2019/04/30 by Denis Davydov, Jean‐Paul Pelteret, Davydov, Denis +5 · 4 citations
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Elasticity and Material Modeling #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1904.13131
The performance of finite element solvers on modern computer architectures is\ntypically memory bound for sufficiently large problems. The main cause for this\nis that loading matrix elements from RAM into CPU cache is significantly slower\nthan performing the arithmetic operations when solving the problem. In order to\nimprove the performance of iterative solvers within the high-performance\ncomputing context, so-called matrix-free methods are widely adopted in the\nfluid mechanics community, where matrix-vector products are computed\non-the-fly.\n To date, there have been few (if any) assessments into the applicability of\nthe matrix-free approach to problems in solid mechanics. In this work, we\nperform an initial investigation on the application of the matrix-free approach\nto problems in quasi-static finite-strain hyperelasticity to determine whether\nit is viable for further extension. Specifically, we study different numerical\nimplementations of the finite element tangent operator, and determine whether\ngeneralized methods of incorporating complex constitutive behavior might be\nfeasible. In order to improve the convergence behavior of iterative solvers, we\nalso propose a method by which to construct level tangent operators and employ\nthem to define a geometric multigrid preconditioner. The performance of the\nmatrix-free operator and the geometric multigrid preconditioner is compared to\nthe matrix-based implementation with an algebraic multigrid preconditioner on a\nsingle node for a representative numerical example of a heterogeneous\nhyperelastic material in two and three dimensions. We conclude that the\napplication of matrix-free methods to finite-strain solid mechanics is\npromising, and that is it possible to develop numerically efficient\nimplementations that are independent of the hyperelastic constitutive law.\n