AGIBC Formulation for Lossy Conductor Modeling Zhi-Guo Qian 1 Mei Song Tong 1 Weng Cho Chew 2 1 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e-mail: zqian2@illinois.edu, meisongt@illinois.edu, tel.: +1 217 3330488, fax: +1 217 244734. 2 Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e-mail: w-chew@illinois.edu, tel.: +1 217 3337309, fax: +1 217 2447345 He is currently on LOA to serve as the Dean of Engineering at The University of Hong Kong Abstract - This paper describes an augmented generalized impedance boundary condition (AGIBC) formulation for accurate and efficient modeling of lossy conductors. It is a surface integral equation method, so that it uses a smaller number of unknowns. The underlying GIBC provides a rigorous way to account for the skin effect. Combining with the novel augmentation technique, the AGIBC formulation works stably in the low-frequency regime. No loop-tree search in required. The formulation also allows its easy incorporation of fast algorithms to enable the solving of large problems with many unknowns. Numerical examples are presented to validate the formulation. 1 INTRODUCTION Conductive materials are widely encountered in real- world electromagnetic problems, such as high-speed interconnect, antenna, scattering, well-logging and subsurface detection. Though various computational electromagnetic methods have matured, it is still nontrivial to model conductors rigorously using a full-wave solver. Due to the skin effect, the electromagnetic fields penetrate deeply into the conductor at low frequencies, but decay rapidly at high frequencies. The small skin depth incurs a large number of unknowns for numerical methods requiring a discretization of conductive medium, so that the computation is prohibitively expensive. Most popular methods thus resort to an impedance boundary condition (IBC) [1], which approximates the effect of the conductive medium to avoid the fine discretization. However, these IBCs are usually derived from a simplified 2-D analysis. They are never rigorous and may cause large error for strongly coupled structures. What is more, different pieces of the conductor require different surface impedance expressions. It causes the corresponding bookkeeping to be very tedious. Surface integral equations (SIEs) can model conductive medium rigorously by treating it as a general complex medium and formulating integral equations for it. The effect of the conductive medium is thus accounted by the lossy Green’s function. To this end, a generalized impedance boundary condition (GIBC) has been introduced in the literature [2]. Numerical examples demonstrate that GIBC can model conductors with good accuracy as well as high efficiency. It also elucidates the relation between the rigorous model and the approximate model using surface impedance with a two-step approximation. Previously, a loop-tree decomposition is needed to remedy the low-frequency breakdown of the dominating EFIE operator. However, the search for loop basis is very difficult for complex structures. To tackle real-world complex problems efficiently, an augmented electric field integral equation (A- EFIE) [3] was proposed to replace the loop-tree decomposition based EFIE. In this paper, we combine GIBC with A-EFIE to formulate an augmented GIBC (AGIBC) method. It can model conductors rigorously and is also stable in the low-frequency regime. 2 FORMULATION First, the notations of integral equations are introduced for the derivation. Second, GIBC is briefly reviewed. At last, the combination of GIBC and A-EFIE is shown in details, and the solution scheme is discussed. 2.1 Notations SIEs only consider the surface of an object. For a general multi-object problem, the notations of the surfaces and regions are illustrated in Fig. 1. Assuming that each surface contains a homogeneous medium, all N surfaces are numbered from 1 S to N S , and the volume enclosed by S ν is indexed as V ν . The outermost region is denoted as 0 V , whose outer boundary is thus 0 S , infinitely far away. In region V ν , the permittivity and permeability are ν ε and ν μ , respectively. The wave impedance is 1 ν ν ν η με - = . The index ν runs from 0 to N . The wave number in free space is k ν ν ω με = . The equivalence currents exist on all surfaces. On surface S ν , the electric current is ν J , and the magnetic current is ν M . There could be excitation in any region. In region V ν , the incident electric field is 978-1-4244-3386-5/09/$25.00 ©2009 IEEE 182