WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

objective
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0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
normalized frequency
Figure 4 - Objective Φ(ω ∗ ) for the air-filled chamber and response for ω ∗ = 0.175. Parameter
values: L = 7, l = 5, H = 1, h = 1.
where the third constraint introduces an optional limit on the amount of solid to be
distributed (V = 1 corresponds to no limit). The optimization problem is solved iteratively
by material redistribution steps, using Svanberg’s MMA [7] as optimizer and the
analytical sensitivities in Eqs. (10)–(11). The optimization formulation considers the
objective for a number of discrete frequencies. Since we are interested in minimizing
the objective in frequency ranges rather than at discrete frequencies we use frequency
sweeps at regular intervals during the optimization in order to identify the most critical
frequencies in the desired interval and then update the target frequencies ωi accordingly.
The frequency sweeps are done fast and accurately using Padé expansions [8].
We now optimize the chamber for a specific set of parameters; L = 7, l = 5,
H = 1, h = 1, ǫ = 0.5. Fig. 4 shows Φ(ω ∗ ) for an air-filled reflection chamber and
the computed pressure field for ω ∗ = 0.175. The frequency is here normalized so that
ω ∗ = 2πcω/H. We choose now to design the chamber so that Φ(ω) is minimized in
the frequency range ω ∗ = 0.15 − 0.20 by distributing maximum 15% solid material
(V = 0.15). The optimized topology, the corresponding response curve, and the pressure
field for ω ∗ = 0.175 are shown in Fig. 5. Clearly, a reduction of Φ is seen in the
designated frequency range. Naturally, a larger reduction can be obtained if the chamber
dimensions are increased. Fig. 6 shows the results for a double length chamber
l = 10 and here a further reduction of the objective is noted.
CONCLUSIONS AND FURTHER WORK
We have demonstrated a general design method for acoustic devices based on topology
optimization.
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