Maria Bayard Dühring - Solid Mechanics

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562 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. <strong>Dühring</strong> 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
y [m] 8 6 4 2 ARTICLE IN PRESS M.B. <strong>Dühring</strong> et al. / Journal of Sound and Vibration 317 (2008) 557–575 563 Sound pressure amplitude for initial guess [dB] 0 0 2 4 6 8 10 12 14 16 18 x [m] 110 100 90 80 70 60 60 24 26 28 30 32 34 36 38 40 frequency, f [Hz] design domain and the natural frequency is now at a lower value than originally. The same tendency is observed for the example in Fig. 2 where the material is distributed at the nodal lines and the natural frequency is moved to a lower value. The response at the two target frequencies f ¼ 9:39 and f ¼ 9:71 Hz for both designs, are given in Table 1 and it is noticed that the objective function is minimized most for both frequencies in the case where the solid material is placed at the nodal plane. So for the target frequency which is smaller than the natural frequency the optimization converges to a solution that is not as good as when the other target frequency is used. The explanation is that a natural frequency, which is originally at one side of the driving frequency, can only be moved to a value on the same side during the optimization, else the objective function would have to be increased during a part of the optimization. It is from this example and the example from Fig. 2 concluded, that when optimizing for a driving frequency close to a natural frequency there is a tendency for the material to be distributed at the nodal planes for the initial design when the natural frequency is moved to a lower value. If the natural frequency is moved to a higher value the material is objective function, Φ [dB] y [m] 130 120 110 100 8 6 4 2 90 80 70 Sound pressure amplitude for optimized design [dB] initial design optimized design target frequency 0 0 2 4 6 8 10 12 14 16 18 x [m] Fig. 2. (a) The optimized ceiling design for target frequency 34.56 Hz; (b) frequency plot for initial and optimized designs; (c) the sound pressure amplitude for initial design; and (d) the sound pressure amplitude for optimized design. Fig. 3. Optimized design for the rectangular room for the higher target frequency f ¼ 4 34:56 Hz. 80 70 60 50 40 30 20
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Optimization of acoustic, optical a
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Title of the thesis: Optimization o
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Resumé (in Danish) Forskningsomr˚
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Publications The following publicat
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vi Contents 6 Design of acousto-opt
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2 Chapter 1 Introduction waves. The
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4 Chapter 2 Time-harmonic propagati
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12 Chapter 3 Topology optimization
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where ∂Φ ∂w R 2,n − i ∂Φ
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finer mesh. For the coupled model i
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Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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