弹塑性大变形问题的改进的无单元Galerkin方法  

The improved element-free Galerkin method for elastoplasticity large deformation problems

作　　者：蔡小杰 彭妙娟[1] 程玉民[2] 

机构地区：[1]上海大学土木工程系,上海200444 [2]上海大学上海市应用数学和力学研究所,上海200072

出　　处：《中国科学：物理学、力学、天文学》2018年第2期53-62,共10页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(编号:11571223)资助项目

摘　　要：基于改进的移动最小二乘法,建立了弹塑性大变形问题的改进的无单元Galerkin方法.改进的移动最小二乘法克服了移动最小二乘法有时形成病态或奇异方程组的缺点.基于改进的移动最小二乘法建立形函数,根据弹塑性大变形问题的Galerkin弱形式建立离散方程,利用罚函数法施加位移边界条件,推导了弹塑性大变形问题的改进的无单元Galerkin方法的公式,采用Newton-Raphson迭代法进行求解.通过数值算例,讨论了权函数、影响域比例参数、罚因子、节点数和迭代步数对计算精度的影响,结果显示,相对于四次和五次样条函数,选取三次样条函数作为权函数具有更高精度;当d_(max)=3.6,α=10^(10)×E时本文方法具有较高精度.考虑不同的节点分布和加载步数,分析了本文方法的收敛性.数值结果验证了本文方法的有效性,说明了该方法具有提高计算效率的优点.Based on the improved moving least-squares （IMLS） approximation, the improved element-flee Galerkin （IEFG） method for elastoplasticity large deformation problems is presented. Compared with the moving least-squares （MLS） approximation, by orthogonalizing the basis function, the IMLS approximation can overcome the disadvantage of ill- conditional or singular equations in the MLS approximation. Compared with the meshless methods based on the MLS approximation, under the similar computational accuracy, the ones using the IMLS approximation have higher computational efficiency. The IMLS approximation is used to form the approximation function, the Galerkin weak form of elastoplasticity large deformation problem is used to form the discretized equation system, and the penalty method is used to apply the displacement boundary conditions, then the formulae of the improved element-free Galerkin （IEFG） method for elastoplasticity large deformation problems are obtained, and Newton-Raphson method is used to obtain the solution of the final equation system. Some numerical examples are given to discuss the influences of the weight function, the scale parameter, the penalty factor, the node distribution and the load step number on the computational precision of the numerical solutions. Considering the relative errors when the cubic, quartic and quintic spline functions are used as weight functions, it is shown that the numerical solution using the cubic spline function has higher computational precision. About the scale parameter of the influence domain and the penalty factor, the numerical results show that dmax=3.6 and a=10^10×E will obtain the solutions with higher precision. The convergence of the method in this paper is analyzed by considering the influence of the node distribution on the computational precision of the solutions, and it is shown that the method in this paper is convergent. And the influence of the total load step number on the computational precision of the solutions is also discussed. N

关 键 词：无网格方法 改进的移动最小二乘法 改进的无单元Galerkin方法 弹塑性大变形 

分 类 号：O241.82[理学—计算数学] O344[理学—数学]