A component decomposition preconditioning for 3D stress analysis problems

A preconditioning methodology for an iterative solution of discrete stress analysis problems based on a space decomposition and subspace correction framework is analysed in this paper. The principle idea of our approach is a decomposition of a global discrete system into the series of subproblems each of which correspond to the different Cartesian co‐ordinates of the solution (displacement) vector. This enables us to treat the matrix subproblems in a segregated way. A host of well‐established scalar solvers can be employed for the solution of subproblems. In this paper we constrain ourselves to an approximate solution using the scalar algebraic multigrid (AMG) solver, while the subspace correction is performed either in block diagonal (Jacobi) or block lower triangular (Gauss–Seidel) fashion. The preconditioning methodology is justified theoretically for the case of the block‐diagonal preconditioner using Korn's inequality for estimating the ratio between the extremal eigenvalues of a preconditioned matrix. The effectiveness of the AMG‐based preconditioner is tested on stress analysis 3D model problems that arise in microfabrication technology. The numerical results, which are in accordance with theoretical predictions, clearly demonstrate the superiority of a component decomposition AMG preconditioner over the standard ILU preconditioner, even for the problems with a relatively small number of degrees of freedom. Copyright © 2002 John Wiley & Sons, Ltd.

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