杂交无单元Galerkin方法施加Dirichlet边界条件研究  

Research on Application of the Dirichlet Boundary Condition Using Hybrid Element-Free Galerkin Method

作　　者：刘燕 程珩 王韦博 LIU Yan;CHENG Heng;WANG Weibo(School of Applied Science,Taiyuan University of Science and Technology,Taiyuan 030024,China)

机构地区：[1]太原科技大学应用科学学院,山西太原030024

出　　处：《中北大学学报(自然科学版)》2025年第1期91-97,共7页Journal of North University of China(Natural Science Edition)

基　　金：山西省青年基金资助项目(20210302124388);山西省创新训练项目(20230712)。

摘　　要：Lagrange乘子法和罚函数法是无网格方法施加边界条件常用的两种方法,为了比较两种方法的优缺点,本文研究了三维Helmholtz方程的杂交无单元Galerkin(Hybrid Element-Free Galerkin,HEFG)方法。引入维数分裂法将控制方程分裂为若干个二维问题,对于每个二维问题,分别采用Lagrange乘子法和罚函数法施加边界条件,建立等价的泛函,并推导相应的积分弱形式。引入改进的移动最小二乘法建立形函数,进而推导二维问题的离散方程。在维数分裂方向采用有限差分法将这些二维离散方程进行耦合,得到原三维Helmholtz方程的离散求解方程。数值算例中对数值解的精度和时间进行对比,分析了两种方法施加Dirichlet边界条件的优缺点,得出采用罚函数法施加边界条件较好的结论。The Lagrange multiplier method and the penalty method are the common methods when applying essential boundary conditions in meshless method.In order to compare the advantage and the disadvantage of two methods,the hybrid element-free Galerkin(HEFG)method was presented for analyzing 3D Helmholtz equation.By introducing the dimensional split method,the governing equation could be split into a few 2D forms,for every 2D problem,the Lagrange multiplier method and the penalty method were used to apply the boundary conditions,and the equivalent functional could be established,thus the corresponding integral weak forms could be derived.By introducing the improved moving least squares(IMLS)approximation to establish shape functions,the discrete equation of 2D forms could be obtained.In dimensional split direction,the finite difference method was selected to couple these 2D equations,thus the final discrete equation of 3D Helmholtz equation was obtained.In numerical examples,by comparing the computational accuracy and computational time of numerical results,the advantages and disadvantages of two methods for applying boundary conditions were analyzed,respectively.It is shown that the penalty method is better than the Lagrange multiplier method when applying essential boundary conditions.

关 键 词：LAGRANGE乘子法 罚函数法 HELMHOLTZ方程 杂交无单元Galerkin方法 

分 类 号：O241.82[理学—计算数学]