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• 1.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
A modiﬁed Schwarz–Christoffel mapping for regions with piecewise smooth boundaries2008In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, Vol. 213, no 1, p. 56-70Article in journal (Refereed)

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.

• 2.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Modified Schwarz–Christoffel mappings using approximate curve factors2009In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 233, no 4, p. 1117-1127Article in journal (Refereed)

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.

• 3.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Numerical Conformal mappings for regions Bounded by Smooth Curves2006Licentiate thesis, monograph (Other academic)

In many applications, conformal mappings are used to transform two-dimensional regions into simpler ones. One such region for which conformal mappings are needed is a channel bounded by continuously differentiable curves. In the applications that have motivated this work, it is important that the region an approximate conformal mapping produces, has this property, but also that the direction of the curve can be controlled, especially in the ends of the channel.

This thesis treats three different methods for numerically constructing conformal mappings between the upper half-plane or unit circle and a region bounded by a continuously differentiable curve, where the direction of the curve in a number of control points is controlled, exact or approximately.

The first method is built on an idea by Peter Henrici, where a modified Schwarz-Christoffel mapping maps the upper half-plane conformally on a polygon with rounded corners. His idea is used in an algorithm by which mappings for arbitrary regions, bounded by smooth curves are constructed.

The second method uses the fact that a Schwarz-Christoffel mapping from the upper half-plane or unit circle to a polygon maps a region Q inside the half-plane or circle, for example a circle with radius less than 1 or a sector in the half--plane, on a region Omega inside the polygon bounded by a smooth curve. Given such a region Omega, we develop methods to find a suitable outer polygon and corresponding Schwarz-Christoffel mapping that gives a mapping from Q to Omega.

Both these methods use the concept of tangent polygons to numerically determine the coefficients in the mappings.

Finally, we use one of Don Marshall's zipper algorithms to construct conformal mappings from the upper half--plane to channels bounded by arbitrary smooth curves, with the additional property that they are parallel straight lines when approaching infinity.

• 4.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Numerical Conformal Mappings for Waveguides2009Doctoral thesis, comprehensive summary (Other academic)

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. Keywords: waveguides, building block method, numerical conformalmappings, 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

• 5.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Numerical Conformal Mappings for Waveguides2010In: Computational Mathematics: Theory, Methods and Applications / [ed] Peter G. Chareton, Hauppauge, NY, USA: Nova Science Publishers, Inc., 2010Chapter in book (Other academic)

A number of newly developed numerical conformal mapping techniques are described. Their purpose is to achieve conformal mappings with good accuracy for regions bounded by smooth or piecewise smooth curves in which the boudary curve direction can be controllod, especially towards infinity in unbounded regions as for example waveguides.

Most of the mappings are variants of the Schwarz-Christoffel mappings.

• 6.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On the curvature of an inner curve in a Schwarz-Christoffel mapping2007Report (Other academic)

In the so called outer polygon method, an approximative conformal mapping for a given simply connected region \Omega is constructed using a Schwarz-­Christoffel mapping for an outer polygon, a polygonal region of which \Omega is a subset. The resulting region is then bounded by a C^\infty -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 half­plane, 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.

• 7.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Schwarz–Christoffel Mappings for Nonpolygonal Regions2008In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 31, no 1, p. 94-111Article in journal (Refereed)

An approximate conformal mapping for an arbitrary region $\varOmega$ bounded by a smooth curve $\varGamma$ is constructed using the Schwarz–Christoffel mapping for a polygonal region in which $\varOmega$ 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 $\varOmega$. $R$ is bounded by a $C^\infty$ curve, and since the mapping function originates from the Schwarz–Christoffel mapping and tangent polygons are used to determine it, important properties of $\Gamma$ 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.

• 8.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
Using a Zipper Algorithm to find a Conformal Map for a Channel with Smooth Boundary2006In: AIP Conference Proceedings: Second Conference on Mathematical Modeling of Wave Phenomena, American Institute of Physics, New York , 2006, p. 378-Conference paper (Refereed)

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.

• 9.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Acoustic Transmission in Ducts of Various Shapes with an Impedance Condition2008In: International Conference on Numerical Analysis and Applied Mathematics 2008, AIP, Melville, USA , 2008, p. 33-36Conference paper (Refereed)

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.

• 10.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Electro-Magnetic Scattering in Variously Shaped Waveguides with an Impedance Condition2009In: Mathematical modelling of wave phenomena: 3rd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, 9 – 13 June 2008, Melville, New York: American Institute of Physics , 2009, p. 36-45Conference paper (Refereed)

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 transformed into 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.

• 11.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
National Institute of Advanced Technology (ENSTA, Paris), France. Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Acoustic waves in a mean flow duct with varying boundary2009In: 15th AIAA/CEAS Aeroacoustics Conference, Miami 11-13May 2009: Paper AIAA-2009-3407, 2009Conference paper (Other academic)

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

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