Generating Efficient Dynamical Models for Microelectromechanical Systems from a
Few Finite-Element Simulation Runs
Elmer S.Hung and Stephen D.Senturia,Fellow,IEEE
Abstract—In this paper,we demonstrate how efficient low-order dynamical models for micromechanical devices can be constructed using data from a few runs of fully meshed but slow numerical models such as tho created by thefinite-element method(FEM).The reduced-order macromodels are generated by extracting global basis functions from the fully meshed model runs in order to parameterize solutions with far fewer degrees of freedom.The macromodels may be ud for subquent simulations of the time-dependent behavior of nonlinear devices in order to rapidly explore the design space of the device.As an example,the method is ud to capture the behavior of a pressure nsor bad on the pull-in time of an electrostatically actuated microbeam,including the effects of squeeze-film damping due to ambient air under the beam.Results show that the reduced-order model decreas simulation time by at least a factor of37with less than2%error.More complicated simulation problems show significantly higher speedup factors. The simulations also show good agreement with experimental data.[399]
Index Terms—Karhunen–Lo`e ve decomposition,macromodels, microelectromechanical simulation,principal component analy-sis,reduced-order models,squeeze-film damping.
I.I NTRODUCTION
T HE development of increasingly complex microelec-tromechanical systems(MEMS)demands sophisticated simulation techniques for design and optimization[1].MEMS devices typically involve multiple coupled energy domains and media that can be modeled using partial differential equations (PDE’s).Often the functionality of the devices can only be captured with time-dependent nonlinear PDE’s. Traditionalfinite-element methods(FEM’s)can be ud for explicit dynamical simulations of PDE’s,but time-dependent FEM’s are usually computationally very intensive,making them difficult to u when a large number of simulations are needed,especially if multiple devices are involved in a system.
Manuscript received November10,1998.This work was supported by the Defen Advanced Rearch Project Agency under contract J-FBI-95-215. Subject Editor,W.N.Sharpe,Jr.
E.S.Hung was with the Department of Electrical Engineering and Computer Science,Massachutts Institute of Technology,Cambridge,MA 02139USA.He is now with the Xerox Palo Alto Rearch Cent
er,Palo Alto, CA94304USA(e-mail:elmer@alum.mit.edu).
S. D.Senturia is with the Department of Electrical Engineering and Computer Science,Massachutts Institute of Technology,Cambridge,MA 02139USA.
Publisher Item Identifier S1057-7157(99)06543-9.Therefore,a major current goal of modeling and simulation rearch is to develop efficient methods of creating accurate reduced-order dynamical models that capture the same infor-mation contained in the original PDE’s(or,equivalently,in a fully meshed dynamic FEM simulation),but in a form that can be ud for fast dynamical simulations in the context of a circuit-or system-level simulation environment.The reduced-order models are often referred to as macromodels. Previous MEMS macromodeling efforts have investigated lumped-parameter techniques[2],[3];however,it is often difficult to construct accurate lumped-element models for continuous systems,especially when arbitrary geometries are involved.It may also be necessary to parately model effects from dissipative and energy-conrving behavior.Another approach us a linear analysis to generate normal modes which are ud as the basis functions for the model.The modes have been ud,for example,to provide a reduced-order t of generalized coordinates for nonlinear capacitance-bad simulations of electrostatically actuated microstructures[4], [5].However,linear modes may not adequately capture all the features of nonlinea
r behavior[6].Also,static modal functions are sometimes not readily available.
In this paper,we describe a procedure in which a few finite-element orfinite-difference simulations are ud to create a reduced-order macromodel that permits fast simulation of MEMS devices while capturing most of the accuracy and flexibility of the full model.Although this approach requires an initial overhead cost to run FEM simulations,once the macromodel is generated,it may be ud to efficiently carry out any further simulations involving the device.This strategy could be ud to greatly speed up multiple simulations of a single device for exploration of a design space,or could be ud as input to system-level simulators for designing systems with many coupled devices.
FEM’s rely on highly localized interpolation functions(or basis functions)to approximate the solution to PDE’s.The local basis functions are generated by meshing the domain of interest and parameterizing the desired solution locally on each mesh element.This parameterized solution converts a continuous(PDE)problem to a coupled system of ordinary differential equations(ODE’s)that can be integrated in time. The resulting ODE system usually has many degrees of freedom(perhaps veral variables per mesh element).If a fine mesh is required,the problem size grows rapidly,with a
1057–7157/99$10.00©1999IEEE
Fig.1.Fixed-fixed beam pull-in experiment tup[14].For analysis pur-pos,we assume that the x and y axes are oriented parallel to the length and width of the beam,respectively,and z is directed up,perpendicular to the substrate.
托运单corresponding rapid growth in computational cost for explicit dynamic simulation.
In contrast to the FEM approach,global basis functions can often be ud to capture the solution with fewer degrees of freedom.For example,Fourier decompositions[7]or modal functions[4]may be ud.However,it is usually difficult to determine,a priori,an optimum t of such functions, particularly when irregular geometries are involved.
This work eks a middle path,focusing on obtaining macromodels bad on global basis functions generation from an approach that is mathematically equivalent to Karhunen-Lo`e ve analysis of a small but reprentative enmble of dynamic FEM runs.Previous work has shown that Karhunen-Lo`e ve decompositions can be ud to generate such basis functions for nonlinear dynamical system simulations for turbulence problems influid mechanics[8],[9],fluid-structure interactions in aerodynamical systems[10],[11],and in chem-ical engineering systems[12].
Here we u singular value decomposition(SVD)to gen-erate global basis functions from explicit FEM results and generate the macromodel by the u of Galerkin method with the original governing PDE’s.We discuss how basis function macromodeling techniques can contribute to MEMS simulation technology and u the method to simulate a problem involving squeeze-film(air)damping of a microbeam. This approach wasfirst prented in[13].
II.C ASE S TUDY E XAMPLE
In order to illustrate the macromodel technique,wefirst introduce a ca study problem that will be referred to through-out the discussion.The device(Fig.1)consists of a deformable elastic beam microstructure that is electrostatically pulled in by an applied voltage waveform.The time it takes for t
he beam to pull in is highly nsitive to the air pressure under the beam.In fact,this structure has been propod for u as a pressure nsor[14].
Simulating the time-dependent dynamics of the device in-volves a nonlinear squeeze-film damping problem with me-chanical,electrostatic,andfluidic components.Efficient simu-lations of squeeze-film damping problems are an active topic of rearch[15],[16]and are important for a variety of applications in order to control moving structures and to determine how fast microstructures can be moved in air. The pull-in time pressure nsor device can be modeled by coupling a1-D elastic beam equation with electrostatic force[(1)]and the2-D compressible isothermal squeeze-film Reynold’s equation[17]for air damping with slipflow[(1)]1
:
(1)
is the electrostatic
force, is the height of the beam above the substrate,
and
m,elastic
modulus
m,
thickness
MPa,
density
,and air
viscosity
kg/(m
(3)
哈尔滨环球雅思国际英语学校
where
is a vector of state variables.For simplicity,we assume that the state
solution are functions of time.
Given,we would like to
determine
of square integrable functions with dot
product
in a parable form as a ries expansion of time varying
coefficients and spatially varying basis
functions
(4) where is the approximation
for
will be addresd in Section IV below.
1The compressible isothermal Reynold’s equation[(2)]can be derived from the Navier-Stokes,continuity,and ideal-gas equations by assuming 1)isothermal conditions,2)a pressure that is constant with z(across the gap),3)negligible inertial effects,and4)negligiblefluid velocity in the z direction(perpendicular to the substrate).Becau the dimensions of interest are beyond the limit where atmospheric air can be modeled accurately as a continuousfluid,slipflow boundary conditions are ud to model the device, parameterized by Knudn’s number K.See[17]for a derivation.
Assuming for the moment that the basis functions are known,the Galerkin method can then be ud to specify the
equations of motion for the
coefficients
.This method requires that the PDE
residual
be orthogonal to each of the basis functions
in
四川大学历年分数线
consists of the
time histories of both
pressure
.Becau of the formulation of the governing PDE’s,we choo to write parate approximations for
pressure
as
follows:
(6)(7)
where
and
are
scalar basis functions for pressure and displacement,respec-tively.
Using the Galerkin method,the PDE’s in (1)and (2)can be reduced to coupled matrix ODE’s in terms of the basis function approximations above.A derivation of the equations appears in Appendix I.The Euler beam equation (1)results in the following
ODE:
(8)
where
indicates integration along the length of the beam.
The Reynolds equation (2)results in the following
ODE:
indicates integration along the beam area.Equations
以夷制夷
(8)and (9)constitute the macromodel formulation.The equa-tions are integrated numerically to simulate dynamic behavior.
Note that the equations are the same general form as tho ud for a standard FEM simulation,but becau of the model-order reduction with global basis functions,constitute a much smaller computational problem.
The main idea behind the macromodel is that the number of ODE’s needed to simulate the system has been reduced from perhaps many thousands in the ca of the full FEM simulation,to just a few basis function coordinates.Thus the macromodel simulation can be very efficient computationally co
mpared to the FEM model.Note also that many of the terms in the matrix ODE’s above can be precomputed once the basis functions are known
[e.g.,
from (8)is sampled in basis function co-ordinates and a rational function approximation
for very efficient during the
numerical integration stage;however,it requires initial effort to generate the rational function approximation.
Note also that instead of using independent basis func-tions for pressure and displacement,it is possible to analyze basis functions for a combined state
vector
.This may produce benefits in reducing
the number of basis functions needed to characterize the dynamics of the system becau it capture
s coupling between displacement and pressure.In this ca,however,independent displacement and pressure basis functions make the Galerkin derivation simpler and also makes n given the physics of the problem.
IV.G ENERATING
THE
B ASIS F UNCTIONS
We now describe how the basis functions are chon.First the system dynamics are simulated using a slow but accurate technique such as finite elements or finite differences.An enmble of runs may be ud to suitably characterize the operating range of the device.
The spatial distributions of each state
variable
corresponds to a particular
“snapshot”in time.
For example,for the microbeam problem,the pull-in dy-namics of the beam are simulated using a finite-difference method for an enmble of step voltages.To determine pres-sure basis functions,we take a ries of
snapshots,
the pressure distributions at various times where the entries in
each
vector correspond to pressures at a different node of the finite-difference mesh.
Now suppo we would like to
pick
,in order to reprent the obrved
state distributions as cloly as possible.One way to do this is to attempt to minimize the following
quantity:幼儿英语话剧剧本
(10)
where onto
subspace
,who columns
are
.The
SVD
of
and
aww
san francisco
minimizing (10)can be chon by
tting are the columns
of
are the eigenvectors
of
and the columns
of
along with substitution back into
a matrix multiplication in order to find the basis functions.However,there are algorithms for computing the SVD
of product directly.Thus,we suggest that computation
of the SVD
of
10node 2-D grid.The
state at each node consists of three
quantities:.
Since
V,
and
and
m).
The accuracy of the macromodel is especially apparent in comparison to the performance of lumped-element methods.For example,linear damping is often ud (e.g.,[14])to simulate the first-order effects of squeeze-film damping in microstructures.Fig.4shows results from a linear damping
(a)
(b)
Fig.3.(a)Comparison of beam deflection versus time simulation results for a10-V step input.M and N are the number of mechanical and pressure
compass
basis functions ud in the macromodel,respectively.Macromodel results are almost indistinguishable fromfinite-element simulation for M 2and N 2.By contrast,the linear damping simulation shows large errors.(b) Macromodel deflection error versus time for the simulation depicted in(a).
Error is measured with respect to thefinite-difference simulation.
simulation of the pull-in time pressure nsor.In this simula-tion,the damping
force
is a damping constant.In this ca,the damping
constant was tuned to match the pull-in time from thefinite-
difference simulation.Fig.4(a)shows displacement versus
time results for a10-V step input(compare with Fig.3),
and Fig.4(b)shows results for a14-V10-kHz sine wave
input(compare with Fig.6).We e that the linear damping
simulation shows large errors compared with the macromodel,
illustrating the type of problems that can result from using
linear models to simulate large-amplitude nonlinear squeeze-
film damping effects.
The results of the macromodel also correspond well to
experimental data from[19](Fig.5).A ries of step input
voltages are applied,and the pull-in time is measured and
plotted for both experiment and simulation.Note that thefinite-
difference results are indistinguishable from the macromodel
for the ries of step input simulations.The relatively small
errors between experiment and simulation may be due
to
(a)
(b)peach怎么读
Fig.4.Beam deflection versus time results for a linear damping simulation
compared to thefinite-difference model.(a)Results for a10-V step input.(b)
Results for a14-V10-kHz sine wave voltage input.The simulations show
thatfirst-order lumped element techniques can result in large errors compared
to the macromodel results(Figs.3and6).
unmodeled effects such as compliant supports and strain-
stiffening of the beam.Note that the order-of-magnitude
impromptuincrea of pull-in time in air and the change in critical pull-
in voltage due to inertial effects are well modeled by both
simulations.This demonstrates that the types of simula-
tions can achieve reasonable accuracy for modeling transient
squeeze-film damping effects.
B.Changing Device Parameters and Input
One question is,howflexible is the basis-function tech-
nique?If the parameters or inputs to a device change,is it
necessary to rerun FEM simulations to generate new basis
functions for the macromodel?This is important becau a pri-
mary motivation for the u of the macromodeling technique
for MEMS devices is that a single macromodel may be ud to
run many simulations,exploring the design space of a device
or system without having to recompute FEM simulations.
To test this,the same macromodel described above,gener-
ated using basis functions from the step voltage input FEM
runs,was ud to simulate the respon to a14-V10-kHz sine
wave voltage input.The results are compared to a fullfinite-
difference solution for the14-V10-kHz sine wave stimulus
(Fig.6).Again,macromodel results
for
and
match thefinite-element run extremely well,while the linear
