246 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 42, NO. 3, AUGUST 2000 The FE-BIE Technique Applied to Some 2-D Problems Relevant to Electromagnetic Compatibility: Optimal Choice of Mechanisms to Take into Account Periodicity Hendrik Rogier, Bart Baekelandt, Frank Olyslager, and Daniël De Zutter Abstract—A study of mechanisms to account for periodicity when modeling two-dimensional (2-D) structures with a hybrid finite-element boundary integral equation (FE-BIE) method is presented. These techniques are either based on the use of Green’s functions or on the application of the Floquet–Bloch theorem as a periodic boundary condition. The described formalism can be used to model very diverse problem geometries as is demonstrated by means of some typical examples. For these configurations, it is shown how an optimal choice can be made between the mechanisms that impose periodicity. Index Terms—Absorbers and shielding, hybrid numerical tech- niques, periodic configurations. I. INTRODUCTION D ESIGNERS concerned with electromagnetic compati- bility (EMC) problems are often confronted with complex configurations that are electrically large. Furthermore, a rig- orous study of the EMC aspect often involves a computationally intensive full-wave analysis. Therefore, much attention is payed to simplify the realistic problem to a simpler and computa- tionally manageable geometry with approximately the same EM behavior as the original structure. In this contribution we will devote our attention to complex structures that have been simplified in two ways: First, we will assume two-dimensional (2-D) configurations. Second, we will consider problems that are infinitely periodic in one direction. This allows to simplify the modeling problem to the study of only one unit cell by invoking the Floquet–Bloch theorem. Hybrid full-wave techniques are well known as flexible tools for modeling complex structures since they combine the finite-element (FE) method for the description of the inhomo- geneous and complex regions with a boundary integral equation (BIE) formalism that takes into account the open and large homogeneous subdomains. These FE-BIE techniques have proven to be very efficient in modeling 2-D periodic problems [1]–[4], as well as three-dimensional (3-D) configurations [5], [6]. They can be seen as the extension of the very popular BIE techniques, as applied to periodic structures [7]–[10]. Manuscript received September 10, 1999; revised April 26, 2000. The work of H. Rogier was supported by a Postdoctoral Grant by the Flemish Institute for the promotion of Scientific and Technological Research in the Industry (IWT). F. Olyslager is a Research Associate of the Fund for Scientific Research—Flanders (Belgium) (FWO-V) of Belgium. The authors are with the Information Technology Department, Ghent Univer- sity, B-9000 Ghent, Belgium. Publisher Item Identifier S 0018-9375(00)06646-1. Fig. 1. Unit cell of general periodic configuration. In this paper, we will extend the more classical 2-D FE-BIE ap- proaches with one inhomogeneous FE subdomain and one open BIE region to a general hybrid formulation in which an arbitrary number of bounded FE subregions and homogeneous BIE do- mains can be present, as discussed in Section II. We will show that our BIE formulation, discussed in Section III, is capable of modeling structures with arbitrary losses. This is a very impor- tant feature when treating problems of EMC. Finally, for the first time in literature, a clear overview is presented in Section IV of the different mechanisms that can be used to take into account the periodicity of the configuration. Several alternative formalisms are given and their computational efficiency is examined in Sec- tion V, by studying two complex sample configurations. II. GEOMETRY OF A GENERAL PERIODIC CONFIGURATION In Fig. 1 a unit cell is shown of a representative 2-D config- uration, of infinite extent and periodic with period along the -direction. The structure is excited by an incident plane wave of the form (1) (2) 0018–9375/00$10.00 © 2000 IEEE