Intelligent Block Iterative Methods 1

SUMMARY We present several preconditioned conjugate gradient methods for the iterative solution of the linear, symmetric systems of equations arising from the nite element method in displacement form, both for the h-version and the p-version. The preconditioners are based on a decomposition of the solution space into overlapping subspaces and solving separately on each subspace. A judicious choice of the subspaces gives good convergence for a uniform mesh. Local adaptive techniques are employed to modify the subspaces for real-world problems with distorted meshes and thin 3D elements. Computational results on workstations are presented for solid and shell aircraft structures with up to over 1,000,000 degrees of freedom and 4GB stiiness matrix. 1. INTRODUCTION Much of the current state of the art in the Finite Element Method (FEM) has been motivated by the necessity to obtain useful practical results from relatively coarse discretizations, thus creating the need for substantial engineering expertise in applying the nite element analysis lest erroneous results are obtained. One way to increase the automatization and reliability of nite element analysis is to use ner and more precise discretizations, in particular high order discretizations, the p-version FEM. (Weaknesses of the mathematical model such as spurious singularities are clearly revealed this way, but that is outside the scope of this article.) Possibly in a not too distant future, it will be possible to dump a CAD model with all geometrical complexity as input to the FEM at a reasonable cost, without worrying about non-physical models and wrong solutions. The ever more powerful computers make the treatment of such discretizations possible. But the classical, direct solution methods for the resulting large, sparse systems of algebraic equations suuer from terminal ll-in and their requirements on storage space and CPU time grow much faster that the number of degrees of freedom. Iterative solvers use the original system data and avoid ll-in, but the performance of most iterative methods is inconsistent and not suitable for industrial type problem. In the last decade, a new class of iterative methods for the numerical solution of partial diierential equations has emerged, the multigrid methods and domain decomposition methods, see, e.g., the surveys 19, 24]. These methods use the properties of the underlying diierential