h, p, hp ADAPTIVE MESHLESS METHOD FOR PLANE CRACK PROBLEM
平面裂纹问题的h, p, hp型自适应无网格方法的研究

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.