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/home/journal/dvi/SMO531-of-SMO.dviTopology 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 topology optimization. The layout of the structure and the elec- trode are simultaneously optimized. A novel, continuous, material-based description of the interface between the structural and electrostatic domains is presented that allows the optimization of the interface topology. The resulting top- ology optimization problem is solved by a gradient-based algorithm. The electromechanical system response is deter- mined by a coupled high-fidelity finite element model and a staggered solution procedure. An adjoint formulation of the coupled electromechanical design sensitivity analysis is introduced, and the global sensitivity equations are solved by a staggered method. The proposed topology optimization method is applied to the design of mechanisms. The opti- mization results show the significant advantages of varying the interface topology and the layout of the electrode ver- sus conventional approaches optimizing the structural layout only.

Keywords Microelectromechanical systems · Topology optimization · 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 early 1980s, microelectromechanical systems (MEMS) have be- come essential components of numerous applications, such

M. Raulli Villanova University, Dept. of Mechanical Engineering, 800 Lancaster Avenue, Villanova, PA 19085, USA E-mail: [email protected]

K. Maute (B) University of Colorado, Dept. of Aerospace Engineering Sciences, Campus Box 429, Boulder, CO 80309, USA E-mail: [email protected]

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 based on the interaction of multiphysics phenomena, such as elec- trothermomechanical coupling and electrostatic-mechanical interaction. Conventionally, these interaction phenomena have been accounted for in the design process by low- fidelity models, which negatively affect the performance and reliability of the system.

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

The potential of topology optimization for the design of MEMS has been studied before. The bulk of this work is concerned with the design of elastic compliant mechanisms for force and displacement amplification purposes (Bruns and Tortorelli 1998; Pedersen et al. 2001). Most often, the physical actuation mechanism is simplified and replaced by a fixed, design-independent, external load. More advanced studies 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 for this class of devices has been lacking and is presented in this study. Considering this type of interaction in the formulation and 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 and electrostatic domains. A major challenge is the appropriate treatment of the electrostatic pressure, as it is governed by the electrostatic field equations and acts on a surface that is not known in advance but evolves in the optimization process. The proposed approach is based on a high-fidelity simulation model of the coupled problem and analytical sen- sitivity analysis embedded into a gradient-based optimiza- tion procedure. The formulations and solution procedures of the topology optimization, analysis, and sensitivity analysis problems are presented and applied to the design of 2D and 3D electrostatically actuated mechanisms.

For this purpose, the paper is arranged in the follow- ing manner. The basic formulations of the electromechanical topology optimization and analysis problems are presented in 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 topology optimization

Recent years have seen the application of topology opti- mization to increasingly complex engineering design prob- lems (Eschenauer and Olhoff 2001). Most often the topology optimization problem is formulated as a material distribu- tion problem in a given design space. Among numerous material interpolation schemes, the Solid Isotropic Material with Penalization (SIMP) model in combination with fil- tering techniques has become popular in the engineering design community (Bendsøe 1989; Zhou and Rozvany 1991; Sigmund 1994). This approach leads to solid-void material distributions 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) = s pE k E0 ; pE > 1 (1)

where ek refers to the kth element in a finite element mesh and sk is an optimization variable that represents the solid- void state of finite element k. The reader is referred to the textbook by Bendsøe and Sigmund (2002) for a thorough in- troduction into topology optimization. The proposed work follows 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) considered thermal loading, and Sigmund (1998, 2001a,b) studied the design of electrothermo-actuated MEMS, as did Yin and Ananthasuresh (2002), including the use of multiple ma- terials. These problems have in common that all physical fields involved are defined for the entire design space and the coupling results in mechanical forces due to thermal strains only. The structural, thermal, and all other fields evolve au- tomatically along with the material distribution in the course of the optimization process.

Multiphysics topology optimization problems with loads acting on the surface of the structure, however, are less adaptable to material topology optimization as the surface is not known in advance. The surface evolves in the course of the optimization process and is not explicitly defined using the material formulation of the topology optimization problem. 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 studies the magnitude of the surface load is given rather than be- ing governed by other physical fields. Though the issues of changing location and magnitude are addressed, the present study proposes to address them in a manner more suitable for modeling multiphysics problems with a varying interface topology. The fully coupled multiphysics problem is not simplified, as in previous design-dependent loading stud-

Topology optimization of electrostatically actuated microsystems

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

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

The allowance of a free interface that evolves in the top- ology optimization process leads to several difficulties in the formulation of the optimization problem and the mod- eling and evaluation of the system response. Figure 1 gives a conceptual idea of fixed versus free interface topology op- timization problems. The fundamental difficulties of a free interface are due to the fact that the spatial extent of the elec- trostatic domain changes, leading to a need to change the computational domain. Also, as indicated by the arrows, the electrostatic 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., and are termed “criteria” herein. The evaluation of these criteria is directly linked to the location and extent of the conducting interface. If the interface is allowed to change, as in Fig. 1b, this has a significant impact on the evaluation of criteria. The evaluation 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 algorithms can efficiently handle a large number of variables. The coupled electromechanical sensitivity analysis is presented in Sect. 2.2. Using gradient-based methods, however, re- quires that the optimization criteria and the system response be sufficiently smooth functions of the optimization vari- ables. Therefore, since the interface is free to evolve, the changes in Fig. 1b must be handled in a continuous fash- ion. The modeling of the free interface is presented in detail in Sect. 3.

2.1 Electromechanical analysis

A broad range of analysis methods are applied to the de- sign of electromechanical MEMS devices. Many studies focus on the design of geometrically simple MEMS and use 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 investigations into the use of a combination of higher- and lower-fidelity- solution techniques. Younis et al. (2003) proposed the use of reduced-order models for simplifying parametric studies of the electromechanical response. A methodology that uses 2D 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 have been developed in the past decade. Senturia et al. (1992) presented a high-fidelity finite element–boundary element solver, for the mechanical and electrostatic analysis, re- spectively (Senturia et al. 1992). A coupled sequential elec- tromechanical solver that uses finite elements for both the mechanical and electrostatic analysis has been developed by Zhulin et al. (2000).

This study uses finite element discretizations for both the structural and electrostatic subproblems. Though boundary elements are the standard for electrostatic analysis, the use of 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 removes the material contribution of nonoptimal elements. In order to achieve a smooth representation of the geometry, a suf- ficiently fine mesh is required that eliminates the use of boundary elements for topology optimization of the full electromechanical domain. Also, since a boundary element method only discretizes the interface, the topology of the electrostatic field cannot be altered. See Sect. 3.2 for further discussion of this idea.

The structural domain is coupled to the electrostatic field because of the electrostatic pressure exerted on the struc- ture. The geometry of the electrostatic domain changes as the structural shape changes due to deformation and opti- mization. The choice of finite elements for the electrostatic discretization necessitates the inclusion of a method that pre- vents the degeneration of the electrostatic mesh due to the shape changes of the electrostatic–structure interface. For boundary element techniques only a surface mesh of the electrostatic domain is necessary and only the geometry of this surface mesh needs to be updated when the structure changes shape. Using a finite element mesh for the elec- trostatic domain, however, requires that the structural shape changes are propagated throughout the entire electrostatic mesh, not just the interface. Subsequently, the mesh up- dating procedure of the electrostatic mesh is referred to as mesh-motion.

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

M. Raulli, K. Maute

The three-field formulation consists of the following residuals, where u is structural displacements, v is electro- static voltages, x is electrostatic mesh displacements, and s is 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-motion equations (6)–(7) are generally Dirichlet boundary condition problems, in which case the only right-hand side in the re- sidual equation results from constraining part of the solution vector. Furthermore, it is assumed that the structure is a per- fect conductor.

S = Ku− fs(v, x) (5)

E = P v +PΓ vΓ −p ; vΓ = v on Γ v (6)

R = {

x Γ

−Tmu (7)

with K and P being the stiffness and permittivity matrices, respectively, in a linear finite element representation. The structural load due to mechanical and electrostatic forces is represented by fs, and the right-hand side for the electro- static problem due to prescribed charge is represented by p and is generally 0. Prescribed voltage boundary conditions are represented by v. The and Γ subscripts represent, respectively, the internal and boundary portions of a par- titioned finite element stiffness matrix. This separation is convenient for the solution of Dirichlet boundary condition problems, as in (6)–(7). The matrix K represents the ficti- cious stiffness of the electrostatic mesh. The mesh-motion stiffness 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 flow problems by Johnson and Tezduyar (1994). The matrix Tm is the transformation for passing structural displacements to the electrostatic mesh. The transpose of this matrix trans- forms electrostatic forces to the structural mesh. This matrix is built with the energy-conserving matching procedure of Farhat et al. (1998), which is applicable to matching and nonmatching meshes.

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

fe = ∫

Te n dΓ E/S (8)

where Γ E/S represents the conducting interface between the electrostatic and structure, n is the outward surface normal on the interface, and Te is the Maxwell stress tensor, defined below:

Te = ε

exey e2 y − 1

2e2 eyez

2e2

(9)

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

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

The conducting interface is discretized with interface fi- nite elements for evaluating (8). The interface elements are linearally 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; therefore the electric field terms present in the Maxwell electrostatic stress tensor (9) are taken directly from the corresponding electrostatic nodes and used to compute the electrostatic forces at the interface element nodes. The elemental forces are then combined to obtain a global electrostatic force vec- tor. Figure 4 illustrates the location of the interface elements for fixed and free interface problems.

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

2.2 Electromechanical sensitivity analysis

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

Previous work in the computation of analytical sensi- tivities for coupled electromechanical problems has been accomplished for a hybrid boundary element–finite element coupled solution technique by Shi et al. (1995). The work of Shi et al. fully couples the fields in analysis but does not fully couple them in the sensitivity analysis. Also, the sensitivities of the electrostatic field are computed by differ- entiating the governing equations and then discretizing the gradients.

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):

x(n) Γ

= Tmu(n) (10)

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

and x(n) Γ .

Step 2: Compute the internal electrostatic state vector (v ), 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 the superscript 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)

, 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 to an optimization variable, si , is computed as follows:

dqj

(14)

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

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

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

M. Raulli, K. Maute

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

and u Γ

. Only the internal portion of dv/ds is considered because 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 discussed in 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 the total state derivatives, obtained from solving (15), for each optimization variable, of which there are ns:

Table 2 Adjoint method algorithm

Step 0: Compute and store ∂qj/∂u, ∂qj/∂v 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 = Tm T (

) (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)

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/∂v

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

follows:

∂fs

Compute av (k) j

with (28) and (22).

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

(k)

∂fs

∂x

T

Step 5: Check convergence:

j

) (24)

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

dqj

(17)

Topology optimization problems, however, lead to a large number 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 large computational problem with this many variables. The com- putational procedure applied to the adjoint equations follows the method of Maute et al. (2003), which was developed for fluid–structure interaction problems.

The adjoint method involves the solution of the transpose of (17) with the solution of the state derivative substituted in:

dqj

−aj

(25)

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

K 0 [ Tm

T K Γ

= 0 (26)

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

Γ is the transpose of K

Γ . The trans-

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

au (k) j = K−1

[ Tm

T (

) − ∂qj

∂u

] (27)

− ∂qj

∂x

(29)

The algorithm used to obtain the adjoint vector is sum- marized in Table 2. An advantage of using this staggered procedure is that it allows the use of direct solvers for the individual domains. Therefore, the factorized stiffness ma- trices from the analysis can be reused with minimal compu- tational cost. The off-diagonal derivative matrices are never stored, due to memory considerations, but are recomputed as a matrix–vector product for each step in the staggered pro- cedure. The matrices only need to be formed on an element level and multiplied by an element vector. This result is then stored in a global vector.

3 Topology optimization model for free interface

3.1 Modified SIMP model for free interface

The classical SIMP model is extended in this study to the electrostatic domain for the purpose of electromechanical topology optimization with a free interface. Figure 1b il- lustrates how the voltage boundary conditions, electrostatic forces, 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. In order to effectively compute the coupled electromechanical response, the analysis of the electrostatic domain must also be considered. The modifications of the SIMP model affect the analysis not only of the system but of the gradients as well. The ∂S/∂si and ∂E/∂si terms of (15) are affected by the SIMP model since structural and electrostatic parameters are linked to optimization variables. The consideration of the above issues is discussed in detail below.

3.1.1 Voltage boundary condition

In the electromechanical problems of this study a voltage differential 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 voltage differential is what drives the electrostatic forcing and the system 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 changes the location of the Dirichlet boundary conditions, as illus- trated in Fig. 2.

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

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

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

Pkk = w(ek) v +P0

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 given electrostatic element is made dependent on an optimization variable such that the enforcement of vk changes in a smooth fashion:

w(ek) v = wv(sk − smin) (32)

wv = wv0Pavg

1− smin (33)

where Pavg is a scalar representing the average value of the entries in P and wv0 is a user-defined value that determines how large the relative weighting factor is. The denomina- tor in wv is used to account for the fact that smin > 0. Since voltage is a nodal quantity, each node with a soft voltage condition is linked to one element to which it is connected. This approximation can be further refined in future work, but for 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 domain but are now void need to be considered as part of the elec-

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

The methodology used in this study is to generate an electrostatic mesh that covers the initial electrostatic do- main (E0) as well as the region occupied by the potential structural domain (Eδ), as indicated on the left-hand side of 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 is accomplished in a continuous manner, rather than an “on– off” manner, by adjusting the permittivity of the electrostatic elements in Eδ.

The permittivity of an element in Eδ is made dependent on an optimization variable such that overlapped electro- static elements that correspond to solid structural elements (sk = 1) behave like conductors (the electrostatic equivalent of 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 to the void structural elements, generally about 1×109 more. This term is necessary since when sk = 1, it should cause the permittivity of that element to go to infinity, resulting in 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 close to void are penalized to more closely represent free space rather 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 the electrostatic 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 and the 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 element should not be considered. Therefore, a weighting term for the electric field computation in a given element is included in the optimization formulation:

w(ek) e = we0

1− smin ; we0 = 1.0 (36)

The denominator in (36) is used such that the elemental weight varies between 0 and 1 even though the optimization variable varies between smin and 1. This SIMP parameter en- sures that the correct electric field is computed where solid and 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, as illustrated 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 forces with (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 representation of the electrostatic forces. The basic methodology employed is that solid structural elements are conducting; therefore, the interface elements should fully compute forces there. For interface elements attached to void structural elements, no electrostatic forces should be computed. The interface elem- ents are modified as follows:

ε (ek) i = ε0(sk − smin)

pi (37)

ε0 = ε0

(38)

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

(ek) i refers only to the permit-

tivity used to compute the electrostatic forces in the interface elements, 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 greater flexibility in the overall design of an electromechanical sys- tem, the topology of the electrostatic domain can also be altered. 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 different minimum value of the variables for the electrostatic elem- ents than the structural elements. This permittivity variation refers only to the part of the electromechanical domain that is always free space, not the part that overlaps the struc- ture. Conceptually, this is the equivalent of putting insulating material in the electrostatic domain in order to prevent the electric field from being transmitted in certain locations. Also, the topology of an electrode can be changed in this manner. Making the permittivity zero in electrostatic elem- ents connected to the electrode is the physical equivalent of removing that part of the electrode. This latter technique is used in the example of Sect. 4.2. The topology optimization of 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 that couple the structural model to the independent permittivity variables, sj .

3.3 Verification of SIMP model behavior

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

The plots in Fig. 6 illustrate the importance of choosing the 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 equivalent at 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 electrostatic meshes for 2D topology representation

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

Another aspect of the SIMP model that requires verifica- tion is that it reproduces a system response consistent with one in which the material distribution is exactly represented by the computational meshes for completely solid and void elements.

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

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

The norms of the nodal vectors for electrostatic and structural quantities, as well as the scaled norm of the dif- ference between the exact and topology solutions, are given in 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 gap Vacuum 8.85×10−12 F/m 100.0 V 0.5/2.5 µm

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

topology vectors are different, since there are more nodes in the topology model than in the model based on the true geometry. However, the additional nodes in the topology models are filtered out such that the differences in solution vectors can be effectively compared. The actual norms of the exact and (unfiltered) topology models are shown to demon- strate that the nodal values filtered out of the topology model are small in comparison to the exact nodal values. It should be noted that the value of the topology norm for the struc- tural displacements still has the nodes filtered out since there are nonzero displacements in the topology model nodes that do not correspond to nodes in the exact model. The associ- ated elements, however, are not significantly contributing to the stiffness. The electrostatic and structural quantities agree well for the exact and topology models.

Fig. 9 Material distribution for topology model

M. Raulli, K. Maute

Fig. 10 Voltage distribution for 2D exact model (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 4.2 presents 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 procedure described in Sect. 2.1. The gradients of the optimization criteria 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 for nanoscale 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). In contrast 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 towards the electrode. Figure 11 gives a schematic of this optimiza- tion example. The desired design will invert the electrostatic pressure acting in the negative y-direction such that point C in 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

(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 at the same node as the displacement objective in order to sim- ulate actuation of a work piece. See Table 6 for the values of the SIMP parameters used in this example.

The mass constraint is applied in order to encourage a ‘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 ) 3.0 Electrostatic permittivity penalization (pε) 6.0 Interface permittivity penalization (pi ) 4.0 Soft voltage weighting factor (wv0 ) 1.0×104

1. The optimization process seeks to maximize the up- ward displacement. Since the electrostatic pressure is what drives the inversion, once the appropriate mech- anisms have been determined by the optimization pro- cess, a larger electrostatic pressure will lead to increased upward 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 pressure becomes too strong, the electromechanical system will become unstable and be pulled into contact with the elec- trode (pull-in), causing the elements in the electrostatic mesh to collapse and the optimization process to stop. To ensure that the optimization process converges to an optimal design and that the design is stable, an energy constraint is enforced such that the structure is limited in its deformations. The value of Π0 is chosen because allowing higher energies consistently leads to pull-in.

2. Since the energy constraint limits the overall strains of the structure, the optimization process is forced to stiffen in 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 full structural and electrostatic meshes used in this example. Ap- plying symmetry boundary conditions, only half of each computational domain is analyzed and optimized. The struc- ture is discretized with four-node quadratic plane stress elements. The electrostatic mesh uses four-node quadratic elements. The physical properties and computational mesh sizes for this optimization problem are given in Tables 7 and 8, 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 of one element; however, the fixed interface problem does not have the freedom to remove the row of interface elements. The same numerical values for mass and energy constraints are used in the fixed and free problems, in order to effec- tively compare the methodologies. The optimization results are summarized in Table 9. The initial displacement in the free interface problem is less because the interface is started at 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 gap Vacuum 8.85×10−12 F/m 2.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Initial width & height Silicon 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 7200 Electrostatic 8133 8133 8072 7920 Mesh-motion 8133 16,266 16,144 7920 Total 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 7080 Initial uc −1.2972×10−12 −4.1089×10−12

Final uc 2.1418×10−9 6.7104×10−10

Displacement change from fixed initial value

−5.2127×104% −1.6332×104%

Energy/Mass constraint ac- tive

Yes/Yes Yes/Yes

Number of iterations 2649 5295 Total 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 the free and fixed interfaces. The methodology that allows for a free interface makes this problem much more flexible, as illustrated by the fact that much of the bottom layer of the structure is removed by the optimization process and the al- lowable material is used to create the necessary mechanisms and 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 entire fixed interface with extra supports, allowing less material to 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 problem is less because there is no need to overlap the electrostatic domain, resulting in a much smaller electrostatic computa- tional problem.

One minor issue is that the interface in the free problem is not entirely solid. This is due to the fact that the prob- lem finds mechanisms that accomplish the desired goal of a force inverter while violating the energy constraint. Rather than changing the mechanisms in order to satisfy the energy constraint, the optimization algorithm reduces the material fraction 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, achieving the same behavior. Forcing the elements at the interface to be completely solid for force inverter problems should be addressed 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, the permittivity in the layer of electrostatic elements above the electrode is also optimized, essentially determining the top- ology of the electrode. Figure 14 gives a schematic of this optimization example. The desired design inverts the elec- trostatic pressure acting in the negative y-direction such that point 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:

Mass ≤ 10% of total

(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 electrostatic domains, respectively. The constraint that prevents the strain energy from decreasing serves the purpose of forc- ing the problem away from the local optimum of a stiff structure. The optimization problem was attempted with- out this constraint, and the results were a stiffer structure with a marginal decrease in the negative y-displacement of point t. With the energy constraint, the optimization algorithm gradually finds the desired optimum of invert- ing the displacement. The mass constraint is used for clarity of the final design. See Table 10 for the values of the SIMP parameters used in this example.

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

Table 10 SIMP parameters for Sect. 4.2

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

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

maxs z(s) = uc subject to:

Mass ≤ 10% of total

(42)

where Π = 4.0 × 10−19, which is approximately one quarter 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 the system from becoming unstable. This optimization for- mulation is run until convergence.

Varying the permittivity independently in the lower layer of electrostatic elements allows the optimization algorithm greater flexibility in determining the electrostatic pressure on 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, resulting in 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 through the insulating layer, leading to negligible voltage gradients and therefore negligible electrostatic pressure. For compar- ison purposes, the optimization problem was run without the 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 and electrostatic meshes used in this example. The structure is discretized with three-node triangular ANDES plate elem- ents (Militello and Felippa 1991). The electrostatic mesh uses eight-node hexahedron elements. The physical prop- erties and computational mesh sizes for this optimization problem 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 gap Vacuum 8.85×10−12 F/m 5.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Init. width & height Silicon 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 6400 Electrostatic 8405 8405 6724 6400 Mesh-motion 8405 25,215 20,172 6400 Total 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 computational domain on one Pentium IV 1.7 GHz processor. The final structural 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 the side of the support is pulled downward, by the electrostatic forces, allowing the curved lever arm, which is connected to node 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 electrostatic mesh. The voltage varies between zero volts (white) and five volts (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 306 Total time 193.5 hours

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

5 Summary

The design of MEMS is continually evolving, with chang- ing parameters and applications. The conceptual design of MEMS 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 methodology for performing topology optimization of MEMS that are electrostatically actuated, without limitation on the interface between 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 the electromechanical response in addition to the fully coupled analysis. Additionally, the classical SIMP model is modified for electromechanical problems. The voltage boundary con- ditions are enforced in an indirect manner in order to allow a flexible interface. Two numerical examples of force invert- ers were presented to show the applicability of the developed methodology.

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 were introduced to prevent pull-in instabilities. In future stud- ies, constraints on pull-in instabilities should be directly accounted for, requiring appropriate prediction and sensi- tivity analysis capabilities of this phenomenon, which are currently lacking.

Acknowledgement The first author would like to acknowledge the support of Sandia National Laboratory under the direction of Jim Allen. Both authors acknowledge the support by the National Science Foundation under Grant DMI-0300539 and the Air Force Office of Scientific Research under Grant F49620-02-1-0037. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring agencies.

References

Abdalla MM, Reddy CK, Faris W, Gurdal Z (2003) Optimal design of an electrostatically actuated microbeam for maximum pull-in voltage. 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 distribution problem. 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 verification of a design methodology for torsion actuators based on a rapid pull-in solver. J Microelectromech Syst 13(1):121–130

Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. 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 symposium on multidisciplinary analysis and optimization, St. Louis, MO, pp 1874–1882

Bustillo JM, Howe RT, Muller RS (1998) Surface micromachining for microelectromechanical 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 Magnet 38(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 mems applications. 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 Optim 27:151–165

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

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

Farhat C, Degand C, Koobus B, Lesoinne M (1998) Torsional springs for two-dimensional dynamic unstructured fluid meshes. Comput Methods 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. Comput Methods Appl Mech Eng 157:95–114

Giunta AA, Sobieszczanski-Sobieski J (1998) Progress towards using sensitivity derivatives in a high-fidelity aeroelastic analysis of a 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 continuum structures subjected to pressure loading. Struct Multidisc Optim 19:85–92

Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite 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 structures by topology optimization. Struct Multidisc Optim 27:27–42

Maute K, Lesoinne M, Farhat C (2000) Optimization of aeroelastic systems 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 design optimization of three-dimensional non-linear aeroelastic systems by 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 plate bending triangles. Comput Methods Appl Mech Eng 91:217–246

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

Rodriques H, Fernandes P (1995) A material based model for top- ology optimization of thermoelastic structures. Int J Numer Methods 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 Syst 1(1):3–13

Shaul D, Sumner T (2004) Estimating accuracy of electrostatic finite elementmodels.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 and Mechanics, Technical University of Denmark, Lyngby, Denmark

Sigmund O (1997) On the design of compliant mechanisms. Mech Struct 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 topology optimization—Part I: One-material structures. Comput Methods Appl Mech Eng 190(49–50):6577–6604

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

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

Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

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

Younis M, Abdel-Rahman E, Nayfeh A (2003) A reduced-order model for 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 Methods Appl Mecha Eng 89(1–3):309–336

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 topology optimization. The layout of the structure and the elec- trode are simultaneously optimized. A novel, continuous, material-based description of the interface between the structural and electrostatic domains is presented that allows the optimization of the interface topology. The resulting top- ology optimization problem is solved by a gradient-based algorithm. The electromechanical system response is deter- mined by a coupled high-fidelity finite element model and a staggered solution procedure. An adjoint formulation of the coupled electromechanical design sensitivity analysis is introduced, and the global sensitivity equations are solved by a staggered method. The proposed topology optimization method is applied to the design of mechanisms. The opti- mization results show the significant advantages of varying the interface topology and the layout of the electrode ver- sus conventional approaches optimizing the structural layout only.

Keywords Microelectromechanical systems · Topology optimization · 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 early 1980s, microelectromechanical systems (MEMS) have be- come essential components of numerous applications, such

M. Raulli Villanova University, Dept. of Mechanical Engineering, 800 Lancaster Avenue, Villanova, PA 19085, USA E-mail: [email protected]

K. Maute (B) University of Colorado, Dept. of Aerospace Engineering Sciences, Campus Box 429, Boulder, CO 80309, USA E-mail: [email protected]

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 based on the interaction of multiphysics phenomena, such as elec- trothermomechanical coupling and electrostatic-mechanical interaction. Conventionally, these interaction phenomena have been accounted for in the design process by low- fidelity models, which negatively affect the performance and reliability of the system.

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

The potential of topology optimization for the design of MEMS has been studied before. The bulk of this work is concerned with the design of elastic compliant mechanisms for force and displacement amplification purposes (Bruns and Tortorelli 1998; Pedersen et al. 2001). Most often, the physical actuation mechanism is simplified and replaced by a fixed, design-independent, external load. More advanced studies 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 for this class of devices has been lacking and is presented in this study. Considering this type of interaction in the formulation and 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 and electrostatic domains. A major challenge is the appropriate treatment of the electrostatic pressure, as it is governed by the electrostatic field equations and acts on a surface that is not known in advance but evolves in the optimization process. The proposed approach is based on a high-fidelity simulation model of the coupled problem and analytical sen- sitivity analysis embedded into a gradient-based optimiza- tion procedure. The formulations and solution procedures of the topology optimization, analysis, and sensitivity analysis problems are presented and applied to the design of 2D and 3D electrostatically actuated mechanisms.

For this purpose, the paper is arranged in the follow- ing manner. The basic formulations of the electromechanical topology optimization and analysis problems are presented in 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 topology optimization

Recent years have seen the application of topology opti- mization to increasingly complex engineering design prob- lems (Eschenauer and Olhoff 2001). Most often the topology optimization problem is formulated as a material distribu- tion problem in a given design space. Among numerous material interpolation schemes, the Solid Isotropic Material with Penalization (SIMP) model in combination with fil- tering techniques has become popular in the engineering design community (Bendsøe 1989; Zhou and Rozvany 1991; Sigmund 1994). This approach leads to solid-void material distributions 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) = s pE k E0 ; pE > 1 (1)

where ek refers to the kth element in a finite element mesh and sk is an optimization variable that represents the solid- void state of finite element k. The reader is referred to the textbook by Bendsøe and Sigmund (2002) for a thorough in- troduction into topology optimization. The proposed work follows 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) considered thermal loading, and Sigmund (1998, 2001a,b) studied the design of electrothermo-actuated MEMS, as did Yin and Ananthasuresh (2002), including the use of multiple ma- terials. These problems have in common that all physical fields involved are defined for the entire design space and the coupling results in mechanical forces due to thermal strains only. The structural, thermal, and all other fields evolve au- tomatically along with the material distribution in the course of the optimization process.

Multiphysics topology optimization problems with loads acting on the surface of the structure, however, are less adaptable to material topology optimization as the surface is not known in advance. The surface evolves in the course of the optimization process and is not explicitly defined using the material formulation of the topology optimization problem. 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 studies the magnitude of the surface load is given rather than be- ing governed by other physical fields. Though the issues of changing location and magnitude are addressed, the present study proposes to address them in a manner more suitable for modeling multiphysics problems with a varying interface topology. The fully coupled multiphysics problem is not simplified, as in previous design-dependent loading stud-

Topology optimization of electrostatically actuated microsystems

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

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

The allowance of a free interface that evolves in the top- ology optimization process leads to several difficulties in the formulation of the optimization problem and the mod- eling and evaluation of the system response. Figure 1 gives a conceptual idea of fixed versus free interface topology op- timization problems. The fundamental difficulties of a free interface are due to the fact that the spatial extent of the elec- trostatic domain changes, leading to a need to change the computational domain. Also, as indicated by the arrows, the electrostatic 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., and are termed “criteria” herein. The evaluation of these criteria is directly linked to the location and extent of the conducting interface. If the interface is allowed to change, as in Fig. 1b, this has a significant impact on the evaluation of criteria. The evaluation 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 algorithms can efficiently handle a large number of variables. The coupled electromechanical sensitivity analysis is presented in Sect. 2.2. Using gradient-based methods, however, re- quires that the optimization criteria and the system response be sufficiently smooth functions of the optimization vari- ables. Therefore, since the interface is free to evolve, the changes in Fig. 1b must be handled in a continuous fash- ion. The modeling of the free interface is presented in detail in Sect. 3.

2.1 Electromechanical analysis

A broad range of analysis methods are applied to the de- sign of electromechanical MEMS devices. Many studies focus on the design of geometrically simple MEMS and use 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 investigations into the use of a combination of higher- and lower-fidelity- solution techniques. Younis et al. (2003) proposed the use of reduced-order models for simplifying parametric studies of the electromechanical response. A methodology that uses 2D 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 have been developed in the past decade. Senturia et al. (1992) presented a high-fidelity finite element–boundary element solver, for the mechanical and electrostatic analysis, re- spectively (Senturia et al. 1992). A coupled sequential elec- tromechanical solver that uses finite elements for both the mechanical and electrostatic analysis has been developed by Zhulin et al. (2000).

This study uses finite element discretizations for both the structural and electrostatic subproblems. Though boundary elements are the standard for electrostatic analysis, the use of 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 removes the material contribution of nonoptimal elements. In order to achieve a smooth representation of the geometry, a suf- ficiently fine mesh is required that eliminates the use of boundary elements for topology optimization of the full electromechanical domain. Also, since a boundary element method only discretizes the interface, the topology of the electrostatic field cannot be altered. See Sect. 3.2 for further discussion of this idea.

The structural domain is coupled to the electrostatic field because of the electrostatic pressure exerted on the struc- ture. The geometry of the electrostatic domain changes as the structural shape changes due to deformation and opti- mization. The choice of finite elements for the electrostatic discretization necessitates the inclusion of a method that pre- vents the degeneration of the electrostatic mesh due to the shape changes of the electrostatic–structure interface. For boundary element techniques only a surface mesh of the electrostatic domain is necessary and only the geometry of this surface mesh needs to be updated when the structure changes shape. Using a finite element mesh for the elec- trostatic domain, however, requires that the structural shape changes are propagated throughout the entire electrostatic mesh, not just the interface. Subsequently, the mesh up- dating procedure of the electrostatic mesh is referred to as mesh-motion.

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

M. Raulli, K. Maute

The three-field formulation consists of the following residuals, where u is structural displacements, v is electro- static voltages, x is electrostatic mesh displacements, and s is 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-motion equations (6)–(7) are generally Dirichlet boundary condition problems, in which case the only right-hand side in the re- sidual equation results from constraining part of the solution vector. Furthermore, it is assumed that the structure is a per- fect conductor.

S = Ku− fs(v, x) (5)

E = P v +PΓ vΓ −p ; vΓ = v on Γ v (6)

R = {

x Γ

−Tmu (7)

with K and P being the stiffness and permittivity matrices, respectively, in a linear finite element representation. The structural load due to mechanical and electrostatic forces is represented by fs, and the right-hand side for the electro- static problem due to prescribed charge is represented by p and is generally 0. Prescribed voltage boundary conditions are represented by v. The and Γ subscripts represent, respectively, the internal and boundary portions of a par- titioned finite element stiffness matrix. This separation is convenient for the solution of Dirichlet boundary condition problems, as in (6)–(7). The matrix K represents the ficti- cious stiffness of the electrostatic mesh. The mesh-motion stiffness 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 flow problems by Johnson and Tezduyar (1994). The matrix Tm is the transformation for passing structural displacements to the electrostatic mesh. The transpose of this matrix trans- forms electrostatic forces to the structural mesh. This matrix is built with the energy-conserving matching procedure of Farhat et al. (1998), which is applicable to matching and nonmatching meshes.

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

fe = ∫

Te n dΓ E/S (8)

where Γ E/S represents the conducting interface between the electrostatic and structure, n is the outward surface normal on the interface, and Te is the Maxwell stress tensor, defined below:

Te = ε

exey e2 y − 1

2e2 eyez

2e2

(9)

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

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

The conducting interface is discretized with interface fi- nite elements for evaluating (8). The interface elements are linearally 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; therefore the electric field terms present in the Maxwell electrostatic stress tensor (9) are taken directly from the corresponding electrostatic nodes and used to compute the electrostatic forces at the interface element nodes. The elemental forces are then combined to obtain a global electrostatic force vec- tor. Figure 4 illustrates the location of the interface elements for fixed and free interface problems.

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

2.2 Electromechanical sensitivity analysis

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

Previous work in the computation of analytical sensi- tivities for coupled electromechanical problems has been accomplished for a hybrid boundary element–finite element coupled solution technique by Shi et al. (1995). The work of Shi et al. fully couples the fields in analysis but does not fully couple them in the sensitivity analysis. Also, the sensitivities of the electrostatic field are computed by differ- entiating the governing equations and then discretizing the gradients.

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):

x(n) Γ

= Tmu(n) (10)

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

and x(n) Γ .

Step 2: Compute the internal electrostatic state vector (v ), 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 the superscript 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)

, 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 to an optimization variable, si , is computed as follows:

dqj

(14)

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

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

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

M. Raulli, K. Maute

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

and u Γ

. Only the internal portion of dv/ds is considered because 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 discussed in 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 the total state derivatives, obtained from solving (15), for each optimization variable, of which there are ns:

Table 2 Adjoint method algorithm

Step 0: Compute and store ∂qj/∂u, ∂qj/∂v 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 = Tm T (

) (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)

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/∂v

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

follows:

∂fs

Compute av (k) j

with (28) and (22).

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

(k)

∂fs

∂x

T

Step 5: Check convergence:

j

) (24)

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

dqj

(17)

Topology optimization problems, however, lead to a large number 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 large computational problem with this many variables. The com- putational procedure applied to the adjoint equations follows the method of Maute et al. (2003), which was developed for fluid–structure interaction problems.

The adjoint method involves the solution of the transpose of (17) with the solution of the state derivative substituted in:

dqj

−aj

(25)

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

K 0 [ Tm

T K Γ

= 0 (26)

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

Γ is the transpose of K

Γ . The trans-

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

au (k) j = K−1

[ Tm

T (

) − ∂qj

∂u

] (27)

− ∂qj

∂x

(29)

The algorithm used to obtain the adjoint vector is sum- marized in Table 2. An advantage of using this staggered procedure is that it allows the use of direct solvers for the individual domains. Therefore, the factorized stiffness ma- trices from the analysis can be reused with minimal compu- tational cost. The off-diagonal derivative matrices are never stored, due to memory considerations, but are recomputed as a matrix–vector product for each step in the staggered pro- cedure. The matrices only need to be formed on an element level and multiplied by an element vector. This result is then stored in a global vector.

3 Topology optimization model for free interface

3.1 Modified SIMP model for free interface

The classical SIMP model is extended in this study to the electrostatic domain for the purpose of electromechanical topology optimization with a free interface. Figure 1b il- lustrates how the voltage boundary conditions, electrostatic forces, 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. In order to effectively compute the coupled electromechanical response, the analysis of the electrostatic domain must also be considered. The modifications of the SIMP model affect the analysis not only of the system but of the gradients as well. The ∂S/∂si and ∂E/∂si terms of (15) are affected by the SIMP model since structural and electrostatic parameters are linked to optimization variables. The consideration of the above issues is discussed in detail below.

3.1.1 Voltage boundary condition

In the electromechanical problems of this study a voltage differential 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 voltage differential is what drives the electrostatic forcing and the system 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 changes the location of the Dirichlet boundary conditions, as illus- trated in Fig. 2.

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

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

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

Pkk = w(ek) v +P0

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 given electrostatic element is made dependent on an optimization variable such that the enforcement of vk changes in a smooth fashion:

w(ek) v = wv(sk − smin) (32)

wv = wv0Pavg

1− smin (33)

where Pavg is a scalar representing the average value of the entries in P and wv0 is a user-defined value that determines how large the relative weighting factor is. The denomina- tor in wv is used to account for the fact that smin > 0. Since voltage is a nodal quantity, each node with a soft voltage condition is linked to one element to which it is connected. This approximation can be further refined in future work, but for 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 domain but are now void need to be considered as part of the elec-

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

The methodology used in this study is to generate an electrostatic mesh that covers the initial electrostatic do- main (E0) as well as the region occupied by the potential structural domain (Eδ), as indicated on the left-hand side of 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 is accomplished in a continuous manner, rather than an “on– off” manner, by adjusting the permittivity of the electrostatic elements in Eδ.

The permittivity of an element in Eδ is made dependent on an optimization variable such that overlapped electro- static elements that correspond to solid structural elements (sk = 1) behave like conductors (the electrostatic equivalent of 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 to the void structural elements, generally about 1×109 more. This term is necessary since when sk = 1, it should cause the permittivity of that element to go to infinity, resulting in 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 close to void are penalized to more closely represent free space rather 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 the electrostatic 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 and the 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 element should not be considered. Therefore, a weighting term for the electric field computation in a given element is included in the optimization formulation:

w(ek) e = we0

1− smin ; we0 = 1.0 (36)

The denominator in (36) is used such that the elemental weight varies between 0 and 1 even though the optimization variable varies between smin and 1. This SIMP parameter en- sures that the correct electric field is computed where solid and 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, as illustrated 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 forces with (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 representation of the electrostatic forces. The basic methodology employed is that solid structural elements are conducting; therefore, the interface elements should fully compute forces there. For interface elements attached to void structural elements, no electrostatic forces should be computed. The interface elem- ents are modified as follows:

ε (ek) i = ε0(sk − smin)

pi (37)

ε0 = ε0

(38)

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

(ek) i refers only to the permit-

tivity used to compute the electrostatic forces in the interface elements, 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 greater flexibility in the overall design of an electromechanical sys- tem, the topology of the electrostatic domain can also be altered. 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 different minimum value of the variables for the electrostatic elem- ents than the structural elements. This permittivity variation refers only to the part of the electromechanical domain that is always free space, not the part that overlaps the struc- ture. Conceptually, this is the equivalent of putting insulating material in the electrostatic domain in order to prevent the electric field from being transmitted in certain locations. Also, the topology of an electrode can be changed in this manner. Making the permittivity zero in electrostatic elem- ents connected to the electrode is the physical equivalent of removing that part of the electrode. This latter technique is used in the example of Sect. 4.2. The topology optimization of 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 that couple the structural model to the independent permittivity variables, sj .

3.3 Verification of SIMP model behavior

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

The plots in Fig. 6 illustrate the importance of choosing the 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 equivalent at 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 electrostatic meshes for 2D topology representation

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

Another aspect of the SIMP model that requires verifica- tion is that it reproduces a system response consistent with one in which the material distribution is exactly represented by the computational meshes for completely solid and void elements.

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

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

The norms of the nodal vectors for electrostatic and structural quantities, as well as the scaled norm of the dif- ference between the exact and topology solutions, are given in 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 gap Vacuum 8.85×10−12 F/m 100.0 V 0.5/2.5 µm

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

topology vectors are different, since there are more nodes in the topology model than in the model based on the true geometry. However, the additional nodes in the topology models are filtered out such that the differences in solution vectors can be effectively compared. The actual norms of the exact and (unfiltered) topology models are shown to demon- strate that the nodal values filtered out of the topology model are small in comparison to the exact nodal values. It should be noted that the value of the topology norm for the struc- tural displacements still has the nodes filtered out since there are nonzero displacements in the topology model nodes that do not correspond to nodes in the exact model. The associ- ated elements, however, are not significantly contributing to the stiffness. The electrostatic and structural quantities agree well for the exact and topology models.

Fig. 9 Material distribution for topology model

M. Raulli, K. Maute

Fig. 10 Voltage distribution for 2D exact model (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 4.2 presents 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 procedure described in Sect. 2.1. The gradients of the optimization criteria 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 for nanoscale 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). In contrast 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 towards the electrode. Figure 11 gives a schematic of this optimiza- tion example. The desired design will invert the electrostatic pressure acting in the negative y-direction such that point C in 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

(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 at the same node as the displacement objective in order to sim- ulate actuation of a work piece. See Table 6 for the values of the SIMP parameters used in this example.

The mass constraint is applied in order to encourage a ‘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 ) 3.0 Electrostatic permittivity penalization (pε) 6.0 Interface permittivity penalization (pi ) 4.0 Soft voltage weighting factor (wv0 ) 1.0×104

1. The optimization process seeks to maximize the up- ward displacement. Since the electrostatic pressure is what drives the inversion, once the appropriate mech- anisms have been determined by the optimization pro- cess, a larger electrostatic pressure will lead to increased upward 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 pressure becomes too strong, the electromechanical system will become unstable and be pulled into contact with the elec- trode (pull-in), causing the elements in the electrostatic mesh to collapse and the optimization process to stop. To ensure that the optimization process converges to an optimal design and that the design is stable, an energy constraint is enforced such that the structure is limited in its deformations. The value of Π0 is chosen because allowing higher energies consistently leads to pull-in.

2. Since the energy constraint limits the overall strains of the structure, the optimization process is forced to stiffen in 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 full structural and electrostatic meshes used in this example. Ap- plying symmetry boundary conditions, only half of each computational domain is analyzed and optimized. The struc- ture is discretized with four-node quadratic plane stress elements. The electrostatic mesh uses four-node quadratic elements. The physical properties and computational mesh sizes for this optimization problem are given in Tables 7 and 8, 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 of one element; however, the fixed interface problem does not have the freedom to remove the row of interface elements. The same numerical values for mass and energy constraints are used in the fixed and free problems, in order to effec- tively compare the methodologies. The optimization results are summarized in Table 9. The initial displacement in the free interface problem is less because the interface is started at 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 gap Vacuum 8.85×10−12 F/m 2.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Initial width & height Silicon 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 7200 Electrostatic 8133 8133 8072 7920 Mesh-motion 8133 16,266 16,144 7920 Total 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 7080 Initial uc −1.2972×10−12 −4.1089×10−12

Final uc 2.1418×10−9 6.7104×10−10

Displacement change from fixed initial value

−5.2127×104% −1.6332×104%

Energy/Mass constraint ac- tive

Yes/Yes Yes/Yes

Number of iterations 2649 5295 Total 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 the free and fixed interfaces. The methodology that allows for a free interface makes this problem much more flexible, as illustrated by the fact that much of the bottom layer of the structure is removed by the optimization process and the al- lowable material is used to create the necessary mechanisms and 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 entire fixed interface with extra supports, allowing less material to 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 problem is less because there is no need to overlap the electrostatic domain, resulting in a much smaller electrostatic computa- tional problem.

One minor issue is that the interface in the free problem is not entirely solid. This is due to the fact that the prob- lem finds mechanisms that accomplish the desired goal of a force inverter while violating the energy constraint. Rather than changing the mechanisms in order to satisfy the energy constraint, the optimization algorithm reduces the material fraction 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, achieving the same behavior. Forcing the elements at the interface to be completely solid for force inverter problems should be addressed 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, the permittivity in the layer of electrostatic elements above the electrode is also optimized, essentially determining the top- ology of the electrode. Figure 14 gives a schematic of this optimization example. The desired design inverts the elec- trostatic pressure acting in the negative y-direction such that point 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:

Mass ≤ 10% of total

(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 electrostatic domains, respectively. The constraint that prevents the strain energy from decreasing serves the purpose of forc- ing the problem away from the local optimum of a stiff structure. The optimization problem was attempted with- out this constraint, and the results were a stiffer structure with a marginal decrease in the negative y-displacement of point t. With the energy constraint, the optimization algorithm gradually finds the desired optimum of invert- ing the displacement. The mass constraint is used for clarity of the final design. See Table 10 for the values of the SIMP parameters used in this example.

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

Table 10 SIMP parameters for Sect. 4.2

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

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

maxs z(s) = uc subject to:

Mass ≤ 10% of total

(42)

where Π = 4.0 × 10−19, which is approximately one quarter 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 the system from becoming unstable. This optimization for- mulation is run until convergence.

Varying the permittivity independently in the lower layer of electrostatic elements allows the optimization algorithm greater flexibility in determining the electrostatic pressure on 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, resulting in 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 through the insulating layer, leading to negligible voltage gradients and therefore negligible electrostatic pressure. For compar- ison purposes, the optimization problem was run without the 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 and electrostatic meshes used in this example. The structure is discretized with three-node triangular ANDES plate elem- ents (Militello and Felippa 1991). The electrostatic mesh uses eight-node hexahedron elements. The physical prop- erties and computational mesh sizes for this optimization problem 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 gap Vacuum 8.85×10−12 F/m 5.0 V 0.5 µm

Structure Elastic modulus Poisson ratio Thickness Init. width & height Silicon 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 6400 Electrostatic 8405 8405 6724 6400 Mesh-motion 8405 25,215 20,172 6400 Total 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 computational domain on one Pentium IV 1.7 GHz processor. The final structural 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 the side of the support is pulled downward, by the electrostatic forces, allowing the curved lever arm, which is connected to node 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 electrostatic mesh. The voltage varies between zero volts (white) and five volts (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 306 Total time 193.5 hours

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

5 Summary

The design of MEMS is continually evolving, with chang- ing parameters and applications. The conceptual design of MEMS 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 methodology for performing topology optimization of MEMS that are electrostatically actuated, without limitation on the interface between 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 the electromechanical response in addition to the fully coupled analysis. Additionally, the classical SIMP model is modified for electromechanical problems. The voltage boundary con- ditions are enforced in an indirect manner in order to allow a flexible interface. Two numerical examples of force invert- ers were presented to show the applicability of the developed methodology.

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 were introduced to prevent pull-in instabilities. In future stud- ies, constraints on pull-in instabilities should be directly accounted for, requiring appropriate prediction and sensi- tivity analysis capabilities of this phenomenon, which are currently lacking.

Acknowledgement The first author would like to acknowledge the support of Sandia National Laboratory under the direction of Jim Allen. Both authors acknowledge the support by the National Science Foundation under Grant DMI-0300539 and the Air Force Office of Scientific Research under Grant F49620-02-1-0037. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring agencies.

References

Abdalla MM, Reddy CK, Faris W, Gurdal Z (2003) Optimal design of an electrostatically actuated microbeam for maximum pull-in voltage. 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 distribution problem. 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 verification of a design methodology for torsion actuators based on a rapid pull-in solver. J Microelectromech Syst 13(1):121–130

Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. 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 symposium on multidisciplinary analysis and optimization, St. Louis, MO, pp 1874–1882

Bustillo JM, Howe RT, Muller RS (1998) Surface micromachining for microelectromechanical 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 Magnet 38(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 mems applications. 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 Optim 27:151–165

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

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

Farhat C, Degand C, Koobus B, Lesoinne M (1998) Torsional springs for two-dimensional dynamic unstructured fluid meshes. Comput Methods 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. Comput Methods Appl Mech Eng 157:95–114

Giunta AA, Sobieszczanski-Sobieski J (1998) Progress towards using sensitivity derivatives in a high-fidelity aeroelastic analysis of a 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 continuum structures subjected to pressure loading. Struct Multidisc Optim 19:85–92

Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite 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 structures by topology optimization. Struct Multidisc Optim 27:27–42

Maute K, Lesoinne M, Farhat C (2000) Optimization of aeroelastic systems 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 design optimization of three-dimensional non-linear aeroelastic systems by 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 plate bending triangles. Comput Methods Appl Mech Eng 91:217–246

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

Rodriques H, Fernandes P (1995) A material based model for top- ology optimization of thermoelastic structures. Int J Numer Methods 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 Syst 1(1):3–13

Shaul D, Sumner T (2004) Estimating accuracy of electrostatic finite elementmodels.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 and Mechanics, Technical University of Denmark, Lyngby, Denmark

Sigmund O (1997) On the design of compliant mechanisms. Mech Struct 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 topology optimization—Part I: One-material structures. Comput Methods Appl Mech Eng 190(49–50):6577–6604

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

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

Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

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

Younis M, Abdel-Rahman E, Nayfeh A (2003) A reduced-order model for 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 Methods Appl Mecha Eng 89(1–3):309–336

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