The investigator develops efficient, accurate and rapidly convergent algorithms for evaluation of the interaction between electromagnetic fields and complex structures. Specifically, the investigator and his collaborators develop a family of algorithms that focus mainly on (1) Fast, high-order numerical solutions for wave-propagation problems in domains that contain geometric singularities, and (2) Scattering problems from penetrable electromagnetic periodic structures, with a particular emphasis on resonant problems relevant to the design of photonic crystals and Negative Index Materials (NIM). The investigator concentrates on development and implementation of a massively parallel computational framework based on integral equation formulations capable of producing fast and high-order solutions of wave scattering problems of realistic complexity. The approach consists of the following main elements: (a) High-order resolution of the singularities of the solutions of the boundary integral equations in non-smooth domains; (b) Pseudodifferential-calculus-based design and analysis of well-conditioned integral equation formulations leading to small numbers of Krylov-subspace iterations for a wide range of electromagnetic transmission problems; and (c) Use of equivalent sources, FFT-based acceleration algorithms, and implementations that take advantage of the newly available Graphic Processing Units (GPUs) computational platforms to dramatically enhance computational times and capabilities.
The algorithms that are developed as part of this project are of fundamental significance to diverse applications such as electromagnetic interference and compatibility (electronic circuits), dielectric/magnetic coated conductors, and composite meta-materials (photonic crystals and Negative Index Materials). The simulation of electromagnetic wave propagation in complex structures gives rise to a host of significant computational challenges that result from non-coercive formulations, oscillatory solutions, geometric singularities, resonances, and ill-conditioning in the high-frequency regime. The recent efforts of the investigator and his collaborators resulted in the development of a highly efficient computational methodology that resolved several of these difficulties and whose extension enables the fulfillment of an ambitious plan: to simulate with high fidelity realistic scattering environments with a high dynamic range.
