DOI 10.1007/s00158-005-0531-3

RESEARCH PAPER

Struct Multidisc Optim (2005)

M. Raulli · K. Maute

Topology optimization of electrostatically actuated microsystems

Received: 11 July 2004 / Revised manuscript received: 15 March 2005 / Published online: 28 July 2005 Springer-Verlag 2005

Abstract This study addresses the design of electrostati-cally actuated microelectromechanical systems by topologyoptimization. The layout of the structure and the elec-trode are simultaneously optimized. A novel, continuous,material-based description of the interface between thestructural and electrostatic domains is presented that allowsthe optimization of the interface topology. The resulting top-ology optimization problem is solved by a gradient-basedalgorithm. The electromechanical system response is deter-mined by a coupled high-fidelity finite element model anda staggered solution procedure. An adjoint formulation ofthe coupled electromechanical design sensitivity analysis isintroduced, and the global sensitivity equations are solvedby a staggered method. The proposed topology optimizationmethod is applied to the design of mechanisms. The opti-mization results show the significant advantages of varyingthe interface topology and the layout of the electrode ver-sus conventional approaches optimizing the structural layoutonly.

Keywords Microelectromechanical systems · Topologyoptimization · Electrostatic-mechanical coupling · Sensitiv-ity analysis · Adjoint formulation

1 Introduction

Since the manufacture of the first polysilicon surface mi-cromachined devices by Bustillo et al. (1998) in the early1980s, microelectromechanical systems (MEMS) have be-come essential components of numerous applications, such

M. RaulliVillanova University, Dept. of Mechanical Engineering, 800 LancasterAvenue, Villanova, PA 19085, USAE-mail: michael.raulli@villanova.edu

K. Maute (B)University of Colorado, Dept. of Aerospace Engineering Sciences,Campus Box 429, Boulder, CO 80309, USAE-mail: maute@colorado.edu

as medical devices, automobiles, locking systems, and com-munications devices, among others. MEMS are used primar-ily for sensing and actuation purposes and allow for minia-turizing at low cost.

The design of MEMS is a challenging task since the fun-damental actuation and sensing mechanisms are often basedon the interaction of multiphysics phenomena, such as elec-trothermomechanical coupling and electrostatic-mechanicalinteraction. Conventionally, these interaction phenomenahave been accounted for in the design process by low-fidelity models, which negatively affect the performance andreliability of the system.

More recently, numerical optimization methods withhigh-fidelity simulation models have been applied to the de-sign of MEMS. In particular, topology optimization is anappealing approach to the design of MEMS for two dis-tinct reasons: (a) MEMS are often used in new contexts andapplications demanding conceptually new designs and (b)the fabrication process allows for arbitrary planar geome-tries without increasing the manufacturing costs. The latteraspect is particularly noteworthy as topology optimizationoften generates designs that are too complex to manufacturein a cost-effective manner for conventional structural appli-cations. This common issue does not apply to the design ofMEMS since it is resolved by their distinct manufacturingtechniques.

The potential of topology optimization for the design ofMEMS has been studied before. The bulk of this work isconcerned with the design of elastic compliant mechanismsfor force and displacement amplification purposes (Brunsand Tortorelli 1998; Pedersen et al. 2001). Most often, thephysical actuation mechanism is simplified and replaced bya fixed, design-independent, external load. More advancedstudies account for thermomechanical and electrothermo-mechanical actuation (Sigmund 1998, 2001a,b).

Most MEMS devices currently in use, however, are elec-trostatically actuated. A topology optimization approach forthis class of devices has been lacking and is presented in thisstudy. Considering this type of interaction in the formulationand solution of the topology optimization problem adds sig-nificant complexity. In contrast to thermal and electrother-

M. Raulli, K. Maute

mal actuation, the electrostatic pressure is not an “internal”force but acts on the interface between the structural andelectrostatic domains. A major challenge is the appropriatetreatment of the electrostatic pressure, as it is governed bythe electrostatic field equations and acts on a surface thatis not known in advance but evolves in the optimizationprocess. The proposed approach is based on a high-fidelitysimulation model of the coupled problem and analytical sen-sitivity analysis embedded into a gradient-based optimiza-tion procedure. The formulations and solution procedures ofthe topology optimization, analysis, and sensitivity analysisproblems are presented and applied to the design of 2D and3D electrostatically actuated mechanisms.

For this purpose, the paper is arranged in the follow-ing manner. The basic formulations of the electromechanicaltopology optimization and analysis problems are presentedin Sect. 2. In Sect. 3, the developments necessary for top-ology optimization without restriction on the conducting in-terface are discussed. Relevant optimization results are pre-sented in Sect. 4 followed by a summary in Sect. 5.

2 Electromechanical topologyoptimization

Recent years have seen the application of topology opti-mization to increasingly complex engineering design prob-lems (Eschenauer and Olhoff 2001). Most often the topologyoptimization problem is formulated as a material distribu-tion problem in a given design space. Among numerousmaterial interpolation schemes, the Solid Isotropic Materialwith Penalization (SIMP) model in combination with fil-tering techniques has become popular in the engineeringdesign community (Bendsøe 1989; Zhou and Rozvany 1991;Sigmund 1994). This approach leads to solid-void materialdistributions and can be applied to the interpolation of var-ious material parameters. For example, the elastic modulus(E) is interpolated as follows:

Fig. 1 Fixed and free interfaces in electromechanical topology optimization

E(ek) = spEk E0 ; pE > 1 (1)

where ek refers to the kth element in a finite element meshand sk is an optimization variable that represents the solid-void state of finite element k. The reader is referred to thetextbook by Bendsøe and Sigmund (2002) for a thorough in-troduction into topology optimization. The proposed workfollows and extends the SIMP-based material topology op-timization approach for electromechanical applications.

Based on the material formulation, topology optimiza-tion has been extended for various multiphysics problems.For example, Rodriques and Fernandes (1995) consideredthermal loading, and Sigmund (1998, 2001a,b) studied thedesign of electrothermo-actuated MEMS, as did Yin andAnanthasuresh (2002), including the use of multiple ma-terials. These problems have in common that all physicalfields involved are defined for the entire design space and thecoupling results in mechanical forces due to thermal strainsonly. The structural, thermal, and all other fields evolve au-tomatically along with the material distribution in the courseof the optimization process.

Multiphysics topology optimization problems with loadsacting on the surface of the structure, however, are lessadaptable to material topology optimization as the surfaceis not known in advance. The surface evolves in the courseof the optimization process and is not explicitly definedusing the material formulation of the topology optimizationproblem. Topology optimization with design-dependent sur-face loads has been studied by Hammer and Olhoff (2000),Chen and Kikuchi (2001), Bourdin and Chambolle (2003),and Du and Olhoff (2004a,b). However, in these studiesthe magnitude of the surface load is given rather than be-ing governed by other physical fields. Though the issues ofchanging location and magnitude are addressed, the presentstudy proposes to address them in a manner more suitablefor modeling multiphysics problems with a varying interfacetopology. The fully coupled multiphysics problem is notsimplified, as in previous design-dependent loading stud-

Topology optimization of electrostatically actuated microsystems

ies. Additionally, the use of parametric surfaces (Hammerand Olhoff 2000; Du and Olhoff 2004a,b) to represent thetopology of the interface is limiting. The methodology pro-posed herein allows for arbitrary interface locations andmaintains the fully coupled multiphysics solution of theelectromechanical system. The downside is, of course,a more complicated methodology and longer computationtimes.

Maute and Allen (2004) applied topology optimizationto the design of aeroelastic structures. The aerodynamicloads act on the fluid–structure interface, that is, the skinof the wing. For practical purposes, only the layout of theinternal structure of the wing was determined by topologyoptimization. The external shape of the structure (the skin)was fixed. While this limitation makes sense for the designof wings, it might considerably restrict the design space forother multiphysics applications with surface loads, such asthe design of electrostatically actuated MEMS.

The allowance of a free interface that evolves in the top-ology optimization process leads to several difficulties inthe formulation of the optimization problem and the mod-eling and evaluation of the system response. Figure 1 givesa conceptual idea of fixed versus free interface topology op-timization problems. The fundamental difficulties of a freeinterface are due to the fact that the spatial extent of the elec-trostatic domain changes, leading to a need to change thecomputational domain. Also, as indicated by the arrows, theelectrostatic force changes in magnitude and location. Fi-nally, the voltage differential is applied at different locations.The change in application of the voltage affects the bound-ary conditions for the electrostatic problem.

The objective and constraints of the optimization prob-lem consist of parameters in the electromechanical problem,such as displacements, stresses, electrostatic forces, etc., andare termed “criteria” herein. The evaluation of these criteriais directly linked to the location and extent of the conductinginterface. If the interface is allowed to change, as in Fig. 1b,this has a significant impact on the evaluation of criteria. Theevaluation of the electromechanical system response is dis-cussed in Sect. 2.1.

Material topology optimization leads to a large num-ber of optimization variables. Gradient-based algorithmscan efficiently handle a large number of variables. Thecoupled electromechanical sensitivity analysis is presentedin Sect. 2.2. Using gradient-based methods, however, re-quires that the optimization criteria and the system responsebe sufficiently smooth functions of the optimization vari-ables. Therefore, since the interface is free to evolve, thechanges in Fig. 1b must be handled in a continuous fash-ion. The modeling of the free interface is presented in detailin Sect. 3.

2.1 Electromechanical analysis

A broad range of analysis methods are applied to the de-sign of electromechanical MEMS devices. Many studiesfocus on the design of geometrically simple MEMS anduse analytical equations for the analysis of the electrome-chanical response, for example, Abdalla et al. (2003) for

microbeams and Bochobza-Degani and Nemirovsky (2004)for torsional actuators. There have also been investigationsinto the use of a combination of higher- and lower-fidelity-solution techniques. Younis et al. (2003) proposed the useof reduced-order models for simplifying parametric studiesof the electromechanical response. A methodology that uses2D finite element models for structural analysis and a sim-plified electrostatic loading has been proposed by Chen et al.(2004).

High-fidelity solvers for electromechanical analysis havebeen developed in the past decade. Senturia et al. (1992)presented a high-fidelity finite element–boundary elementsolver, for the mechanical and electrostatic analysis, re-spectively (Senturia et al. 1992). A coupled sequential elec-tromechanical solver that uses finite elements for both themechanical and electrostatic analysis has been developed byZhulin et al. (2000).

This study uses finite element discretizations for both thestructural and electrostatic subproblems. Though boundaryelements are the standard for electrostatic analysis, the useof finite elements is accurate for electrostatic simulations(Shaul and Sumner 2004) and takes advantage of the exist-ing finite element software. Additionally, when using a fi-nite element discretization, topology optimization removesthe material contribution of nonoptimal elements. In orderto achieve a smooth representation of the geometry, a suf-ficiently fine mesh is required that eliminates the use ofboundary elements for topology optimization of the fullelectromechanical domain. Also, since a boundary elementmethod only discretizes the interface, the topology of theelectrostatic field cannot be altered. See Sect. 3.2 for furtherdiscussion of this idea.

The structural domain is coupled to the electrostatic fieldbecause of the electrostatic pressure exerted on the struc-ture. The geometry of the electrostatic domain changes asthe structural shape changes due to deformation and opti-mization. The choice of finite elements for the electrostaticdiscretization necessitates the inclusion of a method that pre-vents the degeneration of the electrostatic mesh due to theshape changes of the electrostatic–structure interface. Forboundary element techniques only a surface mesh of theelectrostatic domain is necessary and only the geometry ofthis surface mesh needs to be updated when the structurechanges shape. Using a finite element mesh for the elec-trostatic domain, however, requires that the structural shapechanges are propagated throughout the entire electrostaticmesh, not just the interface. Subsequently, the mesh up-dating procedure of the electrostatic mesh is referred to asmesh-motion.

The mesh-motion is treated as a ficticious physical fieldthat is coupled to the structural field via deformations andto the electrostatic field by the determination of the elec-trostatic mesh configuration. The state variables of themesh-motion field are the displacements of the electrostaticmesh due to the structural deformations of the conduct-ing interface. This leads to a three-field formulation of theelectrostatic-mechanical problem, similar to the one intro-duced by Farhat et al. (1998) for fluid–structure interactionproblems.

M. Raulli, K. Maute

The three-field formulation consists of the followingresiduals, where u is structural displacements, v is electro-static voltages, x is electrostatic mesh displacements, and sis the vector of optimization variables:

S(s, u, v, x) = 0 Structure (2)

E(s, v, x) = 0 Electrostatic (3)

R(s, u, x) = 0 Electrostatic mesh (4)

The following state equations define the above residuals.It should be noted that the electrostatic and mesh-motionequations (6)–(7) are generally Dirichlet boundary conditionproblems, in which case the only right-hand side in the re-sidual equation results from constraining part of the solutionvector. Furthermore, it is assumed that the structure is a per-fect conductor.

S = Ku− fs(v, x) (5)

E = PΩΩ vΩ +PΩΓ vΓ −p ; vΓ = v on Γ v (6)

R =

KΩΩ xΩ + KΩΓ xΓ

xΓ

−Tmu(7)

with K and P being the stiffness and permittivity matrices,respectively, in a linear finite element representation. Thestructural load due to mechanical and electrostatic forces isrepresented by fs, and the right-hand side for the electro-static problem due to prescribed charge is represented by pand is generally 0. Prescribed voltage boundary conditionsare represented by v. The Ω and Γ subscripts represent,respectively, the internal and boundary portions of a par-titioned finite element stiffness matrix. This separation isconvenient for the solution of Dirichlet boundary conditionproblems, as in (6)–(7). The matrix K represents the ficti-cious stiffness of the electrostatic mesh. The mesh-motionstiffness matrix K is built using elasticity finite elements,which are topologically identical to the elements in the elec-trostatic mesh. This is similar to the technique used for flowproblems by Johnson and Tezduyar (1994). The matrix Tmis the transformation for passing structural displacements tothe electrostatic mesh. The transpose of this matrix trans-forms electrostatic forces to the structural mesh. This matrixis built with the energy-conserving matching procedure ofFarhat et al. (1998), which is applicable to matching andnonmatching meshes.

The electrostatic force, fe, on the structure, which con-tributes to the total structural load fs in (5), is computed asfollows:

fe =∫

Γ E /S

Te n dΓ E/S (8)

where Γ E/S represents the conducting interface between theelectrostatic and structure, n is the outward surface normalon the interface, and Te is the Maxwell stress tensor, definedbelow:

Te = ε

e2x − 1

2‖e‖2exey exez

exey e2y − 1

2‖e‖2eyez

exez eyez e2z − 1

2‖e‖2

(9)

where ε is the permittivity of free space and e=[ex, ey, ez]

represents the electric field, which is the spatial gradient ofthe voltage.

The conducting interface is discretized with interface fi-nite elements for evaluating (8). The interface elements arelinearally interpolated by two-node linear elements and four-node quadratic elements for 2D and 3D applications, respec-tively. The nodes of the interface elements exactly corres-pond with the nodes of the electrostatic elements; thereforethe electric field terms present in the Maxwell electrostaticstress tensor (9) are taken directly from the correspondingelectrostatic nodes and used to compute the electrostaticforces at the interface element nodes. The elemental forcesare then combined to obtain a global electrostatic force vec-tor. Figure 4 illustrates the location of the interface elementsfor fixed and free interface problems.

The overall coupled system (5)–(7) is solved usinga staggered procedure described in Maute et al. (2000) fora three-field aeroelastic formulation, here extended to elec-tromechanical coupling. The advantage of the staggeredprocedure is that different solution techniques can be usedfor the three fields. All three subproblems are solved withsequential sparse solvers. The solution algorithm is pre-sented in Table 1. Equation (13) assesses the convergenceof the overall staggered procedure. The electrostatic field isassumed to be converged if the structural residual has con-verged since a direct solver is employed for the computationof the electrostatic states.

2.2 Electromechanical sensitivity analysis

The gradients of the optimization criteria are evaluatedby a coupled electromechanical sensitivity analysis. Nu-merical methods, such as finite differencing schemes, arefrequently applied to coupled multiphysics problems asthey require no modifications of existing analysis software(Giunta and Sobieszczanski-Sobieski 1998). However, nu-merical schemes suffer from added computational costs andaccuracy problems. Therefore, in this study, the gradientsof the optimization criteria are computed by solving thecoupled electromechanical sensitivity equations.

Previous work in the computation of analytical sensi-tivities for coupled electromechanical problems has beenaccomplished for a hybrid boundary element–finite elementcoupled solution technique by Shi et al. (1995). The workof Shi et al. fully couples the fields in analysis but doesnot fully couple them in the sensitivity analysis. Also, thesensitivities of the electrostatic field are computed by differ-entiating the governing equations and then discretizing thegradients.

The analytical sensitivity approach used in this study fol-lows the general framework for deriving the global sensitiv-

Topology optimization of electrostatically actuated microsystems

Table 1 Electromechanical solution algorithm

Step 0: Initialize x(0) = u(0) on Γ E /S

For iteration (n):

Step 1: Transfer structural displacements to the electrostatic mesh:

x(n)Γ

= Tmu(n) (10)

Solve the mesh-motion equation (7) to determine x(n)Ω and update the electrostatic mesh configuration using x(n)

Ω and x(n)Γ .

Step 2: Compute the internal electrostatic state vector (vΩ ), using (6).

Step 3: Compute force on structure from electrostatic pressure:

fs(n) = Tm

T fe(n) (11)

The matrix Tm is the transformation matrix of (7), which transfers electrostatic pressure to the structure when transposed (indicated by thesuperscript T ). The electrostatic force vector (fe) is computed with (8) and (9).

Step 4: Solve (5) for u, this is u(n) . Apply a relaxation factor (θ) to u and u(n−1):

u(n) = (1−θ)u(n−1) +θu(n) (12)

Step 5: Check convergence:

∥∥∥S (s, u(n)

, v(n)

, x(n))∥∥∥≤ εem

∥∥∥S (s, u(0)

, v(0)

, x(0))∥∥∥ (13)

where εem is a specified tolerance for the electromechanical analysis. If (13) is satisfied, stop, otherwise go to Step 1.

ity equations of Sobieszczanski-Sobieski (1990). In general,the derivative of an optimization criterion, qj , with respect toan optimization variable, si , is computed as follows:

dqj

dsi= ∂qj

∂si+ ∂qj

∂u

T dudsi

+ ∂qj

∂v

T dvdsi

+ ∂qj

∂x

T dxdsi

(14)

The crux of the gradient calculation in (14) is the computa-tion of the state derivatives with respect to si .

In order to solve for the total derivatives of the stateswith respect to an optimization variable, the state equa-tions (2)–(4) are differentiated. The global sensitivity equa-tions are written as follows:

∂S

∂si

∂E

∂si

∂R

∂si

+

∂S

∂u∂S

∂v∂S

∂x

0∂E

∂v∂E

∂x∂R

∂u0

∂R

∂x

︸ ︷︷ ︸A

dudsi

dvdsi

dxdsi

= 0 (15)

The matrix A is the electromechanical Jacobian matrix,which is the linearization of the electromechanical systemabout the equilibrium solution. The off-diagonal terms in A

do not generally exist in analysis software. However, if theyare neglected, the sensitivities will be incorrect, leading tolonger convergence times or no convergence at all in theoptimization problem. Therefore, in this study these termsare derived by analytically differentiating the associated dis-cretized terms in the finite element models.

Substituting the actual gradients of the terms in A , (15)yields the following:

K − ∂fs

∂vΩ

−∂fs

∂x

0 PΩΩ

∂PΩΩ

∂xvΩ + ∂P

ΩΓ

∂xvΓ

KΩΓ Tm

−Tm

0

KΩΩ 0

0 I

dudsi

dvΩ

dsi

dxΩ

dsi

dxΓ

dsi

+

∂S

∂si

∂E

∂si

∂R

∂si

= 0 (16)

M. Raulli, K. Maute

where I is an identity matrix of appropriate size. The mesh-motion component is divided into internal and boundaryportions, since xΓ = Tmu, not a static value, as with vΓ

and uΓ

. Only the internal portion of dv/ds is consideredbecause the boundary portion dvΓ /ds = 0, since the volt-age is prescribed in this study. Two techniques for solv-ing (16), the direct and adjoint methods, are discussedin Sect. 2.2.1.

2.2.1 Adjoint sensitivity method

The determination of the sensitivities of the optimization cri-teria, of which there are nq , involves the computation of thetotal state derivatives, obtained from solving (15), for eachoptimization variable, of which there are ns:

Table 2 Adjoint method algorithm

Step 0: Compute and store ∂qj/∂u, ∂qj/∂vΩ and ∂qj/∂x; initialize au(0)j = 0, ax

(0)j = 0

Step 1: Compute the pseudoload for the structure and transfer it to the structural mesh (fs).

fs = TmT(

ax(k−1)Γj

− KΓΩ ax(k−1)Ωj

)(18)

Step 2: Determine a(k)uj , which is au

(k)j of (27), with fs substituted in. Apply a relaxation factor (θ):

au(k)j = (1−θ)au

(k−1)j +θa(k)

uj(19)

Step 3: Transfer au(k)j to the electrostatic field.

a(k)uj

= Tmau(k)j (20)

In order to compute (28), the matrix–vector product of(∂fs/∂vΩ

)T au(k)j must be computed. This matrix–vector product is computed as

follows:

∂fs

∂vΩ

T

au(k)j =

(Tm

T ∂fe

∂vΩ

)T

au(k)j = ∂fe

∂vΩ

T

Tmau(k)j (21)

∂fs

∂vΩ

T

au(k)j = ∂fe

∂vΩ

T

a(k)uj

(22)

Compute av(k)Ωj

with (28) and (22).

Step 4: Compute ax(k)j by solving (29) using av

(k)Ω

, a(k)uj and the fact that

∂fs

∂x

T

au(k)j = ∂fe

∂x

T

a(k)uj

(23)

which is expanded for the voltage gradient in (21).

Step 5: Check convergence:

∥∥∥Auj

(au

(k)

j , av(k)

j , ax(k)

j

)∥∥∥≤ εsa∥∥∥Auj

(au

(0)

j , av(0)

j , ax(0)

j

)∥∥∥ (24)

where Au refers to the structural portion of the adjoint global sensitivity equations and εsa is a specified tolerance for the sensitivityanalysis.

dqj

dsi= ∂qj

∂si+[∂qj

∂u∂qj

∂v∂qj

∂x

]

dudsi

dvΩ

dsi

dxΩ

dsi

dxΓ

dsi

(17)

Topology optimization problems, however, lead to a largenumber of optimization variables. In this study, for example,problems with up to 7200 optimization variables are consid-ered. Therefore, the adjoint method is used for the solution

Topology optimization of electrostatically actuated microsystems

of (17), since the direct method is infeasible for a largecomputational problem with this many variables. The com-putational procedure applied to the adjoint equations followsthe method of Maute et al. (2003), which was developed forfluid–structure interaction problems.

The adjoint method involves the solution of the transposeof (17) with the solution of the state derivative substitutedin:

dqj

dsi= ∂qj

∂si−[∂S

∂si

∂E

∂si

∂R

∂si

]A−T

∂qj

∂u∂qj

∂v∂qj

∂x

︸ ︷︷ ︸−aj

(25)

In this form, only nq linear systems need to be solved foreach optimization step. In this study, no more than three cri-teria are used for a given optimization problem. The adjointmethod does present the added difficulty that the inversetranspose ofA is required. In order to obtain the adjoint vec-tor (aj), the following equations need to be solved:

K 0[Tm

T KΓΩ

−TmT]

− ∂fs

∂vΩ

T

PΩΩ 0

−∂fs

∂x

T (∂P

ΩΩ

∂xvΩ + ∂P

ΩΓ

∂xvΓ

)T[

KΩΩ

0

0 I

]

·

au j

avΩ jaxΩ j

axΓ j

︸ ︷︷ ︸aj

+

∂qj

∂u∂qj

∂v∂qj

∂x

= 0 (26)

The diagonal terms in (26) are all symmetric; therefore, theyremain unaffected by the transpose. The off-diagonal terms,however, are not symmetric and their transpose needs to becomputed. The term K

ΓΩis the transpose of K

ΩΓ. The trans-

pose of A is not factorized and stored, but the adjoint statesare computed by solving the linear system in (26), witha staggered Gauss–Seidel procedure, for each optimizationcriteria. Using Gauss–Seidel on the block matrices inA , thefollowing three equations are solved during the process:

au(k)j = K−1

[Tm

T(

ax(k−1)Γj

− KΓΩ ax(k−1)Ωj

)− ∂qj

∂u

](27)

av(k)Ωj

= P−1ΩΩ

[∂fs

∂vΩ

T

au(k)j − ∂qj

∂vΩ

](28)

axΩ

axΓ

(k)

j

=K−1

ΩΩ0

0 I

∂fs

∂x

T

au(k)j −

(∂P

ΩΩ

∂xvΩ + ∂P

ΩΓ

∂xvΓ

)T

·

av(k)Ωj

− ∂qj

∂x

(29)

The algorithm used to obtain the adjoint vector is sum-marized in Table 2. An advantage of using this staggeredprocedure is that it allows the use of direct solvers for theindividual domains. Therefore, the factorized stiffness ma-trices from the analysis can be reused with minimal compu-tational cost. The off-diagonal derivative matrices are neverstored, due to memory considerations, but are recomputed asa matrix–vector product for each step in the staggered pro-cedure. The matrices only need to be formed on an elementlevel and multiplied by an element vector. This result is thenstored in a global vector.

3 Topology optimization model for free interface

3.1 Modified SIMP modelfor free interface

The classical SIMP model is extended in this study to theelectrostatic domain for the purpose of electromechanicaltopology optimization with a free interface. Figure 1b il-lustrates how the voltage boundary conditions, electrostaticforces, and electrostatic domain all change when the con-ducting interface is allowed to evolve during the optimiza-tion process. In order to accurately determine the electrome-chanical response when the interface is evolving and main-tain a smooth relationship between the optimization vari-ables and the system response, the behavior of the electro-static domain is altered for changes in the material distribu-tion. The classical SIMP model only addresses how struc-tural properties are affected by redistribution of material. Inorder to effectively compute the coupled electromechanicalresponse, the analysis of the electrostatic domain must alsobe considered. The modifications of the SIMP model affectthe analysis not only of the system but of the gradients aswell. The ∂S/∂si and ∂E/∂si terms of (15) are affected bythe SIMP model since structural and electrostatic parametersare linked to optimization variables. The consideration of theabove issues is discussed in detail below.

3.1.1 Voltage boundary condition

In the electromechanical problems of this study a voltagedifferential is applied between an electrode and a conduct-ing body (the structure). Therefore, the conducting interface,where the structure interfaces with the electrostatic domain,is subject to a constrained voltage. This prescribed voltagedifferential is what drives the electrostatic forcing and thesystem response. In the computational electrostatic prob-lem, voltage is a Dirichlet boundary condition. Therefore,

M. Raulli, K. Maute

Fig. 2 Changing of voltage boundary condi-tions during topology optimization

changing the location of the conducting interface changesthe location of the Dirichlet boundary conditions, as illus-trated in Fig. 2.

The methodology employed is to enforce all potentialvoltage boundary conditions throughout the topology opti-mization process. This cannot be done by strictly enforcingDirichlet boundary conditions because then changes in volt-age would be in an “on–off” manner, which would lead todiscontinuities.

Therefore, this issue is addressed by enforcing all po-tential voltage boundary conditions indirectly, or “softly.”Rather than eliminating equations in the electrostatic systemfor prescribed voltages, a voltage boundary condition (vk) isenforced by adding a weighting term (w(ek)

v ) to the corres-ponding diagonal entry in the permittivity matrix (Pkk) thatis relatively large. This term will dominate that particularequation, leading essentially to Pkkvk = pk. This allows thevoltage to be enforced in a nonexplicit manner, through thevalues in p, as shown in the following equations:

Pkk = w(ek)v +P0

kk (30)

pk = wvvk (31)

The terms w(ek)v and wv are defined in (32) and (33), re-

spectively. The value of the weighting term for a givenelectrostatic element is made dependent on an optimizationvariable such that the enforcement of vk changes in a smoothfashion:

w(ek)v = wv(sk − smin) (32)

wv = wv0Pavg

1− smin(33)

where Pavg is a scalar representing the average value of theentries in P and wv0 is a user-defined value that determineshow large the relative weighting factor is. The denomina-tor in wv is used to account for the fact that smin > 0. Sincevoltage is a nodal quantity, each node with a soft voltagecondition is linked to one element to which it is connected.This approximation can be further refined in future work, butfor fine meshes the effect is not significant.

3.1.2 Electrostatic mesh

As portions of the initial structural domain become void,regions that were previously part of the structural domainbut are now void need to be considered as part of the elec-

trostatic domain. One potential solution is a remeshing ofthe complete electromechanical domain in order to rede-fine the electrostatic and structural domains. This methodis not chosen because it would require automatic remesh-ing based on the current material distribution, which in itselfcan be difficult to determine due to the high proportion ofpoorly defined “gray” regions in the initial iterations. Addi-tionally, remeshing results in discontinuities and automaticmesh generators typically suffer from robustness problems.

The methodology used in this study is to generate anelectrostatic mesh that covers the initial electrostatic do-main (E0) as well as the region occupied by the potentialstructural domain (Eδ), as indicated on the left-hand sideof Fig. 3. In this way, electrostatic elements become “active”as the corresponding structural elements become void, as il-lustrated on the right-hand side of Fig. 3. This activation isaccomplished in a continuous manner, rather than an “on–off” manner, by adjusting the permittivity of the electrostaticelements in Eδ.

The permittivity of an element in Eδ is made dependenton an optimization variable such that overlapped electro-static elements that correspond to solid structural elements(sk = 1) behave like conductors (the electrostatic equivalentof rigid structures):

ε(ek) = εmax(sk − smin)pε + ε0 (34)

εmax = ε0(smin)−pE (35)

where pE is the exponent penalizing the elastic modu-lus computation, see (1). The maximum permittivity term(εmax) is computed such that conducting electrostatic elem-ents have the same relative magnitude to the free-space elec-trostatic elements as the solid structural elements have tothe void structural elements, generally about 1×109 more.This term is necessary since when sk = 1, it should causethe permittivity of that element to go to infinity, resultingin a poorly conditioned global permittivity matrix. There-fore, εmax is limited in magnitude to avoid numerical prob-lems while still being large enough to represent a conductor.The exponent (pε) is used such that elements that are closeto void are penalized to more closely represent free spacerather than a conductor.

3.1.3 Electrostatic forces

The elements in Eδ also have implications for the compu-tation of the electric field, which is used to compute theelectrostatic forces (9). In the finite element problem, the

Topology optimization of electrostatically actuated microsystems

Fig. 3 Overlap of structural and electrostatic mesh for initial configuration (left) and optimized configuration (right). The structural mesh andthe electrostatic mesh in Eδ are spatially identical but shown offset for illustration purposes

global electric field at a given node is computed by aver-aging the local electric field in all the connected elements.If an element in Eδ corresponds to a solid structural elem-ent (sk = 1), the electric field computation from this elementshould not be considered. Therefore, a weighting term forthe electric field computation in a given element is includedin the optimization formulation:

w(ek)e = we0

1− sk

1− smin; we0 = 1.0 (36)

The denominator in (36) is used such that the elementalweight varies between 0 and 1 even though the optimizationvariable varies between smin and 1. This SIMP parameter en-sures that the correct electric field is computed where solidand void elements meet.

In addition to the effect of the electric field on elec-trostatic force computation, the interface between the elec-trostatic and structural domains is continually changing, asillustrated in Fig. 3. Since electrostatic forces are only com-puted on the conducting interface, the locations of the inter-face finite elements, which compute the electrostatic forceswith (8), are constantly changing. The solution to this prob-lem is handled in a similar manner to the electrostatic elem-ents: interface elements are generated at all element bound-aries in Eδ. Figure 4 shows a conceptual representation of

Fig. 4 Interface elements in standard (left)and SIMP electromechanical models

this approach. This allows for a continuous representationof the electrostatic forces. The basic methodology employedis that solid structural elements are conducting; therefore,the interface elements should fully compute forces there. Forinterface elements attached to void structural elements, noelectrostatic forces should be computed. The interface elem-ents are modified as follows:

ε(ek)i = ε0(sk − smin)

pi (37)

ε0 = ε0

(1− smin)pi

(38)

where pi is the penalization exponent for intermediatevalues of sk and ε0 is the same as is used for building thepermittivity matrix. The term ε

(ek)i refers only to the permit-

tivity used to compute the electrostatic forces in the interfaceelements, see (9).

3.2 Optimization of electrostatic domain topology

As shown in the previous sections, especially in Figs. 1–4,the proposed electromechanical topology model always con-tains some region that is permanently free space (electro-static domain), in addition to the regions where there is

M. Raulli, K. Maute

the potential for structure or free space. To achieve greaterflexibility in the overall design of an electromechanical sys-tem, the topology of the electrostatic domain can also bealtered. This can be accomplished by making the permittiv-ity of electrostatic elements independent optimization vari-ables (sj):

ε(ej ) = εsj ; smin,ε ≤ sj ≤ 1.0 (39)

The term smin,ε indicates that there may be a differentminimum value of the variables for the electrostatic elem-ents than the structural elements. This permittivity variationrefers only to the part of the electromechanical domain thatis always free space, not the part that overlaps the struc-ture. Conceptually, this is the equivalent of putting insulatingmaterial in the electrostatic domain in order to prevent theelectric field from being transmitted in certain locations.Also, the topology of an electrode can be changed in thismanner. Making the permittivity zero in electrostatic elem-ents connected to the electrode is the physical equivalent ofremoving that part of the electrode. This latter technique isused in the example of Sect. 4.2. The topology optimizationof purely electrostatic systems is addressed by Byun et al.(2002).

It is assumed that the changes in permittivity do not af-fect the stiffness of the structural domain in any way; there-fore, there is no need for any additional parameters thatcouple the structural model to the independent permittivityvariables, sj .

3.3 Verification of SIMPmodel behavior

In addition to the two standard structural properties—theelastic modulus and the density—four electrostatic proper-ties have been introduced for smoothly varying the elec-tromechanical interface. The associated SIMP parametersare summarized in Table 3. As multiple material interpo-lation schemes are used, it is important to verify that themodified SIMP model behaves in a manner that encour-ages a “0-1” distribution. The simple example in Fig. 5 isused to verify the behavior of the SIMP model. The stateof the structural/electrostatic element is controlled by oneoptimization variable, s. The value of pE is fixed at 3.0,a common value in topology optimization, and the value ofpi is varied. Figure 6 plots the value of the displacements atthe free structural nodes labeled 1 and 2. The displacementsat the two nodes are identical and are plotted as s is variedbetween 0 and 1 for the different values of pi .

Table 3 Electrostatic SIMP parameters

w(ek )v Controls the ‘soft’ enforcement of voltage boundary

conditions

w(ek )e Adjusts weighting of elemental contribution to electric field

ε(ek) Changes permittivity of overlapped electrostatic elements

ε(ek)i Changes permittivity of overlapped interface elements

Fig. 5 Simple mesh for exponent verification. E: electrostatic elem-ents, S: structural elements

Fig. 6 Displacement relationship to optimization variable, for varyingpi

The plots in Fig. 6 illustrate the importance of choosingthe correct exponential values in order to encourage a “0-1”distribution. Since the goal of the SIMP method is to pe-nalize intermediate variables, thereby obtaining a “0-1” dis-tribution, the curves resulting from pi ≥ 3.0 are desirable.Though the displacement for all values of pi are equivalentat the endpoints, lower values of pi can lead to a nonconvex

Fig. 7 Structural (top) and electrostatic meshes for 2D exact represen-tation

Topology optimization of electrostatically actuated microsystems

Fig. 8 Structural (left) and electrostaticmeshes for 2D topology representation

relationship between displacements and optimization vari-ables that does not encourage a “0-1” distribution. In thisstudy, 3.5 ≤ pi ≤ 4.0. This range is chosen over pi = 6.0because the gradients for s < 0.3 are small and thereby donot strongly force the optimization problem to the lowerbound. This example of the SIMP model behavior is rel-evant for optimization objectives involving displacementsand strain energy, with mass constraints, which is the case inthis study.

Another aspect of the SIMP model that requires verifica-tion is that it reproduces a system response consistent withone in which the material distribution is exactly representedby the computational meshes for completely solid and voidelements.

The meshes used for an exact and a topology optimiza-tion problem are given in Figs. 7 and 8, respectively. Thematerial distribution of the structural mesh in Fig. 8 is ad-justed to the solid–void distribution that corresponds to thegeometry in Fig. 7. The arrows on the structural mesh aredisplacement boundary conditions. The arrowheads on theelectrostatic mesh are strictly enforced voltage boundaryconditions. The physical parameters for the problem aregiven in Table 4.

In order to assess the agreement between the two prob-lems, several physical quantities are compared visually andnumerically. Figure 9 shows the material distribution for thetopology problem, which mimics the geometry of the ex-act problem. Figure 10 shows the voltage distribution in theexact and topology models, which are visually similar.

The norms of the nodal vectors for electrostatic andstructural quantities, as well as the scaled norm of the dif-ference between the exact and topology solutions, are givenin Table 5. It should be noted that the size of the exact and

Table 4 Electrostatic and structural properties for verification of 2D topology approximations

Electrostatic Permittivity Voltage Min/max air gapVacuum 8.85×10−12 F/m 100.0 V 0.5/2.5 µm

Structure Elastic modulus Poisson ratio Plate thickness Width/heightSilicon 1.5×1011 Pa 0.17 1.0×10−7 m 6.0 µm

topology vectors are different, since there are more nodesin the topology model than in the model based on the truegeometry. However, the additional nodes in the topologymodels are filtered out such that the differences in solutionvectors can be effectively compared. The actual norms of theexact and (unfiltered) topology models are shown to demon-strate that the nodal values filtered out of the topology modelare small in comparison to the exact nodal values. It shouldbe noted that the value of the topology norm for the struc-tural displacements still has the nodes filtered out since thereare nonzero displacements in the topology model nodes thatdo not correspond to nodes in the exact model. The associ-ated elements, however, are not significantly contributing tothe stiffness. The electrostatic and structural quantities agreewell for the exact and topology models.

Fig. 9 Material distribution for topology model

M. Raulli, K. Maute

Fig. 10 Voltage distribution for 2D exactmodel (left) and topology model

Table 5 Numerical comparison of exact and topology models

Exact norm (‖e‖) Topology norm (‖t‖) Scaled difference norm(‖e− t‖/‖e‖)

voltage 8.35048712×102 8.35049176×102 1.2662×10−6

x-electric field 4.48090547×108 4.48090344×108 3.1619×10−6

y-electric field 2.16490769×109 2.16490786×109 6.4872×10−7

x-electric force in E 1.65375485×10−9 1.65375030×10−9 2.8157×10−6

x-electric force in S 1.65375000×10−9 1.65375000×10−9 1.0761×10−2

y-electric force in E 1.41106880×10−8 1.41114835×10−8 5.6671×10−7

y-electric force in S 1.41106970×10−8 1.41114938×10−8 1.0761×10−2

x-displacement 7.67808298×10−11 7.76841956×10−11 1.2014×10−2

y-displacement 1.64765068×10−10 1.64072914×10−10 5.7870×10−3

4 Examples

The proposed topology optimization methodology is ap-plied to the design of electrostatically actuated mechanisms.The example in Sect. 4.1 is a 2D force inverter. Section 4.2presents a 3D force inverter. In both examples the optimiza-tion problems are solved by the Method of Moving Asymp-totes (MMA) (Svanberg 1987). The Electromechanical sys-tem response is computed by the computational proceduredescribed in Sect. 2.1. The gradients of the optimizationcriteria are computed by the adjoint approach of Sect. 2.2.Though the geometric dimensions in the following exam-ples are small for MEMS devices and more appropriate fornanoscale devices, the examples demonstrate the effective-ness of this methodology for performing topology optimiza-tion on coupled electromechanical systems.

4.1 Two-dimensional force inverter

The goal of this example is to create a force inverter. A simi-lar design problem has been studied by Sigmund (1997),Pedersen et al. (2001), and Maute and Frangopol (2003). Incontrast to the above problems, the actuation force is not

a given force but the electrostatic pressure acting on the bot-tom edge of the structure, which pulls the structure towardsthe electrode. Figure 11 gives a schematic of this optimiza-tion example. The desired design will invert the electrostaticpressure acting in the negative y-direction such that point Cin Fig. 11 moves in the positive y-direction.

The optimization problem is formulated as follows:

maxs z(s) = uc subject to:

Mass ≤ 10% of total

Π ≤ Π0

1.0×10−3 ≤ sk ≤ 1.0

(40)

where uc is the displacement at point C in Fig. 11. The cur-rent and initial strain energy are represented by Π and Π0,respectively. A static force of −5.0×10−13 N is applied atthe same node as the displacement objective in order to sim-ulate actuation of a work piece. See Table 6 for the values ofthe SIMP parameters used in this example.

The mass constraint is applied in order to encouragea ‘0-1’ distribution using the SIMP model. An energy con-straint is also used in the problem formulation, for two rea-sons:

Topology optimization of electrostatically actuated microsystems

Fig. 11 Schematic of 2D force inverter example

Table 6 SIMP parameters for Sect. 4.1

Elastic modulus penalization (pE ) 3.0Electrostatic permittivity penalization (pε) 6.0Interface permittivity penalization (pi ) 4.0Soft voltage weighting factor (wv0 ) 1.0×104

1. The optimization process seeks to maximize the up-ward displacement. Since the electrostatic pressure iswhat drives the inversion, once the appropriate mech-anisms have been determined by the optimization pro-cess, a larger electrostatic pressure will lead to increasedupward displacement. This is good from the point of

Fig. 12 Structural (left)and electrostatic computa-tional meshes

view of the objective, but if the electrostatic pressurebecomes too strong, the electromechanical system willbecome unstable and be pulled into contact with the elec-trode (pull-in), causing the elements in the electrostaticmesh to collapse and the optimization process to stop.To ensure that the optimization process converges to anoptimal design and that the design is stable, an energyconstraint is enforced such that the structure is limitedin its deformations. The value of Π0 is chosen becauseallowing higher energies consistently leads to pull-in.

2. Since the energy constraint limits the overall strains ofthe structure, the optimization process is forced to stiffenin order to satisfy the constraint. This leads to a more“0-1” material distribution.

This example is treated as purely 2D, in both the struc-tural and electrostatic analysis. Figure 12 shows the fullstructural and electrostatic meshes used in this example. Ap-plying symmetry boundary conditions, only half of eachcomputational domain is analyzed and optimized. The struc-ture is discretized with four-node quadratic plane stresselements. The electrostatic mesh uses four-node quadraticelements. The physical properties and computational meshsizes for this optimization problem are given in Tables 7 and8, respectively.

In addition to running the problem with a free interface,the same problem is run with a fixed interface. Both prob-lems are started with an initial interface with a height ofone element; however, the fixed interface problem does nothave the freedom to remove the row of interface elements.The same numerical values for mass and energy constraintsare used in the fixed and free problems, in order to effec-tively compare the methodologies. The optimization resultsare summarized in Table 9. The initial displacement in thefree interface problem is less because the interface is startedat 80% of solid. The reported times are for running each

M. Raulli, K. Maute

Table 7 Electrostatic and structural properties for Sect. 4.1

Electrostatic Permittivity Voltage Initial air gapVacuum 8.85×10−12 F/m 2.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Initial width & heightSilicon 1.5×1011 Pa 0.17 1.0×10−7 m 5.0 µm

Table 8 Summary of computational problem for Sect. 4.1

Nodes Total DOF Free DOF Elements

Structure 7381 14,762 14,758 7200Electrostatic 8133 8133 8072 7920Mesh-motion 8133 16,266 16,144 7920Total 23,647 39,161 38,974 23,040

Table 9 Summary of optimization problem and results for Sect. 4.1

Free interface Fixed interface

# optimization variables 7200 7080Initial uc −1.2972×10−12 −4.1089×10−12

Final uc 2.1418×10−9 6.7104×10−10

Displacement change fromfixed initial value

−5.2127×104% −1.6332×104%

Energy/Mass constraint ac-tive

Yes/Yes Yes/Yes

Number of iterations 2649 5295Total time 55.4 hours 7.57 hours

computational domain on one Pentium IV 1.7 GHz proces-sor.

The final material distribution is shown in Fig. 13 for thefree and fixed interfaces. The methodology that allows fora free interface makes this problem much more flexible, asillustrated by the fact that much of the bottom layer of thestructure is removed by the optimization process and the al-lowable material is used to create the necessary mechanismsand stiffen the structure such that the energy constraint is not

Fig. 13 Final material distribu-tion for Sect. 4.1, with a free(left) and fixed interface

violated. In the fixed interface problem, the objective can-not improve as much because of the need to brace the entirefixed interface with extra supports, allowing less materialto be used in making the mechanism more efficient. Also,the need to support the fixed interface leads to a more com-plex structure. The computational time for the fixed problemis less because there is no need to overlap the electrostaticdomain, resulting in a much smaller electrostatic computa-tional problem.

One minor issue is that the interface in the free problemis not entirely solid. This is due to the fact that the prob-lem finds mechanisms that accomplish the desired goal ofa force inverter while violating the energy constraint. Ratherthan changing the mechanisms in order to satisfy the energyconstraint, the optimization algorithm reduces the materialfraction at the interface, which effectively reduces the elec-trostatic forcing. If this piece was manufactured, this inter-face could be made solid and the voltage reduced, achievingthe same behavior. Forcing the elements at the interface tobe completely solid for force inverter problems should beaddressed in future work.

4.2 Three-dimensional force inverter

The goal of this example is also to create a force inverter.In addition to the optimization of the structural topology, thepermittivity in the layer of electrostatic elements above theelectrode is also optimized, essentially determining the top-ology of the electrode. Figure 14 gives a schematic of thisoptimization example. The desired design inverts the elec-trostatic pressure acting in the negative y-direction such thatpoint t in Fig. 11 moves in the positive y-direction. This can-tilevered example does not have the advantage of opposing

Topology optimization of electrostatically actuated microsystems

Fig. 14 Schematic of 3D force inverter example

supports to form mechanisms around, as does the 2D ex-ample in Sect. 4.1.

The optimization problem is run as follows.1. The optimization process is started with the following

initial formulation:

maxs z(s) = uc subject to:

Mass ≤ 10% of total

Π ≥ Π0

1.0×10−3 ≤ s(S)i ≤ 1.0

1.0×10−9 ≤ s(E )i ≤ 1.0

(41)

where ut is the displacement at point t in Fig. 14. The su-perscripts S and E indicate optimization variables relat-ing to solid–void status in the structural and electrostaticdomains, respectively. The constraint that prevents thestrain energy from decreasing serves the purpose of forc-ing the problem away from the local optimum of a stiffstructure. The optimization problem was attempted with-out this constraint, and the results were a stiffer structurewith a marginal decrease in the negative y-displacementof point t. With the energy constraint, the optimizationalgorithm gradually finds the desired optimum of invert-ing the displacement. The mass constraint is used forclarity of the final design. See Table 10 for the values ofthe SIMP parameters used in this example.

2. The optimization process slowly minimizes the negativey-displacement until it finally inverts the displacement,indicated by ut becoming positive. The positive displace-ment grows, and eventually the structure reaches pull-inbecause there is no constraint to prevent the increase ofthe structural displacements, causing the simulation pro-

Table 10 SIMP parameters for Sect. 4.2

Elastic modulus penalization (pE ) 3.0Interface permittivity penalization (pi ) 3.5Soft voltage weighting factor (wv0 ) 1.0×104

cedure to fail. At this point, the optimization process isrestarted, with an additional constraint added to the op-timization problem.

maxs z(s) = uc subject to:

Mass ≤ 10% of total

Π ≤ Π

1.0×10−3 ≤ s(S)i ≤ 1.0

1.0×10−9 ≤ s(E )i ≤ 1.0

(42)

where Π = 4.0 × 10−19, which is approximately onequarter the value of the strain energy at the last stable it-eration before pull-in, Πlast1 = 1.7391×10−18, in step 1.In this case, the energy constraint is used to prevent thesystem from becoming unstable. This optimization for-mulation is run until convergence.

Varying the permittivity independently in the lower layerof electrostatic elements allows the optimization algorithmgreater flexibility in determining the electrostatic pressureon the structure. If the permittivity in the electrostatic do-main is not varied, then there will be a large voltage gra-dient everywhere that there is structural material, resultingin significant electrostatic forces. By varying the permit-tivity beneath a location where there is structural material,it is possible to have solid structure without creating elec-trostatic pressure. The voltage is not transmitted throughthe insulating layer, leading to negligible voltage gradientsand therefore negligible electrostatic pressure. For compar-ison purposes, the optimization problem was run withoutthe electrostatic permittivity variations, yielding only an in-significant improvement in the objective.

This example is fully 3D, in both the structural and elec-trostatic analyses. Figure 15 shows the full structural andelectrostatic meshes used in this example. The structure isdiscretized with three-node triangular ANDES plate elem-ents (Militello and Felippa 1991). The electrostatic meshuses eight-node hexahedron elements. The physical prop-erties and computational mesh sizes for this optimizationproblem are given in Tables 11 and 12, respectively.

Fig. 15 Structural and electrostatic mesh for Sect. 4.2

M. Raulli, K. Maute

Table 11 Electrostatic and structural properties for Sect. 4.2

Electrostatic Permittivity Voltage Initial air gapVacuum 8.85×10−12 F/m 5.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Init. width & heightSilicon 1.5×1011 Pa 0.17 1.0×10−7 m 5.0 µm

Table 12 Summary of computational problem for Sect. 4.2

Nodes Total DOF Free DOF Elements

Structure 1681 10,086 9840 6400Electrostatic 8405 8405 6724 6400Mesh-motion 8405 25,215 20,172 6400Total 18,491 43,706 36,736 19,200

Fig. 16 Final material distribution for Sect. 4.2

The optimization results are summarized in Table 13.The computational time is for running each computationaldomain on one Pentium IV 1.7 GHz processor. The finalstructural topology is shown in Fig. 16. A schematic of the

Fig. 18 Permittivity distribution (left) and voltage distribution in displaced mesh

Fig. 17 Schematic of 3D force inverter

final design is given in Fig. 17 to illustrate its functional-ity. The center part of the device, connected to the support,serves as a fulcrum. The part of the device attached to theside of the support is pulled downward, by the electrostaticforces, allowing the curved lever arm, which is connected tonode t, to rotate around the center support.

Figure 18 shows the permittivity distribution in the elec-trostatic mesh, which varies from the permittivity of vac-uum (black) to zero (white). Figure 18 also shows the elec-trostatic voltage distribution in the deformed electrostaticmesh. The voltage varies between zero volts (white) and fivevolts (black). In the permittivity distribution, the front corner

Topology optimization of electrostatically actuated microsystems

Table 13 Summary of optimization problem and results for Sect. 4.2

# optimization variables 1600 (S ); 1600 (E )

Initial ut −2.4799×10−9

Final ut 1.7226×10−6

Displacement change −6.9461×104%Number of iterations 306Total time 193.5 hours

corresponds to the front corner of the voltage distribution.The other corner in the permittivity plot with the solid patchcorresponds to the back corner along the left edge of Fig. 16.The perspective is different because the electrostatic mesh inthe permittivity plot has been flipped upside down to showthe layer of elements on the electrode. In effect, the electrodehas been reduced to two small circular patches, only one ofwhich significantly contributes to the force inverter.

5 Summary

The design of MEMS is continually evolving, with chang-ing parameters and applications. The conceptual design ofMEMS in an automatic fashion through the use of high-fidelity topology optimization is a powerful tool for design-ing new devices. This study has presented a methodologyfor performing topology optimization of MEMS that areelectrostatically actuated, without limitation on the interfacebetween the structural and electrostatic computational do-mains, allowing for greater freedom in the generation of op-timal topologies for various design objectives. This method-ology requires a fully coupled sensitivity analysis of theelectromechanical response in addition to the fully coupledanalysis. Additionally, the classical SIMP model is modifiedfor electromechanical problems. The voltage boundary con-ditions are enforced in an indirect manner in order to allowa flexible interface. Two numerical examples of force invert-ers were presented to show the applicability of the developedmethodology.

The results illustrated the advantages of varying the in-terface topology and the layout of the electrode versus con-ventional approaches optimizing the internal structural lay-out only. In this study, constraints on the strain energy wereintroduced to prevent pull-in instabilities. In future stud-ies, constraints on pull-in instabilities should be directlyaccounted for, requiring appropriate prediction and sensi-tivity analysis capabilities of this phenomenon, which arecurrently lacking.

Acknowledgement The first author would like to acknowledge thesupport of Sandia National Laboratory under the direction of JimAllen. Both authors acknowledge the support by the National ScienceFoundation under Grant DMI-0300539 and the Air Force Office ofScientific Research under Grant F49620-02-1-0037. The opinions andconclusions presented in this paper are those of the authors and do notnecessarily reflect the views of the sponsoring agencies.

References

Abdalla MM, Reddy CK, Faris W, Gurdal Z (2003) Optimal designof an electrostatically actuated microbeam for maximum pull-involtage. In: AIAA/ASME/AHS/AISC 43rd conference on struc-tures, structural dynamics and materials, Norfolk, VA

Bendsøe MP (1989) Optimal shape design as a material distributionproblem. Struct Optim 1:193–202

Bendsøe MP, Sigmund O (2002) Topology optimization, theory,methods and applications. Springer, Berlin Heidelberg New York

Bochobza-Degani O, Nemirovsky Y (2004) Experimental verificationof a design methodology for torsion actuators based on a rapidpull-in solver. J Microelectromech Syst 13(1):121–130

Bourdin B, Chambolle A (2003) Design-dependent loads in topologyoptimization. ESAIM Control Optimisat Calculus Variat 9:19–48

Bruns TE, Tortorelli DA (1998) Topology optimization of geometri-cally nonlinear structures and compliant mechanisms. In: Pro-ceedings of the 7th AIAA/USAF/NASA/ISSMO symposiumon multidisciplinary analysis and optimization, St. Louis, MO,pp 1874–1882

Bustillo JM, Howe RT, Muller RS (1998) Surface micromachining formicroelectromechanical systems. Proc IEEE 86(8):1552–1574

Byun JK, Park IH, Hahn SY (2002) Topology optimization of elec-trostatic actuator using design sensitivity. IEEE Trans Magnet38(2):1053–1056

Chen B-C, Kikuchi N (2001) Topology optimization with design-dependent loads. Finite Elements Anal Des 37:57–70

Chen K-S, Ou K-S, Li L-M (2004) Development and accuracy assess-ment of simplified electromechanical coupling solvers for memsapplications. J Micromech Microeng 14:159–169

Du J, Olhoff N (2004) Topological optimization of continuum struc-tures with design-dependent surface loading—Part I: new com-putational approach for 2d problems. Struct Multidisc Optim27:151–165

Du J, Olhoff N (2004) Topological optimization of continuum struc-tures with design-dependent surface loading—Part II: algorithmand examples for 3d problems. Struct Multidisc Optim 27:166–177

Eschenauer HA, Olhoff N (2001) Topology optimization of continuumstructures: a review. Appl Mech Rev 54(4):331–389

Farhat C, Degand C, Koobus B, Lesoinne M (1998) Torsional springsfor two-dimensional dynamic unstructured fluid meshes. ComputMethods Appl Mech Eng 163:231–245

Farhat C, Lesoinne M, LeTallec P (1998) Load and motion trans-fer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation,optimal discretization and application to aeroelasticity. ComputMethods Appl Mech Eng 157:95–114

Giunta AA, Sobieszczanski-Sobieski J (1998) Progress towards usingsensitivity derivatives in a high-fidelity aeroelastic analysis ofa supersonic transport. In: AIAA 98–4763, 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and op-timization, St. Louis, MO, September 1998, pp 441–453

Hammer VB, Olhoff N (2000) Topology optimization of continuumstructures subjected to pressure loading. Struct Multidisc Optim19:85–92

Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallelfinite element computations of flow problems with moving bound-aries and interfaces. Comput Methods Appl Mech Eng 119:73–94

Maute K, Allen M (2004) Conceptual design of aeroelastic structuresby topology optimization. Struct Multidisc Optim 27:27–42

Maute K, Lesoinne M, Farhat C (2000) Optimization of aeroelasticsystems using coupled analytical sensitivities. In: AIAA 2000–0560, 38th Aerospace Science Meeting and Exhibit, Reno, NV,10–13 January 2000

Maute K, Frangopol D (2003) Reliability-based design of mems mech-anisms by topology optimization. Comput Struct 81:813–824

Maute K, Nikbay M, Farhat C (2003) Sensitivity analysis and designoptimization of three-dimensional non-linear aeroelastic systemsby the adjoint method. Int J Numer Methods Eng 56(6):911–933

M. Raulli, K. Maute

Militello C, Felippa C (1991) The first andes elements: 9-dof platebending triangles. Comput Methods Appl Mech Eng 91:217–246

Pedersen CBW, Buhl T, Sigmund O (2001) Topology optimization oflarge displacement compliant mechanism. Int J Numer MethodsEng 50:2683–2705

Rodriques H, Fernandes P (1995) A material based model for top-ology optimization of thermoelastic structures. Int J NumerMethods Eng 38:1951–1965

Senturia S, Harris R, Johnson B, Kim S, Nabors K, Shulman M,White J (1992) A computer-aided design system for micro-electromechanical systems (memcad). J Microelectromech Syst1(1):3–13

Shaul D, Sumner T (2004) Estimating accuracy of electrostatic finiteelementmodels.CommunicationsNumerMethodsEng20:313–321

Shi F, Ramesh P, Mukherjee S (1995) Simulation methods for micro-electro-mechanical structures (MEMS) with application to a mi-crotweezer. Comput Struct 56:769–783

Sigmund O (1994) Design of material structures using topology opti-mization. PhD thesis, Danish Center for Applied Mathematics andMechanics, Technical University of Denmark, Lyngby, Denmark

Sigmund O (1997) On the design of compliant mechanisms. MechStruct Mach 25:493–524

Sigmund O (1998) Topology optimization in multiphysics problems.In: AIAA 98–4905, Proceedings of the 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimiza-tion, St.Louis, MO, pp 1492–1500

Sigmund O (2001) Design of multiphysics actuators using topologyoptimization—Part I: One-material structures. Comput MethodsAppl Mech Eng 190(49–50):6577–6604

Sigmund O (2001) Design of multiphysics actuators using topologyoptimization—Part II: Two-material structures. Comput MethodsAppl Mech Eng 190(49–50):6605–6627

Sobieszczanski-Sobieski J (1990) Sensitivity of complex, internallycoupled systems. AIAA J 28:153–160

Svanberg K (1987) The method of moving asymptotes—a newmethod for structural optimization. Int J Numer Methods Eng24:359–373

Yin L, Ananthasuresh GK (2002) A novel topology design schemefor the multi-physics problems of electro-thermally actuatedcompliant micromechanisms. Sensors Actuators A 97–98:599–609

Younis M, Abdel-Rahman E, Nayfeh A (2003) A reduced-order modelfor electrically actuated microbeam-based mems. J Microelec-tromech Syst 12(5):672–680

Zhou M, Rozvany GIN (1991) The coc algorithm, Part II: Topological,geometrical and generalized shape optimization. Comput MethodsAppl Mecha Eng 89(1–3):309–336

Zhulin VI, Owen SJ, Ostergard DF (2000) Finite element basedelectrostatic-structural coupled analysis with automated meshmorphing. In: Technical Proceedings of the 2000 internationalconference on modeling and simulation of microsystems, pp 501–504. Computational Publications, Boston