# Efficient mixed-domain analysis of electrostatic MEMS.

**ABSTRACT** We present efficient computational methods for scattered point and meshless analysis of electrostatic microelectromechanical systems (MEMS). Electrostatic MEM devices are governed by coupled mechanical and electrostatic energy domains. A self-consistent analysis of electrostatic MEMS is implemented by combining a finite cloud method-based interior mechanical analysis with a boundary cloud method (BCM)-based exterior electrostatic analysis. Lagrangian descriptions are used for both mechanical and electrostatic analyses. Meshless finite cloud and BCMs, combined with fast algorithms and Lagrangian descriptions, are flexible, efficient, and attractive alternatives compared to conventional finite element/boundary element methods for self-consistent electromechanical analysis. Numerical results are presented for MEM switches, a micromirror device, a lateral comb drive microactuator, and an electrostatic comb drive device. Simulation results are compared with experimental and previously reported data for many of the examples discussed in this paper and a good agreement is observed.

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**ABSTRACT:**Export Date: 17 July 2012, Source: ScopusFinite Elements in Analysis and Design 01/2012; 49(1):28-34. · 1.39 Impact Factor -
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##### Article: Uncertainty quantification of MEMS using a data-dependent adaptive stochastic collocation method

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**ABSTRACT:**This paper presents a unified framework for uncertainty quantification (UQ) in microelectromechanical systems (MEMS). The goal is to model uncertainties in the input parameters of micromechanical devices and to quantify their effect on the final performance of the device. We consider different electromechanical actuators that operate using a combination of electrostatic and electrothermal modes of actuation, for which high-fidelity numerical models have been developed. We use a data-driven framework to generate stochastic models based on experimentally observed uncertainties in geometric and material parameters. Since we are primarily interested in quantifying the statistics of the output parameters of interest, we develop an adaptive refinement strategy to efficiently propagate the uncertainty through the device model, in order to obtain quantities like the mean and the variance of the stochastic solution with minimal computational effort. We demonstrate the efficacy of this framework by performing UQ in some examples of electrostatic and electrothermomechanical microactuators. We also validate the method by comparing our results with experimentally determined uncertainties in an electrostatic microswitch. We show how our framework results in the accurate computation of uncertainties in micromechanical systems with lower computational effort.Computer Methods in Applied Mechanics and Engineering 01/2011; 200(45):3169-3182. · 2.62 Impact Factor

Page 1

Efficient Mixed-Domain Analysis of Electrostatic MEMS

0-7803-7607-2/02/$17.00 ©2002 IEEE

Gang Li and N. R. Aluru

Beckman Institute for Advanced Science and Technology

University of Illinois at Urbana-Champaign

405 N. Mathews Avenue, Urbana, IL 61801

http://www.staff.uiuc.edu/˜aluru/

Abstract

Wepresentefficientcomputationalmethodsforscatteredpoint

andmeshlessanalysisofelectrostaticmicroelectromechanicalsys-

tems(MEMS).ElectrostaticMEMSaregovernedbycoupled me-

chanicalandelectrostatic energydomains. Aself-consistentanal-

ysis of electrostatic MEMS is implemented by combining a finite

cloud method based interiormechanical analysis with a boundary

cloud method based exterior electrostatic analysis. Lagrangian

descriptions are used for both mechanical and electrostatic anal-

yses. Meshless finite cloud and boundary cloud methods com-

bined with fast algorithms and Lagrangian descriptions are flexi-

ble, efficient and attractive alternatives compared to conventional

finiteelement/boundaryelementmethodsforself-consistentelec-

tromechanical analysis. Numerical results are presented for an

electrostatic comb drive device.

keywords: coupled electro-mechanical analysis, meshless,

finite cloud method, boundary cloud method, Lagrangian elec-

trostatics

1Introduction

Although there are many microelectromechanical system de-

signs that use piezoelectric, thermal, pneumatic, and magnetic

actuation, the most popular approach in present day microsensor

and microactuator designs is to use electrostatic forces to move

micromachined parts - referred to as electrostatic MEMS. Com-

putational analysis [1] of electrostatic MEMS requires a self-

consistent solution of the coupled interior mechanical domain

andtheexteriorelectrostaticdomain[2, 3, 4]. Conventionalmeth-

ods for coupled domain analysis, such as FEM/BEM, require

mesh generation, mesh compatibility, re-meshing and interpola-

tion of solution between the domains. Mesh generation can be

difficult and time consuming for complex geometries. Further-

more, mesh distortion can occur for micromechanical structures

that undergo large deformations. To overcome all these difficul-

ties, we propose an efficient approach to perform static analysis

of electrostatically actuated MEMS. The primary contributions

of the paper are as follows: (1) We employ a meshless Finite

Cloud Method (FCM)[5, 6] to solve the interior mechanical do-

main. The Finite Cloud Method is a true meshless method in

which only points are needed to cover the structural domain and

no connectivity information among the points is required. This

method completely eliminates the meshing process and radically

simplifies the analysis procedure. (2) A Boundary Cloud Method

(BCM) [7, 8] is used to analyze the exterior electrostatic domain

to compute the electrostatic forces acting on the surface of the

structures. The BCM utilizes a meshless interpolation technique

and a cell based integration. Besides the flexibility of the cell in-

tegration, the BCM is an excellent match to the FCM for coupled

domain analysis since both of them have meshless interpolations.

(3) A Lagrangian description [9] of the boundary integral equa-

tion is developed and implemented with BCM. Typically, the me-

chanical analysis is performed by a Lagrangian approach using

the undeformed position of the structures. However, the electro-

static analysis is performed on the deformed position of the con-

ductors. The Lagrangian description maps the electrostatic anal-

ysis to the undeformed position of the conductors. Thus, the elec-

trostatic forces and mechanical deformations are all computed on

the undeformed configuration of the structures. The Lagrangian

description eliminates the requirement of geometry updates and

re-computation of the interpolation functions.

2 CoupledAnalysisofElectromechanical Systems

Computational analysis of electrostatically actuated MEMS

requires a self-consistent solution of the coupled mechanical and

electrical equations. Conventionally, a Finite Element Method

(FEM) is employed to perform the mechanical analysis and a

Boundary Element Method (BEM) is employed to compute the

surfaceelectrostaticforces. Themechanicalanalysisisperformed

bydiscretizingthestructuralorthemechanicaldomainintonodes

and elements. A finite element analysis is then performed by

applying the electrostatic pressure as Neumann boundary condi-

tions to compute the structural displacements. Once the displace-

ment is computed, the geometry of the structure or the conductor

is updated. Electrostatic analysis is performed on the updated ge-

ometry by discretizing the surface of the conductor into panels or

elements. A boundary element method is then used to compute

thesurfacechargedensity on each panel. Once thesurfacecharge

densities are known, the new electrostatic pressure is computed.

The mechanical and electrostatic analysis are repeated until an

equilibrium state is computed. Algorithm 1 summarizes the key

steps involved in the self-consistent solution of the coupled elec-

tromechanical problem.

There are several difficulties with the approach described in

Algorithm 1: (1) The structural domain needs to be disctretized

into elements. For structures with complexgeometries mesh gen-

eration can be a complicated and time consuming task. (2) Typ-

ically, the boundary element mesh on the surface of the conduc-

tors does not match with the finite element mesh. In this case, the

electrostatic pressure computed from the BEM analysis needs to

the interpolated to the finite element mesh so that a mechanical

analysiscan be performed. Theinterpolation processcan be cum-

Page 2

Algorithm 1 A procedure for self-consistent analysis of coupled

electromechanical devices

repeat

1. Do mechanical analysis (on the undeformed geometry) to

compute structural displacements

2. Update the geometry of the conductors using the com-

puted displacements

3. Compute surface charge density by electrostatic analy-

sis(on the deformed geometry)

4. Compute electrostatic forces (on the deformed geometry)

5. Transform electrostatic forces to the original undeformed

configuration

until an equilibrium state is reached

bersome and can introduce significant error. One solution to this

problem is to match the finite element nodes on the surface of the

conductors with the boundary element nodes so that no interpo-

lation is involved. However, this can be inefficient as a refine-

ment of either the finite element mesh or the boundary element

mesh would require that the other domain be remeshed. (3) The

need to update the geometry of the conductors before an electro-

static analysis is performed during each iteration also presents

several problems—First, flat surfaces of the conductors in the

initial configuration can become curved due to conductor defor-

mation. This requires the development of complex integration

schemes on curved panels to perform electrostatic analysis. Sec-

ond, when the structure undergoes large deformation, remeshing

the surface may become necessary before an electrostatic analy-

sis is performed. Third, interpolation functions, used in manynu-

merical methods, need to be recomputed whenever the geometry

changes. Each of these issues significantly increases the compu-

tational effort making the self-consistent analysis of electrostatic

MEMS an extremely complex and challenging task.

The combination of the finite cloud method, boundary cloud

method and the Lagrangian electrostatics approach overcomes

the difficultiesmentionedabove. First, a finite cloudmethod does

not require a mesh or elements and mechanical analysis can be

performed by simply sprinkling points with out the need for con-

nectivity information among the nodes or points. Second, using

a boundary cloud method, exterior electrostatic analysis is per-

formed by sprinkling points on the surface of the conductors and

using a background cell structure for integration purpose. Unlike

a boundary element method, the boundary cloud method method

does not require panels or elements. Third, by combining the La-

grangian electrostatics formulation with the total Lagrangian me-

chanical formulation, coupled electromechanical analysis can be

implemented using only the initial configuration. The use of La-

grangian techniques for both mechanical and electrostatic analy-

sis eliminates the need for geometry updates there by simplify-

ing the coupled electromechanical analysis. Algorithm 2 sum-

marizes the Lagrangian approach for efficient scattered point and

self-consistent analysis of coupled electromechanical devices.

3FCM For Mechanical Analysis

Electrostatically actuated microstructures can undergo large

deformations for certain geometric configurations and applied

Algorithm 2 A procedure for self-consistent analysis of coupled

electromechanical devices by using a Lagrangian approach for

both mechanical and electrostatic analysis

repeat

1. Do mechanical analysis (on the undeformed geometry)

by FCM to compute structural displacements

2. Do electrostatic analysis(on the undeformed geometry)

by BCM to compute surface charge density

3. Compute electrostatic pressure (on the undeformed ge-

ometry)

until an equilibrium state is reached

voltages. In this paper, we perform 2-D geometrically nonlin-

ear analysis of microstructures. For electro-mechanical analysis,

the governing equations for an elastic body using a Lagrangian

description are given by

∇

u

P

?

?FS

G

??? B

H

?

0

in

on

on

Ω

(1)

(2)

(3)

?

Γg

Γh

?N

?

whereΩisthemechanicaldomain, Γgistheportionofthebound-

ary on which Dirichlet boundary conditions are specified and Γh

is the portion of the boundary on which Neumann boundary con-

ditions are specified,. The boundary of the mechanical domain

is given by Γ

Γg

second Piola-Kirchhoff stress, B is the body force vector per unit

undeformed volume, u is the displacement, G is the prescribed

displacement, P is the first Piola-Kirchhoffstress tensor given by

PFS, N is the unit outward normal vector in the initial config-

uration and H is the surface traction vector per unit undeformed

area. For electromechanical analysis, H

the surface electrostatic pressure and J is the determinant of F.

We use a meshless Finite Cloud Method to solve the mechan-

ical equations given in Eq. (1-3). The Finite Cloud Method uses

a fixed kernel technique to construct the interpolation functions

and a point collocation technique to discretize the governing par-

tial differential equations. In a 2-D fixed kernel approach, an

approximation ua

?

? Γh. F is the deformation gradient, S is the

?

?

PeJF

?TN, where Peis

?x

? y

? to an unknown u

?x

? y

? is given by

ua

?x

? y

???

?

ΩC

?x

? y

? xk

? s

? yk

? t

? φ

?xk

? s

? yk

? t

? u

?s

? t

? dsdt (4)

where φ is thekernel function centered at

taken as a cubic spline or a Gaussian function. C

t

?xk

? yk

? which is usually

?x

? y

? xk

? s

? yk

?

? is the correction function given by

C

?x

? y

? xk

? s

? yk

? t

???

PT

?xk

? s

? yk

? t

? C

?x

? y

?

(5)

PT

two dimensions, a quadratic basis vector PT

t

is an m

The correction function coefficients are computed by satisfying

the consistency conditions (see [5, 6] for details). The discrete

form of the approximation ua

??? p1

? p2

????????? pm

? is an 1

? m vector of basis functions. In

??? 1

? xk

?

s

? yk

?

?

?xk

? s

?2

?

?xk

? s

?

?yk

? t

???

?yk

? t

?2

?

? m

?

6 is employed. C

?x

? y

?

? 1 vector of unknown correction function coefficients.

?x

? y

? is given by

ua

?x

? y

???

NP

∑

I

? 1

NI

?x

? y

? ˆ uI

(6)

Page 3

where ˆ uI is the nodal parameter for node I, and NI

fixed kernel meshless interpolation function (see [5, 6] for de-

tails). In the mechanical analysis, the displacements u and v are

approximated by using Eq. (6). Consequently, the deformation

gradientFandalltheothermechanicalquantitiescan berewritten

as functions of the approximated displacements uaand va. After

the interpolation functions are constructed, FCM uses a point col-

location technique to discretize the governing equations.

?x

? y

? is the

4Lagrangian Electrostatics

When electrostatic potentials are applied on micro-structures,

electrostaticforcesaregeneratedonthesurfacesofthemicrostruc-

tures. The structures undergo deformation because of electro-

static forces and the surface charge density on the structure redis-

tributes. Typically, the new surface charge density is computed

by updating the geometry of the microstructures and redoing an

electrostatic analysis. The basic idea in Lagrangian electrostatics

is to compute the new surface charge density without updating

the geometry of the microstructures.

The 2D governing equation for electrostatic analysis can be

written in a boundary integral form as [10]

φ

?p

???

?

dωG

?p

? q

? σ

?q

? dγq

? C

(7)

?

dΩσ

?q

? dγq

?

CT

(8)

where p is the source point, q is the field point which moves

along the boundary of the conductors and G is the Green’s func-

tion. In two dimensions, G

the dielectric constant of the medium and

between p and q. CTis the total charge of the system andC is an

unknown variable which can be used to compute the potential at

infinity.

Equations (7) and (8) are defined in the deformed configu-

ration of the conductors, i.e., the surface charge density is com-

putedby solvingtheboundaryintegralequations onthedeformed

geometry of the conductors. We refer to this approach as the de-

formed configuration approach. The need to update the geometry

of the structures in the deformed configuration approach presents

several difficulties as stated in Section 2. In this paper, we em-

ploy a Lagrangian approach [9] to compute the surface charge

density in the undeformed configuration of the conductors. The

Lagrangian form of the boundary integral equations given in Eq.

(7-8) is given by

?p

? q

???

? ln

? p

? q

??

?2πε

? q

? , where ε is

? p

? is the distance

φ

?P

???

?

dΩG

?p

?P

??? q

?Q

??? σ

CT

?q

?Q

??? J

?Q

? dΓQ

? C

(9)

?

dΩσ

?q

?Q

??? J

?Q

? dΓQ

?

(10)

where P and Q are the source and field points in the initial con-

figuration corresponding to the source and field points p and q in

the deformed configuration, J

is the tangential unit vectorat field point Q and C

deformation tensor.

?Q

???

?T

?Q

?

?C

?Q

? T

?Q

???

1

2, T

?Q

?

?Q

? is the Green

5BCM For Electrostatic Analysis

AboundarycloudmethodisemployedtosolvetheLagrangian

description of the electrostatic governing equations (Eq.(9-10)).

In a boundary cloud method, the surface of the domain is dis-

cretized into scattered points. The points can be sprinkled ran-

domly covering the boundary of the domain. Interpolation func-

tions are constructed by centering a weighting function at each

point or node. For the electro-mechanical problem, the potential

φ is prescribed on the structures. The unknown surface charge

density σ in the vicinity of the point t is approximated by either

a Hermite-type interpolation [7] or a varying basis least-squares

approximation [8]. In this paper,we employa varyingbasis least-

squares approach to approximate the unknown quantity, i.e.

σ

?x

? y

???

pT

v

?x

? y

? bt

(11)

where pvis the varying base interpolating polynomial and btis

the unknown coefficient vector for point t. For a point t, the un-

known coefficient vector btis computed by using a least-squares

approach (see [8] for details). The discrete form of the varying

basis approximation for the unknowns is given by

σ

?x

? y

???

NP

∑

I

? 1

NI

?x

? y

? ˆ σI

(12)

The boundary of the structure is discretized into NC cells for in-

tegration purpose. Each cell contains a certain number of nodes

andthenumberofnodescanvaryfromcelltocell. Differentfrom

an element or a panel in boundary-element methods, the cell can

be of any shape or size and the only restriction is that the union

of all the cells equal the boundary of the domain. Substituting the

varying basis approximation for the unknown charge density, the

boundary integral equation for the electrostatic problem given in

Eq. (9-10) can be rewritten in a matrix form as

Mˆ σ

?

φ

(13)

where M is an

ˆ σ are

tively. By substituting the potential on the conductors and the

total charge into Eq. (13), the surface charge density can be com-

puted from Eq. (12) and Eq. (13).

?NC

? 1

???

?NC

? 1

? coefficient matrix and φ and

?NC

? 1

??? 1 right hand side and unknown vector, respec-

6Numerical Results

An electrostatic comb drive discussed in [11] is considered in

this section. The device is simulated with scattered point distri-

butions by using the methods described in the previous sections.

As shown in Figure 1, a center mass with 12 teeth is supported

by fixed-fixed beams. A voltage is applied between the movable

comb and the fixed teeth. The support beams are 1000 µm long,

2.5 µm wide and 4.5 µm thick. The center mass is 98 µm by 98

µm. Each comb tooth is 49 µm long, 2.8 µm wide and 4.5 µm

thick. The gap between the movable teeth and the fixed teeth

is 5.6 µm and the initial overlap at zero volts is 16.8 µm. The

Young’s modulus of the comb structure is 169 GPa and the Pois-

son’s ratio is 0.3. The scattered point distribution for the device is

shown in Figure 2. Figure 3 presents the computed displacement

Page 4

as a function of the applied voltage. Both linear [12] and nonlin-

ear elastostatic theories are employed in this example. As shown

in Figure 3, the comb structure starts to operate in a mechanically

nonlinear regime for an applied voltage of 100 V or higher. The

stiffness of the supporting beams increases quickly as the dis-

placement of the center mass increases. Thus, for MEMS actua-

tors where a large stroke is desired, the fixed-fixed type support

is not advantageous due to the high stiffness of the support. For

this reason, folded supporting beams are widely used in comb

drive applications. The fixed-fixed support, however, provides

a higher stability and a higher resistance to the external forces.

Therefore, for applications such as force sensors the fixed-fixed

support could still be an appropriate choice.

7Conclusions

Wehavepresentedanewcombinedfinitecloud/boundarycloud

method for efficient analysis of microelectromechanical devices.

The FCM/BCM approach requires only a scattered set of points

and no connectivity information or a mesh is necessary. Even

though the electrostatic analysis is coupled to the mechanical

analysis through the same set of boundary nodes, the point distri-

bution for electrostatic analysis can be refined without affecting

the interior mechanical analysis. The Lagrangian electrostatics

formulation combined with the well-known Lagrangian mechan-

ical formulation allows coupled electromechanical analysis with

only the initial configuration, there by eliminating the need for

geometry updates and recalculation of interpolation functions.

Compared to the conventional FEM/BEM approach, the hybrid

FCM/BCM along with Lagrangian electrostatic and mechanical

analysis radically simplifies self-consistent analysis of electro-

static MEMS.

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−500−400−300 −200−1000100200300400500

−50

0

50

100

150

Figure 1: Electrostatic comb drive.

−50050

−80

−60

−40

−20

0

20

40

60

80

100

Figure 2: Scattered point distribution for the

comb drive example

020406080100120140160180200220

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Applied voltage (V)

Displacement (µm)

Linear

Nonlinear

Figure 3: Comparison of the simulation results with

the experimental data.

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Numerical Methods in Engineering, 11, 691-701, 1995.

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12.