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The conformal mapping from an infinite domain bounded by two circular curves onto the infinite domain bounded by two arbitrary closed Jordan curves.

The conformal mapping from an infinite domain bounded by two circular curves onto the infinite domain bounded by two arbitrary closed Jordan curves.

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Figure 1. The conformal mapping from the unit disc onto the infinite...
Figure 2. The conformal mapping from a circular annulus domain onto the...
Figure 3. The conformal mapping from an infinite domain bounded by two...
Figure 4. The mapping points and the curves plotted by different terms.
+3 Figure 6. The conformal mapping from the circular domain onto the...
Numerical Applications of Conformal Mapping with Explicit Formulae Based on the Power Series Method
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  • Feb 2022
A general optimization method, based on the power series method, is presented for computing the conformal mappings with explicit expressions from: (a) the unit disc onto an infinite domain exterior of a closed Jordan curve, (b) the circular annulus domain onto a finite doubly-connected domain bounded by two closed Jordan curves, (c) the infinite do...
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... can be seen in figure 3, and control points are selected anticlockwise to describe the boundaries and , respectively. P fixed at the x-axis corresponds to P fixed at the ξ-axis. ...

Citations

... The problem of finding the mapping function is reduced to solving its coefficients because the form of the mapping function is known. In this paper, the coefficients in z 1 = ω(ζ 1 ) and z 2 = ω(ζ 2 ) are solved using the method described in the literature (Ma et,al. 2022). ...
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Complex variable solution on noncircular and asymmetrical tunnelling embedded by bidirectional conformal mapping incorporating Charge Simulation Method
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  • Jun 2024
Mechanical issues of noncircular and asymmetrical tunnelling can be estimated using complex variable method with suitable conformal mapping. Exsiting solution schemes of conformal mapping for noncircular tunnel generally need iteration or optimization strategy, and are thereby mathematically complicated. This paper proposes a new bidirectional conformal mapping for deep and shallow tunnels of noncircular and asymmetrical shapes by incorporating Charge Simulation Method. The solution scheme of this new bidirectional conformal mapping only involves a pair of linear systems, and is therefore logically straight-forward, computationally efficient, and practically easy in coding. New numerical strategies are developed to deal with possible sharp corners of cavity by small arc simulation and densified collocation points. Several numerical examples are presented to illustrate the geometrical usage of the new bidirectional conformal mapping. Furthermore, the new bidirectional conformal mapping is embedded into two complex variable solutions of noncircular and asymmetrical shallow tunnelling in gravitational geomaterial with reasonable far-field displacement. The respective result comparisons with finite element solution and exsiting analytical solution show good agreements, indicating the feasible mechanical usage of the new bidirectional conformal mapping.
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