WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1624 J. S. JENSEN
h
2h
fcos Ωt
Figure 11. A short cantilever subjected to a harmonic tip-load with frequency .
is to be distributed so that a chosen objective function is minimized. The objective function is
c3 = 1
N
utipūtip
N k=1
so that the response of the tip is chosen as a measure of the dynamic compliance.
A plane stress model with unit thickness is considered and the cantilever is discretized using
80 × 40 bi-linear quadratic elements. The load is applied in three nodes in middle of the right
boundary. Damping coefficients are = 0.05 and = 0.002. The Matlab implementation from
Section 4.1 is used, and a sensitivity filter (filter-size: 1.5×element-size) is included to avoid
mesh-dependent solutions and checkerboard formation [23]. The initial design has 50% material
uniformly distributed in the domain.
Figure 12 shows the cantilever optimized for four different frequency intervals. If the frequency
interval is in the low range ( = 0–0.02) well below the first resonance frequency of the initial
design, then the optimized cantilever (Figure 12(a)) resembles the structure that is obtained for a
static tip-load. The increased static and low-frequency dynamic stiffness is obtained by moving
the first resonance frequency up in the frequency range. Figure 12(b) shows that if the frequency
range is increased to = 0–0.04, the material is slightly redistributed so that the first resonance
frequency is further increased and the high stiffness frequency range broadened. If the optimization
interval is increased to higher frequency ranges, the optimized structures (Figure 12(c), (d)) change
qualitatively.
Figure 13 shows the corresponding responses for the four optimized structures as well as for
the initial design. As mentioned before, the structures in Figure 12(a) and (b) have responses in
which the first resonance frequency is moved up in the frequency range. Similar observations were
made in [22]. However, the structures in Figure 12(c) and (d) obtain a low average response by
a different mechanism: the static and low-frequency stiffness is lower (higher response) but an
anti-resonance is introduced in the optimization interval. At the anti-resonance the tip response is
small whereas the response of the rest of the structure is high. Thus, if a different measure of the
dynamic compliance had been used, e.g. the total energy, these designs would not be optimal.
It should be mentioned that the choice of N (number of expansion terms) is critical for this
example. The initial problem in which the structure is homogeneous is very ill-conditioned and if
N>2 the errors introduced in the PA computation and the corresponding sensitivities become too
large with a double-precision implementation. Therefore, in the shown examples the first iterations
are done with N = 2 and up to eight PA expansions. After a few iterations, say 15–20, N is
increased (up to N = 5) and the number of expansions reduced to one or two.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
(74)