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Numerical Conformal Mappings for WaveguidesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2010 (English)In: Computational Mathematics: Theory, Methods and Applications, Hauppauge NY, USA: Nova Science Publishers , 2010Chapter in book (Other (popular science, discussion, etc.))
##### Abstract [en]

##### Place, publisher, year, edition, pages

Hauppauge NY, USA: Nova Science Publishers , 2010.
##### Series

Computational Mathematics and Analysis Series
##### Keywords [en]

Conformal mapping, Schwarz-Christoffel mapping, Rounded corners, Outer polygon method, Approximate curve factor, Zipper algorithm, Waveguide
##### National Category

Computational Mathematics
##### Identifiers

URN: urn:nbn:se:hj:diva-10510ISBN: 978-1-60876-271-2 (print)OAI: oai:DiVA.org:hj-10510DiVA, id: diva2:241705
#####

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##### In thesis

Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can, if the variations takes place in only one dimension at a time be re-formulated as a two-dimensional scattering problem. 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. We consider in this work acoustic scattering, but the same techniques can be used for both electro-magnetic and some quantum scattering problems. By suppressing the time dependence and by using the Building Block Method, the problem takes the form of the Helmholtz equation 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 channel, 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 properties of the boundary. 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 to round the polygon corners, and we show algorithms by which the parameter problem can be solved after such modifications. It is also possible to use the unmodified Schwarz-Christoffel mapping for regions with smooth boundary, by constructing an appropriate outer polygon to the considered region. Finally, we show how a so-called zipper algorithm can be used for waveguides.

1. Numerical conformal mappings for waveguides$(function(){PrimeFaces.cw("OverlayPanel","overlay263263",{id:"formSmash:j_idt720:0:j_idt724",widgetVar:"overlay263263",target:"formSmash:j_idt720:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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