A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

Abstract

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 redetermined. 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.