<tex>$\mathbf{A}-\Phi$</tex> Formulation Time Domain Integral Equations Free from Interior Resonances

Abstract

The $\mathbf{A}-\Phi$ formulation has been proposed as a new paradigm for deriving computational electromagnetics methods that do not suffer from the various manifestations of low frequency breakdown that have plagued traditional approaches. This formulation utilizes equations developed in terms of the magnetic vector potential (A) and electric scalar potential ( $\Phi$ ), which allows them to overcome many of the limitations inherent to methods based directly on the electric and magnetic fields. The predominant limitation overcome by the $\mathbf{A}-\Phi$ formulation is the aforementioned low frequency breakdown inherent in traditional approaches. Additionally, the $\mathbf{A}-\Phi$ formulation also produces systems which can be effectively and efficiently preconditioned in a simpler manner than those based directly on electric and magnetic field. This formulation is also better suited for integration with quantum physics applications where description of electromagnetic effects is more naturally represented with $\mathbf{A}$ and $\Phi$ . Recent work has developed $\mathbf{A}-\Phi$ formulation time domain integral equations (TDIEs) which were stable over a wide range of problems and could maintain accuracy to very low frequencies. However, these equations were susceptible to being corrupted by interior resonances. The presence of interior resonances lowered the accuracy of the method when extracting frequency domain results near the resonances and also led to the eventual instability of the numerical method if too many interior resonances were excited (exactly synonymous to how the EFIE or MFIE would behave). This precluded the original $\mathbf{A}-\Phi$ formulation TDIEs from being used to analyze multiscale structures that had closed regions approximately a few wavelengths or larger in size. This work proposes a modification to the initial $\mathbf{A}-\Phi$ formulation TDIEs that makes them immune to issues related to interior resonances. This allows the new set of equations proposed in this work to be applied to deeply multiscale problems previously unable to be analyzed with this formulation. Numerical results are used to demonstrate that this new method is free from interior resonances. Further, it is shown that the other favorable qualities of this system are not lost due to the modifications proposed in this work.