WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Reflectance, R 1.0 0.8 0.6 0.4 0.2 0.0 a ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 301 (2007) 319–340 331 0 200 400 600 800 1000 1200 1400 1600 Frequency [Hz] Fig. 6. (a) Optimized distribution of scatter 1 (black) and host material (white) for maximum reflectance in a 25% frequency interval around the center frequency f 0 ¼ 788 Hz; (b) resulting reflectance curves for P and S waves. P wave (solid line), S wave (dashed line). Fig. 7. Wave patterns for the optimized structure computed for the center frequency f 0 ¼ 788 Hz: (a) abs(u) for a P wave and (b) abs(v) for an S wave.
332 It should be emphasized that other local optima can be found with different initial material distributions. The design shown in Fig. 6a is the best structure found after an extensive search but it is not guaranteed to be the global optimum. The problem with many local optima in wave-propagation problems is known and has to the author’s knowledge not yet been solved. Genetic or other evolutionary algorithms are not easily applied to this problem due to the large number of design variables (5000 or more). 6.2. Three-phase design A more effective wave-reflecting structure can be obtained if the design domain length L is increased. This allows for more inclusion layers in the structure and consequently higher reflectance. Also for larger L the optimized design has a combined layered/2D appearance (as in Fig. 6a) so that it is effective in the entire frequency range. For situations where the spatial extent of the design domain is limited higher reflectance can be obtained with increased contrast between the inclusion material and the host. This contrast can be quantified as the ratio between the impedances (cf. Eq. (26)–(27)). Generally, the layout of the optimized designs is different for different contrasts. However, an additional scattering material does not seem to increase the reflectance. Extensive numerical experiments showed in all cases that a design with those two materials having the highest contrast is always better than a three-phase design. This supports previous studies that indicate that maximum contrast is favorable for high wave reflection (cf. Section 4.1). Fig. 8 illustrates this effect. A third material phase is introduced (Table 1) and the corresponding optimized design (Fig. 8a) consists mainly of scatter 1 (gray) and scatter 2 (black) as they have the largest material contrast. However, small areas of host material (white) are seen in the design. If these areas are replaced by scattering material (Fig. 8b), the reflectance (Fig. 8c) is almost identical as for the optimized 3-phase design and actually slightly better averaged over the target range. Thus, it can be concluded that the 3-phase optimized design is a local optimum. Quadratic elements were used for the displacement fields in the last example to increase the convergence and stability of the optimization algorithm. However, although an improvement was noted the final design is still unsymmetrical which is an indication of the instabilities in the optimization procedure. The design is also dominated by small fragmented details. To remedy this problem, different filtering techniques could be applied (e.g. Ref. [15]). This has not been examined further in this work. 7. Numerical example 2 ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 301 (2007) 319–340 In this second example the setting from the first example is kept but now the goal is to maximize the fraction of the input power that is dissipated in the slab. An absorptive material phase with the properties of epoxy (Table 2) is introduced. The damping coefficient is chosen arbitrarily as Z r ¼ 0:05 s 1 and the damping of the other material phases is neglected. The importance of Z r will be investigated for the optimized designs. A third material phase (scatter) is also introduced and it will be investigated how this can improve the performance of the design. The material properties of this material are chosen so that the density is twice and the stiffness 20 times that of the host material. The dissipated power fraction D is computed for the situation with the entire design domain filled up with absorptive material. Fig. 9 shows the dissipation of a P and an S wave with Z r ¼ 0:05 s 1 and also for higher values of Z r. Due to the impedance contrast between the two material phases a part of the incident wave is reflected directly at the input boundary and consequently not all input power is dissipated regardless of the magnitude of Z r. Thus, merely filling up the domain with absorptive material is not optimal for maximum dissipation, although perhaps intuitively attractive. Instead, a good design must ensure a low direct reflection at the input boundary and additionally ensure that the wave is not transmitted but reflected inside the domain and then dissipated. The following sections demonstrate that such designs are generated by the optimization algorithm.
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WAVES AND VIBRATIONS IN INHOMOGENEO
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Denne afhandling er af Danmarks Tek
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List of Thesis Papers [1] J. S. Jen
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Contents Preface i List of Thesis P
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Chapter 1 Introduction This thesis
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eplacemen Phononic and photonic ban
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Optimal material distribution - top
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Chapter 2 The bandgap phenomenon In
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2.1 Band diagram and forced vibrati
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2.3 Experimental demonstration 11 F
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y 2.5 Nonlinearities 13 Response in
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Relations to recent work 15 Amplitu
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Chapter 3 Bandgap structures as opt
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3.1 Vibration-quenching structures
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3.2 Maximizing wave reflection 21 d
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3.4 Maximizing wave dissipation 23
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3.4 Maximizing wave dissipation 25
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3.5 Plate structures 27 a) b) Figur
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3.6 Acoustic design 29 design domai
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Chapter 4 Optimization of photonic
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4.1 Waveguide bends and junctions 3
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4.2 Photonic crystal building block
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4.2 Photonic crystal building block
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4.4 Ridge waveguides 39 w ε =1 ?
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Relations to recent work 41 wave in
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Chapter 5 Advanced optimization pro
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5.1 Padé approximants 45 h 2h A f
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5.3 Space-time topology optimizatio
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5.3 Space-time topology optimizatio
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Relations to recent work 51 Figure
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Chapter 6 Conclusions In the first
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References K. Asakawa, Y. Sugimoto,
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References 57 W. R. Frei, D. A. Tor
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References 59 F. Maestre, A. Münch
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References 61 O. Sigmund, J. S. Jen
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References 63 X. M. Zhou and G. K.
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Dansk resumé 65 i strukturen. En r
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1054 ARTICLE IN PRESS J.S. Jensen /
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1058 fundamental eigenfrequency o
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1060 ðo 2 y;j;k ARTICLE IN PRESS J
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1062 local resonator and splits up
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1064 3.1. Model equations A finite
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1066 FRF (dB) 150 100 50 0 -50 -100
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1068 3.3. Example 2: a 2-D waveguid
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wave-guiding properties show promis
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(a) (b) Acceleration response (dB)
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Page 109 and 110: ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
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Page 134 and 135: Z. Kristallogr. 220 (2005) 895-905
Page 136 and 137: Inverse design of phononic crystals
Page 144: Inverse design of phononic crystals
Page 147 and 148: 320 Optimization of the dissipation
Page 149 and 150: 322 The reflection of the wave is f
Page 151 and 152: 324 which averaged over a wave peri
Page 153 and 154: 326 where the modified stress compo
Page 155 and 156: 328 optimization algorithm is enhan
Page 157: 330 reflectance are also seen (Fig.
Page 161 and 162: 334 b Dissipation, D c Dissipation,
Page 163 and 164: 336 7.2. Three-phase design ARTICLE
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Page 171 and 172: 970 To solve the wave equation with
Page 173 and 174: 972 In the following section, we sh
Page 175 and 176: 974 Design variable, t Design varia
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Page 183 and 184: 982 respect to the higher bound C1
Page 185 and 186: 984 instead vary the value of b. Th
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Page 189 and 190: a thick solid was considered. A few
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Page 205 and 206: The optimization algorithm employs
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J. S. Jensen and O. Sigmund Vol. 22
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Topology optimization and fabricati
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Normalized transmission 1.0 0.8 0.6
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Fig. 4. Scanning electron micrograp
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Broadband photonic crystal waveguid
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2. Design and fabrication Silicon-o
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Fig. 3. Steady-state magnetic field
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Topology optimised broadband photon
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1202 IEEE PHOTONICS TECHNOLOGY LETT
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1204 IEEE PHOTONICS TECHNOLOGY LETT
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11. P. Debackere, S. Scheerlinck, P
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nanoimprinted patterns are transfer
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emoved and the spectra changed dras
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Y-splitter W1 waveguide 60-degree b
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5. Photonic Wire Waveguides Figure
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Figure 5: Optimized designs and cor
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Figure 8: Optimized designs and cor
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[13] Borel P I, Frandsen L H, Harp
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1606 J. S. JENSEN To compute a freq
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1608 J. S. JENSEN Cramer’s rule [
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1610 J. S. JENSEN The following pro
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1612 J. S. JENSEN Table I. Coeffici
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1614 J. S. JENSEN h 2h Α fcos Ωt
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1616 J. S. JENSEN The derivative of
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1618 J. S. JENSEN which is solved f
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1620 J. S. JENSEN Based on the expr
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1622 J. S. JENSEN Response, ln(u ti
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1624 J. S. JENSEN h 2h fcos Ωt Fi
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1626 J. S. JENSEN details and have
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1628 J. S. JENSEN Equation (A10) co
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1630 J. S. JENSEN 7. Coyette J-P, L
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The number of masses in the chain t
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5 Results The optimization algorith
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Figure 9. Response for optimized de
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Space-time topology optimization fo
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good for low to moderate stiffness
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V ¼ 0:3 at a given time instant fo
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The parameters used in this example
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Normalized amplitude 1 0.9 0.8 0.7
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at the Department of Mechanical Eng
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600 J.S. JENSEN techniques for obta
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602 J.S. JENSEN 1 and 2, where the
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604 J.S. JENSEN 4. Numerical analys
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606 J.S. JENSEN Energy, E a) Energy
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608 J.S. JENSEN relaxed somewhat in
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610 J.S. JENSEN when the contrast i
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612 J.S. JENSEN 6. Summary and conc
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614 J.S. JENSEN Sigmund, O. and Jen
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