WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 1075 Fig. 15. The response in points A and B with a path of defects in the periodic structure for a the structure, (a) for vertical path, and (b) for a corner path. The insets show schematics of the paths. Type 1 unit cells with Mx ¼ My ¼ 21: whereas in point A the response remains high indicating that the signal is now confined to the path. Fig. 15b shows the response for the bent path leading to point B. The response is now seen to be normalized in B inside the band gap, whereas in point A the response drops. Contour plots of the response for the two lattice structures with paths are shown in Figs. 16a and b, respectively. The frequency of excitation for both figures is O ¼ 52:1 kHz; i.e., inside the band gap. The figure shows how the paths of defects in the periodic structures effectively isolates the vibrations to the path regions and thus ‘‘leads’’ the vibrations to the structural point A or B, whereas away from the paths the response drops rapidly with distance.
1076 ARTICLE <strong>IN</strong> PRESS J.S. Jensen / Journal of Sound and Vibration 266 (2003) 1053–1078 Fig. 16. Contour plot of the response for O ¼ 52:1 kHz; (a) for a vertical path of defects, and (b) for a corner path of defects. Type 1 unit cells with Mx ¼ My ¼ 21:
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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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Page 49 and 50: 4.4 Ridge waveguides 39 w ε =1 ?
Page 51 and 52: Relations to recent work 41 wave in
Page 53 and 54: Chapter 5 Advanced optimization pro
Page 55 and 56: 5.1 Padé approximants 45 h 2h A f
Page 57 and 58: 5.3 Space-time topology optimizatio
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Page 61 and 62: Relations to recent work 51 Figure
Page 63 and 64: Chapter 6 Conclusions In the first
Page 65 and 66: References K. Asakawa, Y. Sugimoto,
Page 67 and 68: References 57 W. R. Frei, D. A. Tor
Page 69 and 70: References 59 F. Maestre, A. Münch
Page 71 and 72: References 61 O. Sigmund, J. S. Jen
Page 73 and 74: References 63 X. M. Zhou and G. K.
Page 75 and 76: Dansk resumé 65 i strukturen. En r
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Page 81 and 82: 1058 fundamental eigenfrequency o
Page 83 and 84: 1060 ðo 2 y;j;k ARTICLE IN PRESS J
Page 85 and 86: 1062 local resonator and splits up
Page 87 and 88: 1064 3.1. Model equations A finite
Page 89 and 90: 1066 FRF (dB) 150 100 50 0 -50 -100
Page 91 and 92: 1068 3.3. Example 2: a 2-D waveguid
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Page 103 and 104: wave-guiding properties show promis
Page 105 and 106: (a) (b) Acceleration response (dB)
Page 107 and 108: B.S. Lazarov, J.S. Jensen / Interna
Page 109 and 110: ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
Page 111 and 112: [B 2000 ] [B 2000 ] [B 2000 ] 0.25
Page 113 and 114: [B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
Page 115 and 116: 1002 (a) (c) (e) 0. Sigmund and J.
Page 117 and 118: 1004 O. Sigmund and J. S. Jensen wh
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Page 125 and 126: 1012 (a) (b) O. Sigmund and J. S. J
Page 127 and 128: 1014 . ct . _ a) Q^ s^c "3
Page 129 and 130: 1016 O. Sigmund and J. S. Jensen (a
Page 131 and 132: 1018 O. Sigmund and J. S. Jensen 4.
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
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322 The reflection of the wave is f
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324 which averaged over a wave peri
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326 where the modified stress compo
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328 optimization algorithm is enhan
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330 reflectance are also seen (Fig.
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332 It should be emphasized that ot
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334 b Dissipation, D c Dissipation,
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336 7.2. Three-phase design ARTICLE
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338 ARTICLE IN PRESS J.S. Jensen /
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340 ARTICLE IN PRESS J.S. Jensen /
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968 ARTICLE IN PRESS J.S. Jensen, N
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970 To solve the wave equation with
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972 In the following section, we sh
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974 Design variable, t Design varia
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976 Eigenvalue ratio, ω n+1 2 /ωn
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978 to a design parameter te are gi
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980 ARTICLE IN PRESS J.S. Jensen, N
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982 respect to the higher bound C1
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984 instead vary the value of b. Th
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986 References ARTICLE IN PRESS J.S
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a thick solid was considered. A few
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The above expressions provide insta
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In all the examples shown a density
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domain. To the very left the passiv
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Jensen JS, Sigmund O (2005) Topolog
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paper extends on this work. For an
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R 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
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objective 1 0.9 0.8 0.7 0.6 0.5 0.4
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The optimization algorithm employs
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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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