WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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48 Chapter 5 Advanced optimization procedures Figure 5.4 Optimized distribution of the 25 oscillator’s parameters for maximum reflection of a steady-steady sinusoidal wave, top) Natural frequency, middle) Mass ratio, bottom) Nonlinear stiffness. From paper [19].
5.3 Space-time topology optimization 49 input 2m 1m design domain 2m output Figure5.5 Design problemforthespace-time topology optimization example. Thetransmission of a Gauss-modulated sinusoidal pulse is to be minimized by finding an optimized distribution of two materials in a space-time design domain. From paper [20]. 1.5s t design domain 1m Figure 5.6 Left) Indication of the position-time design domain for the optimization problem, right) Optimized distribution of the two materials. From paper [20]. a one-dimensional bandgap structure (a so-called Bragg grating). The optimized static structure is used as a basis for the space-time optimization as illustrated in Fig. 5.6(left). The static bandgap structure is retained in the simulation before the pulse enters the design domain and after the main part of the signal is transmitted through the domain. For a time interval of 1.5s for which the main part of the pulse propagates through the design domain the material is allowed to change in space and time using a corresponding two-dimensional design grid. Fig. 5.6(right) shows the material distribution in a space-time design variable plot. The black (stiffer) material is still distributed in layers but now arranged with a slope in space-time. Fig. 5.7 illustrates the type of dynamic bandgap structure which is created. The plots show the instantaneous material distribution along with the wave motion for two separate time instances. The layered structure moves together with the wave with a speed matching the effective speed of the wave pulse (this speed can be computed from the slope in the space-time design plot in Fig. 5.6). The transmitted wave energy is reduced by approximately a factor three by this dynamic design x
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WAVES AND VIBRATIONS IN INHOMOGENEO
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Denne afhandling er af Danmarks Tek
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
Page 53 and 54: Chapter 5 Advanced optimization pro
Page 55 and 56: 5.1 Padé approximants 45 h 2h A f
Page 57: 5.3 Space-time topology optimizatio
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
Page 77 and 78: 1054 ARTICLE IN PRESS J.S. Jensen /
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
Page 103 and 104: wave-guiding properties show promis
Page 105 and 106: (a) (b) Acceleration response (dB)
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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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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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