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Singh, Modal behavior, cutoff condition and dispersion characteristics of an optical waveguide with a core cross-section bounded by two spirals, Microwave Opt Ž . Technol Lett 21 1999 , 121]124. 21. V. Singh, B. Prasad, and S.P. Ojha, An analytical study of the cutoff conditions and the dispersion curves of a waveguide with a cross-sectional shape resembling an ellipse compressed along the Ž . minor axis, Microwave Opt Technol Lett 1999, in press . Q 1999 John Wiley & Sons, Inc. CCC 0895-2477r99 PERFECTLY MATCHED LAYER TERMINATION FOR FINITE-ELEMENT MESHES: IMPLEMENTATION AND APPLICATION Youssry Y. Botros 1 and John L. Volakis 1 1 Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, Michigan 48109-2122 Recei¤ ed 11 May 1999 ( ) ABSTRACT: Perfectly matched layer PML absorbers deteriorate the condition of the resulting ﬁnite-element sparse systems. Therefore, poor con¤ ergence scenarios are obser¤ ed when an iterati¤ e sol¤ er is employed. In this work, we show that, by choosing the PML parameters in an optimal manner, substantial speedup in the solution con¤ ergence is achie¤ ed without affecting PML absorption. A robust preconditioned sol¤ er with nearly no breakdown possibilities is suggested, implemented, and tested for two microwa¤ e circuit applications. Q 1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 23: 166]172, 1999. Key words: ﬁnite-element methods; matched layer termination; microstrip line; spiral inductor 1. INTRODUCTION wx When introduced by Sacks et al. 1 , the perfectly matched Ž . layer PML absorber was considered a novel, efﬁcient, and Ž . reliable way to terminate ﬁnite-element FE computational domains. They offer ease of implementation coupled with wx excellent absorption characteristics 2. Unlike absorbing Ž . w x boundary conditions ABCs 3, 4 , PML truncation schemes do not require a priori knowledge of propagation constants within the computational domain, and the use of boundary derivatives is avoided altogether. Also, PML termination schemes facilitate de-embedding and parameter extraction wx 5 . Because of these advantages, PML absorbers have been w x extensively employed for truncating FEM domains 6, 7. However, PML absorbers represent active anisotropic materi- als, and their inclusion within the computational domain signiﬁcantly deteriorates the condition of the resulting FEM systems. Consequently, PML-truncated meshes have the un- wx desirable property of being slow to converge 8 . Two steps are proposed for improving solution conver- gence. First, an optimal selection of the PML parameters is given, providing a compromise between convergence and absorption. Second, a robust iterative solver is proposed to solve sparse FEM systems generated when the computational domain is terminated by the PML. Using the designed PML absorbers, we proceed to use them in truncating computa- tional domains for microwave circuit analysis. Two examples are considered: one deals with feed and input impedance characterization of microstrip lines, and the other considers analysis of a spiral inductor with an air bridge. 2. PML PARAMETERS Consider a wave incident upon the interface between two media as shown in Figure 1. Layer 2 is a uniaxial absorber with m and e representing its relative constitutive parame- r r ter tensors a 0 0 2 Ž. 0 b 0 m s e s . 1 2 r r  0 0 0 c 2 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 23, No. 3, November 5 1999 166