有限元法、无单元法及自然单元法之比较研究

无单元法与有限元法的根本区别在于形函数的构造方法不同，因而得到的形函数形式也不同，但可统一于单位分解。另外，采用Galerkin法对偏微分方程求解时需要对弱形式在求解域内积分，研究表明有限元法和无单元法中对弱形式的积分可统一于单位分解积分方法。无单元法解决了有限元法中前处理复杂以及难以有效处理的诸如裂纹扩展和大变形等问题；然而自身也存在着计算量大、边界条件和不连续的场变量及其导数处理上的困难。自然单元法是新近出现的一种求解PDE的数值方法，它兼有无单元的特性和有限元的优点，可以认为是介于两者之间的一种极具发展前途的数值方法。

COMPARATIVE STUDY ON FINITE ELEMENT METHOD,MESHLESS METHOD AND NATURAL ELEMENT METHOD

The fundamental difference between the finite element method(FEM) and the meshless method(MM) lies in the difference of the construction methods of the shape function. But the two kinds of shape functions both serve as a partition of unity. Integration of the weak form over the problem domain is needed when solving the partial differential equation using the Galerkin method. The newly presented quadrature method based on partition of unit provides a unified approach for the integration of the weak form in FEM and MM. MM is advantageous over the traditional FEM in the problems involved with the propagation of crack and large mesh deformation and so on. But the imposition of essential boundary condition and the treatment of discontinuous field variable and the derivative are somewhat troublesome,and the computation cost is large. The natural element method(NEM) is a newly appeared numerical method to solve the PDE. NEM assumes the characteristics of MM and the advantages of finite element method,so it is a most promising method.