Fast low-frequency methods in computational electromagnetics

Abstract

Low-frequency computational electromagnetics (CEM) solvers play an important role in the simulation and advanced modeling for the static/quasi-static field. It can not only capture both inductive and capacitive physics for the circuit design, but also the weak coupling between the electric and magnetic fields. Especially for the modeling with nanometer-scaled objects, the low-frequency CEM (full-wave) methods become indispensable when the electromagnetic interference cannot be considered by the pure static methods. During the past decades, different solutions have been successfully developed such as the loop-star/tree decomposition, Calderon preconditioned EFIE, the augmented EFIE based methods, and the magnetic field integral equation based methods. The perturbation methods are also useful in improving the accuracy of the algorithm at different frequency orders. To solve the real world problems, differential low-frequency fast algorithms have to be employed, such as the mixed-form fast multipole algorithm (FMA), the accelerated Cartesian expansion (ACE), and the fast Fourier transform (FFT). For the acceleration of perturbation-based methods, the kernel independent fast algorithm is preferred, while the FMA-based fast algorithms are only for the methods with the static and dynamic Green's functions. On the other hand, for the integral equation methods at low frequencies, the contribution of the vector potential is swamped by that of the scalar potential. It introduces the nullspace of the divergence operator in the scalar potential term and make the system matrix ill-conditioned. Therefore, some advanced preconditioners have also to be considered in order to improve the eigen spectrum of the system matrix. Here, the latest progress in this research area will be reviewed.