## 留言板

 引用本文: 陈林冲, 李小林. 二维Helmholtz方程的插值型边界无单元法[J]. 应用数学和力学, 2018, 39(4): 470-484.
CHEN Linchong, LI Xiaolin. An Interpolating Boundary Element-Free Method for 2D Helmholtz Equations[J]. Applied Mathematics and Mechanics, 2018, 39(4): 470-484. doi: 10.21656/1000-0887.380202
 Citation: CHEN Linchong, LI Xiaolin. An Interpolating Boundary Element-Free Method for 2D Helmholtz Equations[J]. Applied Mathematics and Mechanics, 2018, 39(4): 470-484.

## 二维Helmholtz方程的插值型边界无单元法

##### doi: 10.21656/1000-0887.380202

###### 作者简介:陈林冲(1988—)，男，硕士(E-mail: 794530653@qq.com)；李小林(1983—),男，教授，博士(通讯作者. E-mail: lxlmath@163.com).
• 中图分类号: O242.2

## An Interpolating Boundary Element-Free Method for 2D Helmholtz Equations

Funds: The National Natural Science Foundation of China(General Program)(11471063)
• 摘要: 针对二维Helmholtz方程的内外边值问题,提出了插值型边界无单元法(interpolating boundary element-free method).在间接位势理论的基础上,利用Laplace方程基本解的特性,建立了求解Helmholtz方程Neumann边值内外问题的正则化形式,有效消除了强奇异积分的计算.其次通过引入全局距离展开成局部距离的幂级数, 详细推导了距离函数的导数和法向导数差值的极限表达式.最后给出了4个插值型边界无单元法的数值算例, 表明了该方法可取得较高的可行性和有效性.
•  [1] ERLANGGA Y A. A robust and efficient iterative method for the numerical solution of the Helmholtz equation[D]. PhD Thesis. Delft: Technische Universiteit Delft, 2005. [2] 嵇醒, 臧跃龙, 程玉民. 边界元法进展及通用程序[M]. 上海: 同济大学出版社,1997.(JI Xing, ZANG Yuelong, CHENG Yumin. The Development of Boundary Element and General Program [M]. Shanghai: Tongji University Press, 1997.(in Chinese)) [3] BELYTSCHKO T, KRONGAUZ Y, ORGAN D, et al. Meshless methods: an overview and recent developments[J]. Computer Methods in Applied Mechanics and Engineering,1996,139(1/4): 3-47. [4] YAGAWA G, FURUKAWA T. Recent developments of free mesh method[J]. International Journal for Numerical Methods in Engineering, 2000,47(8): 1419-1443. [5] LUCY L B. A numerical approach to the testing of the fission hypothesis[J]. The Astronomical Journal, 1977,82: 1013-1024. [6] LANCASTER P, SALKAUSKAS K. Surfaces generated by moving least square methods[J]. Mathematics of Computation, 1981,37: 141-158. [7] NAYROLES B, TOUZOT G, VILLON P. Generalizing the finite element method: diffuse approximation and diffuse elements[J]. Computational Mechanics, 1992,10(5): 307-318. [8] MUKHERJEE Y X, MUKHERJEE S. The boundary node method for potential problems[J]. International Journal for Numerical Methods in Engineering, 1997,40(5): 797-815. [9] WANG Jufeng, WANG Jianfei, SUN Fengxin, et al. An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems[J]. International Journal of Computational Methods,2013,10(6): 1350043. DOI: 10.1142/S0219876213500436. [10] GAO Xiaowei. An effective method for numerical evaluation of 2D and 3D high order singular boundary integrals[J]. Computer Method in Applied Mechanics and Engineering,2010,199(45/48): 2856-2864. [11] 祝家麟, 袁政强. 边界元分析[M]. 北京: 科学出版社, 2009.(ZHU Jialin, YUAN Zhengqiang. The Analysis of Boundary Element [M]. Beijing: Science Press, 2009.(in Chinese)) [12] 王竹溪, 郭敦仁. 特殊函数概论[M]. 北京: 国防工业出版社, 1983.(WANG Zhuxi, GUO Dunren. Introduction to Special Function [M]. Beijing: National Defend Industry Press, 1983.(in Chinese)) [13] 孙新志, 李小林. 复变量移动最小二乘近似在Sobolev空间中的误差估计[J]. 应用数学和力学, 2016,37(4): 416-425.(SUN Xinzhi, LI Xiaolin. Error estimates for the complex variable moving least square approximation in Sobolev spaces[J]. Applied Mathematics and Mechanics,2016,37(4): 416-425.(in Chinese)) [14] TELUKUNTA S, MUKHERJEE S. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners[J]. Engineering Analysis With Boundary Elements, 2004,28(9): 1099-1110. [15] LI Xiaolin, ZHANG Shougui. Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation[J]. Applied Mathematical Modelling,2016,40(4): 2875-2896. [16] 沈杰罗夫Е Л. 水声学波动问题[M]. 何祚镛, 赵晋英, 译. 北京: 国防工业出版社, 1983.(ШЕНДЕРОВ Е Л. Underwater Acoustic Wave Problems [M]. HE Zuoyong, ZHAO Jinying, transl. Beijing: National Defend Industry Press, 1983.(Chinese version)) [17] MA Jianjun, ZHU Jialin, LI Maojun. The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber[J]. Engineering Analysis With Boundary Elements, 2010,34(12): 1058-1063. [18] 贾祖朋, 余德浩. 二维Helmholtz方程外问题基于自然边界归化的重叠型区域分解算法[J]. 数值计算与计算机应用, 2001,22(3): 241-253.(JIA Zupeng, YU Dehao. The overlapping DDM based on nature boundary reduction for 2-D exterior Helmholtz problem[J]. Journal of Numerical Methods and Computer Applications,2001,22(3): 241-253.(in Chinese)) [19] LI Junpu, CHEN Wen, GU Yan. Error bounds of singular boundary method for potential problems[J]. Numerical Methods for Partial Differential Equations,2017,33(6): 1987-2004. [20] LI Junpu, FU Zhuojia, CHEN Wen. Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method[J]. Computers & Mathematics With Applications,2016,71(1): 381-390. [21] LI Junpu, CHEN Wen, FU Zhuojia, et al. Explicit empirical formula evaluating original intensity factors of singular boundary method for potential and Helmholtz problems[J]. Engineering Analysis With Boundary Elements,2016,73: 161-169.

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##### 出版历程
• 收稿日期:  2017-07-20
• 修回日期:  2017-11-27
• 刊出日期:  2018-04-15

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