摘要
无网格方法以其独特的优点：不需“网格”（即节点间的连接信息）划分，特别适合自适应的分析．在分析中只需在高梯度域简单地插入离散点（ｈ型）或保持模型节点数、分布、覆盖大小均不变，只增加高误差覆盖上的覆盖函数的多项式阶次（ｐ型），便可以得到更高精度的数值模型．针对平面弹性问题发展和推导一种显式后验误差指示公式，对平面裂纹实例进行了ｈ型，ｐ型，ｈｐ型三种不同类型的无网格自适应分析．数值分析结果表明了这种自适应无网格方法的有效性．
The meshless methods are attracting more numerical analysis researchers in recent years. Various meshless methods are proposed named SPH, DEM, EFGM, RKPM, MQ, HPCLOUDS, FPM etc. Unlike finite element methods, these meshless methods need only a scattered set of nodes between which no fixed connective information is required. This feature is very useful for many engineering problems such as crack propagation, high impact, and large deformations etc., because remeshing can be avoided. Meshless method is easily used for adaptive numerical analysis due to its special characteristic, which need no mesh (node-connection information). In adaptive analysis, high precision numerical model can be obtained by simply inserting new nodes into high-gradient field (h adaptivity) or only improving cover function polynomial order while model nodal numbers and position, size of cover keep no changing (p adaptivity). In this paper, the meshless method based on cover and a petition of unity is studied. The main idea is to obtain the basic functions by multiplying a partition of unity by cover functions for approximating to field functions. Here, cover functions are defined as polynomials or other appropriate class of functions. Because the moving least squares functions (MLSM) constitute the partition of unity, good properties of the MLSM such as high regularity and compactness are retained. This property allows the easy implementation of h adaptivity, p adaptivity and hp adaptivity as finite element methods but without the burden of a mesh. In this paper, the principles and theories for the design of h adaptivity, p adaptivity and hp adaptivity meshless method for 2-D elastostatic problems have been concerned with. An explicit posteriori error estimation is developed and deduced. The 2-D plane crack problem is analyzed by h, p, hp adaptive meshless method. For h adaptivity, an efficient refinement strategy is used and tested by adding new nodes into high error field but no mesh is needed. After several h steps are performed, the high precision numerical model below the specified error can be obtained. For p adaptivity, cover function polynomial order is increased while model nodal numbers and positive, size of cover is fixed so that the implementation of p adaptivity is easier than the implementation of h or hp adaptivity and p-convergence properties are better than h-convergence properties. For hp adaptivity, its implementation is straightforward after h and p adaptivity has been implemented and the hp adaptive model displays the best convergence properties. To sum up, numerical results show that the adaptive method is effective.
出处
《力学学报》
EI
CSCD
北大核心
2000年第3期308-318,共11页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金!(19672024)
关键词
无网格方法
自适应
后验误差估计
平面裂纹
meshless method, h、p、hp adaptivity, posteriori error estimate, plane crack
