Robust Preconditioners for Linear Elasticity Fem Analyses

This paper deals with two forms of preconditioner which can be easily used with a Conjugate Gradient solver to replace a direct solution subroutine in a traditional engineering finite element package; they are tested in such a package (FINAL) over a range of 2-D and 3-D elasticity problems from geotechnical engineering. Quadratic basis functions are used. A number of modifications to the basic Incomplete Choleski [IC(0)] factorization preconditioner are considered. An algorithm to reduce positive off-diagonal entries is shown in numerical experiments to ensure stability, but at the expense of slow convergence. An alternative algorithm of Jennings and Malik is more successful, and a relaxation parameter o is introduced which can make a further significant improvement in performance while maintaining stability. A heuristic for determining a near-optimal value of o is proposed. A second form of preconditioning, symmetrically scaled element by element, due to Bartelt, is also shown to perform robustly over a range of problems; it does not require assembly of the global stiffness matrix, and has great potential for parallelization.