Keywords: element free method, mapping, three-dimensional elastic problem, essential boundary condition, nodal integration, numerical analysis.

The finite element method (FEM) is a popular and powerful numerical method for solving partial differential equations. However, the mesh generation process required in the FEM is a very burdensome task for today's analyses 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.

The element free Galerkin method (EFGM) proposed by Belytschko et al. [1] has been developed to discretize weak-formed governing equations using only distributed nodal data. In the EFGM, the moving least squares (MLS) approximation is generally used for making interpolation functions and the integrals in the weak form are evaluated by using a rectangular background cell structure which is independent of the field nodes.

The main advantages of the EFGM are that the element mesh is unnecessary, and the approximate solutions and its derivatives are continuous in the entire domain while the derivatives at nodes between elements are discontinuous in the FEM.

One shortcoming of the EFGM based on the MLS is 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. Another shortcoming is that the domain integrals in the weak form are evaluated by using rectangular background cells. The domain shape formed by adding each cell exactly does not agree with the shape of the entire region. Then the integral calculation for a region with curvilinear boundary shape is inaccurate.

In this paper, an element free method using a mapping technique is proposed. The region defined in the real domain is mapped into the computational domain of a 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 a 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 does not need weight functions, that is, not based on MLS, are used in the proposed method. 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.

In general, the numerical integration is performed at integration points in each background cell in order to integrate the weak form of the governing equation. The node-by-node meshless method (NBNM) [2,3] was applied to two-dimensional analyses, in which a nodal integration is utilized using only distributed nodal data for domain integration. The nodal integration method saves computation time and requires less memory to evaluate the stiffness because the total number of nodes is generally smaller than the total number of integration points. Following the idea of NBNM, a nodal integration method for three-dimensional analysis is developed in this paper.

The scheme is applied to three-dimensional elastic problems. 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 effective for analysing boundary value problems with curvilinear boundary shape.

1

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

2

T. Nagashima, "Development of a CAE system based on the node-by-node meshless method", Comput. Methods Appl. Mech. Engng., Vol.187, 1-34, 2000. doi:10.1016/S0045-7825(99)00135-8

3

T. Nagashima, "Node-by-node meshless approach and its applications to structural analyses", Int. J. Numer. Meth. Engng., Vol.46, 341-385, 1999. doi:10.1002/(SICI)1097-0207(19990930)46:3<341::AID-NME678>3.0.CO;2-T

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