基于移动相似中心的比例边界有限元方法  

Scaled boundary element method with moving scaling center

作　　者：张勇 林皋[1] 胡志强[1] 徐喜荣[2] 

机构地区：[1]大连理工大学建设工程学部水利工程学院,大连116024 [2]大连理工大学电子信息与电气工程学部,大连116024

出　　处：《计算力学学报》2012年第5期726-733,共8页Chinese Journal of Computational Mechanics

基　　金：中德科学基金(GZ566);国家自然科学基金(90510018;90915009;51009019;51109134);清华大学水沙科学与水利水电工程国家重点实验室开放基金(sklhse-2010-C-03)资助项目

摘　　要：在传统的比例边界有限元中,相似中心是固定的,难以用其求解关于偏心域的场问题。本文引入移动相似中心的概念,建立新的比例边界坐标变换,并利用加权余量法将控制方程半弱化为关于径向坐标的二阶常微分方程,引入对偶变量,将其降为系数矩阵为Hamilton矩阵的一阶常微分方程。对Hamilton矩阵进行Schur分解,得到微分方程的通解,代入边值条件可得关于积分常数的代数方程。此方法将比例边界有限元扩展到偏心域的边值问题,同时在径向是半解析的,解的精度高;仅需要离散求解域的一个边界,数据量小;在计算中仅需要对Hamilton矩阵进行Schur分解以及求解关于积分常数的代数方程,运算量少。将偏心环形域静电场边值问题的算例与解析解或其他数值方法计算结果的比较,表明此方法具有精度高、数据量小及运算量小的优点。In the traditional scaled boundary finite element method （SBFEM） the scaling centre usually is placed at a fixed point, the concept of moving scaling center is proposed in this paper to extend traditional SBFEM to the problem within eccentric domain. With scaled boundary coordinate transformation based on the concept of moving scaling center, the governing Laplace equation is semi-discritized to scaled boundary ordinary differential equations by the weighted residual approach. To avoid singularity of coef- ficient Hamiltonian Matrix,Schur decomposition is employed and the general solution of electric poten- tial is obtained with integral constants, determined by boundary conditions. As only inner boundary needs to be discretised and semi-analytical property is covered in ODE,it＇s more effective to solve problem and achieves higher accuracy of the results. It＇s shown in several numerical examples that comparison with finite element method with the same discretized level on boundary. It demonstrates that the present method is competent in the problem within eccentric domain and yields excellent results, quick conver- gence rate and less amount of computational time.

关 键 词：移动相似中心 比例边界有限元 偏心环形域 静电场边值问题 SCHUR分解 

分 类 号：TM151[电气工程—电工理论与新技术] O242.21[理学—计算数学]