Simulating the physical behavior of thin structures accurately is central to micro-electromechanical systems (MEMS) CAD. We have implemented an adaptive finite element

method suited for multi-physically active and multi-layered thin MEMS structure simulation. We use a multi-layer Kirchhoff-Love thin structure model that also covers thermomechanical and piezoelectrical effects and which is implemented as an Argyris element.

The adaptivity is carried out by deriving residual error estimates for the thin structure

model together with recursive split patterns for triangular meshes.

Modeling thin structure MEMS

Many mechanical microsystem components are plate-like or beam-like. They are obtained

from semiconductor integrated- circuit manufacturing processes as multi-layer sandwiches

[1].

The bilinear or weak form for thin multi-layer structures can be expressed by means

of a Kirchhoff-Love model

a(U, V )

= (Aˆ0 : (∇u)

S − Aˆ1 : ∇∇w + σ

ext

0

,(∇v)

S

)

−(Aˆ1 : (∇u)

S − Aˆ2 : ∇∇w + σ

ext

1

, ∇∇v)

= (F, V ), (3)

where we have merged the in-plane displacement fields u and the out-of-plane deflection

field w and their variations v and v into the fields U and V respectively. In case the externally imposed prestress σ

ext of the structure is caused by an additional temperature

field or arises from a piezoelectric effect, the corresponding fields also have to be considered

in the weak form. The indices i of the reduced (anisotropic) elastic and prestress tensors

Aˆi

, σi account for the dimensionally reduced solid model and represent the moment order

with respect to the integration that is performed across the structure thickness. The right

hand side of (3) represents the external load acting on the structure. The solution space X

for this problem is some product space of sobolev spaces. We conformingly approximate

this solution space by 21-noded Argyris triangle C

1 finite elements [2].

Error estimation

The bilinear form (3) induces an energy norm in which we are aiming to estimate the

solution error

|kU − Uhk|2 = a(U − Uh, U − Uh). (4)

The starting point for residual error estimation [3] is the inequality

|kU − Uhk| ≤ sup

∈X,|k

k|=1

a(U − Uh, V )

= sup

∈X,|k

k|=1

(F − LUh, V ) (5)

where L denotes the linear differential operator associated with the bilinear form (3)

through

(LW, V ) = a(W, V ) ∀ W, V ∈ X. (6)

The term F − LUh in relation (5) is the residual with respect to the strong form of the

multi-layer plate problem. Calculating the right-hand-side of (5) by exploiting the continuity and ellipticity of the bilinear form (3) and using Cl´ement’s interpolation estimates

[4] yields an a posteriori error estimation

|kU − Uhk| ≤ c

X

T ∈S

η

2

T

(Uh)

!1/2

(7)

where c is some constant and ηT (Uh) is the (local) element error estimator depending

only on the computed solution and the given data. This error estimator consists of

the sum of individual errors that all have a proper physical meaning and can thus be

related to specific simulation and design requirements of the MEMS developper. Major

error contributions consist of non physical body forces k∇(∇Aˆ2 : ∇∇wh)k

2

L2(T)

h

4

T

and

unphysical jumps of shear forces k[n∇Aˆ2 : ∇∇wh]k

2

L2(E)

h

3

E and bending moments k[nAˆ2 :

∇∇wh]k

2

L2(E)

hE across element interfaces (hE and hT are the triangle edge lengths and

the triangle diameters, respectively). The number of error contributions increases in case

of the structure’s multi- layer arrangement or if additional fields are present. In the latter

case a considerable number of terms are caused by the coupling of the various fields. The adaptivity then is carried out using a maximum refinement strategy [3] and a recursive

refinement algorithm for triangles [5]. We will show the formulation of this error estimator,

and demonstrate its use with computed examples from MEMS.

References

[1] C. Hagleitner, A. Hierlemann, D.Lange, A. Kummer, N. Kerness, O. Brand, H. Baltes, Smart Single-Chip Gas Sensor Microsystem, Nature, 414 (2001) pp. 293-296.

[2] S. Taschini, J. Muller, ¨ A. Greiner, M. Emmenegger, H. Baltes, J.G. Korvink, Accurate Modeling and Simulation

of Thermo- mechanical Microsystem Dynamics”, Computer Modeling and Simulation in Engineering 1, (2000), pp.

31-44.

[3] R. Verfurth, ¨ A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (Wiley-Teubner,

Chichester-New York-Stuttgart, 1996).

[4] Ph. Cl´ement, Approximation by finite element functions using local regularization, RAIRO Anal. Numer. 2 (1975),

pp. 77-84.

[5] M.C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Internat. J.

Numer. Meth. Engrg. 20, (1984) pp. 745-756.