1750 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL. 49, NO. 10, OCTOBER2001 A Stable and Efficient Admittance Method Via Adjacence Graphs and Recursive Thresholding Luciano Tarricone and Mauro Mongiardo Abstract—The generalized admittance method is a rigorous full-wave approach for the analysis of waveguide circuits. Unfor- tunately, it may present the risk of ill conditioning, especially when very complex structures are analyzed with a considerably high number of modes. In this paper, the concepts of adjacence graph and recursive thresholding are proposed to solve its numerical problems. By applying the proposed strategy, the linear system representing the core of the analysis is partitioned into many independent and well-conditioned subsystems, thus improving the numerical stability of the approach and its efficiency. The attractive features of the proposed approach are its simplicity and immediate implementation. Results are given, referred to a real industrial case, a complex -plane filter, whose analysis could not be performed via a standard admittance method when a very high number of modes were considered. With the present approach, the ill conditioning is avoided and considerable enhancements in computing times is achieved. Index Terms—Adjacence graphs, generalized admittance method, recursive thresholding, sparse matrices. I. INTRODUCTION T HE complexity of microwave (MW) waveguiding circuits and components is continually increasing. This is due partly to the technological evolution and partly to the rapid progress in computer-aided design (CAD) and computer ma- chines, which allows the analysis of quite complex structures. As a consequence, the development of numerical methods, efficient and accurate, for the analysis of MW circuits, is becoming crucial. Several approaches have been proposed in the literature for a full-wave rigorous solution of the problem. Among them, the mode-matching (MM) approach is extremely attractive and probably the most used for this class of problems for its high efficiency and accuracy. Thus far, many different formulations of this approach have been presented [1]–[3]: they are all accurate and more or less performed depending on the characteristics of the problem. More recently, a new formulation has been proposed based on the use of the generalized admittance matrix (GAM) method [4]–[7], which considers a MW circuit as composed of par- allel-epipedal elementary cells connected one another. The prin- ciple is a “divide and conquer” strategy, in accordance with a common trend in the analysis of very complex circuits, based on a problem segmentation into subdomains [8], [9], possibly analyzed with different methods [10], [11]. This methodology is also open to an efficient and smooth migration toward par- Manuscript received August 14, 2000. The authors are with the Dipartimento di Ingegneria Elettronica e dell’Infor- mazione, Università di Perugia, 06125, Perugia, Italy. Publisher Item Identifier S 0018-9480(01)08669-0. allel or distributed environments, which is often the only way to solve very large and complex circuits in an affordable time given the constraints of industrial design [12]. It has been shown [13] that one of the most noticeable advan- tages of the admittance method is that it allows to use the adjoint network formulation [14] for optimization problems. This re- sults in the generation of a relatively “large” matrix, which con- tains all the relevant information. An open problem in the use of the GAM formulation in industrial software tools for MW engi- neering is the existence of risks of numerical ill conditioning in the solution of a large linear system representing the core of the approach. In fact, for computation accuracy purposes, a large number of localized modes are often considered at each discon- tinuity, thus resulting in the GAM ill conditioning. Unfortunately, the elimination of higher order modes corre- sponds to a rather costly (in terms of efficiency) operation since it requires terminating all the considered higher order modes with their characteristic impedance. In this paper, it is shown that, with the use of the adjacence graph (AG) and recursive thresholding (RT), the latter operation can be avoided and the global linear system can be partitioned into independent well-conditioned subsystems. This way, the complexity of the problem is reduced, with a consequent improvement in computing times. The paper is structured as follows. In Section II, the GAM for- mulation and its numerical properties are briefly resumed and, in Section III, the AG–RT strategy is proposed to solve its numer- ical problems. In Section IV, results are given for a real complex industrial case. Finally, conclusions are drawn. II. GAM FORMULATION The method has been described in several publications [4]–[7] and the reader is referred there for details. The GAM formulation is based on the partitioning of a metallic waveguide complex structure into simple volume elements, to be analyzed independently. The numerical description of the single cell is given in terms of an admittance-type matrix so that the problem of the inter- connection among blocks can be handled in merely circuital terms, by imposing to the equivalent voltages and currents of each cell’s ports the constraints arising from the overall circuit topology. The analysis of a complex structure is, therefore, re- duced to that of the overall equivalent network resulting from the interconnection of its constituent elements [15]. A. GAM of a Resonant Cavity The single-cell analysis, leading to its -matrix characteri- zation, can be performed in a general fashion, by regarding the 0018–9480/01$10.00 © 2001 IEEE