Limited-Memory Deflated Conjugate Gradient for Solid Mechanics

Abstract

The primary computational bottle-neck in finite element based solid mechanics is the solution of the underlying linear systems of equations. Among the various numerical methods for solving such linear systems, the deflated conjugate gradient (DCG) is well-known for its efficiency and simplicity. The first contribution of this paper is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration”. The latter exploits, classic Kirchoff-Love theory for plates and Euler-Bernoulli theory for beams. We demonstrate that curvature-sensitive agglomeration is much more efficient for highly ill-conditioned problems arising from thin structures. The second contribution is a strategy for a limited-memory assembly-free DCG where neither the stiffness matrix nor the deflation matrix is assembled. The resulting implementation is particularly well suited for large-scale problems, and can be easily ported to multi-core architectures. For example, we show that one can solve a 50 million degree of freedom system on a single GPU card, equipped with 3 GB of memory.