Iterative algorithm for the conformal mapping function from the exterior of a roadway to the interior of a unit circle

• Published in: Archive of applied mechanics (1991) Vol. 92; no. 3; pp. 971 - 991
• Main Authors: He, Kai, Chang, Jucai, Pang, Dongdong, Sun, Bingjun, Yin, Zhiqiang, Li, Dong
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer Nature B.V
• Subjects: Algorithms; Classical Mechanics; Coal mines; Conformal mapping; Convergence; Cross-sections; Engineering; Iterative algorithms; Least squares method; Mathematical analysis; Original; Roads & highways; Series (mathematics); Statistical analysis; Stress distribution; Theoretical and Applied Mechanics; Least-squares method; Irregular cross-section; Conformal mapping function; Complex function method; Laurent series
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-021-02087-w

In elastic mechanics, the complex function method proposed by Muskhelishvili can be used to solve the stress distribution of rock surrounding a roadway with an irregular cross-section. However, the solution of the conformal mapping function from the exterior of the roadway to the interior of the unit circle is a prerequisite, but it is difficult to obtain. In this study, based on the Riemann mapping theorem and the boundary correspondence principle, the conformal mapping function was approximated using a Laurent series with finite terms. Assuming that the polar angle of the two corresponding points on the image of the conformal mapping function and the boundary of the roadway cross-section are equal, an iterative algorithm for calculating the conformal mapping function is proposed using the least squares method, and the code was programmed by-using Python. Using the proposed algorithm, two conformal mapping functions were solved for roadways with irregular cross-sections in practical engineering projects, while the parameters and the error were examined. Simultaneously, the performance was analyzed statistically for curved and broken line boundaries. The analysis results has shown that the errors were concentrated on the corners of the roadway. In addition, it was observed that an excessively large number of sample points would not improve the accuracy, but will only extend the algorithm convergence time. When the series includes more terms, the accuracy will be higher and the convergence speed will be slower. However, statistical analysis has shown that the algorithm was converged rapidly, with convergence time of less than 10 ms. Finally, the effectiveness of the algorithm was verified by solving the stress distributions of the rock surrounding two roadways. The algorithm might be used to solve the conformal mapping function from the exterior of a roadway with an irregular cross-section to the interior of the unit circle in coal mines.