Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

The paper presents an overview and reports the recent progress of research on squeeze film air damping in MEMS. The review starts with the governing equations of squeeze film air damping: the nonlinear isothermal Reynolds equation and various reduced forms of the equation for different conditions. After the basic effects of squeeze film damping on the dynamic performances of micro-structures are discussed based on the analytical solutions to parallel plate problems, recent research on various aspects of squeeze film air damping are reviewed, including the squeeze film air damping of perforated and slotted plate, the squeeze film air damping in rarefied air and the squeeze film air damping of torsion mirrors. Finally, the simulation of squeeze film air damping is reviewed. For quick reference, important equations and curves are included.

Squeeze film air damping in MEMS

With the rapid developments in very large-scale integration (VLSI) technology, design and computer-aided design (CAD) techniques, at both the chip and package level, the operating frequencies are fast reaching the vicinity of gigahertz and switching times are getting to the subnanosecond levels. The ever increasing quest for high-speed applications is placing higher demands on interconnect performance and highlighted the previously negligible effects of interconnects such as ringing, signal delay, distortion, reflections, and crosstalk. In this review paper various high-speed interconnect effects are briefly discussed. In addition, recent advances in transmission line macromodeling techniques are presented. Also, simulation of high-speed interconnects using model-reduction-based algorithms is discussed in detail.

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Simulation of high-speed interconnects

In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a "training input." Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewise-linear reduced-order models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed "on the fly," to accelerate simulation of the system response.

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A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices

Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems

This paper describes an algorithm for generating provably passive reduced-order N-port models for RLC interconnect circuits. It is demonstrated that, in addition to macromodel stability, macromodel passivity is needed to guarantee the overall circuit stability once the active and passive driver/load models are connected. The approach proposed here, PRIMA, is a general method for obtaining passive reduced-order macromodels for linear RLC systems. In this paper, PRIMA is demonstrated in terms of a simple implementation which extends the block Arnoldi technique to include guaranteed passivity while providing superior accuracy. While the same passivity extension is not possible for MPVL, comparable accuracy in the frequency domain for all examples is observed.

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PRIMA: passive reduced-order interconnect macromodeling algorithm

Preface. 1. Introduction. 2. The Elmore Delay. 3. Higher-Order RC(L) Delay Metrics. 4. Asymptotic Waveform Evaluation. 5. Moment Generation. 6. Passive Reduced-Order Multiport Models. 7. Interfacing with SPICE. 8. Interfacing Interconnect and Gate-Delay Models. Index.

IC Interconnect Analysis

This paper describes PRIMA, an algorithm for generating provably passive reduced order N-port models for RLC interconnect circuits. It is demonstrated that, in addition to requiring macromodel stability, macromodel passivity is needed to guarantee the overall circuit stability once the active and passive driver/load models are connected. PRIMA extends the block Arnoldi technique to include guaranteed passivity. Moreover, it is empirically observed that the accuracy is superior to existing block Arnoldi methods. While the same passivity extension is not possible for MPVL, we observed comparable accuracy in the frequency domain for all examples considered. Additionally a path tracing algorithm is used to calculate the reduced order macromodel with the utmost efficiency for generalized RLC interconnects.

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Krylov space methods initiated a new era for RLC circuit model order reduction. Although theoretically well-founded, these algorithms can fail to produce useful results for some types of circuits. In particular, controlling accuracy and ensuring passivity are required to fully utilize these algorithms in practice. In this paper we propose a methodology for passive reduction of RLC circuits based on extensions of PRIMA, that is both broad and practical. This work is made possible by uncovering the algebraic connections between this passive model order reduction algorithm and other Krylov space methods. In addition, a convergence criteria based on an error measure for PRIMA is presented as a first step towards intelligent order selection schemes. With these extensions and error criterion, examples demonstrate that accurate approximations are possible well into the RF frequency range even with expansions about s=0.

http://www.cecs.uci.edu/~papers/compendium94-03/papers/1999/iccad99/pdffiles/03d_4.pdf

Practical considerations for passive reduction of RLC circuits

The Elmore delay is the metric of choice for performance-driven design applications due to its simple, explicit form and ease with which sensitivity information can be calculated. However, for deep submicron technologies, the accuracy of the Elmore delay is insufficient. In this paper, we formulate a delay model using a provably stable two pole waveform response that provides a unique mapping between four moments and a specific delay value. Unlike traditional moment matching, this two-pole model permits us to precharacterize the delays, and store them in a table, as a mapped function of three parameters. The model also provides an explicit expression for the peak noise induced on a coupled line as a function of the same three moments. The results indicate runtimes comparable to an Elmore delay calculation but with the accuracy of an AWE approximation.

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Altan Odabasioglu

Papers

PRIMA: passive reduced-order interconnect macromodeling algorithm

IC Interconnect Analysis

PRIMA: passive reduced-order interconnect macromodeling algorithm

Practical considerations for passive reduction of RLC circuits

S2P: A Stable 2-Pole RC Delay and Coupling Noise Metric