极限下限分析的区域光滑径向点插值法  

A Cell⁃Based Smoothed Radial Point Interpolation Method for Lower Bound Limit Analysis

作　　者：陈莘莘[1] 董昊 李庆华[1] CHEN Shenshen;DONG Hao;LI Qinghua(School of Civil Engineering and Architecture,East China Jiaotong University,Nanchang 330013,Jiangxi,China)

机构地区：[1]华东交通大学土木建筑学院,江西南昌330013

出　　处：《力学季刊》2024年第3期697-705,共9页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(12172131,12162014),江西省主要学科学术和技术带头人培养计划(20225BCJ22010)。

摘　　要：极限分析的高效数值计算方法在结构设计和安全评定中具有非常重要的作用.为了更加有效地求解极限分析问题,将区域光滑径向点插值法与二阶锥规划相结合,提出了理想弹塑性结构极限下限分析的一种新方法.将问题域离散为简单的三角形背景单元,每个单元进一步划分成若干个光滑域.为了将复杂的域积分转化为简单的边界积分,并且避免计算形函数的导数,采用广义梯度光滑技术对每个光滑域进行应变光滑处理.由于径向点插值法构造的形函数满足Kronecker delta性质,本质边界条件可以直接施加.依据下限定理,在满足以等效积分弱形式表达的自平衡应力场平衡条件的基础上,用二阶锥规划成功构建了极限分析下限法的计算模型,从而可方便地通过基于原始-对偶内点法的数学规划求解器MOSEK直接求解该问题.数值算例结果表明,本文所提方法有效地克服了维数障碍问题,具有较高的计算精度,并且计算结果对网格畸变十分不敏感.The efficient numerical method of limit analysis plays an important role in engineering design and safety assessment.In order to solve limit analysis problem more effectively,a novel numerical method is proposed to perform lower bound limit analysis of structures made of elasto-perfectly-plastic material making use of the cell-based smoothed radial point interpolation method(CSRPIM)and the second order cone programming.The problem domain is initially discretized by a simple triangular background mesh,and each triangular cell is further divided into several smoothing domains.In order to convert the complex domain integrals into simpler boundary integrals of the smoothing domains and avoid computing the derivative of the shape functions,the generalized gradient smoothing technique is applied to each smoothing domain.The Kronecker delta property of the radial point interpolation method(RPIM)shape functions enables the direct imposition of essential boundary conditions.Based on the lower bound theorem,the mathematical model of lower bound limit analysis is converted into a second-order cone programming problem satisfying equilibrium conditions for the self-equilibrium stress field which are expressed in an equivalent weak form.Then the commercial solver MOSEK with the primal-dual interior point method can be utilized to solve the problem.Numerical examples verify that the proposed method can overcome the dimension obstacle and possesses high computational accuracy.In addition,the computational results of the proposed method are highly insensitive to the mesh distortion.

关 键 词：无网格法 区域光滑径向点插值法 极限下限分析 广义梯度光滑技术 二阶锥规划 

分 类 号：O241[理学—计算数学] O343[理学—数学]