Keywords: elastostatic problems, radial basis function, collocation method, meshfree, global approximation, multiquadric radial basis function.

Conventional methods for the approximate solution of partial differential equations (PDEs) require the use of a mesh: a domain mesh in the case of domain methods such as the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM), or a boundary mesh for boundary methods such as the boundary element method (BEM). The FDM can not handle irregular geometry efficiently and usually involves a rectangular grid system. The generation of a finite element grid with several thousand nodes and various element type and size is a nontrivial task. Using the BEM in nonlinear problems required decomposition of both the domain and the boundary. Because of these drawbacks which are related to the mesh, during recent years, the development of the so-called mesh free methods for approximating the solutions of PDEs has drawn the attention of many researchers in science and engineering and about ten different meshfree methods have been developed such as: the smooth particle hydrodynamics (SPH) method [1], the reproducing kernel particle method (RKPM) [2], the diffuse element method [3], the element free galerkin (EFG) method [4].

In the last decade, there have been some advanced developments in applying the radial basis functions (RBFs) for the numerical solution of various types of partial differential equations (PDEs). The initial development was due to the pioneering work of Kansa [5] who directly collocated the RBFs for the approximate solutions of the equations. This method has many attractive features compared to the other meshfree methods. It does not require numerical integration because it does not involve a weak formulation and gives high rates of convergence. In modelling geometry changes such as crack growth, the inherited merit of meshfree method of not involving a mesh is retained. Special treatment to satisfy the essential boundary conditions are not required. Furthermore, the computational speed for solving problem is suitable.

In this paper a meshfree algorithm based on the radial basis function is proposed and applied to the analysis of typical two-dimensional elastostatic problems. The main advantage of such a two-dimensional meshfree approach is its simplicity in both formulation and implementation. The basic characteristic of the formulation is the definition of a global approximation for the variables of interest from a set of radial basis functions conveniently placed at the boundary and in the domain. A generalized multiquadric radial basis function is considered as an interpolation function in this method.

Several numerical examples are presented and the numerical results are compared with those obtained by the exact solution to demonstrate the performance of the present RBF collocation method for solving elasticity problems. In this paper two types of elasticity problems are considered: problems in a Cartesian coordinate system including a patch test and a cantilever beam, and problems in a polar coordinate system including a cylinder subjected to internal pressure, a cylinder subjected to shear and a plate with a hole.

This study shows that the globally supported RBFs perform excellently in solving elasticity problems with Dirichlet boundary conditions but could result in a significant error in solving elasticity problems with Neumann boundary conditions. A treatment should performed to overcome this problem and it will be investigated in future research.

The performance and robustness of the method for solving Navier equations in elasticity problems may be very interesting for researchers involved in computational mechanics. A lot of work has still to be done on this subject. Several directions of research are now being exploited by a growing number of researchers in particular on themes such as compactly supported RBFs, Hermite type collocation and domain decomposition amongst others.

1

L.B. Lucy, "A numerical approach to the testing of the fission hypothesis", The Astronomical Journal, 82(12), 1013-1024, 1977. doi:10.1086/112164

2

W.K. Liu, Y. Chen, "Wavelet and multiple scale reproducing kernel methods", International Journal for Numerical Methods in Engineering, 21, 901-931, 1995. doi:10.1002/fld.1650211010

3

B. Nayroles, G. Touzot, P. Villon, "Generalizing the finite element method: diffuse approximation and diffuse elements", Journal of Computational Mechanics, 10, 307-18, 1992. doi:10.1007/BF00364252

4

T. Belystcho, Y. Lu, L. Gu, "Element free Galerkin methods", International Journal Numerical Methods in Engineering, 37, 229-256, 1994. doi:10.1002/nme.1620370205

5

E.J. Kansa, "Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics. II. Solution to parabolic, hyperbolic and elliptic partial differential equations", Journal of Computational Mathematical Applied, 19, 147-61, 1990. doi:10.1016/0898-1221(90)90271-K

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