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Schwarz-Christoffel Mappings for Nonpolygonal Regions
Jönköping University, School of Engineering, JTH, Mathematics.
2008 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 31, no 1, p. 94-111Article in journal (Refereed) Published
Abstract [en]

An approximate conformal mapping for an arbitrary region Ω bounded by a smooth curve Γ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which Ω is embedded. An algorithm for finding this so-called outer polygon is presented. The resulting function is a conformal mapping from the upper half-plane or the unit disk to a region R, approximately equal to Ω. R is bounded by a C∞ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of Γ such as direction, linear asymptotes, and inflexion points are preserved in the boundary of R. The method makes extensive use of existing Schwarz–Christoffel software in both the determination of outer polygons and the calculation of function values. By the use suggested here, the capabilities of such well-written software are extended.

Place, publisher, year, edition, pages
Philadelphia: Society for Industrial and Applied Mathematics , 2008. Vol. 31, no 1, p. 94-111
Keywords [en]
numerical conformal mapping, Schwarz-Christoffel mapping, tangent polygon, inner region, outer polygon
National Category
Computational Mathematics Computational Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-6627DOI: 10.1137/070701297OAI: oai:DiVA.org:hj-6627DiVA, id: diva2:113731
Available from: 2008-10-26 Created: 2008-10-26 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Numerical conformal mappings for waveguides
Open this publication in new window or tab >>Numerical conformal mappings for waveguides
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can be re-formulated as a two-dimensional scattering problem, provided that the variations take place in only one dimension at a time. By using the so-called Building Block Method, it is possible to construct the scattering properties of a combination of scatterers when the properties of each scatterer are known. Hence, variations in the waveguide geometry or in the boundary conditions can be treated one at a time.

Using the Building Block Method, the problem takes the form of the Helmholtz equation for stationary waves in a waveguide of infinite length and with smoothly varying geometry and boundary conditions. A conformal mapping is used to transform the problem into a corresponding problem in a straight horizontal waveguide, and by expanding the field in Fourier trigonometric series, the problem can be reformulated as an infinite-dimensional ordinary differential equation. From this, numerically solvable differential equations for the reflection and transmission operators are derived.

To be applicable in the Building Block Method, the numerical conformal mapping must be constructed such that the direction of the boundary curve can be controlled. At the channel ends, it is an indispensable requirement, that the two boundary curves are (at least) asymptotically parallel and straight. Furthermore, to achieve bounded operators in the differential equations, the boundary curves must satisfy different regularity conditions, depending on the boundary conditions.

In this work, several methods to accomplish such conformal mappings are presented. The Schwarz–Christoffel mapping, which is a natural starting point and for which also efficient numerical software exists, can be modified in different ways in order to achieve polygons with rounded corners. We present algorithms by which the parameters in the mappings can be determined after such modifications. We show also how the unmodified Schwarz–Christoffel mapping can be used for regions with a smooth boundary. This is done by constructing an appropriate outer polygon to the considered region.

Finally, we introduce one method that is not Schwarz–Christoffel-related, by showing how one of the so-called zipper algorithms can be used for waveguides.

Place, publisher, year, edition, pages
Växjö: Växjö University Press, 2009. p. 120
Series
Acta Wexionesa, ISSN 1404-4307 ; 177
Keywords
waveguides, building block method, numerical conformal mappings, Schwarz–Christoffel mapping, rounded corners method, approximate curve factors, outer polygon method, boundary curvature, zipper method, geodesic algorithm, acoustic wave scattering, electro-magnetic wave scattering
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-10602 (URN)978-91-7636-661-5 (ISBN)
Public defence
2009-09-25, Weber, Växjö universitet, Växjö, 10:00 (English)
Opponent
Supervisors
Available from: 2009-12-22 Created: 2009-10-10 Last updated: 2009-12-22Bibliographically approved

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Publisher's full texthttp://dx.doi.org/10.1137/070701297

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Andersson, Anders

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