Keywords: microscale continuum modeling, heterogeneous three-dimensional elasticity problems, domain decomposition, preconditioning.

Modern approaches of discretization methods such as the finite element method (FEM) lead to partial differential equations which must be solved numerically. Nowadays in (material) engineering science many investigations of material behaviour in three dimensions, e.g. damage initiation and propagation at different scales, are based on complex and computationally expensive numerical simulations. The most time-consuming part of the solution is the numerical computation of the underlying equation system independent of the kind of the discretized problem. Therefore it is necessary to develop adequate numerical methods for the computation of the linearized problem and to use the available hardware resources in the most efficient way. Starting with the FEM discretization of three-dimensional elasticity problems such as a heterogeneous multiphase specimen the decomposition is based on the separation of domains governed by their fundamentally different material behaviour. For example this will lead to partial equation systems of such domains which should and have to be solved completely in parallel. Thereby based on the separation of elastic-inelastic problems the reduction-condensation of the elastic unknown degree-of-freedom to their boundary degree-of-freedom coupling the elastic with the inelastic domain is enabled using the classical Schur complement method. The iterative computation of the resulting equation system for the nonlinear problem is done using the conjugate gradient method (CG). Furthermore the influence of different preconditioning techniques for the CG computation with respect to computing time is investigated. The parallelization of the preconditioned conjugate gradient method is based on the parallelization of the sparse matrix-vector operations (SPMV) which is distributed to graphical processing units (GPU) such as Nvidia Tesla systems which enable highly scalable SPMV execution. In order to develop a hybrid computation model as a result of the combined SPMV execution on the CPU and GPU hardware devices further improvements related to memory demand and computing time are discussed. Thereby several (CPU-based) OpenMP parallelization techniques as well as different storage formats for the resulting sparse and dense matrices with many millions of nonzero matrix entries on the GPU and CPU are compared.

This paper presents in section 2 the geometry modeling of a heterogeneous specimen. Section 3 reviews the applied constitutive laws. After that section 4 describes the iterative computation model and the preconditioning technique for the final equation system which has to be solved. Section 5 shows some preliminary results based on the methods implemented. In section 6 an outlook for future research activities closes this paper.

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