Numerical Conformal Mapping of Unbounded Multiply Connected Regions onto Circular Slit Regions

Yunus, Arif A. M.; Murid, Ali Hassan Mohamed; Nasser, Mohamed M. S.

doi:10.11113/mjfas.v8n1.120

Cited by 3 publications

(3 citation statements)

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“…Nasser [84][85][86][87][88][89], Delillo and Kropf [90], Al-Hatemi et al [88], and Yunus et al [91] developed the corresponding numerical methods for solving these established integration equations. More numerical methods for conformal mapping are presented by Luo et al [92], Liesen et al [93], Gopal and Trefethen [94], and Trefethen [95].…”

Section: Proposed Conformal Mapping Procedures and Application Casesmentioning

confidence: 99%

Solving Conformal Mapping Issues in Tunnel Engineering

Chen,

Zhang,

Fang

et al. 2024

Symmetry

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The calculation of conformal mapping for irregular domains is a crucial step in deriving analytical and semi-analytical solutions for irregularly shaped tunnels in rock masses using complex theory. The optimization methods, iteration methods, and the extended Melentiev’s method have been developed and adopted to calculate the conformal mapping function in tunnel engineering. According to the strict definition and theorems of conformal mapping, it is proven that these three methods only map boundaries and do not guarantee the mapping’s conformal properties due to inherent limitations. Notably, there are other challenges in applying conformal mapping to tunnel engineering. To tackle these issues, a practical procedure is proposed for the conformal mapping of common tunnels in rock masses. The procedure is based on the extended SC transformation formulas and corresponding numerical methods. The discretization codes for polygonal, multi-arc, smooth curve, and mixed boundaries are programmed and embedded into the procedure, catering to both simply and multiply connected domains. Six cases of conformal mapping for typical tunnel cross sections, including rectangular tunnels, multi-arc tunnels, horseshoe-shaped tunnels, and symmetric and asymmetric multiple tunnels at depth, are performed and illustrated. Furthermore, this article also illustrates the use of the conformal mapping method for shallow tunnels, which aligns with the symmetry principle of conformal mapping. Finally, the discussion highlights the use of an explicit power function as an approximation method for symmetric tunnels, outlining its key points.

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Section: Proposed Conformal Mapping Procedures and Application Casesmentioning

confidence: 99%

Solving Conformal Mapping Issues in Tunnel Engineering

Chen,

Zhang,

Fang

et al. 2024

Symmetry

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“…The right-hand side of the integral equation involves integral with cotangent singularity which is approximated by Wittich's method. The integral equation was discretized by the Nyström method with the trapezoidal rule to obtain a linear system [7][8][9][10][11][12][13]. The method presented in [6][7][8][14][15][16] is based also on the approximate solution of the integral equation.…”

Section: Introductionmentioning

confidence: 99%

The Approximate Conformal Mapping of a Disk onto Domain with an Acute Angle

Ivanshin

Shirokova

2023

Int. J. Appl. Comput. Math

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The method of boundary curve reparametrization is applied to construction of the approximate analytical conformal mapping of the unit disk onto an arbitrary given finite domain with a boundary smooth at every point but fininte number of acute angle points. The method is based on both the Fredholm equation solution and spline-interpolation. This approach consists of approximate solution of a linear system with unknown Fourier coefficients and construction of correction splines. The approximate mapping function has the form of a Cauchy integral. The method presentation is supported by demonstration of some examples. This method is applicable to the case of multiply connected domains with boundary angle points.

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“…Conformal mapping of multiply connected regions can be computed efficiently using the integral equation method. The integral equation method has been used by many authors to compute the one-to-one conformal mapping from multiply connected regions onto some standard canonical regions [3][4][5][6][7][8][9][10][11][12][13][14][15][16] . In [10] the authors presented a fast multipole method, which is fast and accurate method for numerical conformal mapping of bounded and unbounded multiply connected regions with high connectivity and highly complex geometry.…”

Section: 10 Introductionmentioning

confidence: 99%

Integral Equation for the Ahlfors Map on Multiply Connected Regions

Nazar¹,

Murid²,

Sangawi³

2015

Jurnal Teknologi

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This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with where is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of we solve for the boundary correspondence function which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

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Numerical Conformal Mapping of Unbounded Multiply Connected Regions onto Circular Slit Regions

Cited by 3 publications

References 0 publications

Solving Conformal Mapping Issues in Tunnel Engineering

Solving Conformal Mapping Issues in Tunnel Engineering

The Approximate Conformal Mapping of a Disk onto Domain with an Acute Angle

Integral Equation for the Ahlfors Map on Multiply Connected Regions

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