自然单元法研究进展

自然单元法(NEM)是新近出现的一种求解偏微分方程(PDE)的数值方法,它采用自然相邻点插值方法在全域构造近似函数和试函数,该方法基于整个求解域内离散节点的Voronoi结构。NEM形函数的构造简单,形函数及其导数的计算相对容易,由于不涉及到矩阵的运算及其逆运算,与一般的无网格方法相比计算量大大减少。另外由于NEM形函数满足Delta函数的性质,且在边界相邻节点间满足线性插值,从而可以准确地施加边界条件和方便地处理场函数及其导数的不连续性,这是一般的无网格方法所难以实现的。从形函数的构造和性质来看,它兼有无网格的特性和有限单元方法的优点,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。详细介绍了自然单元法的求解过程和最新研究进展,并对目前自然单元法中尚待改进的问题及其相应的解决方案和未来的研究方向进行了初步的探讨。

RESEARCH PROGRESS IN NATURAL ELEMENT METHOD

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.