Keywords: meshfree, meshless, partial differential equations, structural analysis, shape modeling, radial point interpolation method.

The aim of this study is to develop an algorithm for the elastostatic analysis of three-dimensional solids of complex shapes. We assume here that the shape model of the target is constructed using a three-dimensional scanning device such as a laser scanning system. For this purpose, we propose a scheme of stress analysis based on a meshfree method combined with an implicit surface modeling technique for representing the shape of an object.

The element-free Galerkin method (EFG) [1] is one pioneering work for solving partial differential equations in a meshfree manner. The idea of the method is to discretize the weak form of a partial differential equation by shape functions defined using the moving least-squares method. The shape functions depend only on nodes distributed in the problem domain and can be constructed without a mesh structure. EFG has also been successfully applied to a number of problems, including the structural analysis of two- and three-dimensional solids.

The radial point interpolation method (RPIM) [2] is another meshfree method based on the weak form of a partial differential equation. The shape functions for the discretization are constructed using an interpolation technique, which has the advantage that Dirichlet boundary conditions can be simply imposed on the discrete equation. The effectiveness of the method has also been shown through a number of test problems of structural analysis.

A modified version of the RPIM has also been presented in [3]. These shape functions are defined in a slightly different way from those of the original RPIM, which accelerates the process of shape function evaluation. The effectiveness of the modification is shown through application to Poisson's equation [4].

In this paper, an algorithm for elastostatic solid analysis using the modified RPIM in combination with the implicit surface model is presented. The shape of the target is assumed to be modeled using the multilevel partition of unity method [5], one of the most effective and efficient methods for automatic implicit-surface generation, from scanner measurements.

1

T. Belytschko, Y.Y. Lu, L. Gu, "Element-Free Galerkin Methods", Int. J. Numer. Meth. Engng, 37(2), 229-256, 1994. doi:10.1002/nme.1620370205

2

J.G. Wang, G.R. Liu, "A Point Interpolation Meshless Method Based on Radial Basis Function", Int. J. Numer. Meth. Engng, 54(11), 1623-1648, 2002. doi:10.1002/nme.489

3

S. Nakata, "An Efficient Scheme for Meshless Analysis Based on Radial Basis Functions", in "ICNAAM 2006 proceedings", Wiley-VCH, Weinheim, Germany, 263-266, 2006.

4

S. Nakata, "Meshless Analysis with Non-Uniformly Distributed Nodes Using Hierarchical Cell Structure", AIP conference proceedings 936, Melville, NY, 378-381, 2007. doi:10.1063/1.2790156

5

Y. Ohtake, A. Belyaev, M. Alexa, G. Turk, H.P. Seidel, "Multi-Level Partition of Unity Implicits", ACM Transactions on Graphics, 22(3), 463-470, 2003. doi:10.1145/882262.882293

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