# Simulation of shaped comb drive as a stepped actuator for microtweezers application

**ABSTRACT** Finite element analysis is used to simulate electrostatic actuated, shaped comb drives operating under dc conditions (zero actuating frequency). A dynamic multiphysics model is developed using the arbitrary Lagrangian–Eulerian (ALE) formulation. Results show the coupled interaction between the electrostatic and mechanical domains of the transducer. The analysis is based on the evolution of electrostatic force versus comb finger engagement. The relationship between incremental lateral displacement and actuation voltage illustrates the potential for stepped movement for a shaped comb drive. Additionally, through numerical simulations, this project determines an optimum design for a dc-actuated comb drive, which has controllable force output and stable engaging movement.

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**ABSTRACT:**This paper presents a method for fabricating a fundamental MEMS tool—microtweezers. Microtweezers offer an attractive option to meet the increasing need to grasp, manipulate and excise microstructures or biological components. The microtweezers presented here augment a standard micromanipulator, allowing precise positioning in three dimensions. An additional micro-drive control knob, which is affixed to the micromanipulator, allows actuation of the tweezer tips through the use of a tether-cable drive system. This drive actuates the tweezer tips by the reciprocating motion of two microfabricated parts: the tweezers and tweezer box. A simple three-layer planar fabrication scheme allows for a broad range of tweezer styles (straight and serrated tips) and sizes (microns to millimeters). For these studies, 20 µm wide and 10 µm thick nickel beams were developed for the tweezer tips, which could endure 20 mN of force. To demonstrate the concept of microassembly, pick and place operations were performed on 10 µm thick film structures. Additional functionality was achieved by integrating platinum-black microelectrodes into parylene-coated tweezers to allow electrophysiological functions such as cellular stimulation and recording. Ultimately, this unique and simple design affords extraordinarily delicate control that is potentially beneficial for applications in microassembly, electrophysiology and microsurgery.Journal of Micromechanics and Microengineering 04/2008; 18(6):065004. · 1.79 Impact Factor - SourceAvailable from: dspace.uta.edu
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**ABSTRACT:**This paper presents the design and evaluation of a high force density fishbone shaped electrostatic comb drive actuator. This comb drive actuator has a branched structure similar to a fishbone, which is intended to increase the capacitance of the electrodes and hence increase the electrostatic actuation force. Two-dimensional finite element analysis was used to simulate the motion of the fishbone shaped electrostatic comb drive actuator and compared against the performance of a straight sided electrostatic comb drive actuator. Performances of both designs are evaluated by comparison of displacement and electrostatic force. For both cases, the active area and the minimum gap distance between the two electrodes were constant. An active area of 800 × 300 μm, which contained 16 fingers of fishbone shaped actuators and 40 fingers of straight sided actuators, respectively, was used. Through simulation, improvement of drive force of the fishbone shaped electrostatic comb driver is approximately 485% higher than conventional electrostatic comb driver. These results indicate that the fishbone actuator design provides good potential for applications as high force density electrostatic microactuator in MEMS systems.TheScientificWorldJournal. 01/2014; 2014:912683.

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Sensors and Actuators A 123–124 (2005) 540–546

Simulation of shaped comb drive as a stepped actuator for

microtweezers application

Isabelle P.F. Harouche, C. Shafai∗

Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Man., Canada R3T 5V6

Received 4 September 2004; received in revised form 13 March 2005; accepted 25 March 2005

Available online 22 April 2005

Abstract

Finite element analysis is used to simulate electrostatic actuated, shaped comb drives operating under dc conditions (zero actuating

frequency). A dynamic multiphysics model is developed using the arbitrary Lagrangian–Eulerian (ALE) formulation. Results show the

coupled interaction between the electrostatic and mechanical domains of the transducer. The analysis is based on the evolution of electrostatic

forceversuscombfingerengagement.Therelationshipbetweenincrementallateraldisplacementandactuationvoltageillustratesthepotential

for stepped movement for a shaped comb drive. Additionally, through numerical simulations, this project determines an optimum design for

a dc-actuated comb drive, which has controllable force output and stable engaging movement.

© 2005 Elsevier B.V. All rights reserved.

Keywords: Microtweezers; MEMS; Comb drive; FEM; ALE

1. Introduction

Capacitance-based sensors and actuators have been ex-

tensively used in micro electromechanical systems (MEMS)

devices [1,2]. Among different devices, the most commonly

used and analysed is the comb drive [3,4]. The MEMS comb

drive is a laterally driven mechanical actuator activated by

electrostaticinteraction.Thebasicdesignofacombdrivere-

liesonthetheoryofparallel-platecapacitors,whichinturnis

afunctionoftheplates’areaandshape.Inthecaseofacomb

drive, the parallel plates are an array of interdigitated fin-

gers, which are generally rectangular. A typical rectangular-

shaped comb drive design requires simple fabrication steps

(usually only one structural layer) and it is characterized

by low power consumption [5]. The device has a constant

force-to-displacement relationship, which is a function of

the change in capacitance with respect to engagement, rather

than total capacitance. Rectangular comb drives have been

used as actuators for several different applications, including

∗Corresponding author. Tel.: +1 204 474 6302; fax: +1 204 261 4639.

E-mail addresses: harouche@ieee.org (I.P.F. Harouche),

cshafai@ee.umanitoba.ca (C. Shafai).

micro-motors, conveyors, sensing devices and microgripper

devices.

Later, shaped comb drives were introduced as a means to

tailor the rate of change of capacitance with respect to the

lateral displacement. Designs presented in literatures [6,7]

were used to stiffen and weaken resonator springs and hence

offer more controllability over the device operation. These

tunable resonators were introduced as a means to achieve

more linear force-engagement profiles.

The present work discusses a novel move-and-lock mech-

anismbasedonashapedcombdrivedesign.Themainusefor

such device is as a microtweezers actuator for application in

areas such as biological sample handling, MEMS assembly

processes and other activities where precision micromanipu-

lation and force-controlled interaction are required.

2. Design concept

This project introduces an adaptation to the parabolic-

shapedcombdriveanalysed

force–displacement is not linear, but it is also not con-

stant, as in the case of rectangular comb drives. Instead, a

in[6], in that the

0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.sna.2005.03.031

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I.P.F. Harouche, C. Shafai / Sensors and Actuators A 123–124 (2005) 540–546

541

Fig. 1. Illustration of the design for the shaped comb drive as a microtweez-

ers actuator. The blow-up image shows a representation of a jagged-edge

comb drive.

stepwise continuous response of force versus displacement

occurs. Fig. 1 offers a conceptual idea of a possible final

design for the force controllable microtweezers, whereas

the insert depicts a section of the comb drive itself. The

proposed move-and-lock mechanism is based on the change

in the distance between the comb fingers with respect to

engagement, which in turn is a function of the actuation

voltage.

The geometry simulated corresponded to a set of 10 fixed

fingers and 9 movable fingers. Fig. 2 presents all geometries

analysed throughout this paper and should be used as a refer-

ence. The designs had a few dimensions in common, namely

thelengthofeachfinger,whichwas40?mlongwithnotches

at 5?m intervals, and the structural thickness of 2?m. The

set of movable fingers start at a rest position corresponding

to a 20?m engagement. The design of Fig. 2A has minimum

and maximum gap distances between fingers of 3 and 7?m,

respectively. This design complies with the requirements for

the Multi-User MEMS Process (MUMPs) [8]. This report,

however, attains to the actuator design. The gripping pads

and final microtweezers testing are left for future work.

3. Electromechanical principles of the comb drive

The comb drive is a capacitive device with air as the di-

electric material. The region where the problem is defined is

chargefree(ρV=0)andtheelectrostaticproblemisdescribed

by Laplace’s equation (in rectangular coordinates):

∇2V =∂2V

∂X2+∂2V

∂Y2+∂2V

∂Z2= 0(1)

Finite element simulations were carried out to find potential

distributionswhichsatisfyEq.(4)foragivenelectrodegeom-

etryatapredefinedactuationpotentialV0.Sincetherewasno

currentflowinsidethecombdrivedeviceitself,thesurfaceof

the device was assumed equipotential. Potential energy was

a continuous quantity in the domain of interest; hence, the

fundamental requirement of the finite element method was

met. The voltage-dependant continuous field can be approx-

imated by a discrete model composed of a set of piecewise

continuous functions defined over a finite number of subdo-

mains.

OncethescalarfieldVisknow,electricenergyWeiscom-

puted at all elements according to

???

XYZ

We=1

2ε0

|E|2dXdY dZ

(2)

Fig. 2. The five jagged-edge comb designs analysed. Design “A” complies with the MUMPs standard dimensions.

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I.P.F. Harouche, C. Shafai / Sensors and Actuators A 123–124 (2005) 540–546

Fig. 3. Schematic description of the simulated spring; Fesdenotes the load

force, which corresponds to the electrostatic force generated by the comb

drive.Theshuttleisthemountingstructurewherethecombdriveisattached.

Computations for capacitance C and electrostatic force Fes

follows, as shown in Eqs. (3) and (4). The solution of the

electrostatic problem served as the load to the mechanical

problem:

???

XYZ

CTOT=

2

V2

0

WedXdY dZ

(3)

Fes=∂We

The reader is encouraged to find detailed description of the

electrophysics of comb drives elsewhere 0.

∂Y

=1

2

∂C

∂YV2

0

(4)

3.1. Mechanical characterization of the comb drive

The design of the springs followed the specifications in

Fig. 3. The movable set of fingers is attached to a shuttle,

which in turn is the link between the electrostatic force-

generatingdeviceandthetwodouble-foldedcantileverbeam

springs. Hence, the comb displacement was a combination

of electrostatic pulling force Fesand the mechanical spring

restoring force Fmech:

Fmech= k?Y

Thefollowingdefinitionsappliedtothecantileversprings:(a)

thematerialusedinthefabrication(polysilicon)wasassumed

homogeneous and isotropic; (b) the thickness dimension was

small compared to the length; and (c) the stress in the normal

Z-direction was ideally zero. It is nevertheless worth noting

that in non-ideal testing environment the levitation phenom-

ena contributes to a finite Z-component of stress. In any case,

levitation analysis was beyond the scope of this paper. The

stresses and loads were defined in the X–Y plane and so any

other parallel plane has the same stress distribution. Thus,

the springs fell into the characteristic plane stress problem

definition [10].

(5)

4. Numerical analysis

Proper dynamic analysis of the device required coupled

electromechanical analysis. Simulations were performed in

the 2D domain using the finite element method (FEM) in the

Femlab 3.0a software package [11]. Meshes were automati-

callygeneratedbythesoftware.Giventhecomparativenature

of this work, where each simulation run must be analysed

with respect to each other, a mesh analysis was performed

Fig. 4. Mesh density analysis. The arrow points to the Femlab 3.0a stan-

dard “coarser” mesh solution. For this analysis, the coarser mesh returned

approximately 5000 elements.

prior to simulations. The goal was to achieve a mesh density

that could be reasonably matched amongst all runs. Electro-

static simulations were performed for several different mesh

densitiesforlinearandquadraticelements.Thedensitiescho-

sen followed the standard options available in Femlab 3.0a

varying from extremely coarse to extremely fine densities.

Fig. 4 shows that quadratic elements returned similar results

irrespective of mesh density. As for the linear elements, the

range of results varied up to 14%. Based on these results, all

simulations were computed from quadratic elements using

Femlab’s automatically generated coarser mesh.

Basiccombdrivetheoryindicatesthatfringingfieldsinthe

thirddimensionareofsomeimportanceinthecomputationof

electrostatic force [9]. However, due to processing time and

memory requirements, simulations were performed in a 2D

environment.Thatexcludedfieldlinesinthethirddimension.

In any event, discrepancies between 2D- and 3D-capacitance

analysis were investigated as a means to bring model results

into the context of actual fabricated devices. Fig. 5 presents

theresultsofasimplemodeloftwoparallelrectangularplates

Fig. 5. Comparison between 2D and 3D capacitance computations. The

x-coordinates depict the distance between the device boundaries and the

surrounding volume boundaries.

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543

with the same dimensions as the analysed comb fingers. The

plates were kept at a constant 3?m distance from each other.

The distance to the surrounding dielectric volume boundary

was varied. Note that the volume of integration directly af-

fects the amount of fringing fields accounted for in the final

scalar values. The plot in Fig. 5 shows that the total capaci-

tanceofthe3Dsystemisapproximatelyfourtimesthatofthe

2D system. Since the direct relationship between input volt-

age and capacitance is expressed as C = 2We/V2

dimensional examination tells us that the required input volt-

age for 2D simulations should be twice as large as the 3D

case.

0, a quick

4.1. The arbitrary Lagrangian–Eulerian formulation

The generation of successful multiphysics models was

not trivial, due to the complexity of the geometry and the

large deformations at the spring. Difficulties arose in devel-

oping a mesh capable of approximating large deformations.

Additionally, a dynamic parametric simulation was bound

to generate increasingly deformed meshes, which in turn

became unstable and solutions did not converge. The arbi-

trary Lagrangian–Eulerian (ALE) technique is an advanced

method of solving moving boundaries and non-linear prob-

lems in finite element analysis (FEA) [12–14]. The ALE

method was chosen as a means to avoid such lack of con-

vergence. Additionally, ALE can be used as a powerful tool

formultiphysicsanalysis,acommontraitinMEMSdesign.It

has not yet been widely explored in MEMS transducer simu-

lation. Upon extensive investigation, only one other research

paper was found on MEMS simulation using ALE [15].

Basic ALE formulations are available from different

sources [12,16,17], which can be used as an initial template

and adapted to each specific problem. The basis of the tech-

nique is to use an FE mesh that is neither attached to the

material nor fixed in space. An arbitrary motion independent

of the material deformation is assigned to each degree of

freedom of the system. The main advantage of this technique

laysinthefactthatatanypointoftheanalysisasolutionmay

be computed. Both in cases where large and highly localized

deformation of the structure occur, and where unconstrained

flow of material on free boundaries happens.

One characteristic of the ALE formulation is that the so-

lution variables representing structural deformation are only

determined at elements within the structure boundaries. The

couplingwithothersimulationvariableshappensthroughthe

mesh displacement characteristics. Thus the structure defor-

mationpropertiesmustbetransferredtomeshpointsthrough

an updating algorithm. Additionally, prescribed mesh dis-

placements must be assigned for all degrees of freedom of

themesh,ateachiterationofthenumericalsolution[18].The

algorithm performs an automatic grid re-design procedure,

which maps the original domain into the deformed domain

at each displacement instance. Hence, the ALE formulation

specifies which boundaries will move during the simulation

and how they should move.

Fig. 6. Flowchart describing the ALE procedure.

Forthepurposeofthisresearch,theALEformulationwas

used to solve the electrostatic problem defined in the dielec-

tric domain. Both electrostatic and mechanical simulations

were performed concurrently and the solution of the former

served as a boundary condition to the latter. Thus, the ALE

formulations can be described as an algorithm that performs

automatic reasoning [19], as seen in the flow chart in Fig. 6.

The ALE mesh displacements were defined by Poisson’s

equation and solved in integral form, as seen in the following

equations:

?

+(ˆδxXIXy+ˆδxYIYy)δxy]dΩ = 0

Ω

detJ[(ˆδxXIXx+ˆδxYIYx)δxx

(6a)

?

Ω

detJ[(ˆδyXIXx+ˆδyYIYx)δyx

+(ˆδyXIXy+ˆδyYIYy)δyy]dΩ = 0

where (6a) and (6b) represent the x- and y-components in the

computationalALEmeshdomainandthehatsymboldefines

a test function. The term (detJ) is the determinant of the

inverseJacobianwhichmapstheoriginalcoordinatesin(X,Y)

tothedeformedcoordinates(x,y).Itfollowsthatu=x−Xand

v=y−Y. The electrostatic equation starts with Gauss’ law,

(6b)

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I.P.F. Harouche, C. Shafai / Sensors and Actuators A 123–124 (2005) 540–546

presented in integral forming the following equation:

?

+(ˆVXIXy+ˆVYIYy)Vy]dΩ = 0

where Iijcorresponds to the entries in the inverse Jacobian

matrix.

Themechanicalvariablesofinterestweretheglobalspring

displacements (u,v) in the X- and Y-directions. The solution

for the spring problem was derived from the PDE form of

Navier’s equations for the X- and Y-components (Eq. (8)),

where c is a coefficient dependent on the Young’s modulus

E, the Poisson’s ratio ν, ρ is the density of the material. Ad-

ditionally u=(u,v) is the displacement vector:

ε

Ω

detJ[(ˆVXIXx+ˆVYIYx)Vx

(7)

ρ∂2u

∂t2− ∇ · c∇u = K

ThevectorKrepresentstheforceappliedtotheshuttle,which

in turn is connected to the springs and therefore acts as the

load in this system. Hence K is the derived from the elec-

trostatic force computed in the ALE part of the simulation.

Both mechanical and electrostatic solutions combined gave

the desired comb drive dynamic characterization.

(8)

5. Results and discussion

Parametric computations returned the field distribution in

the dielectric for input potentials from 0 to 250V. Forces act-

ing upon the set of movable fingers were computed at each

increment of voltage. Fig. 7a shows a plot of the electro-

static force acting on the comb teeth of design A (shown in

Fig. 2), as a function of the finger displacement due to the

applied voltage. It can be seen that the response is similar to

that expected from a rectangular-shaped comb drive. This is

corroborated by the plot in Fig. 7b, which shows the electro-

Fig. 7. Plots of force and force gradient with respect to finger engagement

fromrestposition.Engagementdistancewascomputedfromtheincremental

application of voltages from 0 to 250V: (a) the forces acting on the jagged-

edge fingers of design A; (b) the forces acting on a set rectangular fingers of

similar dimensions.

Fig. 8. Comb displacement with respect to V0: (a) results from the 8:2?m

gap; (b) results from the 4:1?m gap. The reader should note the difference

in scales in the differential displacement axes.

static force behaviour for a rectangular comb drive with the

samelength,thickness,andminimumgapasthejagged-edge

structure.Thedifferencebetweenthesetworesultsisveryno-

ticeable when observed in terms of the force gradient, shown

in the secondary axes of Fig. 7. The jagged-edge shaped of

design A produces an evident variable rate of change in force

with respect to engagement. The absolute values of the force

gradient are however too small for this design to possess

stepped displacement.

Next, designs B and C of Fig. 2 where investigated. These

two designs both have a 4:1 relationship between maximum

and minimum gap size, but with different dimensions. The

FEMsimulationresults(Fig.8a)showthatdesignB’s8:2?m

gapresultsintwopointsofinflection,whicharemorenotice-

able than the result for the geometry of design A, but still the

absolute displacement values are not enough to justify the

design. Design C’s 4:1?m gap spacing (Fig. 8b) renders an

evidentstepinthetotaldisplacement,buttheexpected“lock-

ing positions” shows some minute slippage. This can be seen

as the plateaus in the curve for differential displacement with

respect to input voltage. The ideal locking position would

have zero displacement gradient with respect to the actua-

tion voltage. In Fig. 9, it can be seen that the force gradient

for design C shows a change of inflection at 4?m displace-

ment from rest position and break points at 6 and 11?m

displacement. These results show that the total electrostatic

force acting upon the movable fingers is seven times larger

than those observed in the 8:2?m gap design.

From the above results it is evident that the key factor in

the jagged-edge shape design is the ratio between maximum

and minimum gaps in conjunction with the actual minimum

gap distance value. From the results, the 4:1 ratio with 1?m

minimum gap is more effective than the 4:1 with 2?m mini-

mum gap. Thus, we deduce that, for increased gap distances,

thedimensionofeachnotchmustbeincreasedaswell.Thatis

sothegaininnotchareabalancestheincreaseingapdistance,

preserving the total electric energy, as seen in the following

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I.P.F. Harouche, C. Shafai / Sensors and Actuators A 123–124 (2005) 540–546

545

Fig. 9. Force and force gradient results for both 4:1?m gap and 8:2?m gap

designs. The values for the 4:1?m gap design is seven times larger than

those seen in Fig. 7.

equation:

We=1

Designs D and E of Fig. 2 have exaggerated ratios between

minimum and maximum distances with respective values of

1 and 7?m gaps. Design E, however, is constructed in such

a way that the gaps are asymmetric with respect to the imag-

inary line between the two fingers.

Resultsforthislastanalysisareshowninthecontextofall

previous simulations. Fig. 10 shows combined plots of dis-

placement gradient with respect to actuation voltage. Note

that the asymmetric 7:1 design shows two clear points of in-

flection. The first step in displacement occurs at about 40V.

By observing the slope of the curve, it is possible to infer an

increase in velocity, which implies that the movable comb

finger accelerates. It then decelerates to a constant, stable

and small displacement. Ideally, the structure should almost

lock in place preventing any Y-direction movement. When

the input potential reaches 150V another surge in acceler-

ation happens. At this point, the total amount of displace-

2

εA

d2V2

0

(9)

Fig. 10. Comparison of differential displacements of all geometries investi-

gated.

Fig. 11. Analysis of the 7:1 symmetric design (shape “D”).

Fig. 12. Analysis of the 7:1 asymmetric design (shape “E”).

ment is larger than in the previous engagement. The general

behaviour of the 7:1 symmetric design was similar, but the

points of inflection are not as accentuated and the trough not

as low, implying more slippage.

Figs. 11 and 12 illustrate the 7:1 designs in detail. The

symmetric design (Fig. 11) has two engagement steps, but

the slippage in the asymmetric design (Fig. 12) is smaller.

It follows that the point of inflection in the force versus dis-

placement plot is more evident in the asymmetric case.

These results illustrate the concept of step movement and

show that, for a given gap ratio, there exists an optimum

design correspondent to an asymmetric geometry. Notable

points of improvement in the design are: (1) optimization of

the jagged notch length with respect to structural thickness,

which balances the influence of gap distance, and (2) consid-

eration of the slippage problem, which prevents total locking

of the device.

6. Conclusions

This paper has introduced a novel application to MEMS

comb drives operating under dc conditions. The jagged-edge

shaped finger was simulated using finite element analysis

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I.P.F. Harouche, C. Shafai / Sensors and Actuators A 123–124 (2005) 540–546

andapplyingthearbitraryLagrangian–Eulerianformulation.

InitialresultsbasedontheMUMPsstandarddimensionswere

not conclusive; consequently, new dimensions were tested.

Upon simulating different jagged-edge designs, it has been

found that the key to achieving the proposed move-and-lock

mechanismisoptimizationofthemaximumtominimumgap

relationship. Based on the designs investigated, the optimum

result came from an asymmetric 7:1 gap ratio, for a device

with 2?m thick structural layer. Further work is required to

define an analytical model for the jagged-edge comb drive.

However, at this point numerical solutions were adequate to

prove the concept of stepped-movement.

Acknowledgements

ThisresearchhasbeenfundedbytheHealthScienceCen-

tre Foundation, Winnipeg, MB, and the Natural Sciences and

Engineering Research Council (NSERC) of Canada. The au-

thors wish to thank Mr. Behraad Bahreyni (Electrical and

Computer Engineering, University of Manitoba) and Mr. Li-

nusAndersson(Comsol,AB)fortheirsuggestionsduringthe

course of this work.

References

[1] C. Lu, M.A. Lemkin, B.E. Boser, A monolithic surface microma-

chined accelerometer with digital output, IEEE J. Solid-State Circuits

30 (1995) 1367–1373.

[2] E.J. Garcia, J.J. Sniegowski, Surface micromachined microengine as

the driver for micromechanical gears, in: Proceedings of the Interna-

tional Conference on Solid-State Sensors and Actuators, Stockholm,

Sweden, 1995, pp. 365–368.

[3] W.C. Tang, T.-C.H. Nguyen, R.T. Howe, Laterally driven polysilicon

resonant microstructures, in: Proceedings of the Technical Digest

IEEE Micro Electro Mechanical Systems Workshop, Salt Lake City,

UT, 1989, pp. 53–59.

[4] W.C. Tang, Electrostatic comb drive for resonant sensor and actuator

application, Ph.D. Dissertation, University of California, Berkeley,

CA, 1990.

[5] G.T.A. Kovacs, Micromachined Transducers Sourcebook, McGraw-

Hill, 1998.

[6] B.J. Jensen, S. Mutlu, S. Miller, K. Kurubayashi, J.J. Allen, Shaped

comb fingers for tailored electromechanical restoring force, J. Mi-

croelectromech. Syst. 12 (3) (2003) 373–383.

[7] W. Ye, S. Mukherjee, N.C. MacDonald, Optimal shape design of

an electrostatic comb drive in microelectromechanical systems, J.

Microelectromech. Syst. 7 (1) (1998) 16–26.

[8] D.A. Koester, R. Mahadevan, B. Hardy, K.W. Markus, MUMPs De-

sign Handbook, Revision 7.0, Cronos Integrated Microsystems, Re-

search Triangle Park, NC, USA, 2001.

[9] W.A. Johnson, L.K. Warne, Electrophysics of mechanical comb ac-

tuators, J. Microelectromech. Syst. 4 (1) (1995) 49–59.

[10] V. Adams, A. Askenazi, Building Better Products with Finite Ele-

ment Analysis, Onword Press, Santa Fe, NM, USA, 1999.

[11] Femlab ReferenceManual,

http://www.comsol.com.

[12] J.-P. Ponthot, Advances in arbitrary Eulerian–Lagrangian finite el-

ement simulation of large deformation processes, in: Proceed-

ings of the Fourth International Conference on Computational

Plasticity (COMPLAS IV), Barcelona, Spain, 1995, pp. 2361–

2372.

[13] J. Wang, M.S. Gadala, Formulation and survey of ALE method

in nonlinear solid mechanics, Finite Elem. Anal. Des. 24 (1997)

253–269.

[14] P. Vachal, R.V. Garimella, M.J. Shashkov, Untangling of 2D

meshes in ALE simulations, J. Comput. Phys. 196 (2004) 627–

644.

[15] A. Beskok, T.C. Warburton, Arbitrary Lagrangian Eulerian analy-

sis of a bidirectional micro-pump using spectral elements, Int. J.

Comput. Eng. Sci. 2 (1) (2001) 43–57.

[16] M.S. Gadala, Recent trends in ALE formulation and its applications

in solid mechanics, Comput. Methods Appl. Mech. Eng. 193 (45–47)

(2004) 4857–4873.

[17] Femlab ALE Models Reference Guide, Comsol, AB, 2004.

http://www.comsol.com.

[18] M.S. Gadala, M.R. Movahhedy, J. Wang, On the mesh motion for

ALE modelling of metal forming processes, Finite Elem. Anal. Des.

38 (2002) 435–459.

[19] J.O. Hallquist, Simplified arbitrary Lagrangian–Eulerian LS-DYNA

Theoretical Manual, Livermore Software, 1998, Chapter 14.

Comsol,AB,2003.

Biographies

Isabelle P.F. Harouche received her BSc in 1998 from the Rio de Janeiro

State University in Rio de Janeiro, Brazil. She is presently pursuing her

MSc in Electrical Engineering at the University of Manitoba in Winnipeg,

Man., Canada. Her current research interests are the design and fabrication

of MEMS devices, as well as the use of finite element analysis in MEMS

simulation.

Cyrus Shafai is an associate professor in the Department of Electrical

and Computer Engineering at the University of Manitoba. He received his

BSc (Electrical Engineering) and MSc (Electrical Engineering) from the

University of Manitoba, Winnipeg, Man., Canada, in 1990 and 1993, re-

spectively. In 1997, he received the PhD degree in Electrical Engineering

from the University of Alberta, Edmonton, Alta., Canada. His princi-

ple research was the development of a micromachined on-chip Peltier

heat pump. His current research includes MEMS-based frequency agile

antenna, phase shifters, micro-sensors.