WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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The optimization problem in Eq. (4) is based on a Galerkin finite element discretization of Eqs. (1)- (3) which leads to the system equation consisting of a set of linear complex equations. A single design variable ̺e is assigned to each finite element within the chosen design domain (Nd elements). This design variable is then used with a SIMP-like model to control the material properties in the corresponding elements: Ae = A1 + ̺e(A2 − A1) (5) Be = B1 + ̺e(B2 − B1). The optimization problem is formulated as a max-min problem aiming to maximize both objectives Φ1 and Φ2. As objective we wish to maximize the wave transmission at the two output ports, so we specify the two objectives as the time-averaged Poynting vector integrated over these ports, which quantifies the corresponding energy flux. The point-wise time-averaged Poynting vector is given as: P(x) = {Px Py} T = 1 2 ωAℜ iu∇u . (6) For further details regarding the discretization of Eq. (6) see [6]. Analytical sensitivity analysis is facilitated by the adjoint method ([4], [6]), and the optimization problem stated in Eq. (4) is solved using the mathematical programming software MMA [5]. 5.2. Penalization - PAMP<strong>IN</strong>G We relate the element design variables to the element material properties using Eq. (5). Since we do not impose a volume constraint, which does not make sense in this problem, we cannot rely on using a penalization factor as in the usual SIMP model to ensure a well defined binary structure. Thus, we use a simple linear interpolation (Eq. (5)) and introduce instead a penalization of intermediate design variables based on artificial dissipation [6]. This is done by adding an extra conduction (or dissipation) term in each element within the design domain: σe ∼ α̺e(1 − ̺e), (7) where α is a scaling factor. Thus, elements with intermediate values of ̺e dissipate energy and consequently reduce the objective function. In this way the element design variables will be forced towards 0 and 1, if α is sufficiently large. We now exemplify results of applying the optimization algorithm to the model problem in Figure 4. 5.3. T-splitter - TM polarization The first example is for TM-polarized waves. We use the setup described in [16], using straight waveguides of width w = 200nm and a dielectric material with refractive index n = 3.2. A similar system was recently studied in [17]. As previously reported the reflection in the wavelength range around 1.55µm is about 25% for a plain simple T-junction. Figure 5 shows the optimized waveguide for a single frequency corresponding to a wavelength of 1.55µm. In this case we have chosen a very small design domain allowing only for small modifications of the material distribution and thus giving limited possibilities for improving the performance. In Figure 5 two different designs are shown along with the corresponding wave patterns. The structure at the top (structure 1) has the better performance and has been obtained by a straightforward implementation of the optimization algorithm. It is noticeable that the designed waveguide is discontinuous and the waves has to cross an air bridge. In the 2D loss-free model this leads to a good performance (see Figure 7), but for a real 3D structure large out-of-plane scattering losses can be expected. To avoid this, we have designed the bottom structure (structure 2) with simulated dissipation in the air. Consequently discontinuities are less favorable since the waves have to travel a distance in the lossy medium. The resulting structure performs worse with the loss-free 2D model, but we anticipate that it will perform better than structure 1 using 3D simulations and also in experiments. This is ongoing work. We now attempt to improve the performance by increasing the design domain. Figure 6 shows two examples of optimized structures with larger design domains. Both designs have been obtained with dissipation in the air. Figure 6 (top) (structure 3) has been obtained by enlarging the design domain in a straightforward manner with a very good performance as a results (see Figure 7), showing practically full transmission near the target frequency. In Figure 6 (bottom) (structure 4) the design domain from 5
Figure 5: Optimized designs and corresponding wave patterns for a small design domain. Top: structure 1 obtained with straightforward optimization and bottom: structure 2 obtained by simulating dissipation in the air. Figure 6: Optimized designs and corresponding wave patterns with enlarged design domains. Top: structure 3 and bottom: structure 4. 6
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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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Page 207 and 208: Appl. Phys. Lett., Vol. 84, No. 12,
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Page 218 and 219: Topology optimization and fabricati
Page 220 and 221: Normalized transmission 1.0 0.8 0.6
Page 222 and 223: Fig. 4. Scanning electron micrograp
Page 224 and 225: Broadband photonic crystal waveguid
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Page 228 and 229: Fig. 3. Steady-state magnetic field
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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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