Modeling of electrically large apertures has traditionally proved computationally prohibitive, so that engineering problems such as antenna design and studies of radar sensitivities have usually been tackled on the basis of significant simplifying assumptions. Accurate electromagnetic modeling of large structures requires the solution of extremely large, ill-conditioned systems of equations—-so challenging, in fact, that the associated matrices are extremely difficult to evaluate, and so large that they cannot even be formed explicitly in computer memory. In this talk, we will present new methodologies for the numerical solution of Maxwell’s equations. Based on a novel Fourier-Continuation (FC) method for the resolution of the Gibbs phenomenon, associated surface-representation methods, and fast high-order methods for evaluation of integral operators, these methodologies give rise to fast, versatile, and highly accurate frequency and time-domain PDE solvers. A variety of applications to linear and nonlinear problems, including design of reflect-array antennas, scattering by complex engineering structures, fluid flow problems, etc., demonstrate the very significant improvements the new algorithms provide over the accuracy and speed resulting from other approaches currently in use.
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