Title: Trefftz Functions in Electromagnetic Analysis: Effective Medium Theory, Slab Geometries, Random Matrices
Abstract:
Trefftz functions, by definition, satisfy (locally) the underlying differential equation of a given problem, along with the relevant interface boundary conditions. Examples include harmonic polynomials for the Laplace equation; plane waves, cylindrical or spherical harmonics for wave problems, and so on.
After a brief review of various applications of Trefftz functions, the talk will focus on two particular examples: non-asymptotic / nonlocal homogenization of periodic structures and simulation of photonic devices with slab geometries.
These examples beg the question: what explains the “unreasonable effectiveness” of Trefftz approximations (to paraphrase Eugene Wigner)? In the mathematical literature, this question has been studied primarily for homogeneous media (e.g. plane wave or cylindrical wave expansions) but needs to be posed much more broadly. We discuss possible mechanisms of the high accuracy of Trefftz approximations – in particular, trigonometric interpolation and a curious connection with the theory of random matrices.
This research was supported in part by the US National Science Foundation grants DMS-1216927 and DMS-1620112. The Guest Professor position at DTU is supported by the Otto Mønsteds Fond.
Biography:
Igor Tsukerman is Professor of Electrical and Computer Engineering at the University of Akron, Ohio, USA, where he has been a faculty member since 1995. His research is focused on the simulation of nanoscale systems, applied electromagnetics and photonics. He teaches a variety of undergraduate and graduate courses (Programming for Engineers, Signals & Systems, Circuits, Electromagnetic Fields, Digital Signal Processing, Random Signal Analysis, Simulation of Nanoscale Systems, and others). Tsukerman has about 180 refereed publications. He has authored the monograph Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer 2008) and co-edited another book, Plasmonics and Plasmonic Metamaterials (World Scientific 2011). Currently he is acting as Editor-in-Chief of a five-volume reference set on electromagnetic analysis and simulation, to be published by World Scientific in 2018. He is also working on the 2nd edition of Computational Methods for Nanoscale Applications.
Before coming to the University of Akron, Tsukerman worked at the Department of Electrical & Computer Engineering, the University of Toronto (1990–1995). Tsukerman’s academic degrees are from St. Petersburg Polytechnic in Russia: a combined B.Sc. / M.Sc. degree (with honors) in Control Systems (1982) and a Ph.D. in Electrical Engineering (1988).