Computational Conformal Mapping

Kythe, Prem K.

doi:10.1007/978-1-4612-2002-2

Cited by 71 publications

(34 citation statements)

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“…There exist many methods for constructing numerical conformal mappings; [7,11], together with [18,16] list a wide collection. However, the required control of the direction of the boundary curve is not achieved by most of these methods.…”

Section: Introductionmentioning

confidence: 99%

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

Andersson

2008

Journal of Computational and Applied Mathematics

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A method where polygon corners in Schwarz-Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so-called tangent points. The Schwarz-Christoffel mapping for that polygon is then constructed and modified to round the corners.Since such a modification causes effects on the polygon outside the rounded corners, the parameters in the mapping have to be redetermined. This is done by comparing side-lengths in tangent polygons to the given curve and the curve produced by the modified Schwarz-Christoffel mapping. The set of equations that this comparison gives, can normally be solved using a quasi-Newton method.The resulting function maps the upper half-plane on a region bounded by a curve that apart from possible vertices is smooth, i.e., one time continuously differentiable, that passes through the tangent points on the given curve, has the same direction as the given curve in these points and changes direction monotonically between them. Furthermore, where the original curve has a vertex, the constructed curve has a vertex with the same inner angle.The method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines, such as channels with parallel walls at the ends. These properties are kept in the region produced by the constructed mapping.

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Section: Introductionmentioning

confidence: 99%

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

Andersson

2008

Journal of Computational and Applied Mathematics

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“…However, exact mapping functions are not known except for some special domains, and the numerical conformal mapping has been an attractive subject in scientific computation [13,16,21,29,48].…”

Section: Introductionmentioning

confidence: 99%

Numerical conformal mappings onto the linear slit domain

Amano

Okano

Ogata

et al. 2012

Japan J. Indust. Appl. Math.

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We propose a numerical method for the conformal mapping of unbounded multiply connected domains exterior to closed Jordan curves C 1 , . . . , C n onto a canonical linear slit domain, which is the entire plane with linear slits S 1 , . . . , S n of angles θ 1 , . . . , θ n arbitrarily assigned to the real axis, respectively. If θ 1 = · · · = θ n = θ then it is the well-known parallel slit domain, which is important in the problem of potential flows past obstacles. In the method, we reduce the mapping problem to a boundary value problem for an analytic function, and approximate it by a linear combination of complex logarithmic functions based on the charge simulation method. Numerical examples show the effectiveness of our method.

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“…Therefore computational method of conformal mappings, that is, numerical conformal mapping, has been an attractive subject in numerical analysis. See Henrici [7], Kythe [9], Nehari [10] and Trefethen [11] for surveys of numerical conformal mappings.…”

Section: Introductionmentioning

confidence: 99%

Numerical conformai mapping of periodic structure domains

Ogata

Okano

Amano

2002

Japan J. Indust. Appl. Math.

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We propose a method of numerical conformal mapping of periodic structure domains onto periodic parallel slit domains. In the method presented here, the mapping problem is reduced to a Dirichlet one for a pair of conjugate harmonic functions with a periodic boundary condition. We modify the charge simulation method for solving periodic boundary value problem and apply it to our mapping problem. Some numerical examples show that our method is efficient. We also show an application of our method to the analysis of potential flow past obstacles in a periodic array.

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Computational Conformal Mapping

Cited by 71 publications

References 0 publications

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

Numerical conformal mappings onto the linear slit domain

Numerical conformai mapping of periodic structure domains

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