【作者】 张希；

【导师】 姚振汉；

【作者基本信息】 清华大学 ， 力学， 2006， 博士

【摘要】 由于网格依赖性,有限元法在计算一些大变形和移动边界问题中遇到了许多困难。一些发展比较成熟的伽辽金型无网格法可以很好地避免这些困难,然而它们需要使用背景网格进行积分,对于大规模问题计算效率较低。本文采用的无网格局部彼得洛夫-伽辽金法(MLPG)具有完全不需要网格、计算速度快和精度高等优点。本文在这种方法的框架内提出了计算非线性材料大变形问题的计算方案,并得到了算例验证,主要工作如下:1.对MLPG法的计算效率和数值稳定性进行了分析。基于对结点搜索、矩阵存储和求解方案的计算效率进行的分析,提出了优化方案,把MLPG法的算法复杂度降低到了O(N)量级;使用谱分解法对MLPG方法的数值稳定性进行了分析,确定了MLPG方法保持数值稳定的参数范围。2.提出了一种基于拉格朗日坐标计算路径无关材料大变形问题的MLPG法计算方案,经过通用的线性化处理得到增量求解格式。以超弹性材料为例对本文提出的计算方案进行了验证,算例表明,非线性MLPG法不需要再做其它处理就可以减轻体积锁死问题,能够比有限元法模拟更大的材料变形,并且可以避免有限元由于网格畸变导致精度下降的问题。3.提出了一种计算路径相关材料大变形问题的MLPG法计算方案。为避免对试探子域和检验子域的更新,基本变量在当前构形的空间坐标中表示,而数值积分在初始构形的物质坐标中进行,对空间坐标的导数通过张量变换转换到物质坐标。为模拟更大的变形采用了极分解的乘法超弹塑性本构模型。作为数值检验,使用这种方法模拟了颈缩、剪切带形成等典型的应变局部化现象。数值算例表明:提出的MLPG计算方案能较精确模拟塑性大变形,可以避免塑性变形体积不变导致体积锁死问题和大变形导致的网格畸变问题;同时,由于无网格形函数可以构造高阶光滑的位移场,MLPG法可以减轻由于应变局部化导致的位移场不连续带来的数值困难。更多还原

【Abstract】 Because of its mesh-dependency the finite element method (FEM) is known to have difficulties in solving the problems with large deformation and moving boundaries. Some well-developed meshless Galerkin methods, such as EFG and RKPM, can solve such problems, but they need background cells to conduct numerical integration, and are inefficient in solving large-scale problems. The meshless local Petrov-Galerkin (MLPG) method has the advantage of“truly meshless”, high efficiency and accuracy. In this paper some numerical schemes for simulating the large deformation of nonlinear materials are presented in the frame of MLPG and verified by numerical examples. The main contributions are:1. The computational efficiency and numerical stability of the MLPG have been analyzed. Based on analyzing the efficiency of the MLPG computation, including the way of nodal query, matrix storage and equation solving, an optimized scheme is presented to make MLPG an O(N) algorithm. A spectral factoring approach is applied to analyze the numerical stability of MLPG, and the parameters which make MLPG numerically stable is obtained.2. A material configuration based numerical scheme of MLPG for the large deformation analysis of path-independent materials is presented, and the incremental formulation is obtained with conventional linearization approach. Some numerical examples of hyperelastic material are given to verify the presented numerical scheme, and the results indicate that the nonlinear MLPG method is capable to ease volume locking without any other treatment, and it can simulate much larger deformation than FEM and avoid the drawback of FEM on inaccuracy caused by mesh distortion.3. A numerical scheme of MLPG for the large deformation analysis of path-dependent materials is presented. In order to avoid the shape updating of trial and test subdomains, variables are represented in the spatial coordinate ofcurrent configuration, while the numerical quadrature is conducted in the material coordinate of initial configuration, and the derivatives of shape functions with respect to the spatial coordinate are converted to material coordinate by tensor transformation. A multiplicative constitutive model of hyperelastoplasticity is adopted to simulate large deformation. To verify the presented numerical scheme, it is used to simulate some typical strain localization phenomenon, such as necking and shear-band forming. The numerical examples have shown that the presented MLPG method is capable to simulate large plastic deformation accurately, and it avoids volume locking caused by volume preserving of plastic strain and mesh distortion by large deformation. Furthermore, the meshless approximation function of MLPG can reproduce high order displacement field, thus it can ease the numerical difficulties induced by the discontinuity line of shear band.更多还原