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Numerical conformal mappings for waveguides
Jönköping University, School of Engineering, JTH, Mathematics.
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 [en]
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: urn:nbn:se:hj:diva-10602ISBN: 978-91-7636-661-5 (print)OAI: oai:DiVA.org:hj-10602DiVA, id: diva2:263263
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
List of papers
1. Numerical Conformal Mappings for Waveguides
Open this publication in new window or tab >>Numerical Conformal Mappings for Waveguides
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]

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.

Place, publisher, year, edition, pages
Hauppauge NY, USA: Nova Science Publishers, 2010
Series
Computational Mathematics and Analysis Series
Keywords
Conformal mapping, Schwarz-Christoffel mapping, Rounded corners, Outer polygon method, Approximate curve factor, Zipper algorithm, Waveguide
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-10510 (URN)978-1-60876-271-2 (ISBN)
Available from: 2009-10-05 Created: 2009-10-05 Last updated: 2009-12-22Bibliographically approved
2. Electro-Magnetic Scattering in Variously Shaped Waveguides with an Impedance Condition
Open this publication in new window or tab >>Electro-Magnetic Scattering in Variously Shaped Waveguides with an Impedance Condition
2009 (English)In: AIP Conference Proceedings: Third Conference on Mathematical Modeling of Wave Phenomena: Växjö, Sweden, 9-13 June, 2008, American Institute of Physics , 2009, p. 36-45Conference paper, Published paper (Refereed)
Abstract [en]

Electro-magnetic scattering is studied in a waveguide with varying shape and crosssection. Furthermore, an impedance or admittance condition is applied to two of the waveguide walls. Under the condition that variations in geometry or impedance take place in only one plane at the time, the problem can be solved as a two-dimensional wave-scattering problems. By using newly developed numerical conformal mapping techniques, the problem is transformedinto a wave-scattering problem in a straight two-dimensional channel. A numerically stable formulation is reached in terms of transmission and reflection operators. Numerical results are given for a slowly varying waveguide with a bend and for one more complex geometry.

Place, publisher, year, edition, pages
American Institute of Physics, 2009
Series
AIP Conference Proceedings, ISSN 0094-243X ; 1106
Keywords
Electro-magnetic scattering, waveguide scattering, impedance, Building Block Method, numerical conformal mappings, Outer Polygon Method, invariant embedding
National Category
Other Physics Topics
Identifiers
urn:nbn:se:hj:diva-8698 (URN)978-0-7354-0643-8 (ISBN)
Available from: 2009-05-11 Created: 2009-05-11 Last updated: 2009-12-22Bibliographically approved
3. A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries
Open this publication in new window or tab >>A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries
2008 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 213, no 1, p. 56-70Article in journal (Refereed) 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.

Keywords
Conformal mapping, Schwarz–Christoffel mapping, Rounding corners, Tangent polygon, Parameter problem
National Category
Computational Mathematics Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-2696 (URN)doi:10.1016/j.cam.2006.12.025 (DOI)
Available from: 2008-01-05 Created: 2008-01-05 Last updated: 2017-12-12Bibliographically approved
4. Modified Schwarz-Christoffel mappings using approximate curve factors
Open this publication in new window or tab >>Modified Schwarz-Christoffel mappings using approximate curve factors
2009 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 233, no 4, p. 1117-1127Article in journal (Refereed) Published
Abstract [en]

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.

In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C1 regularity, C regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.

Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C) boundary curve can be achieved.

Place, publisher, year, edition, pages
Elsevier, 2009
Keywords
Conformal mapping, Schwarz-Christoffel mapping, Approximate curve factor
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-10513 (URN)10.1016/j.cam.2009.09.006 (DOI)
Available from: 2009-10-05 Created: 2009-10-05 Last updated: 2017-12-13Bibliographically approved
5. Schwarz-Christoffel Mappings for Nonpolygonal Regions
Open this publication in new window or tab >>Schwarz-Christoffel Mappings for Nonpolygonal Regions
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
Keywords
numerical conformal mapping, Schwarz-Christoffel mapping, tangent polygon, inner region, outer polygon
National Category
Computational Mathematics Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-6627 (URN)10.1137/070701297 (DOI)
Available from: 2008-10-26 Created: 2008-10-26 Last updated: 2017-12-14Bibliographically approved
6. On the curvature of an inner curve in a Schwarz--Christoffel mapping
Open this publication in new window or tab >>On the curvature of an inner curve in a Schwarz--Christoffel mapping
2009 (English)In: Further Progress in Analysis: Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 2007, World Scientific , 2009, p. 281-290Conference paper, Published paper (Refereed)
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.

Place, publisher, year, edition, pages
World Scientific, 2009
Keywords
Curvature, Schwarz-Christtoffel mapping, inner curve, outer polygon
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:hj:diva-8697 (URN)978-981-283-732-5 (ISBN)981-283-732-9 (ISBN)
Available from: 2009-05-11 Created: 2009-05-11 Last updated: 2009-12-22Bibliographically approved
7. Using a zipper algorithm to find a conformal map for a channel with smooth boundary
Open this publication in new window or tab >>Using a zipper algorithm to find a conformal map for a channel with smooth boundary
2006 (English)In: Mathematical Modeling of Wave Phenomena: 2nd Conference, 2006, p. 378-Conference paper, Published paper (Refereed)
Abstract [en]

The so called geodesic algorithm, which is one of the zipper algorithms for conformal mappings, is combined with a Schwarz–Christoffel mapping, in its original or in a modified form, to produce a conformal mapping function between the upper half-plane and an arbitrary channel with smooth boundary and parallel walls at the end.

Keywords
conformal mapping, Schwarz–Christoffel mapping, smooth boundary curve, zipper, geodesic algorithm
National Category
Computational Mathematics Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-2675 (URN)doi:10.1063/1.2205784 (DOI)0-7354-0325-2 (ISBN)
Available from: 2007-08-02 Created: 2007-08-02 Last updated: 2010-01-12Bibliographically approved

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