WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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28 Chapter 3 Bandgap structures as optimal designs a) b) c) Figure 3.11 a) Sketch of optimization problem for creating a ring wave in a simply supported plate. The black parts indicate fixed steel regions and the arrows indicate in which direction the energy transport is maximized. The two small steel areas indicate where the out-of-phase harmonic forces are applied, b) Optimized material distribution, c) Energy transport in the optimized structure (only vectors larger than 10% of the maximum vector are shown). From paper [8]. be obtained. However, the present setup displays the basic principle. Fig. 3.11b shows the corresponding distribution of the two materials which indicate a relatively well defined structure consisting of steel inclusions (black) in a polycarbonate plate (light gray). Some dark gray areas indicate that design variables with intermediate values exist in the optimized design. The inclusions are observed to act partly to reflect the waves so that they do not propagate toward the edges of the plate and partly so that the waves are guided in the correct direction. 3.6 Acoustic design This chapter is concluded with an example taken from a quite different physical setting. However, optimizationofacoustic wave propagationcanbeperformed using the same optimization scheme that was presented for elastic waves. This can be facilitated merely by changing the material parameters in order to model acoustic instead of elastic media. Fig. 3.12 shows a (half) model of an acoustic reflection chamber. An optimized distribution of reflecting material and air is found in the design domain so that the transmission of steady-state acoustic waves is minimized. Fig. 3.12 presents an example of an optimized material distribution obtained by minimizing the transmission in a finite frequency range. Three inclusions of reflecting material are placed approximately with equal spacing in the design domain which causes reflection of a large part of the incoming wave. This is illustrated in the corresponding transmission plot in Fig. 3.13 showing that a transmission of less than 5% is obtained in the (non-dimensional) frequency range of interest from 0.15−0.20.
3.6 Acoustic design 29 design domain symmetry line Figure 3.12 Bottom: Half model of a channel with two symmetrically placed reflection chambers, Top: Optimized topology in the chamber. Blue: air, red: solid material. From paper [9]. Wave transmission 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 Frequency (non-dimensional) Figure 3.13 Transmission curve for the optimized design corresponding to an optimization frequency range from 0.15−0.20 (non-dimensional frequency). From paper [9]. 0.4 0.5
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Page 3: 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: 3.5 Plate structures 27 a) b) Figur
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 and 58: 5.3 Space-time topology optimizatio
Page 59 and 60: 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
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Page 81 and 82: 1058 fundamental eigenfrequency o
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Page 85 and 86: 1062 local resonator and splits up
Page 87 and 88: 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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1070 ARTICLE IN PRESS J.S. Jensen /
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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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