Keywords: element free method, mapping, elastic problem, essential boundary condition, numerical analysis, finite element method.

The finite element method (FEM) is a useful and powerful tool for use in the numerical solution of partial differential equations. However, the mesh generation process in FEM is a very burdensome task for today's analysis which require large-scale and complicated models. In recent years, the meshless method or element free method has drawn the attention of engineers as an approach which requires no element connectivity data, but only nodal data.

The element free Galerkin method (EFGM) proposed by Belytschko et al. [1] is the one of the practical meshless methods. Many other element free methods have been proposed [2,3,4,5]. In the EFGM, the moving least squares (MLS) approximation is used for making interpolation functions and the integrals in the weak form are evaluated by using a rectangular background cell structure which is independent on the field nodes.

One of advantages of the EFGM is that the approximate solutions and its derivatives are continuous in the entire domain, while the derivatives at node between elements are discontinuous in the FEM. However, the EFGM based on MLS has a disadvantage that the essential boundary conditions can not be enforced as easily as in the FEM because the MLS interpolations do not pass through the nodal values.

In this paper, an element free method using the mapping technique is proposed. The region defined in physical domain is mapped into computational domain of rectangular grid, and the stiffness matrix for the element free method is also transformed and calculated numerically in the mapped computational domain.

By using the mapping technique, the region with curvilinear boundary shape is transformed to the regular rectangular grid. Therefore, the geometry of the region can overlap with the rectangular background cell structure in the mapped domain. In addition, the interpolation functions which do not need weight functions, that is, not based on MLS, are used in the proposed method. The finite Fourier series of relative vectors between two adjacent points [6,7] or Lagrange polynomial are used as interpolation functions in this paper. The approximate values at nodes can be connected directly with the nodal values by using these interpolation functions. Therefore, the essential boundary conditions can be directly evaluated in a manner similar to the ordinary FEM. The boundary value problems with curvilinear boundary shape can be easily treated by calculating in the mapped domain. The proposed method analysing the problems in the mapped domain can be combined with many other grid generation methods.

Two numerical examples are presented; a cantilever beam loaded with the free end and a thick walled tube under uniform pressure. Results are compared with the exact solutions and the ordinary FEM results. It has been shown that the proposed method is sufficiently accurate and effective for analysing boundary value problems with curvilinear boundary shape.

1

T. Belytschko, Y.Y. Lu, L. Gu, "Element-free Galerkin methods", Int. J. Num. Methods Eng., Vol.37, 229-256, 1994. doi:10.1002/nme.1620370205

2

Y.Y. Lu, T. Belytschko, L. Gu, "A new implementation of the element-free Galerkin method", Comp. Methods Appl. Mech. Eng. , Vol.113, 397-414, 1994. doi:10.1016/0045-7825(94)90056-6

3

T. Belytschko, Y. Organ, D. Fleming, "Meshless method: An overview and recent developments" Comp. Methods Appl. Mech. Eng. , Vol.139, 3-47, 1996. doi:10.1016/S0045-7825(96)01078-X

4

H. Noguchi, T. Kawashima, T. Miyamura, "Element free analysis of shell and spatial structures", Int. J. Num. Methods Eng., Vol.47, 1215-1240, 2000. doi:10.1002/(SICI)1097-0207(20000228)47:6<1215::AID-NME834>3.0.CO;2-M

5

Y. Suetake, "Element-free method based on Lagrange polynomial", J. Eng. Mech., ASCE, Vol.128, No.2, 231-239, 2002. doi:10.1061/(ASCE)0733-9399(2002)128:2(231)

6

T. Kusama, T. Ohkami, Y. Mitsui, "Application of The Finite Fourier Series to The Boundary Element Method", Computers and Structures, Vol.32 (6), 1267-1273, 1989. doi:10.1016/0045-7949(89)90304-0

7

T. Kusama, T. Ohkami, "Improvement of Interpolation Function Using Finite Fourier Series", Proc.JSCE, Vol.446/I-19, 167-175, 1992. (in Japanese)

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