760 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 4,NOVEMBER2010 Waveform Relaxation Time Domain Solver for Subsystem Arrays Giulio Antonini, Senior Member, IEEE, and Albert E. Ruehli, Life Fellow, IEEE Abstract—In this paper we present a waveform relaxation ap- proach for the transient analysis of 3-D electromagnetic problems using the partial element equivalent circuit (PEEC) method. Re- lying on weaker couplings among separated systems, a waveform relaxation scheme is proposed to accelerate the transient analysis of large electromagnetic problems. The results are compared with those obtained using a conventional PEEC formulation. They ex- hibit a significant speed-up while preserving the solution accuracy. Index Terms—Partial element equivalent circuit (PEEC) method, partitioning of circuits, transient analysis, waveform relaxation technique. I. INTRODUCTION T ODAY, a large number of classical techniques and solvers are available for the solution of conventional electro- magnetic problems. This has greatly increased the ability to solve relevant design issues for integrated circuit, package and antenna problems. However, remaining challenges in this area are represented by the solution of larger problems in both the time and frequency domain. New techniques exist today for large problems based on integral equations. Most well-known approaches are iterative matrix solution-based [1], [2], mul- tipole techniques [3], [4], and singular value decomposition (SVD) and QR-based reduction methods [5], [6]. There is also a growing need in industry for the solution of large, complex combined circuit and electromagnetic problems. Issues which have limited the solution of such large prob- lems in the past are due to memory size limitations and the per- formance of single processor machines. Fortunately, lower cost multiprocessor chips are now available for parallel processing. Also, the cost of memory is reduced. So we can assume that computers with multicore processors are available for all appli- cations. Unfortunately, many conventional algorithms are not suitable for a parallel processing environment. Hence, new algo- Manuscript received December 31, 2009; revised June 29, 2010; accepted July 18, 2010. Date of publication August 05, 2010; date of current version January 07, 2011. This work was supported by the Italian Ministry of University (MIUR) under a Program for the Development of Research of National Interest, (PRIN grant n. 2006095890). This work was recommended for publication by Associate Editor D. Jiao upon evaluation of the reviewers comments. G. Antonini is with the UAq EMC Laboratory, Dipartimento di Ingegneria Elettrica e dell’Informazione, Università degli Studi dell’Aquila, Monteluco di Roio, 67040, L’Aquila, Italy (e-mail: giulio.antonini@univaq.it). A. E. Ruehli is an Emeritus of the IBM T. J. Watson Research Center, York- town Heights, NY 10598, USA and an Adjunct Professor at the Missouri Uni- versity of S&T in Rolla (e-mail: albert.ruehli@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TADVP.2010.2061229 rithms must be found which are suited for the solution of large problems using parallel processors. Such solutions clearly are based on subdividing a problem into subproblems so that mul- tiple processors can be applied. The partial element equivalent circuit (PEEC) technique, which provides 3-D electromagnetic models in the circuit domain, has evolved over the years. Much progress has been made in the capabilities of electromagnetic PEEC modeling from its origin [7] to numerous extensions including volume and surface-based techniques and non-orthogonal models [8]. It is a natural approach for the solution of combined circuits and electromagnetic problems in the circuit domain. A choice of the most efficient solution techniques is of im- portance for large problems since algorithms often are tailored to a special situation. In this paper, we address a special sub- class of problems that consist of a potentially very large number of subsystems which do not exhibit very strong direct connec- tions. The approach we are applying is of the subclass of domain composition methods called waveform relaxation (WR), which was first developed in [9]. Further, a more detailed description of WR techniques is presented in [10]. Today, new application areas are considered for the WR techniques besides the class of circuit problems which were solved in the early stage of the ap- proach [9]. WR has also been applied to 2-D transmission line problems, e.g., [11]–[14]. Recently, the WR technique has been sucessfully used to model multiple coupled transmission lines by exploiting the partitioning due to weak transverse couplings [15], [16]. Earlier, WR was also applied to analyze a simple an- tenna model in the time domain [17]. Here, we consider the application of WR for electromagnetic problems in the time domain. The approach is directly suit- able for parallel processing. We use the PEEC method, which works for the entire frequency spectrum. Hence, it also provides a dc solution in both the time as well as the frequency domain. This approach utilizes the conventional modified nodal analysis (MNA) circuit solution technique to solve the resultant circuit equations. Further, if a model problem contains nonlinearities, then it is best solved in the time domain. In this paper we con- sider only linear homogeneous problems with a large number of coupled sub-systems (SSy) that are based on a similar circuit topology. However, different models for a SSy can be mixed in the WR approach. The only condition which we need to meet is that the subsystem coupling for the partitioned problem is below an upper bound given by the WR iteration. Fortunately, we will show that this convergence bound can be large from an elec- tromagnetics coupling point of view. In the combined WR-PEEC solution approach ((WR)PEEC), the original system is subdivided into a large number of smaller 1521-3323/$26.00 © 2010 IEEE