IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006 2615 Modeling of 3-D Periodic Multiphase Composites by Homogenization Ouail Ouchetto, Saïd Zouhdi, Senior Member, IEEE, Alain Bossavit, Georges Griso, and Bernadette Miara Abstract—An efficient technique is proposed for analyzing two- and three-dimensional lossy periodic composite materials, combining an asymptotic multiscale method with the unfolding method. The computed effective conductivities for square cylinders and cubes suspended in a host isotropic medium are compared to the Maxwell–Garnett mixing formula predictions. The electro- magnetic field in a finite lattice of round cylinders is compared with the exact electromagnetic field calculated directly in the heterogeneous lattice. Index Terms—Composite materials, effective permittivity, finite- element method (FEM), metamaterials, periodic arrays. I. INTRODUCTION I N RECENT years, there has been an increasing interest in ar- tificial structured materials. Many practical applications for this materials have been suggested in diverse fields, which in- clude: 1) frequency-selective surfaces; 2) filters; 3) integrated optical microwaveguides; 4) thin films; 5) memory devices; and 6) novel antennas [1]–[5]. Artificial materials consist of a large number of metallic or dielectric particles placed in a homoge- nous host medium or background. One is able to engineer or design a material that has a desirable permittivity, permeability, or other electromagnetic characteristics by adjusting the shape, size, material composition, and density of inclusions. Many theories are proposed for predicting the effective electromagnetic proprieties, i.e., conductivity, permittivity, and permeability, of structured electromagnetic materials, when the period of the microstructure is small compared to the wavelength. Some are based on analytical calculation such as Maxwell–Garnett, Bruggeman, Clausius–Mossotti mixing formulas, etc. [6]–[8]. Others are based on computation tech- niques such as the method of moments (MM), finite-difference time-domain method (FDTD), or finite-element method (FEM), in order to solve several types of equations (e.g., partial differ- ential equation, boundary integral equations, etc.) [9]–[12]. Manuscript received October 3, 2005. O. Ouchetto, S. Zouhdi, and A. Bossavit are with the Laboratoire de Génie Electrique de Paris, École Supérieure d’Électricité, 91192 Gif-Sur-Yvette Cedex, France (e-mail: ouchetto@yahoo.fr; sz@ccr.jussieu.fr; Bossavit@lgep.supelec.fr). G. Griso is with the Laboratoire Jacques-Louis Lions, 75252 Paris, France (e-mail: georges.griso@wanadoo.fr). B. Miara is with the School of Engineers, École Supérieure d’Ingénieurs en Électronique et Électrotechnique, 93162 Noisy-le-Grand, France (e-mail: miarab@esiee.fr). Digital Object Identifier 10.1109/TMTT.2006.872928 Fig. 1. (a) Periodic composite material. (b) Same material when the period tends to zero. The main purpose of this paper is to present a novel method to evaluate the effective constitutive parameters of periodic com- posite materials along with an accurate approximate value of the electromagnetic field within the microstructure. The proposed methodology is based on the asymptotic multiscale method as- sociated with the periodic unfolding method. The finite-element technique is used to evaluate the expressions of the constitutive parameters, the electric field within the microstructure, and its associated correctors. In this study, we consider the case of gen- eral lossy composite materials. This paper is organized in the following way. In Section II, we investigate the limit of the solution of Maxwell equations as the periodicity of the microstructure approaches zero. This en- ables us to obtain the expressions of effective constitutive pa- rameters and asymptotic expressions of electromagnetic field within the periodic microstructure. In Section III, we present the numerical validation by comparing the calculated effective permittivity and conductivity with the result of classical mixing formulas (Maxwell–Garnett). In Section IV, the approximate quasi-static electric field developed in this paper, in the case of a finite periodic lattice, is compared to the electric field calculated by the classical FEM in the initial heterogeneous microstructure. II. ANALYSIS OF ARTIFICIAL MATERIAL As shown in Fig. 1, the artificial material is modeled as a triply periodic array of identical inclusion elements suspended in a homogeneous and isotropic background with the permit- tivity and permeability denoted as and . The material is periodic, i.e., it is a collection of identical cubes with side length ( -cell). In the artificial material, the electromagnetic field verifies the Maxwell equations. The constitutive parameters, i.e., permittivity, permeability, and conductivity , the electromagnetic fields , and the excitation sources depend on the period of the lattice. 0018-9480/$20.00 © 2006 IEEE