792 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 3, MARCH 2008 A Warning About Metamaterials for Users of Frequency-Domain Numerical Simulators Giacomo Oliveri, Student Member, IEEE, and Mirco Raffetto Abstract—It is shown that users of frequency-domain numerical simulators should be warned of the danger of using models with adjacent complementary media since such a configuration of mate- rials can be a cause of a numerical instability which cannot be over- come by the presence of other lossy media or absorbing boundary conditions. The physical realizability at any frequency of models considering complementary media with sharp interfaces is then questioned by the contradiction between the deduced result and the one which holds true when the actual dielectric structure of metamaterials is considered. Index Terms—Boundary value problems, convergence of numer- ical methods, finite element methods, frequency domain analysis, metamaterials, numerical stability, waveguide discontinuity problems. I. INTRODUCTION N OWADAYS electromagnetic numerical simulators are very popular and they are widely exploited even from a commercial point of view. The popularity of these computer aided design tools is due to the fact that most of them are able to guarantee a priori several important features. As a matter of fact, most of them are suffi- ciently versatile to allow the simulation of a wide class of elec- tromagnetic problems, and sufficiently efficient to provide the solution of even complicated problems in a reasonable amount of time. Most of commercial simulators are even user friendly. However, the reliability of a simulator is the only feature all users consider as inalienable. As a matter of fact, most of the above indicated features are useless if a simulator is not able to reproduce with a certain level of fidelity what happens in the models of interest. From a practical point of view it is not possible to guarantee the reliability of a simulator in general, without hypotheses re- stricting the class of problems for which such a result can be ob- tained. Fortunately, very general results of reliability are avail- able for some frequency-domain numerical methods [1] and the set of hypotheses under which these results are obtained is usu- ally very simple. For example, if only double positive [2] and linear materials are involved, for simulators based on the finite element method it is sufficient that some losses are present in the dielectric media occupying just a subregion of the domain of numerical investigation or that absorbing boundary conditions Manuscript received April 24, 2007; revised October 4, 2007. The authors are with the Department of Biophysical and Electronic Engi- neering, University of Genoa, I-16145 Genoa, Italy (e-mail: oliveri@dibe. unige.it; raffetto@dibe.unige.it). Digital Object Identifier 10.1109/TAP.2008.916955 are present on some part of the boundary [1]. As a consequence, if only double positive and linear materials are involved, a very basic knowledge of electromagnetics is sufficient to avoid the cases in which the method is not reliable and even non-expert users can obtain satisfactory results from electromagnetic nu- merical simulators in the design of complex electromagnetic systems. The recent advent of metamaterials [3], [2], [4] has partially modified the previously depicted situation. As a matter of fact, metamaterials are studied by considering two approaches. In one of these all inhomogeneities are considered and, even though it usually results in extremely complicated numerical problems, all previous considerations on the reliability of simu- lators apply since only double positive media are involved. On the contrary, with the second approach, based on the definition of effective constitutive parameters, the advantage usually obtained in terms of complexity of the numerical problems to be dealt with is paid with the lack of very general results of reliability. There are, actually, some results in this direction for frequency-domain simulators based on the finite element method [5] but they are obtained under hypotheses which are not sufficiently general to cover cases of interest such as those studied in [6] or [7]. On the contrary, there are some recent experimental results [8], [9], [10], [11], which suggest that the reliability and the efficiency of time-harmonic finite element based numerical simulators could be questioned. However, even a large number of experiments cannot be used to deduce that the simulator considered is not able to provide converging sequences of approximations. A much stronger indication that the situation could be less satisfactory when metamaterials described as effective media are present than in cases involving just double positive media can be deduced from [12]. In that paper some time-harmonic electromagnetic boundary value problems involving complementary (or conjugate) media [13] are theoretically shown to be ill-posed [12] and, as a by-product, it is shown that they cannot be reliably simulated by most nu- merical simulators. However, there could still be some hope to obtain sufficiently general and simple reliability results for frequency-domain numerical simulators since the negative results in [12] have been obtained by considering only lossless double positive and double negative media, with matched impedance boundary conditions on the two ports, meaning that both materials are thought of as effective homogeneous media of infinite extensions. These hypotheses could be considered as unreasonably idealistic. This paper shows that our hope to retain the validity of simple and physically sound results of reliability, as the one reported above concerning the sufficiency of the presence of some losses 0018-926X/$25.00 © 2008 IEEE