LI Guo-qing. A Conjugate Boundary Element Method for Complex Analysis of Analytic Function[J]. Applied Mathematics and Mechanics, 2017, 38(8): 863-876. doi: 10.21656/1000-0887.370315
Citation: LI Guo-qing. A Conjugate Boundary Element Method for Complex Analysis of Analytic Function[J]. Applied Mathematics and Mechanics, 2017, 38(8): 863-876. doi: 10.21656/1000-0887.370315

A Conjugate Boundary Element Method for Complex Analysis of Analytic Function

doi: 10.21656/1000-0887.370315
Funds:  The National Natural Science Foundation of China(10972083)
  • Received Date: 2016-10-17
  • Rev Recd Date: 2016-12-08
  • Publish Date: 2017-08-15
  • An analytic function is composed of 2 real conjugate harmonic functions, of which the complex analysis plays an important role in the fields of applied mathematics and mechanics. A set of weighted residual equations were proposed and proved to be equivalent to the approximate solution to the original problem involving 2 governing equations in the domain, the boundary condition and the CauchyRiemann equation at the boundary. 2 conventional direct boundary integral equations at the boundary collocation points were deduced from 2 of the weighted residual equations, and 1 finite difference equation was deduced from the rest one. The mathematical problem arising from the illconditioned linear equations was solved and the Cauchy integral equation was adopted for numerical calculation of the fields at the internal points inside the domain. Finally, the proposed conjugate boundary element method with constant elements was completely established. 3 examples demonstrate that, the proposed method is valid for analytic functions in terms of the power function, the exponential function and the logarithmic function in interior or exterior domains, and the error estimation of the proposed method is at the same order as that of the boundary element method for 2D potential problems.
  • loading
  • [1]
    Jeans J H. The Mathematical Theory of Electricity and Magnetism [M]. 4th ed. London: Cambridge University Press, 1920.
    Kolosov G V. Application of Complex Variables in the Theory of Elasticity [M]. Moscow-Leningrad, 1935.(in Russian)
    Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of Elasticity [M]. Moscow: Nauka, 1966.(in Russian)
    England A H. Complex Variable Method in Elasticity [M]. New York: Wiley, 1971.
    毛翎, 姚伟岸, 高强, 等. 空间各向异性弹性问题的二十节点理性单元[J]. 应用数学和力学, 2014,35(6): 589-597.(MAO Ling, YAO Wei-an, GAO Qiang, et al. 20 node rational elements for 3D anisotropic elastic problems[J]. Applied Mathematics and Mechanics,2014,35(6): 589-597.(in Chinese) )
    胡元太, 李国清, 蒋树农, 等. 具有刚性双边裂纹的压电介质中的电荷相互作用分析[J]. 应用数学和力学, 2005,26(8): 911-920.(HU Yuan-tai, LI Guo-qing, JIANG Shu-nong, et al. Interaction of electric charges in a piezoelectric with rigid external cracks[J]. Applied Mathematics and Mechanics,2005,26(8): 911-920.(in Chinese))
    周伟建, 陈伟球. 表面效应对偏场下介电高弹体表面波传播的影响[J]. 应用数学和力学, 2015,36(2): 119-127.(ZHOU Wei-jian, CHEN Wei-qiu. Surface effect on propagation of surface waves in a dielectric elastomer half space subject to biasing fields[J]. Applied Mathematics and Mechanics,2015,36(2): 119-127.(in Chinese))
    Brebbia C A. The Boundary Element Method for Engineers [M]. London: Pentech Press, 1978.
    Hromadka II T V, Lai C. The Complex Boundary Element Method in Engineering Analysis[M]. New York: Springer, 1987.
    Whitley R J, Hromadka II T V. Theoretical developments in the complex variable boundary element method[J]. Engineering Analysis With Boundary Elements,2006,30(12): 1020-1024.
    禤启沃, 吴兹潜. 二维位势边界元技术的共轭函数法[J]. 计算物理, 1989,6(2): 191-196.(XUAN Qi-wo, WU Ci-quian. The method of conjugate function for boundary element technique[J]. Chinese Journal of Computational Physics,1989,6(2): 191-196.(in Chinese))
    M·拉夫连季耶夫, B·沙巴特. 复变函数论方法[M]. 第6版. 施祥林, 夏定中, 吕乃刚, 译. 北京: 高等教育出版社, 2006.(Lavrentieff M A, Shabat B. Methods of Functions of a Complex Variable [M]. 6th ed. SHI Xiang-lin, XIA Ding-zhong, Lü Nai-gang, tansl. Beijing : Higher Education Press, 2006.(Chinese version))
    徐次达. 加权残数法解固体力学问题[J]. 力学与实践, 1980,2(4): 12-20.(XU Ci-da. Weighted residuals method for solid mechanics[J]. Mechanics in Engineering,1980,2(4): 12-20.(in Chinese))
    Branski A, Borkowski M, Borkowska D. A comparison of boundary element methods based on inverse variational formulation[J]. Engineering Analysis With Boundary Elements,2012,36(4): 505-510.
    Wearing J L, Sheikh M A. A regular indirect boundary element method for thermal analysis[J]. International Journal for Numerical Methods in Engineering,1988,25(2): 495-515.
    姚振汉, 王海涛. 边界元法[M]. 北京: 高等教育出版社, 2010.(YAO Zhen-han, WANG Hai-tao. Boundary Element Method [M]. Beijing: Higher Education Press, 2010.(in Chinese))
    Jacobsen M, Hansen P C, Saunders M A. Subspace preconditioned LSQR for discrete ill-posed problems[J]. Bit Numerical Mathematics,2003,43(5): 975-989.
    Miao X Y, Li G Q. Analysis of piezoelectric plates with a hole using nature boundary integral equations and domain decomposition[J]. Engineering Analysis With Boundary Elements,2014,40: 71-77.
  • 加载中


    通讯作者: 陈斌,
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (791) PDF downloads(1413) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint