consider several intricate structural analysis problems, and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities. Copyright ? 1999 John Wiley & Sons, Ltd. KEY WORDS: domain decomposition; heterogeneities; preconditioning; scalability 1. INTRODUCTION Iterative solvers, have slowly but nally made their debut in commercial nite element structural and thermal software. 1–3 The key factors for this change in culture have been (a) the pressing need for higher-delity nite element structural models with as many as ve million degrees of freedom 4 and, for such large problems, the extreme demands placed on computer resources by direct solvers, (b) the signicant progress achieved during the last decade in the development of fast and robust iterative algorithms for the solution of solid mechanics and plate and shell structural problems, and (c) the advent of commercial parallel hardware and the fact that iterative methods are more amenable to parallel processing than direct algorithms. Early work on preconditioning the conjugate gradient method 5 has focused on incomplete Cholesky factorization procedures 6; 7 to provide a bridge that spans the gap between direct and itera- tive methods. Then, attention shifted to element-by-element preconditioning techniques 8 * Correspondence to: Charbel Farhat, Department of Aerospace Engineering Sciences, and Center for Aerospace Structures, Campus Box 429, University of Colorado at Boulder, Boulder, CO 80309-0429, U.S.A. E-mail: charbel@boulder.colorado.edu. Contract=grant sponsor: Fonds National de la Recherche Scientique Contract=grant sponsor: Sandia National Laboratories CCC 0029–5981/99/040489–28$17.50 Received 5 December 1997 Copyright ? 1999 John Wiley & Sons, Ltd. Revised 28 April 1998