Enhancement of the numerical stability of the adaptive integral method at low frequencies through a loop-charge formulation of the method-of-moments approximation

The performance of integral-equation-based fast iterative solution schemes such as the adaptive integral method (AIM) and the conjugate gradient (CG) fast Fourier transform (FFT) algorithm can be substantially improved at low frequencies if the electric current and electric charge densities are treated as two separate unknown quantities. The representation of the current density in terms of solenoidal expansion functions (loops) and the charge density in terms of pulse basis functions provides for an exact decomposition of the original electromagnetic (EM) boundary-value problem into its magnetostatic and electrostatic forms at zero frequency. This formulation allows for accurate EM modeling down to very low frequencies free of numerical instabilities, while the spectral properties of the matrix equation are substantially improved compared to the standard method of moments formulation in either loop-tree or loop-star basis. The AIM and the CG FFT algorithms can be appropriately adjusted to accommodate for the use of loop-charge basis functions, thus leading to efficient solvers with O(N log N) solution complexity and O(N) memory requirements for two-and-one-half-dimensional and penetrable three-dimensional (3-D) structures. For general 3-D objects, the CPU time and memory of the algorithms scale as O(N^1.5 log N) and O(N^1.5), respectively. The new Implementation of the AIM is discussed and demonstrated through its application for the broad-band simulation of complex interconnect and electronic packaging structures.