Keywords: GPU computing, meshfree method, solid mechanics, large deformation analysis, nonlinear analysis, radial point interpolation method.

Large deformation analysis of three-dimensional solids is a major problem in solid mechanics. The aim of this paper is to develop a parallel algorithm for the large deformation analysis using a meshfree approach. The problems we consider in this paper are quasi-static problems with large deformation of solids and we formulate the problem so that the computation can be performed effectively using hierarchically allocated processors on the graphics processing units (GPUs). We show that the meshfree method called the modified radial point interpolation method (RPIM [1]) is suitable for parallel computation on the GPUs. In our algorithm, the equation of motion is described using the total Lagrangian formulation [1], i.e., physical quantities at a particular time is expressed using the coordinate system at the initial configuration and the integration is performed using the initial configuration. The following are the assumptions commonly required for the computation of a three-dimensional solid analysis using the meshfree methods discussed above.

Nodes located on the surface and inside the solid are required in the computations. We call these boundary and interior nodes, respectively. The density of the nodes is a primary factor in the accuracy of the solution. There are several approaches for the generation of the nodes, e.g. a set of nodes can be located randomly or uniformly in and on the solid, or nodes can be located adaptively based on error estimate.

The problem domain must be divided into smaller regions if the method requires background cells for numerical integration over the problem domain. This requirement stems from the weak-form-based formulation of partial differential equations. Note that, although the background cells can be regarded as a type of mesh structure, they are independent of the nodes, and the process of creating the background cells is simpler than mesh generation in the finite element method.

In this paper, we propose a parallel algorithm of meshfree large deformation analysis for three-dimensional solids. For the parallel computation, we hierarchically divide the most costly parts of the computation, generation of the shape function matrices and the tangent stiffness matrices, into small processes appropriately to the structure of the GPUs supporting the compute unified device architecture (CUDA).

1

J.G. Wang, G.R. Liu, "A point interpolation meshless method based on radial basis function", Numerical Methods in Engineering, 54(11), 1623-1648, 2002.

2

K.J. Bathe, "Finite element procedures", Prentice-Hall, New Jersey, 1996.

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