Parallel element-free Galerkin method algorithm with application to three-dimensional nonlinear adaptive analysis in solid mechanics

Abstract

Most of the real world solid mechanics problems are three-dimensional with material and geometrical nonlinearities and their numerical solution is computationally very expensive, therefore it is more convenient to solve these with parallel adaptive analysis. One of the most prominent meshless method, The element-free Galerkin method (EFGM) [1] is used in this research, which has distinct advantages over the conventional finite element methods (FEMs) in large deformation and adaptive analysis. The maximum entropy shape functions are used instead of the conventional moving least squares (MLS) to facilitate the imposition of the essential boundary conditions.
Existing two-dimensional, linear, elasto-static error estimation procedure [2] is adapted for the current adaptive analysis. A parallel computer code is developed based on the distributive memory computer architecture with FORTRAN as a programming language and message passing interface (MPI) library as a communications protocol between the processors. The Durham University’s, 228 nodes high performance computing cluster (Hamilton cluster) is used in the current research. Two main computationally expensive parts in the code, the computation and assembly of the stiffness matrix and the solution of the final linear system of equations are considered as candidates for parallel programming. METIS - a multilevel graph partitioning algorithm [3] is used for the first part to intelligently divide the problem domain to different processers, In this case the partition is based on the integration cells. MUltifrontal massive parallel solver (MUMPS) [4] is used for the second part for the solution of the final system of linear equations in parallel.

Keywords: EFGM; Nonlinear adaptive analysis; Parallel computation

[1].Belytschko, T. and Lu, Y. Y. and Gu, L. (1994). Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 37 p. 229-256.

[2].Chung, H. -J. and Belytschko, T. (1998). An error estimate in the EFG method. Computational Mechanics, 21 p. 91-100.

[3].Karypis, G. Kumar, V. (2011). METIS - Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 5.0. http://glaros.dtc.umn.edu/gkhome/views/metis

[4].MUMPS (2011): a MUltifrontal Massively Parallel Solver, version 4.10.0 http://graal.ens-lyon.fr/MUMPS/

Full text

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