Stabilizing the magnetic field integral equation at low frequencies for multiply connected scatterers

Michael O'Neil

Courant Institute, NYU

A classical problem in electromagnetics concerns the solution of the time-harmonic Maxwell equations in the low-frequency and static regimes. When solving scattering problems from a simply connected body, standard integral equation methods (analytical and numerical) provide sufficient means with which to calculate the scattered field. However, when the scatterer is multiply connected, topology plays a fundamental role and calculating the scattered field is not so straightforward. In fact, for multiply connected conductors, at zero frequency the standard boundary conditions on the tangential components of the incoming magnetic field do not uniquely determine the induced surface current, and thus do not uniquely determine the scattered field. With this in mind, we will describe a new consistency condition (independent of gauge) on the vector potential that overcomes this non-uniqueness and resolves a long-standing difficulty in inverting the Magnetic Field Integral Equation (MFIE). Numerical examples of this stabilizing consistency condition in axisymmetric geometries will be shown.

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