Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions

Nasser, Mohamed M. S.

doi:10.1016/j.jmaa.2012.09.020

Cited by 28 publications

(27 citation statements)

References 26 publications

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“…Some approaches to derive the mapping function have already been proposed for some multiply connected regions32333435. Here, we describe a general method to determine the appropriate mapped structure for several typical situations of multiply connected region with several terminals.…”

Section: Mapping Methodsmentioning

confidence: 99%

Conformal mapping for multiple terminals

Wang

et al. 2016

Sci Rep

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Conformal mapping is an important mathematical tool that can be used to solve various physical and engineering problems in many fields, including electrostatics, fluid mechanics, classical mechanics, and transformation optics. It is an accurate and convenient way to solve problems involving two terminals. However, when faced with problems involving three or more terminals, which are more common in practical applications, existing conformal mapping methods apply assumptions or approximations. A general exact method does not exist for a structure with an arbitrary number of terminals. This study presents a conformal mapping method for multiple terminals. Through an accurate analysis of boundary conditions, additional terminals or boundaries are folded into the inner part of a mapped region. The method is applied to several typical situations, and the calculation process is described for two examples of an electrostatic actuator with three electrodes and of a light beam splitter with three ports. Compared with previously reported results, the solutions for the two examples based on our method are more precise and general. The proposed method is helpful in promoting the application of conformal mapping in analysis of practical problems.

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Section: Mapping Methodsmentioning

confidence: 99%

Conformal mapping for multiple terminals

Wang

et al. 2016

Sci Rep

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“…We next show how to compute from the knowledge of and . Note that the Ahlfors map is related to the Szegö kernel and the Garabedian kernel by [18] The Szegö kernel and Garabedian kernel are related on as so that (4) (6) and (9), we can find the derivative of Szegö kernel from (8). We next show how to find .…”

Section: Immentioning

confidence: 99%

“…For solving the integral equation (6) numerically, the reliable procedure is by using the Nyström method with the trapezoidal rule with n equidistant nodes in each interval , [5,6,7,8,9]. The trapezoidal rule is the most accurate method for integrating periodic functions numerically [25, pp.134-142].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Using the integral equation method, most of the conformal mapping of multiply connected regions can be computed efficiently. 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 [1, 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. In [9], Nasser et al 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: Introductionmentioning

confidence: 99%

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The computation of zeros of Ahlfors map for doubly connected regions

Nazar

Murid

Sangawi³

2016

AIP Conference Proceedings

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Abstract. The Ahlfors map and Szeg kernel are both classically related to each other. Ahlfors map can be computed using Szegö kernel without relying on the zeros of Ahlfors map. The Szegö kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. The exact zeros of the Ahlfors map are unknown except for the annulus region. This paper presents a numerical method for computing the zeros of the Ahlfors map of any bounded doubly connected region. The method depends on the values of Szegö kernel, its derivative and the derivative of boundary correspondence function of Ahlfors map. Using combination of Nyström method, GMRES method, fast multiple method and Newton's method, the numerical examples presented here prove the effectiveness of the proposed method.

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“…For some other approaches of conformal mappings of multiply connected regions, e.g. [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Various applications of conformal mappings in science and engineering are considered in, e.g., [22,[32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions

2014

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This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy's-type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method.

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Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions

Cited by 28 publications

References 26 publications

Conformal mapping for multiple terminals

Conformal mapping for multiple terminals

The computation of zeros of Ahlfors map for doubly connected regions

Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions

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