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

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 and A are the system matrices, B is the input matrix, and C is theoutput matrix. The vector u comprises inputs functions. The output matrix specifies particular linearcombinations of x(t) that of interest to an engineer.

The difference from conventional model reduction is that the system matrix A contains some parameters to bepreserved during model reduction. In the linear case, one can partition the original matrix as follows

A=A0+∑ikiAi,(2)

where ki is the parameter to be preserved. The transfer function of (1) with the condition (2)

H(s)=C{sE (A0+∑ikiAi)} 1B,(3)

in addition to Laplace variable s contains the parameter ki.

Model reduction is based on an assumption that there exists a low-dimensional subspace V that accuratelyenough captures the dynamics of the state vector x(t):

x≈Vz.(4)

In the moment matching approach [5][6], the subspace V is found in such a way as to preserve moments of thetransfer function (3) in respect to the Laplace variable s around some value s0 chosen a priori. Yet, thesemoments depend on ki and, as result, conventional subspace V does not work well in the case of Eq (2).In order to make V not depend on ki, the transfer function (3) can be treated as a function in many variables (sand ki) and one can perform its multivariate expansion. The next step is to find V so that to match multivariatemoments [11][12][13]. This way, V does not depend on parameters in Eqs (2) and (3).

Provided V is known, one obtain a low-dimensional model by projecting (1) and (2) on V as follows

VTEVdx(t)={VTA0V+dt

y(t)=CVz(t)∑ikiVTAiV}z(t)+VTBu(t).(5)

Eq (5) preserves the original parameters ki and as a result we call this approach as parametric model reduction.

3. MEMS Case Studies

We have chosen three important MST applications that require parametric model reduction expressed by Eq (1)and (2). They are described below.

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

A. Compact thermal models for electro-thermal simulation

In the development of integrated circuits and microsystems, thermal management is always essential [14][15].There are several electro-thermal and thermo-electric coupling effects but probably the biggest concern is aboutthe Joule heating, which generates heat during conduction of the electric current through a resistor. In anintegrated circuit, one has to remove the generated heat to keep the board temperature within acceptable limits.In microsystems, the Joule heating is often employed to keep a designated part (hotplate) at a given elevatedtemperature. In any case, the right temperature regime is crucial for the correct system functioning and itsreliability.

The finite element method allows us to make an accurate model to describe heat transfer but its highdimensionality prevents engineers to employ it during system-level simulation. Hence, an important practicalquestion is how one can make accurate but low-dimensional thermal models.

The two European projects, DELHPI and PROFIT have addressed this need: to produce an accurate but a smallthermal model of a chip [16][17][18]. The DELHPI project has identified a number of requirements for acompact thermal model, one of the most important being that the compact model must be boundary conditionindependent. This means that a chip producer does not know conditions under which the chip will be used andhence the chip compact thermal model must allow an engineer to research on how the change in the environmentinfluences the chip temperature. The chip benchmarks representing boundary condition independentrequirements have been described [18].

Recently, modern methods of model order reduction have been successfully applied to automatically generate acompact thermal model [19][20][21][22], however they do not meet the criterion of boundary conditionindependence.

It should be noted that the term "boundary condition independence" is a bit ambiguous as different engineeringgroups refer to different requirements. Eqs (1) and (2) fit the case when thermal engineers use convectionboundary conditions to separate the thermal model from environment. This assumes that the normal heat flux q⊥is proportional to the temperature difference between the boundary T and the bulk temperature Tbulk of theneighboring bulk phase

q⊥=k(T Tbulk),(6)

where k is the film coefficient that depends on the flow condition. As flow conditions may change at systemlevel simulation, the film coefficient is to be preserved during model reduction.

The thermal model usually has different film coefficients for different surfaces. After the discretization in spaceof the heat transfer equation, we obtain Eq (1) and (2) where ki is the film coefficient for the i-th boundary.Another important engineering requirement for this case is that the film coefficient can be changed in a hugerange of values up to nine orders of magnitude [18].

B. Electrochemical scanning microscopy

Electrochemical scanning microscopy [23] involves the measurement of the current through anultramicroelectrode when it is moved in electrolyte in the vicinity of a substrate. It allows us to studyhomogeneous and heterogeneous reactions, for high resolution imaging of chemical reactivity and topography ofvarious interfaces [24], especially for biological systems [25]. In addition it can also be used for nano-patterning

[26].

An experimental design of SECM eliminates many nonlinear effects such as convection [27]. A finalmathematical model is a combination of linear multi-species diffusion equation with the Butler-Volmer equationto describe the reaction at the electrode. The latter is expressed as linear mixed boundary conditions

ci,⊥=k(U)=beazU,i(7)

where ci,⊥ is the normal flux of the i-th species, zi is the species charge, U is the applied voltage and a and bare some constants.

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

A typical run includes cyclic voltammogram when current is measured as a function of the applied voltage U.After the discretization, the voltage enters the system matrices and this leads to similar requirements as in theprevious section: the reduced model must preserve the voltage in the symbolic form.

C. Flow meter (Anemometer)

A common way to measure the flow rate by means of MST technology is a combination of a heater withthermocouples positioned from both sides of the heater [28][29]. The flow influences the temperaturedistribution and, thus, the thermocouple signals can be converted to flow rate. An engineering requirement in thiscase is a compact flow meter model that allows us to use flow velocity as a parameter.

The complete model is quite involved as it is necessary to solve Navier-Stokes equations coupled with the heattransfer through the surrounding. Yet, it can be simplified in the case of a given velocity profile with anassumption that a flow meter does not influence significantly the flow profile. In this case, a mathematical modelis a linear convection-diffusion heat transfer [30]. The system matrix depends linearly on the flow velocity v asfollows

A=A0+vA1,(8)

4. Results

In this section, we review our results obtained by the multivariate Pade-type approximation for the models fromthe previous section. More information can be found in the original papers.

We have modified a microthruster benchmark in order to make it similar to a chip model with different filmcoefficients [31] (see Fig. 1). The model is a generic example of a device with a single heat source when thegenerated heat dissipates through the device to the surroundings. The exchange between surrounding and thedevice is modeled by convection boundary conditions. From this viewpoint, it is quite similar to a chip model

used as a benchmark in [18].

Fig. 1. A 2D-axisymmetrical model of the microthruster unit (not scaled). The axis of symmetry is on the left

side. A heater is shown by the red block.

Result for parametric model reduction, when the same film coefficient is assumed for all surfaces, are describedin Ref [32][33] and shown in Fig. 2. The dimension of the original model is 4725. We have made two reducedmodels with a dimension 33 and 49 respectively. Fig. 2 shows a relative error between transient simulation of thefull scale and reduced models as a function of the film coefficient. Note the logarithmic scale for the filmcoefficient. The approximation error is within 1% for a very wide parameter range. Such an agreement is verygood for engineering applications [18].

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

Fig. 2. The error of transient simulation for a reduced model as a function of the film coefficient. Note that the

film coefficient changes by nine orders of magnitude.

A model for electrochemical simulation related to scanning electrochemical microscopy is described in [34] andschematically shown in Fig. 3. The computation domain under a 2D-axisymmetrical approximation includes theelectrolyte under the electrode. We assume that the concentration does not depend on the rotation angle. A singlechemical reaction takes place on the electrode:

Ox+e Redkb kf(9)

A feature of the electrochemistry model is nonzero initial conditions as the concentration of all species cannot be

zero. Our way to deal with this is presented in [34].

Fig. 3. The layout of electrochemistry simulation.

Parametric model reduction for simulation of voltammograms is presented in [35]. Fig. 4 shows simulationresults for the original model of a dimension 16912 and a reduced model of a dimension 202. The figures displaythe current as a function of voltage (not in time) as this is the usual way to represent voltammograms. The solidline is the result computed by full simulation of the original large model, the dashed line is the result computedby the reduced model. It should be noted that the results of the reduced model are accurate for a wide range ofthe dynamic behavior when the voltage rate (du/dt) changes by three orders of magnitude (0.0005-0.5) [35].

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

Fig. 4. Simulation results of the cyclic voltammogram when du/dt=±0.5.

An anemometer is a flow meter that consists of a heater and temperature sensors before and after the heater inthe direction of the flow [36] (see Fig. 5). The flow influences the temperature field and thus leads to atemperature difference between the sensors. This temperature difference is measured and used to determine the

fluid flow.

Fig. 5. Model of the anemometer. The flow enters from the left. A heater is surrounded by two thermosensors.Parametric model reduction has been performed in [36]. Fig. 6 shows the temperature difference between sensorsfor steady state solutions with different velocities. The small graph within the figure shows the deviationbetween the reduced model of a dimension 102 and the original model of a dimension 29008. This deviation lies

below 1% of the difference signal. The reduced model captures the dynamic behavior as well [36].

Fig. 6. Steady state solution for the temperature difference between the sensors for different flow velocities for

the original and reduced models.

5. Discussion

We have presented three MEMS engineering problems that require parametric model reduction and reviewed ourresults obtained within the framework of multivariate Pade-type approximants. In all case studies we were ableto reduce the model dimension considerably and at the same time to preserve the accuracy within engineeringrequirements. However, we should stress that the models considered were relatively simple, as they havecontained only one or two parameters to be preserved. The generalization of the approach to engineering modelswith more parameters, unfortunately, may not be straightforward. Below we briefly list potential problems.

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

Methods in [12][13] use moments directly to construct the projection subspace and this may lead to the rapidaccumulation of rounding errors. We have developed a Krylov-subspace based approach in [33] but itsgeneralization to many parameters seems to be difficult.

A usual problem for moment matching methods is how to choose the number of moments to match. In the caseof multivariate matching, the situation is much worse as there are many different types of derivatives. Forexample, in the case of a single parameter, we have derivatives of the transfer function in respect to the Laplacevariable, derivatives in respect to a parameter, and then mixed derivatives. The more parameters are included inthe transfer function, the more different derivative types we have. A straightforward approach to generate all themoments up to a given order does not scale well with the number of parameters [13]. For example, if we chooseto preserve four film coefficients then a reduced model made from all first derivatives has the dimension of 6, areduced model made from all second derivatives has the dimension of 21, and a reduced model from all thirdderivatives already has the dimension of 56 (see Appendix F in [13]). At the same time, we may need derivativesof higher order than three to describe accurately the transient behavior of the original model.

In order to treat the problem, we have suggested a heuristic procedure in Ref [37]. It is based on 1) neglecting themixed moments; 2) employing the local error control to chose the right number of moments along the Laplacevariable and each parameter. This has been applied for the first device to preserve three film coefficientsindependently and results of numerical experiments were promising. However, more research is needed to see towhat extent this can be used in other applications.

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