WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Dissipation, D 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) 319–340 335 0 200 400 600 800 1000 1200 1400 1600 Frequency [Hz] Fig. 10. (a) Optimized distribution of absorptive material (black) and loss-free host material (white) for maximum dissipation in a 10% frequency interval around f 0 ¼ 788 Hz; (b) corresponding dissipation of P and S waves. P wave Z r ¼ 0:05 s 1 (solid), S wave Z r ¼ 0:05 s 1 (dash), P wave Z r ¼ 0:75 s 1 (dot), S wave Z r ¼ 0:75 s 1 (dash–dot). Fig. 11. Point-wise dissipated power for the optimized design for f 0 ¼ 788 Hz: (a) a P wave and (b) an S wave. Z r ¼ 0:05 s 1 .
336 7.2. Three-phase design ARTICLE <strong>IN</strong> PRESS The extra scattering material is now included in an attempt to further improve the performance. The hypothesis is that with a highly reflecting material available a larger part of the wave can be reflected leading to higher wave amplitudes and higher dissipation. Fig. 12a shows the optimized design with gray as absorptive material and black as scattering material. Qualitatively, the design is similar to the two-phase design. The major difference is that the inclusion near the output boundary is made of the scattering material as the higher impedance contrast leads to increased reflection. Additionally, two small reflecting inclusions are seen in the inner part of the structure. Fig. 12b shows the dissipated power fraction for the three-phase design. The improvement of the performance is quite limited (from about 32% to about 35% in average) and although the details of the twophase and three-phase designs are different the performance in the target range is similar. Thus, it seems possible to create many good optimized designs (local optima) with similar overall features but with different structural details such as precise size and placement of the absorptive inclusions. The effect of a refinement of the computational model and the use of higher-order finite elements is illustrated in Fig. 13. Fig. 13a shows the optimized design with quadratic elements for the displacement fields instead of linear elements. The higher-order elements stabilize the optimization algorithm (as for the reflection example) and the small asymmetries in the design in Fig. 12a vanish. Apart from this the two designs are very similar. Fig. 13b shows an optimized design with 100 100 linear elements in the design domain (80 80 used in the other examples) and with 15 optimization frequencies in the target range (instead of 10 used in the other examples). The overall features of the design are unchanged but the details are different. The numerical instabilities are more severe and lead to a very unsymmetrical design. However, as illustrated in Fig. 13c, these differences in the design are not reflected in the performance which is very similar in the target range. b Dissipation, D 0.5 0.4 0.3 0.2 0.1 0.0 J.S. Jensen / Journal of Sound and Vibration 301 (2007) 319–340 a 0 200 400 600 800 1000 1200 1400 1600 Frequency [Hz] Fig. 12. (a) Optimized distribution of absorptive material (gray), host material (white), and scattering material (black) for maximum dissipation in a 10% frequency interval around f 0 ¼ 788 Hz; (b) corresponding dissipation of P and S waves compared to the two-phase design in Fig. 10. P wave-3 phase (solid), S wave-3 phase (dash), P wave-2 phase (dot), S wave-2 phase (dash–dot). Z r ¼ 0:05 s 1 .
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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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B.S. Lazarov, J.S. Jensen / Interna
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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 and 158: 330 reflectance are also seen (Fig.
Page 159 and 160: 332 It should be emphasized that ot
Page 161: 334 b Dissipation, D c Dissipation,
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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
Page 177 and 178: 976 Eigenvalue ratio, ω n+1 2 /ωn
Page 179 and 180: 978 to a design parameter te are gi
Page 181 and 182: 980 ARTICLE IN PRESS J.S. Jensen, N
Page 183 and 184: 982 respect to the higher bound C1
Page 185 and 186: 984 instead vary the value of b. Th
Page 187 and 188: 986 References ARTICLE IN PRESS J.S
Page 189 and 190: a thick solid was considered. A few
Page 191 and 192: The above expressions provide insta
Page 193 and 194: In all the examples shown a density
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Page 205 and 206: The optimization algorithm employs
Page 207 and 208: Appl. Phys. Lett., Vol. 84, No. 12,
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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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