Large-scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients

Abstract

The major costs of large implicit finite element calculations, particularly in three dimensions, arise from computing solutions to systems of linear equations. Direct methods, i.e., those based upon Gaussian elimination, can easily require prohibitively large amounts of both CPU time and storage, even on current supercomputers. Iterative procedures avoiding the formation and factorization of a global system of equations can circumvent these difficulties. The element-by-element (EBE) preconditioned conjugate gradients (PCG) algorithm is presented in the context of a vectorized implementation within the production nonlinear stress analysis code nike3 d. Due to continued confusion as to the ease of vectorizing finite element procedures, we include examples of the main EBE subroutines in their entirety. The concept of a fractal dimension of a finite element mesh is introduced, and proves useful in characterizing the efficiency of this iterative algorithm with respect to a variable band, active column direct method. Sample calculations on a Cray X-MP/48 with solid-state storage device (SSD) illustrate the economy and range of applicability of EBE/PCG. Asymptotic cost formulae derived for two linear problems underscore differences between the direct and iterative algorithms for large problems and lead to predictions of problem size limitations imposed by the computing environment.