Abstract
In this study, we present a novel strategy to the method of finite elements (FEM) of linear elastic problems of very high resolution on graphic processing units (GPU). The approach exploits regularities in the system matrix that occur in regular hexahedral grids to achieve cache-friendly matrix-free FEM. The node-by-node method lies in the class of block-iterative Gauss-Seidel multigrid solvers. Our method significantly improves convergence times in cases where an ordered distribution of distinct materials is present in the dataset. The method was evaluated on three real world datasets: An aluminum-silicon (AlSi) alloy and a dual phase steel material sample, both captured by scanning electron tomography, and a clinical computed tomography (CT) scan of a tibia. The caching scheme leads to a speed-up factor of ×2-×4 compared to the same code without the caching scheme. Additionally, it facilitates the computation of high-resolution problems that cannot be computed otherwise due to memory consumption.
Highlights
MethodsOur method solves the following standard linear elasticity problem
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Assume that the boundary of the elastic body is divided into two disjoint sets ΓD (Dirichlet boundary) and ΓN (Neumann boundary) and assume that a system of body forces f : O ! R3 and surface tractions gN : GN ! R3 act on the body
Summary
Our method solves the following standard linear elasticity problem. On the other part ΓD of the boundary, the body is partially fixed in space. Under the assumption of small displacements, the displacement u = (ui) i 3 satisfies the following problem: À X3 j1⁄41 @sij @xj ðuÞ 1⁄4 fi in. O; ui 1⁄4 0 on GD; X3 sijðuÞnj 1⁄4 gi on GN; j1⁄41 where n = (ni) i 3 is the unit outward normal to the boundary ΓD, fi and gi are the components of the forces f and gN, and σij(u) is the stress tensor, cf [38]. The problem is discretized in the standard way in finite element analysis. Standard notations for Lebesgue and Sobolev spaces are used. Let ( , ) denote the inner product in (L2(O))d,d 1. Find u2H1(O) such that aðu; vÞ 1⁄4 hF; vi for all v 2 HD1 ðOÞ3; with the scalar product
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