576 PIERS Proceedings, Cambridge, USA, July 2–6, 2008 Simulation of Multiscale Circuit Problems Using Equivalence Principle Algorithm Mao-Kun Li 1 , Weng Cho Chew 2 , and Zhi Guo Qian 3 1 Schlumberger-Doll Research, Cambridge, MA, USA 2 The University of Hong Kong, Hong Kong, China 3 University of Illinois at Urbana-Champaign, Urbana, IL, USA Abstract— The equivalence principle algorithm is introduced in this paper to solve multiscale problems in the electromagnetic modeling of circuit problems. In these problems, low frequency circuit physics coexists with high frequency wave physics that makes the matrix equation ill- conditioned. As a domain-decomposition based algorithm, the equivalence principle algorithm casts the entire domain into different parts based on their physical properties. Each part is solved individually by introduction of virtual equivalence surfaces. Moreover, the interactions among domains are still modeled accurately that captures the electromagnetic wave propagation. With this algorithm and powered up with multilevel fast multipole algorithm, large multiscale problems can be solved on a single computer efficiently. 1. INTRODUCTION As technological devices become smaller, faster, and more complex, computer simulations of phys- ical phenomena become an increasingly important part of the design process. This makes compu- tational electromagnetic (CEM) simulations indispensable. However, many challenges arise when applying CEM solvers to microelectronic structures. Multiscale phenomena are some of them. A single device can be very tiny and dominated by circuit physics, while thousands of devices can extend to several wavelengths in high frequencies that are dominated by wave physics. The existence of both wave and circuit physics introduces large and small eigenvalues simultaneously resulting an ill-conditioned matrix equation. One kind of methods uses multi-resolution basis set to describe different physics such as [1]. The other approach separates different physics via domain decomposition such as [2]. The references here are far more from complete due to space limitation. Equivalence principle algorithm (EPA) is essentially a domain decomposition scheme based on equivalence principle and integral equations. By the introduction of virtual equivalence surfaces to enclose the regions with fine features, low frequency physics is isolated from high frequency physics. This results in a better conditioned matrix equation with fewer unknowns. Compared with other integral-equation-based domain decomposition scheme, such as boundary element tearing and interconnecting methods, EPA still allows the interactions between any two basis functions, hence it can catch wave propagation with less dispersion error. To model continuous current flow in and out of the equivalence surface, a tap basis scheme was introduced. This scheme avoids the computation of current singularities at the cut and can be incorporated with EPA naturally. In the errors of EPA, a main factor is the high-frequency noise in field projection onto equiv- alence surfaces. This noise comes from the discretization of the equivalent currents and becomes dominant when objects are very close to the equivalence surfaces. To recover the accuracy, the high-order quadrature point-sampling scheme was used on the equivalence surfaces. This scheme samples the currents directly at points on the equivalence surfaces and integrates using high-order quadrature rules. By using this, EPA is shown to be accurate. The translation procedure in EPA can be accelerated using attached unknown accelerations when equivalence surfaces contact with each other. The current interactions on the touched regions can be computed using local interpo- lations instead of field integrations. This saves the time in both matrix filling and matrix-vector multiplications. Moreover, the efficiency of EPA can be further improved when accelerated using multilevel fast multipole algorithm (MLFMA). With a proper preconditioning scheme and a careful balancing of the equation, large multiscale problems of over 3 million unknowns can be computed on a personal computer. 2. EQUIVALENCE PRINCIPLE ALGORITHM EPA is an integral equation solver derived from the equivalence principle. The basic notations and formulations can be found in [3]. There are mainly two operators used in EPA: the equivalence