Bidimensional spline fitting procedure for rapid evaluation of Green´s function for multilayered media

One of the key steps in the Method of Moments (MoM) is the computation of the Green´s functions needed to fill the impedance matrix. When the object is located in a stratified medium, the corresponding spatial domain Green´s functions are expressed by some Sommerfeld Integrals (SI´s). The numerical evaluation of SI is extremely time consuming because of the oscillating and slowly decaying nature of its integrand. In the past, the discrete complex image method (DCIM) has been used most commonly to circumvent this difficulty. With the DCIM, a factor of the SI integrand is approximated by a sum of complex exponential functions, thereby transforming the SI into a series of complex image contributions. These complex images are independent of the transversal separation (rho) between the source and observation points. It is not necessary to evaluate the SI repeatedly for each p and, hence, considerable time saving can be achieved. It is well known, however, that the DCIM is inefficient when the Green´s function has to be evaluated for a set of sources/field points with different vertical coordinates. A separate complex exponential fitting has to be performed and a new set of complex images must be determined each time the source z´ and/or observation z is changed. To circumvent these problems, a new bidimensional spline fitting procedure for a faster Green´s function evaluation has been developed and is presented in this paper. This new approach enables us to accurately approximate the representation of each element of the Green´s function dyad without computing all of them directly and thus results in considerable time-saving. In the following, we first briefly review the DCIM to identify the difficulty. We then discuss the proposed solution and illustrate the performance by some numerical examples.