## 1. Introduction

[2] Complex electromagnetic configurations, which combine inhomogeneous features with an open three-dimensional simulation space, can be described efficiently with hybrid techniques that combine a rigorous boundary integral equation (BIE) for the open domain with the finite element method (FE) or the finite difference-time domain (FDTD) technique for the inhomogeneous part of the problem. On the one hand, the FE-BIE approaches have become mature by now. In electromagnetic (EM) literature, they have proven their usefulness for a wide range of applications [*Volakis et al.*, 1997; *Eibert et al.*, 1999; *Gedney et al.*, 1992; *McGrath and Pyati*, 1996; *Rogier et al.*, 1996, 2000a]. On the other hand, FDTD-BIE formalisms proved to be more involved. When modeling multiple subregions with a combination of FDTD with a time domain integral equation [*Johnson and Rahmat-Samii*, 1997; *Bretones et al.*, 1998; *Naishadham and Esselle*, 1998], a careful implementation is required in order to avoid late-time instability and time dispersion, linked with the dependency of the integral equation formulation on the time history of the fields. Yet, combining FDTD with other full-wave modeling techniques can give rise to additional benefits, since the FDTD is the most memory-efficient simulation technique. Moreover, the FDTD approach allows an easy incorporation of small details in its Yee-cell mesh, through the use of subcell techniques, and the easy incorporation of discrete element models, e.g., for resistors, inductors or capacitors. Finally, FDTD even allows to include materials with time-dependent constitutive relations.

[3] In *Rogier et al.* [1998b, 2000d, 1998a, 2000b, 2000c], we presented a new hybrid formalism that combines the FDTD and BIE techniques by mainly working in the frequency domain and by constructing interaction matrices _{BIE}, for each homogeneous subregion, and _{FDTD}, for each inhomogeneous and bounded subdomain. More specifically, _{BIE} is found by applying the Poggio and Miller integral expressions, whereas the excitation of a rooftop basis function on the boundary of an inhomogeneous subdomain in an FDTD simulation yields one column of that subregion's system matrix. Obviously, the construction of _{FDTD} for each inhomogeneous subregion requires a number of FDTD simulations that is equal to the number of basis functions that expand the tangential electric and magnetic fields on the boundary of that subdomain, which results in high CPU-times. Different techniques to alleviate this problem have been presented in the past [*Rogier et al.*, 1998b, 2000d, 2000b]. One particular method [*Rogier et al.*, 2000c] to drastically reduce the number of FDTD runs consists of applying a biconjugate gradient (BiCG) solution mechanism in a special way. Instead of calculating _{FDTD} explicitly for each FDTD subregion, the FDTD calculations are performed to find their contribution to the matrix-vector products needed in each iteration of the BiCG formalism that solves the global matrix system. The required number of FDTD runs per FDTD subregion then equals the number of iterations needed for the BiCG algorithm to converge.

[4] In this paper, we present a new hybrid formalism that combines both the finite element method and FDTD with a BIE approach and an efficient solution scheme by using a BiCG solver. Schematically, the new technique proceeds as follows. In Section 2, the simulation domain is decomposed into a number of subregions, and the tangential fields at the interfaces between the different subdomains are chosen as unknowns. After expanding these unknown fields into rooftop basis functions, for each homogeneous subregion, system matrices _{BIE} are constructed, following Section 3.1, and for some of the inhomogeneous subdomains either the system matrix is calculated using the FE technique with the frontal elimination algorithm, following Section 3.2, or the system matrix _{FDTD} is computed by performing FDTD simulations, following Section 3.3. Under the assumption that a similar matrix _{FDTD} characterizes the remaining inhomogeneous subdomains, it is shown in Section 3.4 how the global matrix system is assembled and inverted by a BiCG algorithm. We then explain in Section 3.5 how one calculates the contribution of the remaining FDTD subregions to the matrix-vector products needed in the BiCG method by performing FDTD simulations. Finally some representative examples are given in Section 4 to elucidate the new technique. In Section 4.2, for example, an IC package in the neighborhood of a multilayered dielectric substrate is modeled in order to show the versatility of the FDTD technique in the new hybrid formalism.