CAJ | 학술논문

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。
장복잡개알단면위변계적역학문제전화성단위원위변계적역학문제，시이용복변함수이론개전지하수동위암역학분석적전제。종려만존재정리출발，건립료이락랑급수유한항표시적단위원외역도임의개알단면수동외역적공형영사함수。근거변계대응원리，장역도역적공형영사문제전화성단위원주선도임의개알단면수동변계선적공형영사문제。종삼각삽치이론출발，채용기우삽치점반복상호질대대영사함수적구해개전료연구。위쾌속지진행질대계산，채용법선핍근법진행계산점（영사점）조정。제출료이영사변계선여실제개알동형변계선지간적절대오차작위질대계산적수렴조건，보증료공형영사정도。급출료3개개알단면수동영사함수적구해산례，병장연구결과여문헌중적방법진행료비교。결과표명，제출적단위원외역도임의개알단면수동외역공형영사적계산방법구유조작간단、정도고화수렴쾌등특점。
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.