Abstract—In this paper, we describe the algorithms used in FastImp, a program for accurate analysis of wide-band electromagnetic effects in very complicated geometries of conductors. The program is based on a recently developed surface integral formulation and a precorrected fast Fourier transform (FFT) accelerated iterative method, but includes a new piecewise quadrature panel integration scheme, a new scaling and preconditioning technique as well as a generalized grid interpolation and projection strategy. Computational results are given on a variety of integrated circuit interconnect structures to demonstrate that FastImp is robust and can accurately analyze very complicated geometries of conductors. Index Terms—Fast integral equation solver, panel integration, parasitic extraction, preconditioner, surface integral formulation, wide-band analysis. I.

In order to optimize high-speed systems, designers need tools that automatically generate reduced-order SPICE compatible models from geometric descriptions of interconnect and packaging. In this paper, we consider structures small compared to a wavelength, and use a discretized integral formulation combined with an Arnoldi-based model-order reduction strategy to compute efficiently accurate reduced-order models from three-dimensional (3-D) structures. Several issues are addressed including: 1) formulation to insure passivity in the reduced-order models; 2) efficient reduction using preconditioned inner-loop iterative methods; 3) expansion about multiple s-domain points.

A new method, Fourier model reduction (FMR), for obtaining stable, accurate, low-order models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discrete-time Fourier coefficients of the large system transfer function are calculated and used to construct the Hankel matrix of an intermediate system with guaranteed stability. Explicit balanced truncation formulae are then applied to obtain the final reduced-order model, whose size is determined by the Hankel singular values of the intermediate system. In this paper, the method is applied to two computational fluid dynamic systems, which model unsteady motion of a twodimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, guarantees the stability of the reduced-order model, uses both input and output information, and is valid over a wide range of frequencies.

Interconnect structures including dielectrics can be modeled by an integral equation method using volume currents and surface charges for the conductors, and volume polarization currents and surface charges for the dielectrics. In this paper we describe a mesh analysis approach for computing the discretized currents in both the conductors and the dielectrics. We then show that this fully meshbased formulation can be cast into a form using provably positive semidefinite matrices, making for easy application of Krylovsubspace based model-reduction schemes to generate accurate guaranteed passive reduced-order models. Several printed circuit board examples are given to demonstrate the effectiveness of the strategy.

The ability to compute the parasitic inductance of the interconnect is critical to the timing verification of modern VLSI circuits. A challenging aspect of inductance extraction is the solution of large, dense, complex linear systems of equations via iterative methods. Accelerating the convergence of the iterative method through preconditioning is made difficult due to the non-availability of the system matrix. This paper presents a novel algorithm to solve these linear systems by restricting current to a discrete solenoidal subspace in which Kirchoff's law is obeyed, and solving a reduced system via an iterative method such as GMRES. A preconditioner based on the Green's function is used to achieve near-optimal convergence rates in several cases. Experiments on a number of benchmark problems illustrate the advantages of the proposed method over FastHenry.

As very large scale integration (VLSI) circuit speeds and density continue to increase, the need to accurately model the effects of three-dimensional (3-D) interconnects has become essential for reliable chip and system design and verification. Since such models are commonly used inside standard circuit simulators for time or frequency domain computations, it is imperative that they be kept compact without compromising accuracy, and also retain relevant physical properties of the original system, such as passivity. In this paper, we describe an approach to generate accurate, compact, and guaranteed passive models of RLC interconnects and packaging structures. The procedure is based on a partial element equivalent circuit (PEEC)-like approach to modeling the impedance of interconnect structures accounting for both the charge accumulation on the surface of conductors and the current traveling in their interior. The resulting formulation, based on nodal or mixed nodal and mesh analysis, enables the application of existing model order reduction techniques. Compactness and passivity of the model are then ensured with a two-step reduction procedure where Krylov-subspace moment-matching methods are followed by a recently proposed, nearly optimal, passive truncated balanced realization-like algorithm. The proposed approach was used for extracting passive models for several industrial examples, whose accuracy was validated both in the frequency domain as well as against measured time-domain data.

...................................................................................................................................... viii List of publications .......................................................................................................................ix Chapter 1: Introduction..................................................................................................................1 1.1 Scope and overall research contribution..............................................................................1 1.2 Motivation............................................................................................................................2 1.2.1 The interconnect problem .............................................................................................2 1.2.2 Capabilities and limitations of electrical interconnects................................................4 1.2.3 Advantages of optical interconnects ......................................

The ability to compute the parasitic inductance of the interconnect is critical to the timing verification of modern VLSI circuits. A challenging aspect of inductance extraction is the solution of large, dense, complex linear systems of equations via iterative methods. Accelerating the convergence of the iterative method through preconditioning is made difficult due to the non-availability of the system matrix. This paper presents a novel algorithm to solve these linear systems by restricting current to a discrete solenoidal subspace in which Kirchoff’s law is obeyed, and solving a reduced system via an iterative method such as GMRES. A preconditioner based on the Green’s function is used to achieve near-optimal convergence rates in several cases. Experiments on a number of benchmark problems illustrate the advantages of the proposed method over FastHenry. Categories and Subject Descriptors B.7.2 [Integrating Circuits]: Design Aids—placement and routing, simulation, verification