无摩擦弹性接触问题的自适应交替方向乘子法  

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems

作　　者：袁欣 张守贵 YUAN Xin;ZHANG Shougui(School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,P.R.China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《应用数学和力学》2023年第8期989-998,共10页Applied Mathematics and Mechanics

基　　金：国家自然科学基金项目(11971085);重庆市自然科学基金项目(cstc2020jcyj-msxmX0066);重庆市研究生教育教学改革研究项目(yjg213071)。

摘　　要：对一类无摩擦的弹性接触问题,得到了求其数值解的自适应交替方向乘子法.由该问题导出相应的变分问题,引入辅助变量将原问题转化为一个基于增广Lagrange函数表示的鞍点问题,并采用交替方向乘子法求解;为了提高算法性能,提出了利用边界迭代函数自动选取合适罚参数的自适应法则.该算法的优点是每次迭代只需计算一个线性变分问题,同时显式计算了辅助变量和Lagrange乘子.对算法的收敛性进行了理论分析,最后用数值结果验证了该算法的可行性和有效性.A self-adaptive alternating direction multiplier method was designed for frictionless elastic contact problems.An augmented Lagrange function was introduced for the variational formulation of the problem with an auxiliary variable,to deduce a minimization problem and an equivalent saddle-point problem.Then the alternating direction multiplier method was used to solve the problem.To enhance the performance of the algorithm,a self-adaptive rule based on the iterative function on the boundary was proposed to automatically select the proper penalty parameter.The advantage of this algorithm is that,each iteration only needs to solve a linear variational problem and explicitly calculate the auxiliary variable and the Lagrange multiplier.The convergence of the algorithm was analyzed theoretically.The numerical results illustrate the feasibility and effectiveness of the proposed method.

关 键 词：弹性接触问题 交替方向乘子法 自适应法则 增广Lagrange 

分 类 号：O221.6[理学—运筹学与控制论]