IEEE Proof IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL. 52, NO. 3, MARCH 2004 1 Generalized Poisson–Neumann Polygonal Basis Functions for the Electromagnetic Simulation of Complex Planar Structures Luc Knockaert, Senior Member, IEEE, Jeannick Sercu, Member, IEEE, and Daniel De Zutter, Fellow, IEEE Abstract—Rooftop functions are commonly used for the dis- cretization of planar currents in electromagnetic (EM) simulators. We describe the generalization of the rectangular and triangular rooftop functions to arbitrary polygonally shaped subdomains. It is shown that these generalized basis functions are solutions to a pertinent Neumann–Poisson problem, and we derive the integral equations satisfied by these basis functions. The new generalized polygonal functions allow for a more efficient meshing of com- plex geometrical structures in terms of polygonally shaped cells. They naturally model the current flow in the polygonal cells, sat- isfy the current continuity relation, and significantly enhance the EM simulation performance for complex geometrical structures. The increased simulation performance is demonstrated for a com- plex radio-frequency board interconnection layout and a spiral in- ductor on a silicon substrate. Index Terms—Basis functions, complexity reduction, electro- magnetic (EM) simulation, planar structures, Poisson–Neumann problem. I. INTRODUCTION O VER THE past decade, planar electromagnetic (EM) sim- ulators 1 have been extensively used for the time-harmonic characterization of planar structures in radio-frequency (RF) board microwave circuit and antenna applications. The EM be- havior of the planar structure is governed by an integral equation in the unknown surface currents flowing on the planar metalliza- tion patterns. This integral equation is solved numerically by ap- plying the method of moments (MoM). The planar structure is therefore typically subdivided or discretized into a mesh of rect- angular and/or triangular cells. The fundamental reason for the triangular and/or rectangular subdivision scheme is that the cur- rents can then be expanded in a basis of simple linear divergence and curl-conforming vector functions [1], also called rooftop functions or surface doublets [2]–[4], which satisfy the perti- nent continuity equations and exhibit a locally constant charge distribution. The most important current continuity condition, Manuscript received June 30, 2003; revised Ocotber 1, 2003. This work was supported by the Flemish Institute for the Promotion of Scientific-Technological Research in the Industry. L. Knockaert and D. De Zutter are with the Department of Information Technology–Interuniversity Microelectronics Centre, University of Ghent, B-9000 Gent, Belgium (e-mail: eknokaert@intec.UGent.be). J. Sercu is with the EEsof Electronic Design Automation Division, Agilent Technologies, 9000 Gent, Belgium Digital Object Identifier 10.1109/TMTT.2004.823577 1 Examples include em (Sonnet Software, Liverpool, NY), Momentum (Agi- lent Technologies, Palo Alto, CA), IE3D (Zeland Software, San Francisco, CA), and Ensemble (Ansoft Corporation, Pittsburgh, PA). in order for the currents to locally satisfy Kirchoff’s law, is for the normal component of the current to be continuous across the boundaries of adjacent cells in the mesh. The above-mentioned rooftop functions associated with rect- angular or triangular cells [5], [6] can be briefly described as follows. One vector function, which models the normal com- ponent of the current flowing across the cell side, is associated with each side of the cell. This vector function is constant along the corresponding side and varies linearly to zero along the ad- jacent sides of the cell. Rooftop functions with a rectangular subdomain have only one component in the direction normal to the correponding cell side. Rooftop functions with triangular support, however, also have a component tangential to the cell side. This component is necessary to obtain the continuity of the normal current across the adjacent cell sides in the triangular cell. The next step, after triangles and rectangles, would seem to be quadrilaterals. Unfortunately enough, for general quadri- laterals, there do not exist linear rooftop functions with locally constant charge distribution that satisfy all of the relevant con- tinuity requirements [7], [8]. In this paper, we describe the generalization of the rectan- gular and triangular rooftop functions to arbitrary polygonally shaped subdomains. In mathematical terms, it is shown that these generalized basis functions are solutions to a pertinent Neumann–Poisson problem. These generalized polygonal vector functions naturally model the current flow in a polyg- onal cell and by definition satisfy the normal current continuity condition across the edges of the cell. When used as current basis functions in a MoM numerical solution algorithm, they significantly enhance the simulation performance for complex geometrical structures, as will be demonstrated in Sections III and IV. Parts of the results of this paper were earlier very succinctly presented in [9], [10]. II. MATHEMATICAL FRAMEWORK The commonly used triangular and rectangular rooftop func- tions are locally curl-free with a locally constant charge den- sity [1]–[4]. In order to generalize these functions to cells with a more general shape, the pertinent question is: can we find a curl-free current density with constant divergence over a general simply connected domain such that its flux has pre- assigned values on its piecewise smooth boundary ? In other words, we need (1) 0018-9480/04$20.00 © 2004 IEEE