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hadaptivity analysis based on multiple scale reproducing kernel particle method




 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... 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 (nodeconnection information). In adaptive analysis, high precision numerical model can be obtained by simply inserting new nodes into highgradient 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 2D elastostatic problems have been concerned with. An explicit posteriori error estimation is developed and deduced. The 2D 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 pconvergence properties are better than hconvergence 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.  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．  Borrowing an idea from wavelet theory, the shape function of Reproducing Kernel Particle Method (RKPM) and resultant structural responses are decomposed into different multiple scales. Twoscale decomposition on 2D linear stress concentration problems is performed in this paper. The obtained highest scale components indicate the high gradient solution and are used as an indicator for hadaptivity analysis. Furthermore, a new strategy for node refinement based on"four quadrants" criterium is proposed.... Borrowing an idea from wavelet theory, the shape function of Reproducing Kernel Particle Method (RKPM) and resultant structural responses are decomposed into different multiple scales. Twoscale decomposition on 2D linear stress concentration problems is performed in this paper. The obtained highest scale components indicate the high gradient solution and are used as an indicator for hadaptivity analysis. Furthermore, a new strategy for node refinement based on"four quadrants" criterium is proposed. The numerical results verified the feasibility of this method and also the resultant convergence history demonstrated the convergence of the hadaptivity.  借助于“小波”理论,再生核质点方法RKPM(ReproducingKernelParticleMethod)可以将形状函数及求得的结构响应分解为多个尺度。本文对线弹性二维应力集中问题进行了双尺度分解,并由各应力分量计算得到的高梯度点作为误差指示,实现了该方法的h型自适应分析。并且提出了一种新的方法———“四象限法”对高梯度区域进行加密,计算结果表明自适应后的解的精度更高,从而证明了这种自适应无网格方法的有效性。  An h_adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two nodes refinement strategies were constructed using searching_neighbor_nodes (SNN) and local_Delaunay_triangulation (LDT) techniques, which were suitable and effective for h_adaptivity analysis on 2_D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h_adaptivity analyses on 2_D linear elastostatics and bending plate problems demonstrate that the improper... An h_adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two nodes refinement strategies were constructed using searching_neighbor_nodes (SNN) and local_Delaunay_triangulation (LDT) techniques, which were suitable and effective for h_adaptivity analysis on 2_D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h_adaptivity analyses on 2_D linear elastostatics and bending plate problems demonstrate that the improper high_gradient indicator will reduce the convergence property of the h_adaptivity analysis, and that the efficiency of the LDT node refinement strategy is better than SNN, and that the presented h_adaptivity analysis scheme is provided with the validity, stability and good convergence property.  基于多尺度再生核质点法(RKPM)的多分辨分析特性,给出了适于求解二维问题的h型自适应分析方法,讨论了不同高梯度指标对检测高梯度区的影响,同时还提出了基于邻节点搜索(SNN)及局部Delaunay三角分解(LDT)技术的适于任意节点分布的无网格自适应分析节点加密技术.通过对二维线弹性平面应力及平板弯曲问题的h型自适应分析,说明了不合适的高梯度指标将对自适应分析的收敛造成不良影响,且LDT节点加密技术较SNN技术的效率更高.数值算例的结果说明自适应分析的有效性、稳定性及良好的收敛特性.  
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