Online proceedings - EDA Publishing Association

11-13 May 2011, Aix-en-Provence, France
V.
MODEL VALIDATION WITH A DIRECT FEM 4.
Our validation consists in comparing the results of our
model with results obtained using a direct FEM for the
three-dimensional elasticity system. We have carried our
computation with a small number due to long computing
time of FEM simulations. We compare the modal structure of
our model to the modal structure of a FEM in
three-dimensional elasticity. The eigenvalues of the
three-dimensional elasticity equations also constitute an
increasing positive sequence that accumulates at infinity. As
for the two-scale model, its density distribution exhibits a
number of concentration points and also some isolated
values. Bunch sizes are still equal to the number 2 of
cantilevers for low eigenvalues (log ), see Figure
3 representing eigenmode distributions for , and
Fig.4. Eigenmode density distributions for finite element model and
for the two-scale model
We remark that a number of eigenvalues in the FEM
spectrum have not their counterparts in the two-scale model
spectrum. We have checked that the missing elements
correspond to the modes which have membrane
displacement in some local cells and torsion in the
cantilevers. These cases are not modeled in the current
simple two-scale model. We also compare the eigenmodes
and especially those belonging to bunches of eigenvalues. By
visual inspection, we observe that the deformed shape of
cantilevers from FEM model and from our model are similar
for identical eigenvalues, see Figure 5.
Fig. 3. Eigenmode Density Distributions for Finite Element Model
Extrapolating this observation, we derive that when the
number of cantilevers increases bunch size increases
proportionally. Since the two-scale model is an
approximation in the sense of an infinitely large number of
cantilevers, this explains why the two-scale model exhibits
mode concentration with infinite number of elements. This
remark provides guidelines for operating mode selection in
the two-scale model. In order to determine an approximation
of the spectrum for an -cantilevers array, we suggest to
operate a truncation of the mode list so that to retain a simple
infinity of eigenvalues ij i , ,
2 and j
We stress on the
fact that
2 - eigenvalue bunches are generally not
corresponding to a single column of the truncated matrix ij .
We consider a silicon array of 10 by 10 cantilevers, see
Figure 1 (a). Computing the eigenvalues associated to the
two-scale model, we observe that they are grouped in
bunches of size 100 accumulated around each clamped-free
cantilever eigenvalue. A number of eigenvalues are isolated
far from these bunches. We compare the modal structure of
our model with the one of a FEM based on three-dimensional
elasticity system for the same configuration. Densities of
square root of eigenvalues in logarithm are reported in Figure
(a)
(c)
(d)
Fig. 5. (a) First base mode in two-scale model (b) First mode in FEM
model (c) First cantilever mode in two-scale model (d) Matched mode
in FEM model
In a future work, we will develop a numerical test, as in the
paper [18] related to one-dimensional arrays of cantilevers,
so that to eliminate modes corresponding to physical effects
not modeled by our model. It will be applied on transverse
displacement only. We will also conduct FEM calculations
for larger (more than 10) on a more powerful computing
system in order to complete the convergence analysis of the
solution to the FEM towards the solution of our model.
In order to compare the distribution of the spectrum for a
-cantilever array, we operate a truncation of mode list. It
corresponds to the range [ 6 of log in Figure 4.
(b)
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