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On the curvature of an inner curve in a Schwarz--Christoffel mapping
Högskolan i Jönköping, Tekniska Högskolan, JTH, Matematik.
2009 (Engelska)Ingår i: Further Progress in Analysis: Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 2007, World Scientific , 2009, s. 281-290Konferensbidrag, Publicerat paper (Refereegranskat)
Abstract [en]

In the so called outer polygon method, an approximative conformal mapping for a given simply connected region Ω is constructed using a Schwarz–Christoffel mapping for an outer polygon, a polygonal region of which Ω is a subset. The resulting region is then bounded by a C-curve, which among other things means that its curvature is bounded.In this work, we study the curvature of an inner curve in a polygon, i.e., the image under the Schwarz–Christoffel mapping from R, the unit disk or upper half–plane, to a polygonal region P of a curve inside R. From the Schwarz–Christoffel formula, explicit expressions for the curvature are derived, and for boundary curves, appearing in the outer polygon method, estimations of boundaries for the curvature are given.

Ort, förlag, år, upplaga, sidor
World Scientific , 2009. s. 281-290
Nyckelord [en]
Curvature, Schwarz-Christtoffel mapping, inner curve, outer polygon
Nationell ämneskategori
Matematisk analys
Identifikatorer
URN: urn:nbn:se:hj:diva-8697ISBN: 978-981-283-732-5 (tryckt)ISBN: 981-283-732-9 (tryckt)OAI: oai:DiVA.org:hj-8697DiVA, id: diva2:216633
Tillgänglig från: 2009-05-11 Skapad: 2009-05-11 Senast uppdaterad: 2009-12-22Bibliografiskt granskad
Ingår i avhandling
1. Numerical conformal mappings for waveguides
Öppna denna publikation i ny flik eller fönster >>Numerical conformal mappings for waveguides
2009 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Växjö: Växjö University Press, 2009. s. 120
Serie
Acta Wexionesa, ISSN 1404-4307 ; 177
Nyckelord
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
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:hj:diva-10602 (URN)978-91-7636-661-5 (ISBN)
Disputation
2009-09-25, Weber, Växjö universitet, Växjö, 10:00 (Engelska)
Opponent
Handledare
Tillgänglig från: 2009-12-22 Skapad: 2009-10-10 Senast uppdaterad: 2009-12-22Bibliografiskt granskad

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Andersson, Anders

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