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Computing periodic conformal mappings

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CATW03 - Computational complex analysis

Conformal mappings are used to model a range of phenomena in the physical sciences. Although the Riemann mapping theorem guarantees the existence of a mapping between two conformally equivalent domains, actually constructing these mappings is extremely challenging. Moreover, even when the mapping is known in principle, an efficient representation is not always available. Accordingly, we present techniques for rapidly computing the conformal mapping from a multiply connected canonical circular domain to a periodic array of polygons. The boundary correspondence function is found by solving the parameter problem for a new periodic Schwarz—Christoffel formula. We then represent the mapping using rational function approximation. To this end, we present a periodic analogue of the adaptive Antoulas—Anderson (AAA) algorithm to obtain the relevant support points and weights. The procedure is extremely fast; evaluating the mappings typically takes around 10 microseconds. Finally, we leverage the new algorithms to solve problems in fluid mechanics in periodic domains.

This talk is part of the Isaac Newton Institute Seminar Series series.

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