| 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 |
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| 引用本文： | 祝江鸿,杨建辉,施高萍,王 珺,蔡建平.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. |
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| 作者姓名： | 祝江鸿 杨建辉 施高萍 王 珺 蔡建平 |
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| 作者单位： | 1. 浙江科技学院 建工学院，杭州 310023；2. 浙江水利水电专科学校，杭州 310018； 3. 北京科技大学 土木与环境工程学院，北京 100083 |
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| 基金项目： | 浙江省科技厅面上基金资助项目(No. 2008C23009)。 |
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| 摘 要： | 将复杂开挖断面为边界的力学问题转化成单位圆为边界的力学问题,是利用复变函数理论开展地下隧洞围岩力学分析的前提。从黎曼存在定理出发,建立了以洛朗级数有限项表示的单位圆外域到任意开挖断面隧洞外域的共形映射函数。根据边界对应原理,将域到域的共形映射问题转化成单位圆周线到任意开挖断面隧洞边界线的共形映射问题。从三角插值理论出发,采用奇偶插值点反复相互迭代对映射函数的求解开展了研究。为快速地进行迭代计算,采用法线逼近法进行计算点(映射点)调整。提出了以映射边界线与实际开挖洞形边界线之间的绝对误差作为迭代计算的收敛条件,保证了共形映射精度。给出了3个开挖断面隧洞映射函数的求解算例,并将研究结果与文献中的方法进行了比较。结果表明,提出的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具有操作简单、精度高和收敛快等特点。
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| 关 键 词： | 任意开挖断面 共形映射 三角插值 法线逼近法 收敛条件 |
| 收稿时间： | 2013-04-06 |
Calculating method for conformal mapping from exterior of unit circle to exterior of cavern with arbitrary excavation cross-section |
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| ZHU Jiang-hong,YANG Jian-hui,SHI Gao-ping,WANG Jun,CAI Jian-ping.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. |
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| Authors: | ZHU Jiang-hong YANG Jian-hui SHI Gao-ping WANG Jun CAI Jian-ping |
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| Institution: | 1. School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou 310023, China; 2. Zhejiang Water Conservancy and Hydropower College, Hangzhou 310018, China; 3. School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China |
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| Abstract: | It is the essential prerequisite for mechanical analysis of the surrounding rock around underground cavern with the complex variable theory that the mechanical boundary must be transformed from the actual excavation cross-section to the unit circle. According to the Riemann’s existence theorem, the mapping function is established with the finite Laurent series, by which the exterior of unit circle can be conformally mapped to the exterior of carven with the arbitrary excavation cross-section. Based on the boundary correspondence principle, the conformal mapping of source and target domains is transformed into that of the boundary lines of unit circle and cavern. Research on solving the mapping function is carried out with the triangle interpolation theory, by which even and odd interpolation points are repeatedly iterated each other. In order to accelerate the iterative calculation, the calculating points are adjusted by the normal approximation method. Precision of the conformal mapping is ensured when the absolute error between the mapping boundary and the actrual excavation boundary is set as the convergence condition. Moreover, examples of the calculating mapping functions are provided for caverns with three excavation cross-sections. Compared to other methods in the literatures, the conformal mapping of the exteriors of unit circle and cavern with the arbitrary excavation cross-section is achieved more simply, accurately and rapidly by the method in this study. |
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| Keywords: | arbitrary excavation cross-section conformal mapping triangle interpolation normal approximation method convergence condition |
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