A Simple Procedure to Optimize Small Radius Rounded Corners Obtained From Schwarz–Christoffel Conformal Transformations

Both conformal mapping via Schwarz-Christoffel (SC) formulas and finite element methods (FEM) can provide accurate results in analyzing 2-D electric or magnetic fields. In the presence of curved boundaries with small radius of curvature, the first are normally constrained to introduce piecewise straight lines. The original contribution of this paper consists of presenting a reliable but simple procedure to smooth several sharp vertices of a polygonal boundary by the formula for rounded corners, and comparing the results with those obtained by replacing sharp corners with piecewise straight lines. Differences are perceived only in close proximity, and this quantitatively explains the similar results obtained from maps and from FEMs and provides reliable assessment of the obtainable results. Two approaches are followed. According to the first, sharp corner geometries are directly mapped into rounded corner ones, accepting small changes in the whole structure, and negligible only for small corner radii. According to the second, the simple optimization procedure of the SC prevertices allow us to match the original geometry in all details.