Maria Bayard Dühring - Solid Mechanics

4.2 Results 23
To obtain better numerical scaling in the optimization process the logarithm is
taken to the objective function, and the sensitivities hence have to be adjusted by
dividing them with Φ. The volume constraint imposes a limit on the amount of
material distributed in Ωd given by the volume fraction β in order to save material.
The initial guess for the optimization is a uniform distribution of solid material in Ωd
with the volume fraction β. A small amount of mass-proportional damping is added
to the problem and the morphology-based filter described in section 3.5 is applied.
Second order Lagrange elements are used for the complex pressure amplitude and
the design variables are discretized by first order Lagrange elements.
4.2 Results
In the first example, the sound barrier is optimized for the octave band center frequency
f = 125 Hz (where f = ω/2π) and the volume fraction β = 0.9. The
performance of the optimized design obtained from the topology optimization is
compared to the performance of a straight and a T-shaped barrier. The two conventional
barriers are centered in the design domain and they both have the same
height as Ωd and the T-shaped barrier has the same width. The increase of the
objective function ∆Φ for the T-shaped barrier and the optimized design is given
in table 4.1 compared to Φ for the straight barrier. It is observed that the Tshaped
barrier reduces the noise slightly more than the straight barrier, which is in
agreement with the literature. The table shows that the objective function for the
optimized design is reduced with more than 9 dB compared to the straight barrier.
The optimized barrier is indicated in figure 4.2(a) and is close to a 0-1 design. The
distribution of the sound pressure amplitude for the optimized design is seen in figure
4.2(b). The sound pressure amplitude is high in the cavity connected to the
right surface, which is acting as a kind of a Helmholtz resonator.
It is also possible to design the sound barrier for a frequency interval using Padé
approximations as described in section 3.3. The next example is thus optimized
for the interval between the two center band frequencies [63;125] Hz with 7 target
frequencies. The volume factor is again β = 0.9 and the objective function Ψ is
reduced from 71.4 dB for the initial guess to 64.4 dB for the optimized design. The
barrier is seen in figure 4.3(a) and it has a cavity on both vertical edges that can act
as Helmholtz resonators. In figure 4.3(b), the value of Φ is plotted as a function of
the frequency f for the optimized design as well as for the straight and the T-shaped
Table 4.1 The increase of the objective function ∆Φ for the T-shaped and the optimized
barrier compared to Φ for the straight barrier for f = 125 Hz.
frequency straight T-shape optimized
f [Hz] Φ [dB] ∆Φ [dB] ∆Φ [dB]
β = 0.90
125 70.02 -0.90 -9.13