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矩形到任意多边形区域的Schwarz-Christoffel变换数值解法

王玉风 姬安召 崔建斌

王玉风, 姬安召, 崔建斌. 矩形到任意多边形区域的Schwarz-Christoffel变换数值解法[J]. 应用数学和力学, 2019, 40(1): 75-88. doi: 10.21656/1000-0887.390050
引用本文: 王玉风, 姬安召, 崔建斌. 矩形到任意多边形区域的Schwarz-Christoffel变换数值解法[J]. 应用数学和力学, 2019, 40(1): 75-88. doi: 10.21656/1000-0887.390050
WANG Yufeng, JI Anzhao, CUI Jianbin. Numerical Solution of Schwarz-Christoffel Transformation From Rectangles to Arbitrary Polygonal Domains[J]. Applied Mathematics and Mechanics, 2019, 40(1): 75-88. doi: 10.21656/1000-0887.390050
Citation: WANG Yufeng, JI Anzhao, CUI Jianbin. Numerical Solution of Schwarz-Christoffel Transformation From Rectangles to Arbitrary Polygonal Domains[J]. Applied Mathematics and Mechanics, 2019, 40(1): 75-88. doi: 10.21656/1000-0887.390050

矩形到任意多边形区域的Schwarz-Christoffel变换数值解法

doi: 10.21656/1000-0887.390050
基金项目: 甘肃省自然科学基金(1606RJZM092;17JR5RM355);甘肃省高等学校科研项目(2017B-61)
详细信息
    作者简介:

    王玉风(1986—),女,讲师,硕士(通讯作者. E-mail: yinyu413@163.com).

  • 中图分类号: O242.1

Numerical Solution of Schwarz-Christoffel Transformation From Rectangles to Arbitrary Polygonal Domains

  • 摘要: 运用Schwarz-Christoffel变换方法,建立多边形区域到带状区域共形映射数学模型.对于模型中的约束条件和奇异积分问题,根据Riemann(黎曼)原理,建立复参数与实参数互逆变换,消除非线性系统的约束条件;经过合理积分路径的确定,模型中的奇异积分转化为Gauss-Jacobi(高斯雅可比)型积分;采用Levenberg-Marquardt算法对非线性系统模型进行求解.根据第一类椭圆函数性质,建立了矩形区域到带状区域共形映射数学模型,通过复参数椭圆函数的计算,得到矩形边界与带状区域边界的关系.最后,对8点对称多边形区域与27点不规则条带状区域计算,将不规则封闭区域边界映射到矩形区域边界,矩形区域内的正交网格,通过变换之后在多边形区域内依然满足正交性,为研究不规则区域到规则区域映射的数值计算奠定基础.
  • [1] 王刚, 许汉珍, 顾王明, 等. 数值许瓦尔兹-克力斯托夫变换与数值高斯-雅可比型积分[J]. 海军工程大学学报, 1994,1(2): 25-34.(WANG Gang, XU Hanzhen, GU Wangming, et al. Numerical Schwarz-Christoffel transformation and numerical Gauss-Jacobi quadrature[J]. Journal of Naval Academy of Engineering,1994,1(2): 25-34.(in Chinese))
    [2] 王刚, 陆小刚, 顾王明. 槽形内域中的数值许瓦尔兹-克力斯托夫保角变换[J]. 海军工程大学学报, 1995,1(4): 16-23.(WANG Gang, LU Xiaogang, GU Wangming. Numerical Schwarz-Christoffel conformal mapping in channel region[J]. Journal of Naval Academy of Engineering,1995,1(4): 16-23.(in Chinese))
    [3] 祝江鸿. 隧洞围岩应力复变函数分析法中的解析函数求解[J]. 应用数学和力学, 2013,34(4): 345-354.(ZHU Jianghong. Analytic functions in stress analysis of the surrounding rock for caverns with the complex variable theory[J].Applied Mathematics and Mechanics,2013,34(4): 345-354.(in Chinese))
    [4] 祝江鸿, 杨建辉, 施高萍, 等. 单位圆外域到任意开挖断面隧洞外域共形映射的计算方法[J]. 岩土力学, 2014,35(1): 175-183.(ZHU Jianghong, YANG Jianhui, SHI Gaoping, et al. Calculating method for conformal mapping from exterior of unit circle to exterior of cavern with arbitrary excavation cross-section[J]. Rock and Soil Mechanics,2014,35(1): 175-183.(in Chinese))
    [5] 皇甫鹏鹏, 伍法权, 郭松峰. 基于边界点搜索的洞室外域映射函数求解法[J]. 岩石力学, 2011,32(5): 1418-1424.(HUNAGFU Pengpeng, WU Faquan, GUO Songfeng. A new method for calculating mapping function of external area of cavern with arbitrary shape based on searching points on boundary[J]. Rock and Soil Mechanics,2011,32(5): 1418-1424.(in Chinese))
    [6] 朱大勇, 钱七虎, 周早生. 复杂形状洞室映射函数的新解法[J]. 岩石力学与工程学报, 1999,18(3): 279-282.(ZHU Dayong, QIAN Qihu, ZHOU Zaosheng. New method for calculating mapping function of opening with complex shape[J].Chinese Journal of Rock Mechanics and Engineering,1999,18(3): 279-282.(in Chinese))
    [7] 王润富. 一种保角映射法及其微机实现[J]. 河海大学学报, 1991,19(1): 86-90.(WANG Runfu. A method of conformal mapping and its computer implementation[J]. Journal of Hohai University,1991,19(1): 86-90.(in Chinese))
    [8] 徐趁肖, 朱衡君, 齐红元. 复杂边界单连通域共形映射解析建模研究[J]. 工程数学学报, 2002,19(4): 135-138.(XU Chenxiao, ZHU Hengjun, QI Hongyuan. Analytically modeling of complicated boundary simply connected region conformal mapping[J]. Journal of Engineering Mathematics,2002,19(4): 135-138.(in Chinese))
    [9] 王志良, 申林方, 姚激. 浅埋隧道围岩应力场的计算复变函数求解法[J]. 岩土力学, 2010,31(1): 86-90.(WANG Zhiliang, SHEN Linfang, YAO Ji. Calculation of stress field in surrounding rocks of shallow tunnel using computational function of complex variable method[J]. Rock and Soil Mechanics,2010,31(1): 86-90.(in Chinese))
    [10] 王振武, 牛铮铮, 冯秀苓. 地下矩形洞室应力分布的复变函数解[J]. 北华航天工业学院学报, 2010,20(4): 86-90.(WANG Zhenwu, NIU Zhengzheng, FENG Xiuling. A semi-analytical elastic stress solution for perimeter stresses of rocks around a rectangular[J]. Journal of North China Institute of Aerospace Engineering,2010,20(4): 86-90.(in Chinese))
    [11] 李明, 茅献彪. 基于复变函数的矩形巷道围岩应力与变形粘弹性分析[J]. 力学季刊, 2011,32(2): 195-202.(LI Ming, MAO Xianbiao. Based on the complex variable functions of rectangular roadway surrounding rock stress and deformation viscoelastic analysis[J]. Chinese Quarterly of Mechanics,2011,32(2): 195-202.(in Chinese))
    [12] 袁林, 高召宁, 孟祥瑞. 基于复变函数法的矩形巷道应力集中系数黏弹性分析[J]. 煤矿安全, 2013,44(2): 196-200.(YUAN Lin, GAO Zhaoning, MENG Xiangrui. Viscoelastic analysis of stress concentration coefficient in rectangular roadway based on complex variable function[J]. Safety in Coal Mines,2013,44(2): 196-200.(in Chinese))
    [13] 施高萍, 祝江鸿, 李保海, 等. 矩形巷道孔边应力的弹性分析[J]. 岩土力学, 2014,35(9): 2587-2601.(SHI Gaoping, ZHU Jianghong, LI Baohai, et al. Elastic analysis of hole-edge stress of rectangular roadway[J]. Rock and Soil Mechanics,2014,35(9): 2587-2601.(in Chinese))
    [14] 陈凯, 唐治, 崔乃鑫, 等. 矩形巷道围岩应力解析解[J]. 安全与环境学报, 2015,15(3): 124-128.(CHEN Kai, TANG Zhi, CUI Naixin, at el. Analytical solution of rectangular roadway surrounding rock stress[J]. Journal of Safety and Environment,2015,15(3): 124-128.(in Chinese))
    [15] 何峰, 唐治, 朱小景, 等. 矩形巷道围岩应力分布特征[J]. 数学的实践与认识, 2015,45(20): 128-134.(HE Feng, TANG Zhi, ZHU Xiaojing, et al. Stress distribution characteristics of rectangular roadway surrounding rocks[J]. Mathematics in Practice and Theory,2015,45(20): 128-134.(in Chinese))
    [16] 赵凯, 刘长武, 张国良. 用弹性力学的复变函数法求解矩形硐室周边应力[J]. 采矿与安全工程学报, 2007,24(3): 361-365.(ZHAO Kai, LIU Changwu, ZHANG Guoliang. Solution for perimeter stresses of rocks around a rectangular chamber using the complex function of elastic mechanics[J]. Journal of Mining and Safety Engineering,2007,24(3): 361-365.(in Chinese))
    [17] HOWELL L H, TREFETHEN L N. A modified Schwarz-Christoffel transformation for elongated regions[J]. Society for Industrial and Applied Mathematics,1990,11(5): 928-949.
    [18] 拉夫连季耶夫 М А, 沙巴特 Б В. 复变函数论方法[M]. 6版. 施祥林, 夏定中, 吕乃刚, 译. 北京: 高等教育出版社, 2006.(ЛАВРЕНТЬЕВ М А, ШАБАТ Б В.Methods of the Theory of Complex Function [M]. 6th ed. SHI Xiangling, XIA Dingzhong, Lü Naigang, transl. Beijing: Higher Education Press, 2006.(Chinese version))
    [19] COSTAMAGNA E. A new approach to standard Schwarz-Christoffel formula calculations[J]. Microwave and Optical Technology Letters,2002,32(3): 196-199.
    [20] DRISCOLL T A. Algorithm 843: improvements to the Schwarz-Christoffel toolbox for MATLAB[J]. ACM Transactions on Mathematical Software,2005,31(2): 239-251.
    [21] 崔建斌, 姬安召, 鲁洪江, 等. Schwarz Christoffel变换数值解法[J]. 山东大学学报(理学版), 2016,51(4): 104-111.(CUI Jianbin, JI Anzhao, LU Hongjiang, et al. Numerical solution of Schwarz Christoffel transform[J]. Journal of Shandong University(Natural Science),2015,51(4): 104-111.(in Chinese))
    [22] 崔建斌, 姬安召, 王玉风, 等. 单位圆到任意多边形区域的Schwarz Christoffel变换数值解法[J]. 浙江大学学报(理学版),2017,44(2): 161-167.(CUI Jianbin, JI Anzhao, WANG Yufeng, et al. Numerical solution method for Schwarz-Christoffel transform from unit circle to arbitrary polygon area[J]. Journal of Zhejiang University(Science Edition),2017,44(2): 161-167.(in Chinese))
    [23] NATARAJAN S, BORDAS S, MAHAPATRA D R. Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping[J]. International Journal for Numerical Methods in Engineering,2009,80(1): 103-134.
    [24] CROWDY D. The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains[J]. Proceedings Mathematical Physical & Engineering Science,2005,146(2061): 2653-2678.
    [25] 刘浩. 大规模非线性方程组和无约束优化方法研究[D]. 博士学位论文. 南京: 南京航空航天大学, 2008.(LIU Hao. Research on methods for large-scale nonlinear equations and unconstrained optimization[D]. PhD Thesis. Nanjing: Nanjing University of Aeronautics and Astronautics, 2008.(in Chinese))
    [26] 姚征. 椭圆函数的精细积分改进算法[J]. 数值计算与计算机应用, 2008,29(4): 251-260.(YAO Zheng. The improved precise integration method for elliptic functions[J]. Journal on Numerical Methods and Computer Applications,2008,29(4): 251-260.(in Chinese))
    [27] ABRAMOWITZ M, STEGUN I A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables [M]. Washington DC: Dover Publications, 1996.
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出版历程
  • 收稿日期:  2018-01-29
  • 修回日期:  2018-04-23
  • 刊出日期:  2019-01-01

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