WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Optimization of space-time material layout for 1D wave propagation 605 ρ = 1 E = 1 t < t 0 ρ = ρ0 t > t0 E = E0 Figure 1. Propagation of a sine-modulated gaussian pulse in a homogeneous medium with an instant change of material properties at t = t0. up into a forward and a backward travelling wave. It can be shown analytically that the relative change in wave energy at the moment of change of material properties is given as: ∆E E 1 = 2 (E0 + 1 ) − 1. (20) ρ0 In Fig. 2 the wave energy is plotted as a function of time. Both plots in the figure correspond to the case where the material properties are changed at t0 = 0.85 s. In the first plot the material properties are ρ0 = 1 and E0 = 1.5, which correspond to a relative energy jump of 0.25 and as appears from the plot, this jump is accurately predicted by the time-discontinuous Galerkin procedure but also with a normal central difference scheme. In the second plot ρ0 = 2 and E0 = 1.5 are chosen and thus zero energy jump should occur. From the plot we can see that the time-discontinuous scheme correctly captures the behavior as opposed to the central difference scheme. It should be mentioned that the time-discontinuous scheme is computationally more expensive than the straightforward central-difference scheme, since it involves inner loop iterations. The computational overhead depends on the specific value of the tolerance set for the inner-loop iterations (see Wiberg and Li, 1999, for more details). It is possible that more efficient schemes could be developed. 5. Example: pulse compression The optimization algorithm is now demonstrated on the particular design problem illustrated in Fig. 3. A single gaussian pulse is sent through a onedimensional structure and the transmitted wave is recorded. Wave propagation in the bar is simulated by applying a time-dependent load at the left boundary and adding absorbing boundary conditions in the form of simple dampers at both ends.
606 J.S. JENSEN Energy, E a) Energy, E b) 1000 800 600 400 200 1000 800 600 400 200 0 0 0.5 1 1.5 2 Time, t central diff. time-disc. 0 0 0.5 1 1.5 2 Time, t central diff. time-disc. Figure 2. Simulation values of the wave energy in the homogeneous structure with an instantaneous change of material properties at t0 = 0.85 s. a) ρ0 = 1 and E0 = 1.5 and b) ρ0 = 2 and E0 = 1.5. design domain Figure 3. Design problem. A single gaussian pulse is to be compressed when propagating through the design domain by a suitable stiffness and mass density distribution in space and time.
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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
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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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