不规则板弯曲问题的重心插值正则区域法研究  

A regular domain collocation method based on barycentric interpolation for solving bending problems of irregular thin plate

作　　者：庄美玲[1] 王兆清[1] 纪思源 张磊[1] 

机构地区：[1]山东建筑大学力学研究所,山东济南250101

出　　处：《山东建筑大学学报》2015年第4期345-350,共6页Journal of Shandong Jianzhu University

基　　金：国家自然科学基金项目(51379113)

摘　　要：正则区域法可将不规则区域嵌入规则矩形区域中,能有效解决复杂区域上的偏微分方程的边值问题。文章利用重心插值正则区域法研究不规则板的弯曲问题,分析了在不规则板嵌入的矩形区域上的离散不规则板弯曲问题的控制方程和边界条件,建立了矩形区域内的离散代数方程组;采用最小二乘法对离散代数方程组进行求解,并对不规则板上任意节点的挠度值与其对应的解析解进行了比较与分析。结果表明:重心插值正则区域法的计算公式简单,程序实施方便;利用重心有理插值法可插值得到不规则区域内任意节点函数值,求出不规则板上任意点的挠度值;利用重心插值正则区域法解决任意不规则板的弯曲问题上的计算精度可高达10-8以上。The bending problems of irregular thin plate can be modeled by boundary problem of partial differential equation in complex domain. This paper describes a regular domain collocation method based on the barycentric rational interpolation for solving the bending problems of irregular thin plate. Embedded the irregular domain into a rectangular domain, the governing equation and boundary condition equation of bending plate in a rectangular domain is discretized by the differentiation matrix derived from barycentric rational interpolation to form a system of algebraic equations. The paper uses the least square method to solve the equations and get the node displacement values in rectangular domain. The deflection values of arbitrary point on irregular plate are solved by using interpolation of the barycentric interpolation to get function values of any node in the irregular plate. The calculation formula of the method is simple and program of it is convenient. By comparing and analyzing the numerical error of the deflection values of arbitrary point on irregular plate by a regular domain collocation method based on the barycentric rational interpolation with the analytical solutions of the deflection values of irregular plate according to the numerical examples in this paper, we can conclude that the method is a new method with high efficiency and calculation precision of over 10^ -8 for solving bending problems of any irregular thin plate.

关 键 词：不规则板 重心有理插值法 配点法 正则区域法 微分矩阵 

分 类 号：O241[理学—计算数学]