求解三维弹性力学问题的快速耦合方法  

The Fast Hybrid Method for Solving 3D Elasticity Problems

作　　者：程珩 梁冬琼 刘燕 彭飘飘 程玉民[3] CHENG Heng;LIANG Dongqiong;LIU Yan;PENG Piaopiao;CHENG Yumin(School of Applied Science,Taiyuan University of Science and Technology,Taiyuan 030024,Shanxi,China;School of Electric Power and Architecture,Shanxi University,Taiyuan 030006,Shanxi,China;School of Mechanics and Engineering Science,Shanghai University,Shanghai 200072,China)

机构地区：[1]太原科技大学应用科学学院,山西太原030024 [2]山西大学电力与建筑学院,山西太原030006 [3]上海大学力学与工程科学学院,上海200072

出　　处：《力学季刊》2023年第4期793-802,共10页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(12271341);山西省青年基金(202203021212482)。

摘　　要：将多种数值方法耦合,充分利用各种方法的优点建立新的数值方法,是求解三维复杂问题的有效途径之一.本文将无单元Galerkin(Element-Free Galerkin,EFG)方法、有限元法和维数分裂法耦合,提出了求解三维弹性力学问题的快速耦合方法(Fast Hybrid Method,FHM).将三维弹性力学问题分裂为若干个二维平面问题,对于每个二维问题采用罚函数法施加边界条件,并推导其相应的积分弱形式,引入Shepard基函数的移动最小二乘法建立形函数,进而推导二维平面问题的离散方程.第三个方向上采用有限元法将这些二维离散方程进行耦合,可以得到原三维弹性力学问题的快速耦合方法数值解的求解公式.通过数值算例验证了本文快速耦合方法求解三维弹性力学问题的收敛性,将数值解与解析解对比,说明了本文方法求解三维弹性力学问题的有效性.Coupling multiple numerical methods and fully utilizing their advantages to establish new methods is one of the effective ways to solve 3D complex problems.In this paper,by coupling the element-free Galerkin(EFG)method,the finite element method(FEM)and the dimension splitting method,we propose the fast hybrid method(FHM)for solving 3D elasticity problems.The 3D elasticity problem can be split into a few 2D forms.For each 2D problem,the boundary condition is imposed by the penalty method,and the corresponding integral weak form is derived.By introducing the moving least squares(MLS)approximation based on Shepard basic function to establish the shape functions,we can obtain the discrete equations of 2D form.In the third direction,FEM is used to couple these 2D equations,and final discrete equation the expressions of the FHM numerical solutions for the original 3D elasticity problems are obtained.The convergence of the proposed method for solving 3D elasticity problem is demonstrated through numerical examples,and the effectiveness of the proposed method in solving 3D elasticity problem is verified by comparing the numerical solutions with the analytical ones.

关 键 词：无单元Galerkin方法 Shepard基函数 维数分裂法 有限元法 弹性力学 

分 类 号：O241.82[理学—计算数学]