基于扩展单元插值法二维弹性问题的边界元法分析  

Boundary Element Analysis for 2D Elasticity Problems Based on Expanding Element Interpolation Method

作　　者：钟玉东 侯俊剑 谢贵重 张见明[2] 李源[3] ZHONG Yu-dong;HOU Jun-jian;XIE Gui-zhong;ZHANG Jian-ming;LI Yuan(Henan Key Laboratory of Mechanical Equipment Intelligent Manufacturing, Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou 450000, China;College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China;College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China)

机构地区：[1]郑州轻工业大学机电工程学院,河南省机械装备智能制造重点实验室,郑州450000 [2]湖南大学机械与运载工程学院,长沙410082 [3]河南师范大学计算机与信息工程学院,新乡453007

出　　处：《科学技术与工程》2020年第30期12290-12296,共7页Science Technology and Engineering

基　　金：国家自然科学基金(11602229,11702087);上海交通大学舰船设备噪声与振动控制技术国防重点学科实验室开放课题基金(VSN201903);河南省科技攻关重点项目(192102210227)。

摘　　要：为了提高边界元法中物理变量的插值精度,通过一种新型的插值方法-扩展单元插值法,研究了二维弹性问题的边界元求解问题。扩展单元是在原非连续单元两端添加虚节点,将非连续单元变成阶次更高的连续单元。原非连续单元的内部点被称为源节点,其形函数用来构建源节点和虚节点之间的关系,称为RawShape。扩展单元的形函数是由源节点和虚节点构造,用于边界物理变量的插值,称为FineShape。扩展单元继承了连续和非连续单元的优点,同时克服了它们的缺点;既可以插值连续场,也可以插值非连续场;在不改变方程自由度的前提下(边界积分方程只在源点处配置),把插值精度提高了至少两阶,最大限度发挥了边界积分方程试函数可以不连续的特性。最后通过数值算例来验证,结果表明本文方法获得的计算结果的精度和收敛性明显比连续和非连续单元的结果要好。To improve the interpolation accuracy of physical variables in boundary element method and seek for boundary element solution to the 2D elasticity problems,a new interpolation method,i.e.,the expanding element interpolation method was proposed,in which an expanding element is obtained by adding virtual nodes along the perimeter of the original discontinuous element and turning the discontinuous elements into higher-order continuous elements.The internal nodes of the original discontinuous element are called source nodes and its shape function are referred as the raw shape function that used to construct the relationship between virtual nodes and source nodes.The shape functions of expanding element were constructed on both source nodes and virtual nodes named as fine shape functions,and used to interpolate boundary physical variables.The expanding element inherits the advantages of both the conti-nuous and discontinuous elements while overcomes their disadvantages.With the expanding element,both continuous and discontinuous fields on the domain boundary can be accurately approximated.Without changing the degrees of freedom of the equation(the boundary integral equations are only collocated at source nodes),the interpolation accuracy increases by two orders compared with the original discontinuous element.In addition,the expanding elements take full advantages of the characteristic that the trial function of the boundary integral equation can be discontinuous.At last,a few numerical examples are presented to verify the proposed method.Therefore,the computational accuracy and convergence obtained by the proposed method are obviously better than those of continuous and discontinuous elements.

关 键 词：扩展单元 边界元法 边界积分方程 弹性问题 

分 类 号：O343.1[理学—固体力学]