Iterative algorithm for the conformal mapping from the unit disk to domains with regular boundaries

Abstract

Conformal mapping functions have significant applications in mechanics and other fields, and their computation methods have drawn considerable attention. We propose an iterative algorithm to compute the conformal mapping from the unit disk to physical domains with regular boundaries, defined by having only prime ends of the first kind. The mapping function is expanded into a Laurent series and use its truncated partial sum as an approximation. The Schwarz–Christoffel mapping formula provides the initial estimates for the series coefficients, which are then iteratively optimized. This algorithm efficiently handles complex domain shapes, such as winding orifices and slits, with high computational speed. Moreover, it offers valuable insights for designing algorithms to solve other types of conformal mapping problems and has practical significance in applications involving conformal mappings.