Maria Bayard Dühring - Solid Mechanics

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When the mesh size is decreased the optimization will in general result in mesh-dependent solutions with
small details which make the design inconvenient to manufacture. To avoid these problems a morphologybased
filter is employed. Such filters make the material properties of an element depend on a function of the
design variables in a fixed neighborhood around the element such that the finite design is mesh-independent.
Here a close-type morphology-based filter is chosen [30], which has proven efficient for wave-propagation type
topology optimization problems. The method results in designs where all holes below the size of the filter
(radius rmin) have been eliminated. A further advantage of these filter-types is that they help eliminating gray
elements in the transition zone between solid and air regions.
3. Results
In this section results are presented for rooms in 2D and 3D as well as for outdoor sound barriers in 2D.
3.1. Optimization of a rectangular room in 2D
ARTICLE IN PRESS
M.B. Dühring et al. / Journal of Sound and Vibration 317 (2008) 557–575
The average of the squared sound pressure amplitude is minimized in the output domain Oop by distributing
material in the design domain Od in the rectangular room as shown in Fig. 1. The maximum volume fraction is
chosen to b ¼ 0:15 and the initial design for the optimal design is a uniform distribution of 15% material in
the design domain. The vibrational velocity of the pulsating circle is U ¼ 0:01 m s 1 and the target frequency is
f ¼ 34:56 Hz, f ¼ o=2p, which is a natural frequency for the room with the initial material distribution. The
modeling domain is discretized by triangular elements with maximum side length hmax ¼ 0:3 m and the filter
radius is rmin ¼ 1:0hmax. An absolute tolerance of 0.01 on the maximum change of the design variables is used
to terminate the optimization loop. The optimized design was found in 281 iterations and the objective
function was reduced from 110.9 to 76.1 dB. Fig. 2 shows the optimized design as well as the sound pressure
amplitude for the initial and optimized designs. It is clearly seen that in comparison to the initial design the
redistributed material in the design domain is influencing the sound pressure in the room such that it has a
very low value in the output domain Oop with a nodal line going through it. The material is placed at the nodal
lines for the initial design which is an observation that will be elaborated on later. On the top right of Fig. 2 the
frequency response for the initial design and the optimized design are shown, where F is plotted as function of
the frequency f. In comparison to the initial design, the natural frequencies for the optimized design have
changed and the natural frequency, which was equal to the driving frequency for the initial design, has been
moved to a lower value. It is noted that the solid material forms small cavities and that there is a tendency for
the sound pressure amplitude to be higher in these cavities than outside them. The cavities resemble Helmholtz
resonators (see Ref. [2] for a description of a Helmholtz resonator). It should be noted, that even though the
filter is used and the value of rmin is varied it is difficult to obtain fully mesh-independent solutions due to
many local minima.
In the previous example a low frequency has been used to obtain an optimized design. In the next example
the room is optimized for the frequency f ¼ 4 34:56 Hz, and the quantities b ¼ 0:5, hmax ¼ 0:2 m, and
rmin ¼ 1:0hmax. The optimized design is seen in Fig. 3 where the objective function is decreased from 95.7 to
62.1 dB in 478 iterations. Compared to the design for the lower frequency the design is now a complicated
structure with many small features. The reason is that for increasing frequencies the distribution of the sound
pressure amplitude in the room gets more complex and the design needed to minimize the objective function
will naturally also consist of more complicated details. For higher frequencies well defined designs can still be
obtained, but it is hard to get a mesh-independent design as it is very sensitive to discretization, filtering radius,
starting guess as well as local minima.
In the next example the optimization is done for f ¼ 9:39 and 9.71 Hz which is less and higher, respectively,
than the first natural frequency 9.55 Hz for the room with the initial design with b ¼ 0:15. The corresponding
mode shape has a vertical nodal line in the middle of the room and high sound pressure amplitude along the
walls. The quantities used in both cases are hmax ¼ 0:3 m and rmin ¼ 0:75hmax. The optimized designs and the
corresponding frequency response are seen in Fig. 4. For the case with f ¼ 9:39 Hz the solid material is
distributed at the corners with the high pressure amplitude and the natural frequency is moved to a higher
frequency. However, in the case with f ¼ 9:71 Hz the material is placed at the nodal plane in the middle of the