WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Phononic band-gap optimization 2, . . . . . . . . 80 - I.~ -20I~,,~ I{11 1 I I I I I - V I I I I I I I I I -10 ( ' ' ' ' ( 40 60) 80 100 frequ6ency. /' (k t l/) 1009 Figure 5. Response of a 12 cm x 12 cm square structure subjected to periodic loading on the left boundary, absorbing boundary conditions on the left and right edges and out-of-plane modelling. High-contrast case. ing two material phases in the design domain such that an objective function is extremized subject to a number of constraints. The design variables are material 'densities' Xe C [0, 1] that interpolate the material properties for each element used to discretize the structure. If the design variable takes a zero value we should have pure phase 1 material, and if the design variable takes a value of one we should have pure phase 2 material in the element. We choose a linear interpolation between the phases, given as p(Xe) = (1 - Xe)p + XeP2, (3.1) /1(Xe) = (1 - Xe)Pl + XeL2, (3.2) A(xe) = (1 - xe)Al + XeA2, (3.3) where subscripts '1' and '2' denote the properties in materials 1 and 2, respectively. The choice of linear interpolation stems from the observation that there appears to be no need for penalizing intermediate 'densities', i.e. values of Xe other than zero or one in the final design. The reason for this is believed to originate in the nature of the band-gap phenomenon, where large contrasts between the involved material phases is favoured. (a) Material optimization An obvious goal for the optimization of band-gap materials is to maximize the relative band-gap size. In this way the range of prohibited frequencies will be wider and more signals may be sent through a waveguide based on defects in the band-gap material. The goal of the optimization is to maximize the relative band-gap size between bands j and j + 1, i.e. maximize the lowest value of the overlying bands and minimize the maximum value of the underlying bands. This can written as a (double) max-min objective w2 - ) M ink1) 2 (J+k,x) - maxk w2(k,x) 3 max: c(x)-= () - . x (3.4) W2 m(x) 0~~~~~)+ mink Wi (2 (k,x) j +maxk W j(k,x) Phil. Trans. R. Soc. Lond. A (2003)
1010 O. Sigmund and J. S. Jensen This is a 'dirty' objective function in the sense that it is a max-min problem with varying critical points (the k-vector(s) for the critical frequencies may change dur- ing the optimization) and it may have several multiple eigenvalues. Interestingly, however, there is no need for a volume fraction constraint in the problem, since pure phase 1 will give no band gap and pure phase 2 will give no band gap (cf. figure 3a, b). Somewhere in between there must be a volume fraction that results in the biggest band gap. The optimization problem may be written as Aw2(x) max 2() s.t.: (K(k)- M) u = 0, k E [F-X-M-r], x woW 0
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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
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)
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
Page 119 and 120: 1006 (a) - - - - - - _ - - - - I I
Page 121: 1008 O. Sigmund and J. S. Jensen we
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
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 and 162: 334 b Dissipation, D c Dissipation,
Page 163 and 164: 336 7.2. Three-phase design ARTICLE
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Page 171 and 172: 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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