Parameter Preserving Model Reduction for MEMS System-level S

Abstract. Model reduction is a very helpful tool to generate compact models for system-level simulation. Quite often however, system matrices depend on design parameters and the new goal is not only to reduce the original system but also to preserve system

Parameter Preserving Model Reduction for MEMS System-level Simulation and Design

E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink

IMTEK, Freiburg University, Germany

Corresponding Author: J. G. Korvink

Lab of Simulation, Department of Microsystems Engineering (IMTEK), University of Freiburg

Georges-Köhler-Allee 102 00 086, 79110 Freiburg, Germany

Phone: +49 761 203 7436, Fax: +49 761 203 7437

email: korvink@imtek.uni-freiburg.de

Abstract. Model reduction is a very helpful tool to generate compact models for system-level simulation. Quiteoften however, system matrices depend on design parameters and the new goal is not only to reduce the originalsystem but also to preserve system parameters in the symbolic form during model reduction. We introducemultivariate moment matching as a possible solution to this problem. We consider several examples fromMEMS to demonstrate the feasibility of the approach: a device cooled by airflow, a microhotplate, a flow meter(anemometer) and a microelectrode. Finally, we discuss problems that should be overcome in order to use thistechnique in software for engineering design applications.

1. Introduction

The microelectronic industry enjoys tremendous productivity levels due to its high level of design automation(EDA). This is possible because the industry has agreed on how design should progress, and what the futurepriorities are in terms of necessary achievements. In microsystem and nanosystem development this has not yethappened [1][2][3], mainly because the industry is not yet mature enough, but especially because themicrosystem design automation industry is in its infancy.

The key to success in design automation is an accurate compact model of the MEMS/NEMS device. Yet, ithappens that conventional compact modeling does not work well for the MST area where the number of differentdevices is too big to hope that one can apply a simple empirical approach. Here a community working on aparticular device just does not have researchers with enough experience and intuition to develop compactmodels. And when the compact model is finally developed, it well may be that the interested parties have alreadyswitched to another technology.

Model order reduction is a rapidly developing interdisciplinary area [4][5][6]. There is considerable progress inthe application of modern model reduction to MST for the last five years and, in our opinion, model reducitoncan be considered as Compact Modeling on Demand.

It so happens that a high-dimensional ordinary differential equation system, as generated from e.g. asemidiscretized finite element model, possesses an inherent mathematical property that allows us to drasticallyreduce its dimension without sacrificing the precision of solution. Mathematically speaking, this is due the rapiddecay of the system Hankel singular values [4]. There is much evidence that this is the case for most discretizedMEMS models and the savings in computational speed are dramatic.

Model reduction of a linear system of ODEs can be considered as almost a solved problem. In this case, modelreduction gradually becomes a common practice among engineering groups. There are good chances that thisfeature will be available in commercial tools in a few years.

However, conventional model reduction fails to preserve parameters during model reduction process. This limitsseverely its applicability for the design flow and system-level simulation. In the present paper we consider howone can overcome this limit. We start by a short overview of available approaches from the literature. Then, wereview our results scattered over several conference papers. We present three engineering MEMS applicationsthat require us to preserve parameters in a compact model and review results on parametric model reduction forthese devices. After that, we discuss how to choose moments to include into the reduced model automatically.

2. Overview of Parametric Model Reduction

Because of its semi-empirical nature, compact modeling allows us to include some parameters in the symbolicform. For example, compact transistor models include some geometry parameters [7]. This is possible to some

Abstract. Model reduction is a very helpful tool to generate compact models for system-level simulation. Quite often however, system matrices depend on design parameters and the new goal is not only to reduce the original system but also to preserve system

extent because compact modeling always includes a parameterization step when numerical values of unknowncoefficients are found based on data fitting to experimental curves. Unfortunately, the process is hard to use inpractice as it is inherently based on intuition.

A formal approach related to parametric model reduction is a reduced-basis method [8][9][10]. An idea is toobtain several solutions distributed in the parameter space and then to use them to estimate a solution for anarbitrary point in the parameter space. Yet, the method is limited to a stationary problem.

In our view, the best choice is the multivariate Pade-type approximation that is a natural generalization of themoment matching method in conventional model reduction [5][6]. It was first suggested in [11] for anelectromagnetic problem and then employed for interconnect modeling in [12][13].

Let us consider the last approach in more detail. The discretization in space (for example, by the finite elementmethod) leads to a system of ordinary differential equations as follows

Edx(t)=Ax(t)+Bu(t),dt

y(t)=Cx(t)(1)

where x(t) is the vector of unknowns. E …… 此处隐藏：26390字，全部文档内容请下载后查看。喜欢就下载吧 ……