Maria Bayard Dühring - Solid Mechanics

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564 objective function, Φ [dB] 130 120 110 100 90 80 70 8 9 10 11 12 frequency, f [Hz] ARTICLE IN PRESS M.B. <strong>Dühring</strong> et al. / Journal of Sound and Vibration 317 (2008) 557–575 optimized design target frequency resonance frequency objective function, Φ [dB] 70 8 9 10 11 12 distributed at the high sound pressure amplitudes. The intuitive explanation of this phenomenon is that if a natural frequency has to be decreased it must be made possible for the room to resonate at a lower frequency. Thus the material from the high pressure amplitudes is moved to the nodal planes. If instead the natural frequency has to be increased the system has to be made stiffer at the critical places. In this case the material is removed from the nodal planes and distributed at the high pressure amplitudes. It is difficult to say in general if one of these designs is best for all the frequencies close to the natural frequency and it looks like it depends on how far away the natural frequency can be moved in one of the directions from the considered frequencies. Similar effects have been observed for design of plates subjected to forced vibration [31]. The optimization problem is now changed such that the optimization can be done for an entire frequency interval. The objective is to minimize the sum of responses for a number of target frequencies oi in the interval considered as in Eq. [27]. The chosen interval is divided into M equally sized subintervals and the target frequency in each subinterval, which results in the highest value of F, is determined. The room is then optimized for the new objective function C which is the sum of F evaluated at the determined target frequencies and divided by the number of intervals M to get the average value P maxoi2Ii ðFðoiÞÞ o1;:::;oM minx C ¼ ; I 1 ¼½o1; o2½; :::; I M ¼ŠoM; oMþ1Š. (18) M Here oMþ1 o1 is the entire frequency interval and I i are the equally sized subintervals. By this optimization procedure F is minimized at all the target frequencies and these are updated at regular intervals during the 130 120 110 100 90 80 frequency, f [Hz] optimized design target frequency resonance frequency Fig. 4. Optimized design and frequency response after the optimization for two frequencies close to the natural frequency f ¼ 9:55 Hz for the room with the initial design: (a) optimized design for f ¼ 9:39 Hz; (b) optimized design for f ¼ 9:71 Hz; (c) frequency response for the optimized design for f ¼ 9:39 Hz; and (d) frequency response for the optimized design for f ¼ 9:71 Hz. Table 1 The value of the objective function for the two frequencies 9.39 and 9.71 Hz for the two designs from Fig. 4 Frequency f (Hz) F for optimized design for f ¼ 9:39 Hz (dB) F for optimized design for f ¼ 9:71 Hz (dB) 9.39 99.7 96.4 9.71 120.1 90.8
optimization by approximating the objective function F as function of the frequency using Pade´ expansions, see Ref. [32]. The room from Fig. 1 is then optimized for the frequency interval [18;23] Hz using five target frequencies where the target frequencies are updated every 25th iteration step. The quantities used in the optimization are b ¼ 0:85, hmax ¼ 0:3 m and rmin ¼ 1:5hmax. The optimized design obtained after 415 iterations is illustrated in Fig. 5 together with the response curve for the initial design and the optimized design. The objective function is reduced from 111.2 to 75.7 dB. It is seen from the two response curves that the objective function is minimized in the entire interval for the optimized design. Two of the high peaks have been moved out of the interval and the last one has been significantly reduced. The solid material in the optimized design is distributed such that a kind of Helmholtz resonator is formed. 3.2. Optimization of a rectangular room in 3D ARTICLE IN PRESS M.B. <strong>Dühring</strong> et al. / Journal of Sound and Vibration 317 (2008) 557–575 565 Fig. 5. (a) The optimized design for the frequency interval [18;23] Hz and (b) the frequency response for the initial design and the optimized design. The optimization problem is now extended to 3D problems and a rectangular room with the geometry shown in Fig. 6 is considered. We optimize two examples for intervals around the first natural frequency f ¼ 42:92 Hz for the room with a vertical nodal plane at x ¼ 2 m and high sound pressure amplitude at the end walls at x ¼ 0 and 4 m. The quantities used are b ¼ 0:5, hmax ¼ 0:4 m and rmin ¼ 0:5hmax and the target frequencies are updated for each 15 iterations. For the first example the optimization is done for the interval [41.5;44.5] Hz and one target frequency. After 252 iterations the objective function C is reduced from 115.2 to 81.6 dB. The optimized design and the response for the initial design and the optimized design are seen in Fig. 7. It is observed that the solid material is distributed at the walls with high sound pressure amplitude and that the natural frequency has been moved to a higher value outside the interval. For the next example the frequency interval is extended to [40.5;45.5] Hz and four target frequencies are used. C is minimized from 96.6 to 46.9 dB in 218 iterations and the results are illustrated in Fig. 8. In this case most of the solid material is distributed at the nodal plane around x ¼ 2 m rather than at the two shorter walls opposite to the previous example. From the response curve in Fig. 8 it is seen that the first natural frequency is not contained in the interval after the optimization and instead a natural frequency has appeared at a lower value. These two examples show that if the frequency interval is slightly changed, the optimization can converge to two very different, in fact opposite designs. The optimized designs here depend on what side of the interval the first natural frequency is moved to after the optimization. The best design in this case is the design where the solid material is placed at low pressure amplitude for the initial design. The two examples here can thus be interpreted as an extension to 3D of the examples in Fig. 4.
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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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10 Chapter 2 Time-harmonic propagat
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12 Chapter 3 Topology optimization
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22 Chapter 4 Design of sound barrie
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24 Chapter 4 Design of sound barrie
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where ˜pijkl are the rotated strai
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where ∂Φ ∂w R 2,n − i ∂Φ
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[12] J.S. Jensen and O. Sigmund, To
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Energy storage and dispersion of su
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Publication [P7] Improving surface
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FIG. 1: The geometry of the acousto
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finer mesh. For the coupled model i
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FIG. 4: Results for the finite mode
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FIG. 6: Results for the acousto-opt
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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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