IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 2, MARCH 2004 557

Finite-Element Simulation ofMicro-Electromechanical Systems (MEMS) by

Strongly Coupled Electromechanical TransducersMiklos Gyimesi, Senior Member, IEEE, Ilya Avdeev, and Dale Ostergaard, Member, IEEE

Abstract—The paper introduces a new methodology for thestrongly coupled electrostatic-structural finite-element (FE) simu-lation of micro-electromechanical systems (MEMS) using a noveldistributed electromechanical transducer that is compatible withregular solid and lumped FE. Its application range is as general asthe sequentially coupled procedures, but converges more robustlywith about an order of magnitude smaller number of iteration.The new transducers apply automatic internal mesh morphing.The morphing allows the mesh density to be different across thedomain permitting accurate modeling of fringing effects.

Index Terms—Finite-element methods, micro-electromechanicaldevices, modeling, transducers.

I. INTRODUCTION

THE increasing functionality of micro-electromechan-ical systems (MEMS) leads to complex geometrical

configurations of MEMS [1]–[3], which requires efficientfinite-element (FE) modeling techniques to solve coupledelectromechanical problems. Lumped models are no longerapplicable for devices where fringing electrostatic fields aredominant such as combdrives [1] or electrostatic motors [3].There have been several numerical methods proposed for thetreatment of electromechanical systems: FE or boundary-el-ement methods using sequential physics coupling [4], [5];strongly coupled but reduced order methods using fully lumpedor mechanically distributed but electrically lumped 1-D [6],multidimensional [7], or modal-space transducers [8]. All thesemethods need some extra meshing or morphing, introduce sim-plifying assumptions, and may not be convenient to use. Thispaper introduces a distributed strongly coupled electromechan-ical transducer finite element with internal meshing/morphingfor full and accurate modeling of the underlying physicalphenomena. The new transducer can be used in an FE modelwith classical lumped and/or solid mechanical elements.

II. HISTORICAL REVIEW

Sequential coupling between electrical and mechanical FEphysics domains for MEMS analysis was introduced in [4].That allows the most general treatment of individual physics

Manuscript received July 1, 2003.M. Gyimesi and D. Ostergaard are with ANSYS, Inc., Canonsburg, PA 15317

USA (e-mail: miklos.gyimesi@ansys.com; dale.ostergaard@ansys.com).I. Avdeev is with the Department of Mechanical Engineering, University of

Pittsburgh, Pittsburgh, PA 15261 USA (e-mail: ivast@pitt.edu).Digital Object Identifier 10.1109/TMAG.2004.824592

domains. However, it cannot be applied to small signal modaland harmonic analyses because a total system eigenfrequencyanalysis requires matrix coupling. Moreover, sequential cou-pling generally converges slowly and not robustly.

To eliminate these shortcomings, a strongly coupled lumpedtransducer element was introduced in [4] and [6]. Coupling be-tween electrostatic forces and mechanical forces can be charac-terized by mapping the capacitance as a function of the devicemotion. Modeling the fully coupled system, the lumped trans-ducer stores electrostatic energy as well as converts it into me-chanical energy and vice versa. The lumped transducer is a lineelement with voltage and structural displacements degrees offreedom as across variables and current and force as throughvariables. Its input is a capacitance-stroke relationship that canbe derived from electrostatic field solutions. The electrostaticmesh is removed from the problem domain and replaced by adistributed set of lumped transducers hooked to the mechan-ical and electrical FE model providing a highly efficient re-duced order model (ROM) of a coupled electrostatic-structuralsystem. The transducer handles the difficult hysteric pull-in/re-lease analysis well [5], [7]. However, the lumped transducer isapplicable only to simple geometries when fringing effects areinsignificant.

The paper introduces a distributed transducer, which does nothave these geometrical restrictions, whereas inheriting its ex-cellent convergence behavior, due to the strong coupling of me-chanical and electrostatic physics domains.

III. FORMULATION

The full FE model of an electromechanical system, most typi-cally a MEMS device such as a combdrive shown in Fig. 4, con-sists of classical mechanical elements (lumped or distributed)and the new distributed transducer elements. The mechanicalelements are used to model the electrodes and support struc-ture and store mechanical energy (elastic and kinetic). The spacebetween the electrodes is meshed by the transducers, shown inFig. 1 (2-D for the sake of simplicity), that store electrostatic en-ergy (magnetic energy is neglected). The transducers have elec-trical potential and structural displacement as nodal degreesof freedom. The reactions are electrical charge and mechan-ical force .

In a standard electrostatic FE model, the mesh is constant. Inthe new transducer, the mesh morphs during solution. In Fig. 1,the solid lines show the original element and the dashed linesshow the morphed element. The new morphed node location

0018-9464/04$20.00 © 2004 IEEE

558 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 2, MARCH 2004

Fig. 1. Original (solid) and deformed (dashed) transducer element.

is computed as the sum of the original node location and thedisplacement vector . Element quantities, suchas energy and electric field, are the functions of and .

The formulation of the new transducer is based onRitz–Galerkin variational principles. The total system energyis the sum of mechanical and electrostatic energies. The energychange due to electric potential and displacement changesproduces reactions: electrical charges and mechanical forces.At equilibrium, these reactions balance each other at the in-ternal nodes and balance applied external forces at the externalnodes. That ensures the compatibility of transducer elementswith regular mechanical elements. The electrostatic energy ofa transducer element is given by

(1)

where is the permittivity, is the element volume, is theelectric field vector, and is the electrostatic potential. For alinear triangular or tetrahedral element with a constant permit-tivity, the integral in (1) can be evaluated analytically

(2)

where is the number of element faces, is the altitude,is the normal to the th face, and is the nodal potential.

Assembling all transducer elements together, we obtain the totalelectrostatic energy of the system

(3)

where is the number of transducers, and is the systemcapacitance matrix. This is similar to classical electrostatic FEcapacitance matrix. The difference is that the transducers aremorphing, the capacitance depends on nodal displacements; thederivatives of are used in the strong coupling formulation.The total energy of the system is

(4)

The nodal forces and charges can be calculated using the vir-tual work principle

(5)

(6)

The mesh morphs and the potential distribution develops suchthat at equilibrium, forces between transducers and mechanicalelements balance each other. In most practical cases, the systemof equilibrium equations is linear from an electrical point ofview but highly nonlinear from a mechanical point of view. Thenonlinear system of static equilibrium equations can be solvedusing the Newton–Raphson method

(7)

where and are the increments of nodal displacementsand potentials, and are the out-of-balance nodal forcesand charges, and the blocks of the tangent stiffness matrix aregiven by

(8)

(9)

(10)

(11)

The mesh, including the structural domain and the air re-gion, deforms to reach force equilibrium. The original nodesare constantly updated according to the electrostructural forceequilibrium. This procedure is highly nonlinear and huge dis-placements are allowed for an arbitrary uneven mesh. For ex-ample, in Fig. 4, combdrive fingers can significantly penetrate,dragging the fine mesh at the corners while keeping the coarsemesh elsewhere.

For the transient, modal or harmonic coupled-field FE anal-ysis, in addition to the stiffness matrix given by (8)–(11), themass and damping matrices of the mechanical domain have tobe calculated using standard mechanical finite-element method.The transducer element is also compatible to network electricalelements [4], which allows for static, transient, modal, and har-monic network analyses.

IV. EXAMPLE PROBLEMS

A. Dielectric Slab Capacitor

The purpose of the following example is to illustrate the vir-tual work principle used in the transducer formulation [9]. Thesystem consists of two anchored capacitor electrodes with con-stant charges, , and a dielectric slab sliding between the elec-trodes, suspended with a spring (see Fig. 2). The air permittivityis denoted as while the permittivity of the slab is denoted as

GYIMESI et al.: FINITE-ELEMENT SIMULATION OF MICRO-ELECTROMECHANICAL SYSTEMS 559

Fig. 2. Dielectric slab capacitor (illustration of energy formulation).

. The displacement of the slab is denoted by . The total po-tential energy is

(12)where is the electrode thickness, is the electrode width, isthe gap, is the spring stiffness, and is the capacitance.The total force acting upon a slab is

(13)

where is the potential difference between the electrodes. Atequilibrium, the total force is equal to zero, from which the equi-librium displacement of the slab is

(14)

The mechanical system stiffness is equal tothe spring stiffness because the electrostatic energy is a linearfunction of the displacement .

B. Thin Electrode Combdrive

This example (Fig. 3) is an idealization of the combdrive(Fig. 4): it has one pair of fingers (electrodes), which are in-finitesimally thin and extended to the infinity. The overlap be-tween the fingers is 16 m, and the gap is 2 m. The spacearound the electrodes is filled with air. The potential differenceis 4 V. Because of the geometry, the fringing field effects aresignificant (the equipotential lines are shown in Fig. 3). Thevalues of computed forces are N and

N. Without fringing fields, according tothe analytical formula of a plate capacitor, the forces wouldbe: N and N. Theagreement between results is reasonable. However, to reach thisagreement the mesh had to be substantially refined around theelectrode edges, since 99.9% of the horizontal force was con-centrated near the electrode tips [10].

Fig. 3. Potential field distribution generated by two overlapped electrodes.

Fig. 4. Potential field distribution between combdrive fingers (see data in [1]).

C. Combdrive Transducer

Fig. 4 shows the potential distribution in the air between thepulled-in fingers of a realistic FE model of the combdrive trans-ducer [1]. The comb fingers are meshed by regular mechanicalFE. The air region between the comb fingers is meshed by thetransducers in the original position (not shown). The mesh is re-fined near the edges and corners of comb fingers in order to ac-curately calculate fringing fields (see example B). The large ge-ometrical change of the air domain with highly unevenly meshis properly captured by transducer morphing. The computedvalues of driving force match within 1% margin with literatureresults [1].

D. Clamped Beam Pull-In/Release Bending

In this example, the new transducer element is used to effi-ciently solve the difficult pull-in/release hysteric beam-bendingcontact problem [5]. This difficulty is caused by the stabilityproblems due to the bifurcation of static equilibrium. The max-imum vertical beam deflection (in micrometers) versus poten-tial difference between the beam and the ground is depicted inFig. 5. Lacking analytical solution, the new strongly coupled re-sults are compared against results obtained by sequentially cou-pled procedures [4], [5]. There is a good agreement regarding

560 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 2, MARCH 2004

Fig. 5. Hysteresis loop: pull-in with contact and release (see data in [5]).

Fig. 6. Transient response of the current-pulse driven switch (see data in [11])The difference between two curves, less than 1%, cannot be seen on a graph ofthis size.

the accuracy whereas the new transducers converging more ro-bustly and quickly (ten times faster).

E. Current Pulse Driven RF Switch

The new transducer is applied for a transient coupled-fieldanalysis of an RF switch driven by a current pulse [11]. Theswitch is modeled as a variable gap capacitor with parallelsquare electrodes. One electrode is fixed in space and anotherone is attached to a spring-damper system. The transducerelements are used to model the air gap between the electrodes.The results of FEA (stroke versus time) agree very well with theresults obtained using Runge–Kutta time integration scheme[11] (see Fig. 6). The transducer works robustly with charge,voltage, force, or displacement loads.

REFERENCES

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[2] R. T. Howe, B. E. Boser, and A. P. Pisano, “Polysilicon integrated mi-crosystems: technologies and applications,” Sens. Actuators A, vol. 56,pp. 167–177, 1996.

[3] A. J. Sangster and V. D. Samper, “Accuracy assessment of 2-D and 3-DFE models of a double-stator electrostatic wobble motor,” J. Microelec-tromechan. Syst., vol. 6, pp. 142–150, June 1997.

[4] ANSYS Release 5.6 User Manuals, 1999.[5] J. R. Gilbert, G. K. Ananthasuresh, and S. D. Senturia, “3D modeling

of contact and hysteresis in coupled electro-mechanics,” in Proc.MEMS’96, 1996, pp. 127–132.

[6] M. Gyimesi and D. Ostergaard, “Electro-mechanical transducer forMEMS in ANSYS,” in Proc. 1999 Int. Conf. Modeling/SimulationMicrosystems, Apr. 1999.

[7] I. Avdeev, M. Gyimesi, M. Lovell, and O. Onipede, “Beam modelingfor simulation of MEMS transducers using strong coupling approach,”in 6th U.S. Nat. Congr. Comput. Mech., Dearborn, MI, Aug. 1–4, 2001.

[8] M. Gyimesi and D. Ostergaard, “FE based ROM of micro electro me-chanical systems (MEMS),” in Int. Conf. MSM 2000, San Diego, CA,Mar. 27–29, 2000.

[9] A. E. Fitzgerald, C. Kingsley, Jr., and S. D. Umans, Electric Ma-chinery. New York: McGraw-Hill, 1990.

[10] I. Vago and M. Gyimesi, Electromagnetic Fields. Budapest, Hungary:Akademia Kiado, 1998.

[11] L. Castaner, J. Pons, R. Nadal-Guardia, and A. Rodriguez, “Analysis ofthe extended operation range of electrostatic actuators by current-pulsedrive,” Sens. Actuators A, vol. 90, pp. 181–190, 2001.