Koren B., Vuik K. (Editors) Advanced Computational Methods in Science and Engineering

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Numerical Modeling of the Electromechanical Interaction in MEMS

Table 1 Computation times.

Method Computation Error tolerance

time (s) (relative)

Voltage block GS with path following 277 10

−3

DIPIE 296 5 ×10

−2

Charge block Gauss-Seidel 76 10

−3

Voltage monolithic path following 181 10

−3

1 2 3 4 5 6 7

-4

-3

-2

-1

Voltage block GS with path following

DIPIE

Charge block GS

Voltage with monolithic path following

Error

Number of iterations

Fig. 18 Error curves for the simple beam

4 Coupled eigenfrequencies

One of the earliest applications of electromechanical MEMS is the microresonator

[40]. An advantage of these electromechanical resonators is that the eigenfrequency

depends on the bias voltage that is applied [29, 1, 11]. This can simply be explained

with the 1D example from section 1.2. In that section static equilibrium was deﬁned.

The dynamic equation of motion can be written as [37]:

m¨x + k(x −x

) +

= 0 . (53)

It is assumed that the voltage is ﬁxed and the mass moves only with small dis-

placements around the static equilibrium equilibrium position x = x

+∆x, and thus

¨x = ∆¨x. Equation (53) is linearized as follows:

m∆¨x +

∂



k(x −x

) +





x=x

∆x = 0 , (54)

335

S.D.A. Hannot and D.J. Rixen

m∆¨x + k∆x −

∆x = 0 , (55)

The eigenfrequency of this linear differential equation is then by deﬁnition:

−

, (56)

which shows that the eigenfrequency can be changed by giving the device a bias

voltage.

4.1 Staggered method

For the mechanical system the eigenfrequency of a multiple dof system is well de-

ﬁned [16]. Once the inertia terms are included in the mechanical equilibrium equa-

tion (2) this results in:

∂

+ F

−

¨u

= 0 , (57)

where ¨u

is the acceleration in direction j and

is the density of the material. After

discretization one obtains the classical equation of motion for a system of multiple

dofs:

u+ K

u = f

ext

, (58)

where M

is the mass matrix. The linear eigenfrequencies

and eigenmodes x

are then deﬁned as the solutions of the eigenvalue problem:

−

+ K

= 0 . (59)

Of these eigenfrequencies only the lowest few are of practical interest. For instance

a tunable microresonator normally operates at it lowest eigenfrequency.

However for an electromechanically coupled problem the external force f

ext

de-

pends on u:

u+ K

u = f

ext

(u,V) . (60)

In a staggered model, this effect is not incorporated in the mechanical stiffness ma-

trix, therefore not in the eigenfrequencies. In the literature a method exists to com-

pute the inﬂuence of this dependency on the eigenfrequency [30]. The idea starts

with projection of the non-linear equation on the lowest mechanical eigenmode. In

that case it is assumed that the displacement vector around an equilibrium can be ap-

proximated by u = u

∆u. When the non linear equation of motion is projected

on the ﬁrst eigenmode one ﬁnds

∆¨u + x

∆u + x

= x

ext

+ x

∆u,V) , (61)

336

Numerical Modeling of the Electromechanical Interaction in MEMS

Since at equilibrium

= f

ext

,V) , (62)

one can also write

∆¨u + x

∆u = x



ext

+ x

∆u,V) −f

ext

,V)



, (63)

which reduces the set of equations to a single differential equation:

∆¨u + k

∆u = ∆f

∆u,V) , (64)

where m

is the modal mass, k

the modal stiffness and f

the modal participation of

the force.

An estimate of the coupled eigenfrequency can then be found as:

−

∂

∆f

∂

∆u

, (65)

where the term

∂

∆f

∂

∆u

can be obtained by a ﬁnite difference approximation. Of course

this method only works properly if the electric forces do not change the modeshape

too much.

4.2 Monolithic method

Monolithically the computation is much easier. To obtain the linearized equation

of motion only inertia terms are added to the linearized equilibrium equations (27)

[37]:



∆



+ K



∆u

∆v





∆f

∆q



, (66)

where M is the coupled inertia matrix. Since the electric domain was assumed to be

electrostatic, there are no inertia terms for the potentials, hence:

M =



0 0



. (67)

The full stiffness matrix, as deﬁned by equation (27), is evaluated at the static

equilibrium position. Therefore the eigenvalue problem becomes:



0 0





−







= 0 , (68)

which will give the correct eigenfrequencies and modeshapes since the full elec-

tromechanical coupling is accounted for. The only problem is that the inertia matrix

is singular, hence there are inﬁnitely large eigenfrequencies. However the static dofs

337

S.D.A. Hannot and D.J. Rixen

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10

−5

1.2

1.4

x 10

−5

Fig. 19 First mechanical eigenmode of the beam

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10

−5

1.2

1.4

x 10

−5

Fig. mechanical eigenmode of the

beam

can be condensed out, in other words the electric dofs can be condensed out. The

bottom line of eq. (68) deﬁnes the relation between x

and x

. If this relation is

substituted into the upper line of (68) it results in

−



−K

−1



= 0 , (69)

which deﬁnes an eigenvalue problem that can be solved exactly like the purely me-

chanical eigenvalue problem.

4.3 Example

These algorithms were tested on the model of a simple beam in ﬁg. 15. The density

used was 5 ×10

kg/m

. The ﬁrst and third purely mechanical eigenmodes of the

beam are shown in ﬁgs. 19 and 20. After the computation of each static solution

shown in ﬁg. 16 the ﬁrst and second eigenfrequency and eigenmode around this

static solution were determined. The ﬁrst eigenmode is very similar to the shape of

the static deﬂection shown in ﬁg. 17, thus the modeshape will not be changed very

much by the electric ﬁeld. Therefore it is expected that the staggered method will be

accurate for the ﬁrst eigenfrequency. The second mode shape is more complex. The

electrostatic load might affect the modeshape, thus the eigenfrequency. This will

affect the accuracy of the results obtained with the staggered method, but not with

the monolithic method.

The curves for the ﬁrst eigenfrequency shown in ﬁg. 21 conﬁrm that the staggered

method computes the same curve as the monolithic method. It also shows that the

frequency goes to zero at the pull-in point, similarly to what is found in limit point

buckling. Both methods compute the same curve, however the curve obtained with

the staggered method is less smooth. A disadvantage of the staggered method is that

the accuracy depends heavily on the size of the generalized displacement step that

is used for the ﬁnite difference approximation.

The curves for the second eigenfrequency are shown in ﬁg. 22. As expected the

curves are very similar, but there is a small discrepancy due to the geometrical com-

plexity of the mode which induces more inﬂuence of the electrostatic forces on the

20 Second

338

Numerical Modeling of the Electromechanical Interaction in MEMS

0 5 10 15 20 25 30 35 40 45

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Potential (V)

Frequency (Hz)

Monolithic

Staggered

Fig. 21 First eigenfrequency as a function of applied potential

5 10 15 20 25 30 35 40 45

2.15

2.2

2.25

2.3

2.35

2.4

2.45

x10

Potential (V)

Frequency (Hz)

Monolithic

Staggered

Fig. 22 Second eigenfrequency as a function of applied potential

mode shape. It is interesting to note that the second eigenvalue is not zero at pull-in

which is the case for all the higher modes.

339

S.D.A. Hannot and D.J. Rixen

5 Summary and conclusions

We have presented an overview of important issues in the numerical modeling of

electromechanical coupling as seen with MEMS. In particular the solution tech-

niques have been classiﬁed in staggered and monolithic solvers. Several different

staggered and monolithic algorithms for ﬁnding the static load displacement curve

were outlined and illustrated. Finally the computation of linearized eigenfrequencies

around the static equilibria was discussed.

From these discussions it can be concluded that staggered methods can be advan-

tageous. Staggered schemes can easily couple different discretization procedures for

different problems, such as the often seen combination of structural FEM with elec-

tric BEM. When the proper algorithm is used (charge loading) these algorithms can

be faster than monolithic methods.

An advantage of monolithic methods is that knowledge of the full tangent stiff-

ness matrix enables eigenvalue and eigenfrequency analysis. Deﬁnitely for the

higher modes these eigenvalues can only be properly computed with monolithic

methods. However these computations can be performed with hybrid methods,

methods that compute the pull-in curve staggered, and assemble the full tangent

stiffness matrix to do the eigenvalue analysis only when the solution of an incre-

ment has converged.

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342

Simulation of Progressive Failure in Composite

Laminates

F.P. van der Meer and L.J. Sluys

Abstract Fiber reinforced polymers are materials with excellent mechanical prop-

erties and relatively much design freedom. However, because complex failure mech-

anisms originating from the microstructure of the material may occur, realistic sim-

ulation of the failure process is still a challenge. Two alternative models for the

modeling of failure in composite laminates are presented. The ﬁrst is a continuum

damage model that is supposed to cover all ply failure mechanisms. A limitation of

the continuum approach with respect to the modeling of matrix failure is illustrated.

Therefore, a discontinuous model has been developed for matrix failure speciﬁcally.

1 Introduction

Fiber reinforced polymers are promising structural materials, because of their excel-

lent strength-to-weight and stiffness-to-weight ratios. The materials are composed

of ﬁbers with high strength and longitudinal stiffness (e.g. glass or carbon) held to-

gether with a polymer matrix (e.g. epoxy). An advantage of these materials is that

there is design freedom in choosing the ﬁber direction such that the stiffness and

strength are highest in the most critical direction. To exploit this design freedom,

proper predictive tools are needed. However, the development of reliable numer-

ical models is a challenging task, because the failure process is complex, involv-

ing different processes such as splitting, ﬁber/matrix debonding, delamination, ﬁber

breaking, and ﬁber pull out (see e.g. [14, 16,35]).

This contribution is restricted to composite laminates with unidirectional plies,

i.e. each layer of the laminate contains long straight ﬁbers which are all oriented in

a single direction, and the strength and stiffness of the laminate as a whole are con-

Faculty of Civil Engineering and Geosciences, Section of Structural Mechanics,

Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands

e-mail: f.p.vandermeer@tudelft.nl

343

Lecture Notes in Computational Science and Engineering 71,

DOI 10.1007/978-3-642-03344-5_12,

B. Koren and C. Vuik (eds.), Advanced Computational Methods in Science and Engineering,

F.P. van der Meer and L.J. Sluys

Microlevel Mesolevel Macrolevel

Fig. 1 Three scales levels of observation of laminated composites.

trolled by stacking layers with different ﬁber direction. In analyzing these laminates,

three levels of observation may be distinguished (see Fig. 1):

• the microlevel, on which the ﬁber, the matrix and the ﬁber-matrix interface are

explicitly present,

• the mesolevel, on which each ply is considered to be a homogeneous continuum,

and

• the macrolevel, in which the laminate is considered as a plate with a single set of

properties after through-thickness homogenization.

Fig. 2 Complex failure in a composite laminate (Green et al. [14], used with permission from

Elsevier).

In Fig. 2, a complex failure mechanism is shown for a laminate with a circular

hole loaded in uni-axial tension. It can be observed that there is much delamina-

tion, which allows the different plies with different ﬁber orientation to fail each in

a different failure plane. In the ±45

◦

-plies, splitting occurs, i.e. the failure plane is

parallel to the ﬁber. Also in the 90

◦

-plies, the failure plane is located between the

ﬁbers. Only in the 0

◦

-plies, ﬁbers break and an irregular failure pattern is observed.

While Fig. 2 is typical for the mesolevel approach, Fig. 3 shows a schematic

representation on the microlevel of the processes that constitute ply failure under

tension in ﬁber direction. Also on this level, there are different processes involved

such as, matrix cracking, ﬁber/matrix debonding, ﬁber failure and ﬁber pull-out.

How to represent these different mechanisms in a single computational frame-

work that is generic with respect to geometry, loading conditions and ply layup?

That is the central question in the research presented here. Is it possible to build a

model with which the complex laminate failure mechanisms can be simulated and

344