Keywords: material point method, arbitrary Lagrangian-Eulerian formulation, granular flow, plastic forming, parallel processing, OpenMP.

The material point method (MPM) is a computational approach which is an efficient tool of analysis of problems of solid mechanics with large strains, [1,2]. Two kinds of space discretisation are used in the method: the Lagrangian and Eulerian ones. Beside the division of the region representing an analysed body into a set of subregions, a computational finite element mesh, covering the virtual position of the body, is introduced. Each subregion is represented by one of its points called a material point. The motion of the material points is traced together with the values of state variables using the interpolation functions and their derivatives defined by the use of the computational mesh which can remain constant or be changed during the computations. As the computational mesh can be defined in an arbitrary way, the problem of element distortions, which appears in the purely Lagrangian formulation of the finite element method, is avoided. The material point method can be regarded as the finite element method formulated in an arbitrary Lagrangian-Eulerian description of motion. Due to its features, the method is well-suited for modelling the large strain engineering problems like granular flow and plastic forming problems [2].

The analyses of dynamic, large strain problems, to which the material point method can be applied, are time consuming. Therefore, there is a need to use parallel computations for such tasks. A parallelism technique used in the present paper is based on the OpenMP programming model [3,4]. As, in many cases, OpenMP may require only small modifications of the serial code, it is very attractive especially due to the fact that, at present, PC computers and notebooks allow the use of multicore and hyper threading technologies. In the present paper, a part of the existing finite element program related to the material point analysis has been parallelised using loop-level parallelism in the following procedures:

• assembly of global matrices and vectors (the loop over the elements),
• updating stresses (the loop over the elements),
• searching elements in which the material points are located (the loop over the material points),
• solution of the diagonalised global system of equations (the loop over equations).

The parallelised version of MPM has been utilised for problems of granular flow and plastic forming problems. Using a four processor computer, the speedup factor of up to 2.6 has been obtained in the calculations.

1

D. Sulsky, H.L. Schreyer, "Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems", Computer Methods in Applied Mechanics and Engineering, 139, 409-429, 1996. doi:10.1016/S0045-7825(96)01091-2

2

Z. Wieckowski, "The material point method in large strain engineering problems", Computer Methods in Applied Mechanics and Engineering, 193, 4417-4438, 2004. doi:10.1016/j.cma.2004.01.035

3

R. Chandra, L. Dagum, D. Kohr, D. Maydan, J. McDonald, R. Menon, "Parallel Programming in OpenMP", Academic Press, 2001.

4

B. Chapman, G. Jost, R. van der Pas, "Using OpenMP. Portable Shared Memory Parallel Programming", The MIT Press, 2008.

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