重心插值配点法求解BIOT固结问题  

Barycentric interpolation collocation method for solving Biot's consolidation problems

作　　者：赵晓伟[1] 王兆清[2] 李广惠[1] 商丽华 ZHAO Xiaowei;WANG Zhaoqing;LI Guanghui;SHANG Lihua(Scientific research department Shandong Jianzhu University,Jinan 250101,China;School of science Shandong Jianzhu University,Jinan 250101,China;Shanghe County Housing and Urban Rural Construction Service Center,Jinan 251699,China)

机构地区：[1]山东建筑大学科研处,山东济南250101 [2]山东建筑大学理学院,山东济南250101 [3]商河县住房和城乡建设服务中心,山东济南251699

出　　处：《山东建筑大学学报》2023年第5期121-126,共6页Journal of Shandong Jianzhu University

基　　金：山东省住房与城乡建设厅项目(2021-K3-3)。

摘　　要：比奥(BIOT)固结微分方程组在数学上用解析法求解很困难。文章采用重心拉格朗日(Lagrange)插值近似未知函数,利用配点法离散BIOT固结问题的控制方程,实现了BIOT固结问题控制偏微分方程初边值问题的统一格式数值计算,其控制方程可以利用微分矩阵直接表达成矩阵方程的形式,初始边界条件也可通过重心Lagrange插值和微分矩阵离散成代数方程的形式,通过附加方程法施加初边界条件,得到求解BIOT固结问题的代数方程组,进而求解该方程组得到BIOT固结问题的数值解。结果表明:所提出的求解BIOT固结问题的数值方法,具有计算格式简单和易于程序实现,且所得数值精度高、稳定性好的特点。It is difficult to solve the consolidation differential equations given.This paper uses the barycentric Lagrange interpolation to approximate the unknown function,and uses the collocation method to discretize the governing equations of BIOT's consolidation problems,to make a unified format numerical calculation of the initial-boundary values of the partial differential equations governed by BIOT's consolidation problems.The governing equations can be directly expressed as matrix equations by the use of differentiation matrix.The initial-boundary conditions can be discretized as algebraic equations by barycentric Lagrange interpolation and differentiation matrix.The algebraic equations for solving BIOT's consolidation problems can be obtained by applying the initial-boundary conditions with the additional equation method,and then the numerical solution of BIOT's consolidation problems can be obtained by solving the equations.The numerical results show that the proposed method for solving BIOT's consolidation problems has the advantages of simple calculation format,easy program implementation,high numerical calculation precision and good stability.

关 键 词：微分矩阵 配点法 重心Lagrange插值 BIOT固结问题 

分 类 号：TU443[建筑科学—岩土工程]