The “dottrick TDEFIE”: a DC stable integral equation for analyzing transient scattering from PEC bodies

Marching on in time, time domain integral equation solvers represent an increasingly appealing avenue for analyzing transient electromagnetic interactions with large and complex structures. Compared to their differential equation counterparts, these solvers automatically impose radiation conditions, do not require unknown fields to be discretized throughout homogeneous volumes, and are highly immune to numerical dispersion. Among their many incarnations, marching on in time, time domain electric field integral equation (MOT-TDEFIE) solvers remain the most widely used. Unfortunately MOT-TDEFIE solvers often are plagued by DC instabilities, i.e. constant and linear-in-time solutions of MOT-TDEFIE systems that reside in the null space of the (non-causal) TDEFIE operator. In the past, these instabilities have been partially cured by using loop-tree decompositions [1] and by enforcing boundary conditions on normal magnetic field components [2]. Unfortunately, neither technique completely annihilates the static null space of the TDEFIE operator, nor guarantees that MOT-TDEFIE solutions are free of DC remnants. In this paper, a modified TDEFIE that resolves static and linear-in-time currents is presented. The equation is obtained by leveraging the time domain Calderon identities in conjunction with the ldquodot-trickrdquo, viz. a careful rearrangement of temporal derivative operators appearing in sequences of TDEFIE operators. The effectiveness of the dottrick TDEFIE for both open and closed structures is demonstrated theoretically, proving the resonance free behavior of the dottrick TDEFIE, and numerically.