WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Title of the thesis: Waves and Vibrations in Inhomogeneous Structures – bandgaps and optimal designs Author: Jakob Søndergaard Jensen E-mail: jsj@mek.dtu.dk Address: Department of Mechanical Engineering Technical University of Denmark Nils Koppels Allé, Building 404 DK-2800 Kgs. Lyngby, Denmark Copyright © 2011 Jakob Søndergaard Jensen ISBN 978-87-90416-48-5
Denne afhandling er af Danmarks Tekniske Universitet antaget til forsvar for den tekniske doktorgrad. Antagelsen er sket efter bedømmelse af den foreliggende afhandling. Kgs. Lyngby, den 2. marts 2011 Lars Pallesen Rektor /Martin P. Bendsøe Dekan This thesis has been accepted by the Technical University of Denmark for public defence in fulfilment of the requirements for the degree of Doctor Technices. The acceptance is based on an evaluation of the present dissertation. Kgs. Lyngby, 2 March 2011 Lars Pallesen Rector /Martin P. Bendsøe Dean
Page 1: WAVES AND VIBRATIONS IN INHOMOGENEO
Page 7 and 8: List of Thesis Papers [1] J. S. Jen
Page 9 and 10: Contents Preface i List of Thesis P
Page 11 and 12: Chapter 1 Introduction This thesis
Page 13 and 14: eplacemen Phononic and photonic ban
Page 15 and 16: Optimal material distribution - top
Page 17 and 18: Chapter 2 The bandgap phenomenon In
Page 19 and 20: 2.1 Band diagram and forced vibrati
Page 21 and 22: 2.3 Experimental demonstration 11 F
Page 23 and 24: y 2.5 Nonlinearities 13 Response in
Page 25 and 26: Relations to recent work 15 Amplitu
Page 27 and 28: Chapter 3 Bandgap structures as opt
Page 29 and 30: 3.1 Vibration-quenching structures
Page 31 and 32: 3.2 Maximizing wave reflection 21 d
Page 33 and 34: 3.4 Maximizing wave dissipation 23
Page 35 and 36: 3.4 Maximizing wave dissipation 25
Page 37 and 38: 3.5 Plate structures 27 a) b) Figur
Page 39 and 40: 3.6 Acoustic design 29 design domai
Page 41 and 42: Chapter 4 Optimization of photonic
Page 43 and 44: 4.1 Waveguide bends and junctions 3
Page 45 and 46: 4.2 Photonic crystal building block
Page 47 and 48: 4.2 Photonic crystal building block
Page 49 and 50: 4.4 Ridge waveguides 39 w ε =1 ?
Page 51 and 52: 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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[B 2000 ] [B 2000 ] [B 2000 ] 0.25
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[B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
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1002 (a) (c) (e) 0. Sigmund and J.
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1004 O. Sigmund and J. S. Jensen wh
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1006 (a) - - - - - - _ - - - - I I
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1008 O. Sigmund and J. S. Jensen we
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1010 O. Sigmund and J. S. Jensen Th
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1012 (a) (b) O. Sigmund and J. S. J
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1014 . ct . _ a) Q^ s^c "3
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1016 O. Sigmund and J. S. Jensen (a
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1018 O. Sigmund and J. S. Jensen 4.
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Z. Kristallogr. 220 (2005) 895-905
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Inverse design of phononic crystals
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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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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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