自然单元法研究进展

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Abstract The natural element method (NEM) is a newly appeared numerical method to solve the partial differential equation (PDE). In the natural element method，the trial and test function are constructed using the natural neighbor interpolation method and the interpolation is constructed with respect to the Voronoi tessellation of the scattered points in the problem domain. The computation of the NEM shape function and its derivatives are relatively trivial. Because the matrix operation and inverse operation are not involved，the computation cost is small in contrast to common meshless methods. The NEM shape function satisfies the property of Delta function and the linear interpolation property is observed，so the essential boundary conditions can be exactly imposed，and the discontinuity in the field and its derivatives can be treated easily，which are troublesome in common meshless methods. Considering the construction and the properties of its shape function，the natural element method possesses the characteristics of meshless method and the advantages of finite element method，so it is the most promising numerical method and can be viewed as a midpoint between meshless method and finite element method. The solving process and the new research progresses are addressed. Some problems to be solved in the present NEM are pointed out and relating schemes are presented. Future development of this method and their applications are discussed.

Key words： rock and soil mechanics natural element method natural neighbor interpolation partition of unit numerical integration stress recovery

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https://rockmech.whrsm.ac.cn/EN/Y2004/V23/IS1/4319

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