This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Signal Integrity Analysis of Integrated Circuits by Using Embedded Domain Decomposition Method Jiaqing Lu and Jin-Fa Lee, Fellow, IEEE Abstract— In integrated circuit designs, adjustments and replacements of components are frequently required for analyz- ing or improving signal integrity. However, the modifications of geometry will cause remeshings and resimulations in the full-wave analysis. Consequently, the design efficiency can be impaired due to repeated discretizations. In this paper, we introduce an embedded domain decomposition approach to address such repetitive simulations. A problem is first decomposed into a simpler background subdomain and multiple embedded subdo- mains. The geometrical details of interest can be placed into the embedded subdomains while the background subdomain covers the rest. Afterward, the communications between the subdomains are built by imposing coupling sources that represent field continuities, material polarizations, and surface currents on perfect electric conductor and port. The meshes between the subdomains are completely nonconformal, and the modification of an embedded subdomain will not affect the geometries and discretizations in the other subdomains. Such flexibility can be exploited in simulating complicated 3-D printed circuit board structures or investigating different components in an electronic system, as illustrated in the numerical examples. Index Terms— Domain decomposition method (DDM), embedded meshes, finite-element method (FEM), integrated circuits (ICs), signal integrity (SI). I. I NTRODUCTION T HE past several decades have witnessed a continuing downscaling of integrated circuit (IC) feature sizes. To take advantage of this advancement, IC designers are integrating more components and functional blocks into a single electronic system in the forms of various 3-D stacking technologies. The ever-increasing device densities and com- plexities, together with higher clock frequencies, impose grow- ing challenges in the efficient prediction of signal integrity (SI) in IC designs and analyses. Equivalent circuit simulation [1]–[3], such as SPICE [4] and behavioral modeling [5], plays an essential role in IC engineering because it runs fast and is easy to use. However, electromagnetic (EM) field effects in high frequencies, such as the skin effect and signal crosstalk, cannot be accurately captured by pure circuit modeling. This is especially true when Manuscript received May 2, 2018; revised August 8, 2018; accepted September 19, 2018. This work was supported by Ansys, Inc. Pittsburgh, PA, USA. This paper is an expanded version from the IEEE MTT-S International Microwave Symposium (IMS2018), Philadelphia, PA, USA, June 10–15, 2018. (Corresponding author: Jiaqing Lu.) The authors are with the ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212 USA (e-mail: lu.987@osu.edu; lee.1863@osu.edu). 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/TMTT.2018.2874639 3-D device packaging and dispersive materials are used in circuit designs. In contrast, full-wave simulation inherently includes high-frequency phenomena such as EM coupling, radiation, and attenuation, thus giving a better description of the SI performance of a complicated physical design. Among the full-wave numerical methods for IC and pack- age simulations, the finite-difference method [6]–[9] and the finite-element method (FEM) [10]–[13] have been well studied and documented. Finite-difference method has been extensively used due to its efficiency and simplicity. How- ever, it suffers from numerical dispersion errors and low- order approximation of complicated objects. By contrast, FEM offers the robust and accurate modeling of complex geometries and, therefore, is preferable when sophisticated 3-D systems need to be solved. Nevertheless, conventional FEM requires a globally conformal mesh to be constructed before simulation. Because printed circuit boards (PCBs), packages, and chips usually contain numerous fine fea- tures such as signal traces and 3-D interconnects, it will demand tremendous efforts to generate a good-quality mesh for all the multiscale details. To alleviate the burden in discretizing complex geometries, the FEM-based nonover- lapping domain decomposition method (DDM) [14]–[22] has been developed in the past several years. The method gains its efficiency by breaking a problem into several disjoint and more manageable subproblems. The subproblems are meshed and solved independently, and adjacent subproblems are coupled through transmission conditions (TCs) on their touching interfaces. Owing to its inherent parallelism and flexibility, nonoverlapping DDM has been successfully applied to IC problems in [23]. In the designs of PCBs and electronic packages, to meet certain technical requirements, it is often necessary to per- form parameter optimizations for the shapes and materials of some components, identify the functionalities of certain struc- tures, or add/replace a functional block. As a consequence, we have to conduct the simulations of an electronic sys- tem repetitively. Consider a PCB-package-chip model shown in Fig. 1. The model contains a variety of small geometrical entities such as via holes, soldering balls, and bonding wires. To ensure a good signal transmission quality from the PCB side (port 1) to the chip side (port 2), or vice versa, we may need to adjust the diameters and shapes of the bonding wires, change the distance between the traces or add some extra vias. Unfortunately, in the conventional FEM, a simple geometrical modification will break the global conformity of the mesh, resulting in model remeshing and system matrix 0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.