Tobin A. Driscoll
Abstract
The Schwarz-Christoffel transformation and its variations yield formulas for conformal maps from standard regions to the interiors or exteriors of possibly unbounded polygons. Computations involving these maps generally require a computer, and although the numerical aspects of these transformations have been studied, there are few software implementations that are widely available and suited for general use. The Schwarz-Christoffel Toolbox for MATLAB is a new implementation of Schwarz-Christoffel formulas for maps from the disk, half-plane, strip, and rectangle domains to polygon interiors, and from the disk to polygon exteriors. The toolbox, written entirely in the MATLAB script language, exploits the high-level functions, interactive environment, visualization tools, and graphical user interface elements supplied by current versions of MATLAB, and is suitable for use both as a standalone tool and as a library for applications written in MATLAB, Fortran, or C. Several examples and simple applications are presented to demonstrate the toolbox's capabilities.
Supplemental Material
Available for Download
Software for "A MATLAB Toolbox for Schwarz-Christoffel Mapping"
- DAPPEN, H.D. 1988. Die Schwarz-Christoffel-Abbildung fiir zweifach zusammenhangende Gebiete mit Anwendungen. Ph.D. thesis, ETH Zfirich.]]Google Scholar
- DAVIS, R.T. 1979. Numerical methods for coordinate generation based on Schwarz-Christoffel transformations. In Proceedings of the 4th AIAA Computational Fluid Dynamics Conference (Williamsburg, Va.). AIAA, Easton, Pa., 1-15.]]Google Scholar
- DELILLO, T. K. AND ELCRAT, A.R. 1993. Numerical conformal mapping methods for exterior regions with corners. J. Comput. Phys. 108, 199-208.]] Google ScholarDigital Library
- DENNIS, J. E. AND SCHNABEL, R.B. 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, N.J.]] Google ScholarDigital Library
- FLORYAN, J.M. 1986. Conformal-mapping-based coordinate generation method for flows in periodic configurations. J. Comput. Phys. 62, 221-247.]] Google ScholarDigital Library
- FLORYAN, J. M. AND ZEMACH, C. 1987. Schwarz-Christoffel mappings: A general approach. J. Comput. Phys. 72, 347-371.]] Google ScholarDigital Library
- GORDON, C., WEBB, D., AND WOLPERT, S. 1992. Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. lie, 1-22.]]Google Scholar
- HENmCI, P. 1974. Applied and Computational Complex Analysis. Vol. 1. John Wiley and Sons, New York.]]Google Scholar
- HOEKSTRA, M. 1986. Coordinate generation in symmetrical interior, exterior, or annular 2D domains, using a generalized Schwarz-Christoffel transformation. In Numerical Grid Generation in Computational Fluid Mechanics, J. Hauser and C. Taylor, Eds. Pineridge Press, Swansea, U.K.]]Google Scholar
- HOUGH, D.M. 1990. User's guide to CONFPACK. ETH Ziirich IPS Res. Rep., ETH Ziirich, Switzerland, 90-11.]]Google Scholar
- HOWELL, L. H. 1993. Numerical conformal mapping of circular arc polygons. J. Comput. Appl. Math. 46, 7-28.]] Google ScholarDigital Library
- HOWELL, L.H. 1994. Schwarz-Christoffel methods for multiply-elongated regions. In Proceedings of the 14th IMACS World Congress on Computation and Applied Mathematics. North-Holland, Amsterdam]]Google Scholar
- HOWELL, L. H. AND TREFETHEN, L.N. 1990. A modified Schwarz-Christoffel transformation for elongated regions. SIAM J. Sci. Stat. Comput. 11,928-949.]] Google ScholarDigital Library
- McGRATTAN, K. B., REHM, R. G., AND BAUM, H.R. 1994. Fire-driven flows in enclosures. J. Comput. Phys. lie, 285-291.]] Google ScholarDigital Library
- NAJM, H. M. AND GHONIEM, A.F. 1991. Numerical simulation of the convective instability in a dump combustor. AIAA J. 29, 911-919.]]Google ScholarCross Ref
- PEARCE, K. 1991. A constructive method for numerically computing conformal mappings for gearlike domains. SIAM J. Sci. Stat. Comput. 12, 231-246.]]Google ScholarCross Ref
- REPPE, K. 1979. Berechnung von Magnetfeldern mit Hilfe der konformen Abbildung dutch numerische Integration der Abbildungsfunktion von Schwarz-Christoffel. Siemens Forsch. u. Entwickl. Ber. 8, 190-195.]]Google Scholar
- SRIDHAR, K. P. AND DAVIS, R.T. 1985. A Schwarz-Christoffel method for generating twodimensional flow grids. J. Fluids Eng. 107, 330-337.]]Google ScholarCross Ref
- STARKE, G. AND VARGA, R.S. 1993. A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations. Num. Math. 64, 213-240.]]Google ScholarDigital Library
- TREFETHEN, L.N. 1980. Numerical computation of the Schwarz-Christoffel transformation. SIAM J. Sci. Stat. Comput. 1, 82-102.]]Google ScholarDigital Library
- TREFETHEN, L.N. 1984. Analysis and design of polygonal resistors by conformal mapping. Z. Angew. Math. Phys. 35, 692-704.]]Google ScholarCross Ref
- TREFETHEN, L.N. 1989. SCPACK user's guide. MIT Numerical Analysis Rep. 89-2. MIT, Cambridge, Mass.]]Google Scholar
- TREFETHEN, L.N. 1993. Numerical construction of conformal maps. In Fundamentals of Complex Analysis for Mathematics, Science, and Engineering. Prentice-Hall, Englewood Cliffs, N.J.]]Google Scholar
- WOODS, L. C. 1961.The Theory of Subsonic Plane Flow. Cambridge University Press, Cambridge, Mass.]]Google Scholar
Index Terms
- Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping
Recommendations
Schwarz-Christoffel Mappings for Nonpolygonal Regions
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-...
Revisiting the Crowding Phenomenon in Schwarz-Christoffel Mapping
We address the problem of conformally mapping the unit disk to polygons with elongations. The elongations cause the derivative of the conformal map to be exponentially large in some regions. This crowding phenomenon creates difficulties in standard ...
A modified Schwarz-Christoffel mapping for regions with piecewise smooth boundaries
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 ...
Comments