二维Helmholtz方程的插值型边界无单元法

陈林冲; 李小林

doi:10.21656/1000-0887.380202

二维Helmholtz方程的插值型边界无单元法

doi: 10.21656/1000-0887.380202

重庆师范大学数学科学学院, 重庆 401331

基金项目: 国家自然科学基金(面上项目)(11471063)；重庆市基础科学与前沿技术研究重点项目(cstc2015jcyjBX0083)

详细信息

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
计量
- 文章访问数: 967
- HTML全文浏览量: 109
- PDF下载量: 563
- 被引次数: 0
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

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

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China

Funds: The National Natural Science Foundation of China(General Program)(11471063)

摘要

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.
- 内外边值问题 /
- 插值型边界无单元法 /
- 正则化 /
- 强奇异积分 /
- 幂级数
Abstract: An interpolating boundary element-free method was presented for solving interior and exterior boundary value problems of 2D Helmholtz equations. According to the indirect potential theory and the characteristics of the fundamental solution of Laplace’s equation, a regularized boundary integration equation formulation was established to avoid the computation of the strongly singular integration. Besides, through expansion of the global distance into power series in the form of the local distance, the limit expressions of the distance derivative and the difference between 2 normal derivatives were deduced in detail. Finally, 4 numerical examples were given to show the feasibility and efficiency of the proposed method.
- interior and exterior boundary problem /
- interpolating boundary element-free method /
- regularization /
- strongly singular integration /
- power series

HTML全文

参考文献(21)

[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.