nonlinear mechanics of mems plates with a total lagrangian approach

Computers and Structures 83 (2005) 758–768

www.elsevier.com/locate/compstruc

Nonlinear mechanics of MEMS plates with a totalLagrangian approach

Subrata Mukherjee a,*, Zhongping Bao a, Max Roman b, Nadine Aubry b

a Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USAb Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA

Accepted 27 August 2004

Available online 1 January 2005

Abstract

The subject of this paper is a fully Lagrangian approach for coupled (mechanical deformations caused by applied

electric fields) analysis of Micro-Electro-Mechanical (MEM) plates. The analysis is carried out by employing the

two-dimensional (2-D) Finite Element Method (FEM) to analyze mechanical deformations in the plate and the

three-dimensional (3-D) Boundary Element Method (BEM) to obtain the electric field (and then tractions on the plate

surface) in the region exterior to the plate. Self-consistent solutions to the coupled problem are obtained by the relax-

ation method. Such simulations, especially their dynamic version, has many potential applications such as to under-

stand and design synthetic microjets.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Boundary element methods; MEMS; Plates; Coupled BEM/FEM; Lagrangian formulation

1. Introduction

The field of Micro-Electro-Mechanical Systems

(MEMS) is a very broad one that includes fixed or mov-

ing microstructures; encompassing micro-electro-

mechanical, microfluidic, micro-opto-electro-mechanical

and micro-thermo-mechanical devices and systems.

MEMS usually consists of released microstructures that

are suspended and anchored, or captured by a hub-cap

0045-7949/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.compstruc.2004.08.023

* Corresponding author. Present address: Department of

Theoretical and Applied Mechanics, Kimball Hall, Cornell

University, Ithaca, NY 14853, USA. Tel.: +1 607 255 7143; fax:

+1 607 255 2011.

E-mail addresses: [email protected] (S. Mukherjee),

[email protected] (Z. Bao), [email protected] (M. Roman),

[email protected] (N. Aubry).

structure and set into motion by mechanical, electrical,

thermal, acoustical or photonic energy source(s).

Typical MEMS structures consist of arrays of thin

beams with cross-sections in the order of microns (lm)and lengths in the order of ten to hundreds of microns.

Sometimes, MEMS structural elements are plates. Of

interest in this work are small rectangular silicon plates

with sides in the order of mm and thicknesses of the

order of microns, that deform when subjected to electric

fields. Owing to their small size, significant forces and/or

deformations can be obtained with the application of low

voltages (�10 V). Examples of devices that utilize vibra-tions of such plates are synthetic microjets ([1,2]––for

mixing, cooling of electronic components, micropropul-

sion and flow control), microspeakers [3] etc.

Numerical simulation of electrically actuated MEMS

devices have been carried out for around a decade or so

ed.

mailto:[email protected]




S. Mukherjee et al. / Computers and Structures 83 (2005) 758–768 759

by using the Boundary Element Method (BEM––see,

e.g. [4–8]) to model the exterior electric field and the Fi-

nite Element Method (FEM––see, e.g. [9–11]) to model

deformation of the structure. The commercial software

package MEMCAD [12], for example, uses the commer-

cial FEM software package ABAQUS for mechanical

analysis, together with a BEM code FASTCAP [13]

for the electric field analysis. Other examples of such

work are [14–16]; as well as [12,17] for dynamic analysis

of MEMS.

The coupled BEM/FEM methods employed in the

references cited above perform a mechanical analysis

on the undeformed configuration of a structure

(Lagrangian approach) and an electrical analysis on

the deformed configuration (Eulerian approach). A

relaxation method is then used for self-consistency be-

tween the two domains. Therefore, the geometry of the

structure must be updated before an electrical analysis

is performed during each relaxation iteration. This pro-

cedure increases computational effort and introduces

additional numerical errors since the deformed geometry

must be computed at every stage. Li and Aluru [18] first

proposed a Lagrangian approach for the electrical anal-

ysis as well, thus obviating the need to carry out calcu-

lations based on the deformed shapes of a structure.

Two and three-dimensional (2-D and 3-D) quasi-static

Lagrangian exterior BEM analysis was addressed in

[18,19]; while a fully coupled 2-D quasi-static MEMS

analysis has been carried out in [20]. A fully-coupled

2-D dynamic Lagrangian MEMS analysis has been re-

cently carried out by De and Aluru [21]. Additional

advantages of the fully Lagrangian approach, for dy-

namic analysis of MEMS, are described in [21], in which

a Newton method has been developed and compared

with the relaxation scheme. The reader is referred to

Bathe [22] for a comprehensive discussion of the

Lagrangian approach in mechanics.

The present paper is concerned with 3-D quasi-static

analysis of MEMS using a fully Lagrangian approach.

The geometry under consideration is a rectangular

MEMS plate together with the region exterior to the

plate. The relaxation method is employed here. Excellent

B∂

B

V

Electrostatic Force

y

Fig. 1. A deformable cantilever pla

work on BEM analysis of thin structures has been re-

ported by Liu and his group (see, e.g. [23,24]). Their

work involves accurate computation of solid angles in

nearly-singular situations by converting surface integrals

into line integrals using Stokes� theorem. The present

work, however, only involves the kernel 1/r (where r is

the distance between the source and field points). The

weakly singular case involving this kernel has been ad-

dressed before by many authors (see, e.g. [25,26]). The

nearly (also called quasi) weakly singular case, along

with other nearly singular integrals of various orders,

can be effectively evaluated by employing a cubic poly-

nomial transformation due to Telles [27] and Telles

and Oliveira [28]. Several other authors have also con-

sidered similar problems (e.g. [29])––many of these refer-

ences are available in [27,25,28,29] and are not repeated

here in the interest of brevity. A new simple approach

for evaluation of nearly weakly singular integrals is pre-

sented in Section 5.1 of the present paper.

The present paper is organized as follows. Coupled

BEM/FEM analysis of the MEMS problem in 3-D,

using a Lagrangian approach, is presented first. The

FEM model for the plate is nonlinear and allows mod-

erately large transverse deformations (of the order of

the plate thickness––see [30,2]). The BEM model allows

consideration of thin plates. Some numerical results

complete the paper.

2. Electrical problem in the exterior domain

Fig. 1 shows (as an example of a MEMS device) a

deformable cantilever plate over a fixed ground plane.

(This schematic figure is very similar to Fig. 1 in [21].)

The undeformed configuration is B with boundary oB.

The plate deforms when a potential V is applied between

the two conductors, and the deformed configuration is

called b with boundary ob. The charge redistributes on

the surface of the deformed plate, thereby changing

the electrical force on it and this causes the plate to de-

form further. A self-consistent final state is reached,

and this state is computed in the present work by a

b∂

b

V

E lectrostatic Force

te over a fixed ground plane.

760 S. Mukherjee et al. / Computers and Structures 83 (2005) 758–768

relaxation scheme. An alternative is to use a Newton

scheme [21].

2.1. Boundary integral equations in the deformed

configuration

The governing Boundary Integral Equation (BIE), in

current (deformed) coordinates, is [31]:

/ðpÞ ¼ZobGðp; qÞrðqÞdsðqÞ ð1Þ

Also:

hðqÞ ¼ � r2ðqÞ2�

n ð2Þ

Here /(p) is the potential at a surface source point p, q isa surface field point, and r and h are the charge density

and force, respectively, per unit deformed surface area.

Also, � is the dielectric constant of the external medium,n is an unit normal to ob (pointing inwards into the

plate) at a point on it where ob is locally smooth, and

ds is an infinitesimal surface element on ob. Finally,

the Green�s function for 3-D problems is:

Gðp; qÞ ¼ 1

4p�rðp; qÞ ð3Þ

where r is the Euclidean distance between p and q.

2.2. Boundary integral equations in the undeformed

configuration

The BIE, in a Lagrangian framework, is presented

next [19]. From Nanson�s law [32]:

nds ¼ JN � F�1 dS ð4Þ

where n and N are unit normal vectors to ob and oB, at

generic points q and Q, respectively, F ¼ oxoXis the defor-

mation gradient, J = det(F) and dS is an area element on

oB. Here, X and x denote coordinates in the undeformed

and deformed configurations, respectively. From (4):

ds ¼ J jN � F�1jdS ð5Þ

Using (5), (1) becomes:

/ðpðP ÞÞ ¼ZobGðpðP Þ; qðQÞÞrðqðQÞÞJ jN � F�1jdSðQÞ

ð6Þ

Define R, the charge density per unit undeformed sur-

face area, as:

R ¼ JrjN � F�1j ð7Þ

Using (7), (6) becomes:

/ðpðP ÞÞ ¼Z

GðpðP Þ; qðQÞÞRðQÞdSðQÞ ð8Þ
oB
Finally, from:

ZoBHdS ¼

Zobhds ð9Þ

(where H is the force per unit undeformed surface area),

and (2), (4) and (7), one gets:

H ¼ � Jr2N � F�1

2�¼ � R2

2J�N � F�1

jN � F�1j2ð10Þ

The Green�s function in undeformed coordinates has thesimple form (see (3)):

GðpðPÞ; qðQÞÞ ¼ 1

4p�RðP ;QÞ ð11Þ

where:

rðpðP Þ; qðQÞÞ � RðP ;QÞ ¼ xðqðQÞÞ � xðpðPÞÞj j¼ XðQÞ þ uðQÞ � XðP Þ � uðPÞj j ð12Þ

with u denoting the displacement at a point in B.

3. Mechanical problem in the elastic plate

Nonlinear deformation of plates, without initial in-

plane forces, are discussed in this section. The plates

are square (side = L), linearly elastic, and are of uniform

rectangular cross-section (thickness h). The boundary

condition considered here is a plate with all edges

clamped. Also, the edges are immovable, i.e. u = v = 0

on all edges of the plate. Here u(x,y) and v(x,y) are

the in-plane and w(x,y) the transverse displacement of

the mid-plane of the plate. The force distribution (per

unit area) H(x,y) is applied to the plate.

3.1. The model

The kinematic equations adopted here are those for

a von Karman plate [33,30]:

�xx

�yy

cxy

264

375 ¼

u;x þ 12ðw;xÞ2

v;y þ 12ðw;yÞ2

u;y þ v;x þ w;xw;y

264

375;

jxx

jyy

jxy

264

375 ¼

�w;xx

�w;yy

�2w;xy

264

375ð13Þ

where [�] = [�xx,�yy cxy]T are the in-plane strains (mea-

sured at the mid-plane), and [j] = [jxx,jyy, jxy]T are

the curvatures and the twist.

The constitutive equations are:

½N � ¼ H ðIÞ½C�½��; ½M � ¼ H ðOÞ½C�½j� ð14Þ

where [N] = [Nxx,Nyy,Nxy]T = h[rxx, ryy,rxy]

T are the in-

plane forces per unit length, rij are the components of

stress and [M] = [Mxx, Myy,Mxy]T are the bending and

twisting moments. Also:


C ¼1 m 0

m 1 0

0 0 12ð1� mÞ

264

375; H ðIÞ ¼ Eh

1� m2;

H ðOÞ ¼ Eh3

12ð1� m2Þ ð15Þ

with m the Poisson�s ratio of the plate material.

The membrane strain energy EðIÞ, the bending strain

energy EðOÞ, and the work done W are [34]:

EðIÞ ¼ 1

2

ZD½Nx�xx þ Ny�yy þ Nxycxy �dxdy ð16Þ

EðOÞ ¼ 1

2

ZD½Mxjxx þMyjyy þMxyjxy �dxdy ð17Þ

W ¼Z

D½Hxuþ Hyvþ Hzw�dxdy ð18Þ

where D is the area of the plate surface.

Using Eqs. (13)–(15), the energy expressions (16) and

(17) can be written in terms of the plate parameters E, m,h and the displacement derivatives. These expressions

are available in [34] on pages 313 and 95, respectively.

3.2. FEM model for plates with immovable edges

3.2.1. FEM discretization

Each plate element has four corner nodes with 6 de-

grees of freedom at each node. These are u, v, w, w,x, w,y,

w,xy. For each element, one has:

u

v

w

264

375 ¼ N ðIÞ 0

0 N ðOÞ

" #qðIÞ

qðOÞ

" #ð19Þ

with:

½N ðIÞðx; yÞ� ¼N 1 0 N 2 0 N 3 0 N 4 0

0 N 1 0 N 2 0 N 3 0 N 4

�;

½N ðOÞðx; yÞ� ¼ ½P 1; P 2; . . . ; P 16�ð20Þ

½qðIÞ� ¼ ½u1; v1; . . . ; u4; v4�T;½qðOÞ� ¼ ½w1; ðw;xÞ1; ðw;yÞ1; ðw;xyÞ1; . . . ;w4;

ðw;xÞ4; ðw;yÞ4; ðw;xyÞ4�T

ð21Þ

Here Nk and Pk are bilinear interpolation functions [9]

and [q(I)] and [q(O)] contain the appropriate nodal de-

grees of freedom.

Define:

½D� ¼w;x 0

0 w;y

w;y w;x

264

375; ½G� ¼

N ðOÞ;x

N ðOÞ;y

" #ð22Þ

½BðIÞ� ¼N 1;x 0 N 2;x 0 N 3;x 0 N 4;x 0

0 N 1;y 0 N 2;y 0 N 3;y 0 N 4;y

N 1;y N 1;x N 2;y N 2;x N 3;y N 3;x N 4;y N 4;x

264

375;

½BðOÞ� ¼ �N ðOÞ

;xx

N ðOÞ;yy

2N ðOÞ;xy

2664

3775 ð23Þ

Substituting the interpolations (19) into the expressions

(16)–(18), and minimizing the potential energy, results

in the element level equations:

KðIÞ 0

0 KðOÞ

" #qðIÞ

qðOÞ

" #þ 0 KðIOÞ

2KðIOÞT KðNIÞ

" #qðIÞ

qðOÞ

" #¼ ½P �

ð24Þ

The various submatrices and vector in (24) are:

½KðIÞ� ¼ H ðIÞZ

DðeÞ½BðIÞ�T½C�½BðIÞ�dxdy;

½KðOÞ� ¼ H ðOÞZ

DðeÞ½BðOÞ�T½C�½BðOÞ�dxdy

ð25Þ

½KðIOÞ� ¼ H ðIÞ

2

ZDðeÞ

½BðIÞ�T½C�½D�½G�dxdy;

½KðNIÞ� ¼ H ðIÞ

2

ZDðeÞ

ð½D�½G�ÞT½C�½D�½G�dxdyð26Þ

P½ � ¼Z

DðeÞ

N ðIÞ 0

0 N ðOÞ

" #T Hx

Hy

Hz

264

375dxdy ð27Þ

where D(e) is the area of a finite element.

The global version of (24) is now obtained in the

usual way.

Note that the in-plane and out-of-plane (bending)

matrices [K(I)] and [K(O)] are / h and h3, respectively,

the matrix [K(IO)] / Ah represents coupling between

the in-plane and out-of-plane displacements, and the

matrix [K(NI)] / A2h arises purely from the nonlinear

axial strains.

It is well known that for the linear theory

[K(O)] [K(I)] as h ! 0. It is very interesting, however,

to note that if A/h remains Oð1Þ, the bending matrix

[K(O)], which arises from the linear theory, and the ma-

trix [K(NI)] from the nonlinear theory, remain of the

same order as h ! 0. This fact has important conse-

quences for the modeling of very thin plates [2].

4. Lagrangian relaxation scheme for the coupled problem

Consider, for simplicity, a thin conducting plate

with a ground plane with V = 0. Fig. 2 shows a

B b

∂B ∂b

V

t

V1

V2

t 1 t2

∆V1

(a) (b)

Fig. 2. (a) Deformation of body; (b) voltage history.


schematic of the applied voltage history on this plate.

The deformation history u(x, t) of this plate is obtained

by a combined BEM/FEM approach as described

below.

The voltage history V(t) is first decomposed into a

series of steps—V1,V2 = V1 + DV1, . . .,Vn+1 = Vn + DVn.

Consider the first step with applied voltage V1. The

BEM problem is first solved in the region exterior to

the plate, and the charge density Rð0Þ1 and traction H

ð0Þ1

are obtained on the plate surface. The FEM problem

with applied traction Hð0Þ1 is next solved for the plate,

resulting in the calculation of the displacement field

uð0Þ1 in the plate. The BEM problem is next solved in

the region exterior to the deformed plate by the

Lagrangian approach (i.e. using the undeformed plate

surface). This calculation yields the next iterate of the

charge density and traction, Rð1Þ1 and H

ð1Þ1 , respectively,

on the plate surface. The next iterate of the displacement

field in the plate, uð1Þ1 , is obtained next by solving the

FEM problem in the plate with applied traction Hð1Þ1 .

This iterative process is repeated until convergence.

The converged values of the traction on the plate, and

displacement field in the plate, at time t1, are called H1

and u1, respectively.

The next task is to proceed from time t1 to t2. To

this end, the voltage increment DV1 is first applied

to the deformed configuration of the plate at time

t1. Solution of the corresponding BEM problem

(using, again, the Lagrangian approach), yields the

incremental charge density DRð0Þ1 and incremental

traction DHð0Þ1 . The displacement field u

ð0Þ2 is

obtained next by solving the FEM problem in the

(undeformed) plate with the traction Hð0Þ2 ¼

H1 þ DHð0Þ1 . The BEM problem is next solved with

V2 = V1 + DV1. The result is the charge density Rð1Þ2

and the traction Hð1Þ2 ; followed by the FEM solution

for uð1Þ2 . Again, this iterative process is continued until

the converged values u2 and H2 are obtained at

time t2.

The time step t2–t3 is considered next, and so on, un-

til the final time tn+1 is reached.

The algorithm employed for solving the coupled

problem is outlined below:

1. Apply V1 to B.

Solve BEM problem on oB

Get charge density Rð0Þ1 and traction H

ð0Þ1

2. Solve FEM problem in B with traction Hð0Þ1

Get displacement uð0Þ1 on oB

3. Set k = 0

Repeat

4. Update oB, obðkÞ1 ¼ oBþ uðkÞ1

Solve BEM problem on oB for obðkÞ1 with V1

Get charge density Rðkþ1Þ1 and traction H

ðkþ1Þ1

5. Solve FEM problem in B with traction Hðkþ1Þ1

Get displacement uðkþ1Þ1 on oB

6. Update k = k + 1

Until convergence i.e.jwðkþ1Þ

1�wðkÞ

1j

jwðkÞ1

j� 100 < tol and

jrðkþ1Þ1

�rðkÞ1

jjrðkÞ1

j� 100 < tol

Now t= t1.Get convergedvalues u1, ob1=oB+u1,H1

7. Apply DV1 to b1Solve BEM problem on oB for ob1Get charge density DRð0Þ

1 , traction DHð0Þ1

8. Solve FEM problem in B with traction Hð0Þ2 ¼

H1 þ DHð0Þ1

Get displacement uð0Þ2 on oB

9. Set k = 0

Repeat

10. Update oB, obðkÞ2 ¼ oBþ uðkÞ2

Solve BEM problem on oB for obðkÞ2 with V2 =

V1 + DV1

Get charge density Rðkþ1Þ2 and traction H

ðkþ1Þ2

11. Solve FEM problem in B with traction Hðkþ1Þ2

Get displacement uðkþ1Þ2 on oB

12. Update k=k + 1

Until convergence i.e.jwðkþ1Þ

2�wðkÞ

2j

jwðkÞ2

j� 100 < tol and

jrðkþ1Þ2

�rðkÞ2

jjrðkÞ2

j� 100 < tol

13. Now t = t2. Get converged values u2, ob2 =

oB + u2,H2

14. Proceed until tn+1


5. Numerical implementation

3x

h

2x

(0,6,0)

(2,0,0)

..

p (0,0,h)

5.1. Boundary integral equations for thin plates

MEMS applications typically employ very thin

plates––with thicknesses of the order of microns. The

BIE for the 3-D region exterior to thin plates needs spe-

cial care. Typically, weakly singular, and nearly weakly

singular integrals must be dealt with.

5.1.1. Proposed new method for the accurate evaluation

of nearly weakly singular integrals

Consider a source point p on the top face of a plate

and its image point p̂ on the bottom face in Fig. 3(a).

Two kinds if singular (Oð1=rÞ) integrals arise––a weaklysingular integral on the boundary element D on the top

face of the plate that contains p, and, since h is small, a

nearly weakly singular integral on the boundary element

D̂ (the image of D) on the bottom face of the plate that

contains p̂. The weakly singular integral is evaluated by

employing the mapping method outlined in [25,26]. This

method transforms such integrals over triangular

(curved or flat) domains into regular two-dimensional

ones. Integrals over curved quadratic or flat linear trian-

gles are further reduced to regular line integrals that can

be easily evaluated to desired accuracy by standard

Gaussian quadrature.

A new simple method is presented below for the

accurate and efficient evaluation of nearly weakly singu-

lar integrals. This approach transforms a nearly weakly

singular integral into a weakly singular one; which is

then evaluated by the method described in [25,26]. A

nearly weakly singular integral of interest here has the

form:

IðpÞ ¼Z

D̂

rðq̂Þdsðq̂Þ4p�rðp; q̂Þ ð28Þ

p

p

h α

α h

p̂ ˆ q̂

p̂ ˆ q̂

(a)

(b)

r

r

r

r

||

||

∆

∆̂

Fig. 3. Singular integrals.

The integrand above is multiplied by r̂=r̂ with the result:

IðpÞ ¼Z

D̂

½rðq̂Þðr̂=rÞ�dsðq̂Þ4p�r̂ðp̂; q̂Þ ð29Þ

Since r̂=r is O(1) and !0 as q̂ ! p̂ (i.e. as r̂ ! 0), the

integrand in (29) is weakly singular, of Oð1=r̂Þ as

r̂ ! 0. Therefore, the integral (29) can be evaluated by

employing the methods described in [25,26].

The source point p may also lie on a side face of the

plate as shown in Fig. 3(b). The same idea (29) can be

applied in this case as well.

5.1.2. Performance of new method

The performance of the new method is compared

with that of standard Gauss integration. Fig. 4 shows

the source point and region of integration (a triangle)

for (a) the out of plane and (b) the in plane cases, respec-

tively. The triangle is purposely chosen to be fairly elon-

gated. Numerical results (for r = 1 and � = 1) appear

in Fig. 5. It is seen that for h < 1/100, standard Gauss

3x

h

2x

1x

(0,6,0)

(2,0,0)

. .p

(-h,0,0)

1x (a)

(b)

Fig. 4. Numerical integration over a triangle. Source point (a)

out of plane (b) in plane.

10 10 10 10 10 10 10

1

2

3

4

5

6

7

8

7 Gauss points19 Gauss pointsNew method

Out of plane distance h

Perc

enta

ge e

rror

______ 6 5 4 3 2 1

104

103

102

101

10

1

2

3

4

5

6

7

8

7 Gauss points19 Gauss pointsNew method

Perc

enta

ge e

rror

In plane distance h

____

(b)

(a)

Fig. 5. Errors in numerical integration over a triangle. Source

point (a) out of plane (b) in plane.


integration, even with 19 Gauss points, cannot reduce

the error below around 4%. The new method is seen to

take care of these nearly weakly singular integrals very

well, even for very small values of h.

5.2. Nonlinear finite element analysis

It is important to point out that Eq. (24) is nonlinear

due to the fact that the matrices [K(IO)] and [K(NI)] con-

tain the gradient of w (see (22) and (26)). Newton itera-

tions are used to solve (24) once the transverse

displacement w becomes significant.

5.3. Coupling of BEM and FEM

Consider three sets of points:

B: nodes and Gauss points on boundary elements over

the entire bounding surface of a plate.

F: Gauss points on square finite elements on the mid-

plane of a plate.

S: points corresponding to those in F on the top and

bottom (square) surfaces of a plate.

Data transfer between the BEM and the FEM, and

vice versa, proceeds as follows. The BIE (8) (with

u = 0) is first solved for the charge density R. Next, (10)(with F = I––here I is the identity tensor) is used to find

the traction H on the set of pointsS, and then the resul-

tant forces R are obtained on the set F. These resultant

forces are transferred from the BEM to the FEM.

Once the FEM calculations are completed, the dis-

placement u is determined at all points on the set B

and the displacement gradient F is obtained on the set

S. The quantities u and F are then transferred from

the FEM to the BEM.

The FEM calculation, of course, delivers the mid-

plane displacements [u,v,w]. From these, it is easy to

get the actual displacements ½�u;�v; �w� at any point in the

plate, followed by the deformation gradient. The rele-

vant equations are given below.

�uðx; y; zÞ ¼ uðx; yÞ � zw;xðx; yÞ

�vðx; y; zÞ ¼ vðx; yÞ � zw;yðx; yÞ

�wðx; y; zÞ ¼ wðx; yÞ ð30Þ

F ij ¼ dij þ �ui;j ð31Þ

F ¼1þ �u;x �u;y �u;z�v;x 1þ �v;y �v;zw;x w;y 1

264

375 ð32Þ

In the above, the new symbol dij denotes the elements ofthe Kronecker delta.

It is clear that (32) requires the second derivatives

w,xx, w,xy and w,xy. These quantities are obtained from

the interpolation functions Pk,k = 1,2, . . ., 16 and the no-dal variables q(0) (see (19)–(21)).

5.4. Discretization

Standard T6 boundary elements are used in the BEM

for the program verification problems described in Sec-

tion 6.2. However, T3 boundary elements are used for

the (coupled analysis of the) thin plate problem, with

one layer of T3 boundary elements placed on each of

the four rectangular side faces of a plate. The primary

reason for this is to avoid having additional unknowns

on the sides of a thin plate. Use of higher order elements,

of course, increases computational effort by adding extra


degrees of freedom, but such elements are much more

accurate than, say, piecewise constant BEM elements

that were used in [19]. Quadrilateral (here square) plate

bending elements are used to discretize a plate mid-plane

for the FEM (Section 3.2), with identical uniform

squares being used on the square plate faces for both

the methods. (Of course, for the BEM, each square is

further divided into two triangles).

It is well known that the charge density is singular on

the edges of a plate. Therefore, nonconforming or singu-

lar boundary elements should be used whenever an ele-

ment edge coincides with a plate edge. This procedure,

however, makes the BEM code somewhat cumbersome

and expensive. For simplicity, regular T6 (for Section

6.2) or T3 (for Section 6.3) elements are used on plate

faces, with collocation points placed everywhere includ-

ing on plate edges.

6. Numerical results

6.1. Material properties

Material properties used for Silicon conductors in

free space are [35,36]:

E ¼ 169 GPa; m ¼ 0:22; � ¼ 8:85� 10�12 F=m

ð33Þ

silicon

insulator

silicon

V/2

V/2_

Fig. 6. MEMS plates.

Table 1

Central displacement of plate as a function of applied voltage

V (V) w0/h from several steps w0/h from one step

10 0.1775 0.1775

20 0.5825 0.5825

30 0.9726 0.9728

40 1.2966 1.2989

50 1.5752 1.5781

6.2. Program verification

The FEM program has been carefully verified in [2].

The BEM program has been verified by solving two

problems. In the first, Laplace�s equation is solved in a

region exterior to a unit sphere, with a prescribed uni-

form value of the potential / on its surface. With 80

T6 elements on the sphere surface, the percentage L2

error in the numerical solution for s = o//on, comparedto the exact one, is 1.34 %. The L2 error for Dirichlet

problems is here defined as:

� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNB

i¼1ðsðexactÞi � sðnumÞi Þ2

sffiffiffiffiffiffiNB

pjsðexactÞi jmax

� 100% ð34Þ

where NB is the total number of boundary nodes.

The second problem is a Dirichlet problem in the re-

gion exterior to a unit cube, with uniform unit potential

prescribed on the surface of the cube. The capacitance

(the total charge on the cube surface divided by the volt-

age) (with � = 1) from the BEM is 8.28 while that from

FastCap [37] is 8.3. The BEM mesh for this problem

has an 8 · 8 array of squares on each face, each divided

into two T6 triangles, for a total of 768 boundary

elements.

6.3. MEMS plates

6.3.1. The problem

Deformation of a silicon MEMS plate (the silicon is

doped so that it is a conductor), subjected to a progres-

sively increasing electrostatic field, is simulated here by

the coupled BEM/FEM. Each plate is clamped around

its edges and two plates are used in order to have a zero

voltage ground plane (the plane of symmetry) midway

between them (Fig. 6). Each plate is square of side

L = 3 mm and thickness h = 0.03 mm, and the gap g be-

tween them is 1 mm. Both plates are allowed to deform.

6.3.2. Discretization

The mesh used here is as follows. The top and bottom

(square bounding) faces of each plate has an 8 · 8 arrayof squares, with each square divided into two T3 trian-

gles, thus yielding 128 boundary elements on each face.

Each side face has a single row of rectangles, with each

rectangle divided into two T3 triangles, yielding 16 ele-

ments on each side face. Thus, there is a total of 320

boundary elements. Of course, there are only 64 square

finite elements on the mid-plane of each plate.

6.3.3. Convergence of algorithm

The voltage is applied in steps of 10 V and tol = 1 for

the algorithm outlined in Section 4. Table 1 is a check of

the algorithm. The second column is the value of w0/h

(where w0 is the transverse displacement at the center

of the top plate (positive downwards) obtained by apply-

ing the voltage in several ten volt steps (i.e. 20 V in 2


steps, 50 V in 5 steps etc.) The last column is the same

quantity obtained by applying the entire voltage in one

step. The corresponding values in the two columns show

excellent agreement. Convergence is rapid with about 2

iterations needed for a 10 volt and 3 for a 50 volt step.

6.3.4. Results

Fig. 7 shows the central deflection of the top plate as

a function of the square of the applied voltage, for rela-

tively small values of the applied voltage. This time, each

voltage step is 1 volt and tol = 1. As expected (see, e.g.

Fig. 8 in [15]) the first part of the curve (up to around

10 V) is linear; beyond which membrane stiffening starts

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

V (Volt )

w /h 0

22

Fig. 7. Central displacement of top plate as a function of

square of applied voltage for relatively small values of applied

voltage.

0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

3

x 106

V^ 2

w /h 0

Fig. 8. Nondimensional central displacement of top plate as

a function of square of nondimensional applied voltage.

to become apparent. (Please see the next paragraph for

more discussion of this phenomenon.)

Numerical results for the nondimensional central

displacement of the top plate, as a function of the square

of the nondimensional applied voltage, for a larger

value of the applied voltage (Vmax � 100 V), appear in

Fig. 8. The nondimensionalization used here is:

V̂2 ¼ �L5V 2

2g3EIð35Þ

where the moment of inertia of the plate cross-section,

I = Lh3/12. For the values of parameters used here,

V̂2 ¼ ð0:9426� 10�9ÞV 2: This time, the voltage is ap-

plied in steps of 10 V and tol = 1 in the algorithm in Sec-

tion 4. The membrane stiffening of the clamped plate,

for larger and larger values of the applied voltage (hence

applied force), is clearly evident. (See [10] for the corre-

sponding phenomenon in the pressure-deflection curve

of a thin plate subjected to increasing pressure.) No such

stiffening is observed in Fig. 8 of [15] since in that case

the mechanical problem for a cantilevered beam has

no membrane forces and is geometrically linear. The re-

sults in Figs. 7 and 8 of [15], on the other hand, demon-

strate softening near the instability around 78 V. No

such softening due to an approaching instability is

apparent in Fig. 8 of the present paper because this sim-

ulation stops far short of the point of instability.

7. Discussion

This work presents a first attempt at a fully Lagrang-

ian approach for the analysis of coupled 3-D MEMS

problems. The Lagrangian approach uses only the

(usually simple) undeformed configuration of a plate

for both the electrical and mechanical analyses––thus

obviating the need to discretize any deformed configu-

ration.

The hybrid BEM/FEM approach is able to handle

thin plates (with h/L = 1/100) efficiently. Convergence

is achieved for relatively large voltage steps with only

a few iterations. The proposed simple new approach

for accurate evaluation of nearly weakly singular inte-

grals works well. (It is noted that although the idea is

illustrated in Section 5.1 for flat plates, it is also success-

fully employed when the plates become curved due to

deformation). It is seen from Fig. 8 that the nonlinear

membrane stiffening effect can be very significant. There-

fore, the nonlinear FEM model, employed in this work,

is of crucial importance.

The two-plate MEMS example solved in this paper

has h/L = 1/100 and g/L = 1/3. In practice [1], one can

have h/L = 1/1000 with g � h. Simulation of such prob-

lems requires further advancement in BEM technology.

Research along such lines is currently in progress.


The next step is 3-D dynamic analysis of this prob-

lem. Dynamic MEMS analysis with a fully Lagrang-

ian approach is expected to provide many additional

benefits relative to the traditional approach that uses

Lagrangian mechanical but Eulerian electrical analysis

(see [21] where the 2-D dynamic problem has been stud-

ied recently). Eventually, the motion of the fluid (such as

air) between the plates must be modeled in the problem

discussed in [1]. This motion would cause the voltages to

be increased substantially, especially at higher frequen-

cies, due to the damping and compressibility effects of

the fluid between the plates.

Acknowledgement

This research has been partially supported by Grant

# EEC-0303674 of the National Science Foundation to

Cornell University, and by Grant # 01-2042-007-25 of

the New Jersey Commission on Science and Technology

to New Jersey Institute of Technology through the New

Jersey Center for Micro-Flow Control.

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