Fast integral equation solution techniques for planar-3D structures in multilayered media

In this contribution, a hybrid space/spectral domain integral equation approach is presented for the characterization of quasi-3D structures in multilayered media. The vertical conductors for the quasi-3D components like vias, through holes, airbridges, finite dielectric regions etc. can cross an arbitrary number of dielectric layers with arbitrary vertical discretization. The electrodynamic behavior of the vertical currents is incorporated by extended spectral domain Green's functions derived by analytical integrations over the vertical direction and by a summation over the Cartesian wavenumbers which must be computed only once and stored in a database. For short lateral distances, the couplings are computed with a subtraction technique based on asymptotic representations of the extended Green's functions, whereas for larger distances and for group coupling computations in context with fast matrix vector product evaluations, adaptive integration path deformations with extended Laguerre quadrature methods are applied. For the solution of the linear systems of equations, a GMRES solver was implemented together with a diakoptic preconditioner based on the group decomposition of larger structures, combined with a reversed Cuthill McKee reordering applied to the submatrix pattern of the system matrix. With all these techniques, the analysis of complex quasi-3D structures can be carried out with the same performance than pure planar microstrip and/or slotline structures.