A mixed-form thin-stratified medium fast-multipole algorithm for both low and mid-frequency problems

A mixed-form thin-stratified medium fast-multipole algorithm is proposed for fast simulation of general microstrip structures at both low and mid-frequencies. The newly developed matrix-friendly formula of layered medium Green's function is applied in this algorithm. For well-separated interactions, the path deformation technique is implemented to achieve a smoother and exponentially convergent integral. The two-dimensional addition theorem is then incorporated into the integrand to expedite the matrix-vector product. In our approach, multipole expansion (low-frequency fast-multiple algorithm) as well as the plane wave expansion (mid-frequency fast-multipole algorithm) of the translational addition theorem are combined into a single multilevel tree to capture quasi-static physics and wave physics simultaneously. The outgoing wave is represented first in terms of multipole expansion at leafy levels, and then switched to plane wave expansion automatically at higher levels. This seamless connection makes the algorithm applicable in simulations, where subwavelength interaction and wave physics both exist.