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A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries
Högskolan i Jönköping, Tekniska Högskolan, JTH, Matematik.
2008 (Engelska)Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 213, nr 1, s. 56-70Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

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 re-determined. 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.

Ort, förlag, år, upplaga, sidor
2008. Vol. 213, nr 1, s. 56-70
Nyckelord [en]
Conformal mapping, Schwarz–Christoffel mapping, Rounding corners, Tangent polygon, Parameter problem
Nationell ämneskategori
Beräkningsmatematik Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:hj:diva-2696DOI: doi:10.1016/j.cam.2006.12.025OAI: oai:DiVA.org:hj-2696DiVA, id: diva2:33516
Tillgänglig från: 2008-01-05 Skapad: 2008-01-05 Senast uppdaterad: 2017-12-12Bibliografiskt 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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