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Andersson, Andersorcid.org/0000-0001-9200-7143

Open this publication in new window or tab >>Fourier methods for harmonic scalar waves in general waveguides### Andersson, Anders

### Nilsson, Börje

### Biro, Thomas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Journal of Engineering Mathematics, ISSN 0022-0833, E-ISSN 1573-2703, Vol. 98, no 1, p. 21-38Article in journal (Refereed) Published
##### Abstract [en]

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

Springer, 2016
##### Keywords

Building block method, Conformal mappings, Dirichlet-to-Neumann operators, Fourier series, Harmonic scalar waves, Normal surface admittance
##### National Category

Other Mathematics
##### Identifiers

urn:nbn:se:hj:diva-27685 (URN)10.1007/s10665-015-9808-8 (DOI)000376643300003 ()2-s2.0-84938152325 (Scopus ID)
#####

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Available from: 2015-08-11 Created: 2015-08-11 Last updated: 2018-09-21Bibliographically approved

Jönköping University, School of Engineering, JTH, Physics and Mathematics and Chemical Engineering.

Linnaeus university, Vaxjö, Sweden.

Jönköping University, School of Engineering, JTH, Physics and Mathematics and Chemical Engineering.

A set of semi-analytic techniques based on Fourier analysis is used to solve wave-scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are the newly developed conformal mapping methods, wave splitting, Fourier series expansions in eigenfunctions to non-normal operators, the building block method or the cascade technique, Dirichlet-to-Neumann operators, and reformulation in terms of stable differential equations for reflection and transmission matrices. For an example, the results show good correspondence with a finite element method solution to the same problem in the low- and medium-frequency domains. The Fourier method complements finite element analysis as a waveguide simulation tool. For inverse engineering involving tuning of straight waveguide parts joining complicated waveguide elements, the Fourier method is an attractive alternative including time aspects. The prime motivation for the Fourier method is its added physical understanding primarily at low frequencies.

Open this publication in new window or tab >>Numerical Conformal Mappings for Waveguides### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 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

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)
#####

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#####

Available from: 2009-10-05 Created: 2009-10-05 Last updated: 2009-12-22Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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.

Open this publication in new window or tab >>Acoustic waves in a mean flow duct with varying boundary### Nilsson, Börje

### Augey, Romain

### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2009 (English)In: 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), Reston, Va.: < American Institute of Aeronautics and Astronautics , 2009Conference paper, Published paper (Refereed)
##### Abstract [en]

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

Reston, Va.: < American Institute of Aeronautics and Astronautics, 2009
##### National Category

Fluid Mechanics and Acoustics
##### Identifiers

urn:nbn:se:hj:diva-10719 (URN)1-56347-974-5 (ISBN)978-1-56347-974-8 (ISBN)
#####

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#####

Available from: 2009-10-21 Created: 2009-10-21 Last updated: 2010-07-30Bibliographically approved

International Centre of Mathematical Modelling, Växjö University, Sweden.

National Institute of Advanced Technology (ENSTA, Paris), France.

Jönköping University, School of Engineering, JTH, Mathematics.

The problem of modelling sound waves in a two-dimensional wave-guide of general shape carrying a mean flow is addressed. The mean flowmay be inhomogeneous but is irrotational. A convective wave equation forthe velocity potential is derived. It is in a form suitable for generalizingan earlier developed theory for a stable modelling of acoustic waves inquiescent waveguides with complicated geometry to also include a meanflow. The theory is illustrated with numerical results for reflection andtransmission demonstrating the effectiveness of the method for low andmedium frequencies.

Open this publication in new window or tab >>Electro-Magnetic Scattering in Variously Shaped Waveguides with an Impedance Condition### Andersson, Anders

### Nilsson, Börje

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2009-05-11 Created: 2009-05-11 Last updated: 2009-12-22Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

International Centre for Mathematical Modelling, Växjö University.

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.

Open this publication in new window or tab >>Modified Schwarz-Christoffel mappings using approximate curve factors### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2009-10-05 Created: 2009-10-05 Last updated: 2017-12-13Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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 *C*^{1} 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.

Open this publication in new window or tab >>Numerical conformal mappings for waveguides### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2009 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

### Benedicks, Michael

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##### Supervisors

### Nilsson, Börje

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Available from: 2009-12-22 Created: 2009-10-10 Last updated: 2009-12-22Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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.

Mathematics department, Royal Institute of Technology, Stockholm.

School of Mathematics and Systems Engineering, Växjö Univerity, Växjö.

Open this publication in new window or tab >>On the curvature of an inner curve in a Schwarz--Christoffel mapping### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2009-05-11 Created: 2009-05-11 Last updated: 2009-12-22Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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.

Open this publication in new window or tab >>A modiﬁed Schwarz-Christoffel mapping for regions with piecewise smooth boundaries### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2008-01-05 Created: 2008-01-05 Last updated: 2017-12-12Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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.

Open this publication in new window or tab >>Acoustic Transmission in Ducts of Various Shapes with an Impedance Condition### Andersson, Anders

### Nilsson, Börje

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2008 (English)In: International Conference on Numerical Analysis and Applied Mathematics 2008, Melville: American Institute of Physics , 2008, p. 33-36Conference paper, Published paper (Refereed)
##### Abstract [en]

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

Melville: American Institute of Physics, 2008
##### Series

AIP Conference Proceedings, ISSN 0094-243X ; 1048
##### Keywords

acoustic scattering, waveguide scattering, impedance, Building Block Method, numerical conformal mapping, Outer Polygon Method, invariant embedding
##### National Category

Fluid Mechanics and Acoustics Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-6650 (URN)10.1063/1.2990928 (DOI)978-0-7354-0576-9 (ISBN)
#####

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Available from: 2008-10-28 Created: 2008-10-28 Last updated: 2009-01-28Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

International Centre for Mathematical modelling, Växjö University.

Propagation^{ }of acoustic waves in a two-dimensional duct with an impedance^{ }condition at the boundary, is studied. The duct is assumed^{ }to have two ends at infinity being asymptotically straight, but^{ }otherwise to be arbitrarily shaped.The so called Building Block Method^{ }allows us to synthesize propagation properties for ducts with complicated^{ }geometries from results for simpler ducts. Conformal mappings can be^{ }used to transform these simple ducts to straight ducts with^{ }constant cross-sections.By using recently developed techniques for numerical conformal mappings,^{ }it is possible to construct a transformation between an infinite^{ }strip and an arbitrarily shaped duct with smooth or piecewise^{ }smooth boundary, keeping both smoothness and the well controlled boundary^{ }direction towards infinity that the above mentioned method requires.To accomplish^{ }a stable formulation of the problem, we express it in^{ }terms of scattering operators. The resulting differential equation is solved^{ }using wave splitting and invariant embedding techniques. We expand the^{ }involved functions in Fourier series, and hence, it is possible^{ }to give the operators a matrix representation. Numerical results are^{ }produced using truncated matrices.

Open this publication in new window or tab >>Schwarz-Christoffel Mappings for Nonpolygonal Regions### Andersson, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2008-10-26 Created: 2008-10-26 Last updated: 2017-12-14Bibliographically approved

Jönköping University, School of Engineering, JTH, Mathematics.

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.