求解非齐次方程的直线积分边界元法研究  

Research on line integration boundary element method for solving nonhomogeneous equations

作　　者：卢云杉 王桥[1] 周伟[1] 常晓林[1] 陈辉煌 LU Yunshan;WANG Qiao;ZHOU Wei;CHANG Xiaolin;CHEN Huihuang(State Key Laboratory of Water Resources Engineering and Management,Wuhan University,Wuhan 430072,China;China Railway Construction Group Zhongnan Construction Co.,Ltd.,Wuhan 430080,China)

机构地区：[1]武汉大学水资源工程与调度全国重点实验室,湖北武汉430072 [2]中铁建设集团中南建设有限公司,湖北武汉430080

出　　处：《武汉大学学报(工学版)》2025年第4期525-530,共6页Engineering Journal of Wuhan University

基　　金：国家重点研发计划项目(2022YFC3005504);国家自然科学基金项目(51979207,U2040223)。

摘　　要：边界元法作为求解偏微分方程的重要数值方法之一,仅对计算边界划分网格,在热传导问题、力学问题和电磁问题等领域得到了极大的发展。然而,边界元法在求解非齐次方程(如泊松方程)时,得到的积分方程包含域内积分项,若直接计算域积分,需要对求解域划分体网格,背离了传统边界元法仅对计算边界划定网格的基本原则。为了更好地解决这一问题,提出了一种只需要划分边界单元来计算域内积分的边界单元法——直线积分边界元法,该方法在传统边界元法的基础上,最终可将非齐次项导致的域内积分转换为相互平行的直线上的积分,避免了划分域内网格。同时,可以通过背景网格将积分直线分段,提高计算效率。通过与解析解的对比可知,该方法计算结果与解析解吻合良好。As one of the important numerical methods for solving partial differential equations,the boundary element method only needs to divide the computational boundary into grids,which has been greatly developed in the fields of heat conduction,mechanical and electromagnetism.However,when the boundary element method is used to solve non-homogeneous equations,such as Poisson equation,the integral equation obtained contains integral terms in the domain.If the domain integration is directly calculated,the solution domain needs to be divided into volume grids,which deviates from the basic principle that the traditional boundary element method only delimits the grid for the calculation boundary.In order to solve this problem better,this paper proposes a boundary element method,linear integral boundary element method,which only needs to divide the boundary into elements to calculate the domain integral.Based on the traditional boundary element method,this method can finally convert the domain integral caused by non-homogeneous terms into the integrals on parallel straight lines,avoiding dividing the grid in the domain.At the same time,the integral lines can be segmented through the background grid to improve the calculation efficiency.By comparing with the analytical solution,it is found that the results obtained by the proposed method agree well with the analytical solutions.

关 键 词：边界元法 域内积分 直线积分边界元法 

分 类 号：TU502[建筑科学—建筑技术科学]