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Control and Cybernetics vol. 39 (2010) No. 3 Optimization of space-time material layout for 1D wave propagation with varying mass and stiffness parameters ∗ by Jakob S. Jensen Department of Mechanical Engineering, Solid Mechanics Technical University of Denmark Nils Koppels Allé, Building 404, Denmark e-mail: jsj@mek.dtu.dk Abstract: Results are presented for optimal layout of materials in the spatial and temporal domains for a 1D structure subjected to transient wave propagation. A general optimization procedure is outlined including derivation of design sensitivities for the case when the mass density and stiffness vary in time. The outlined optimization procedure is exemplified on a 1D wave propagation problem in which a single gaussian pulse is compressed when propagating through the optimized structure. Special emphasis is put on the use of a time-discontinuous Galerkin integration scheme that facilitates analysis of a system with a time-varying mass matrix. Keywords: dynamic structures, topology optimization, wave propagation, transient analysis. 1. Introduction The method of topology optimization is a popular method for obtaining the optimal layout of one or several material constituents in structures and materials (Bendsøe and Kikuchi, 1988; Bendsøe and Sigmund, 2003). The methodology has within the last two decades evolved into a mature and diverse research field involving advanced numerical procedures and various application areas such as fluids (Borrvall and Petersson, 2003), waves (Sigmund and Jensen, 2003), electromagnetism (Cox and Dobson, 1999), as well as various coupled problems such as e.g. fluid-structure interaction (Yoon, Jensen and Sigmund, 2007). Additionally, industrial applications in the automotive and aerospace industries are established and widespread. The success has been facilitated by the large design freedom inherently associated with the concept, but also by efficient numerical techniques such as adjoint sensitivity analysis for rapid computation of gradients (Tortorelli and Michaleris, 1994), various penalization and regularization ∗ Submitted: June 2009; Accepted: June 2010.
600 J.S. JENSEN techniques for obtaining both meaningful and useful designs (Sigmund and Petersson, 1998) and the close integration with mathematical programming tools, such as the method of moving asymptotes (MMA) (Svanberg, 1987). Recently it was suggested to apply (and somewhat extend) the standard topology optimization framework to design a 1D structure in which the stiffness could change both in space and time (Jensen, 2009). As an example, an optimized “dynamic structure” that prohibits wave propagation was designed and manifested itself as a moving bandgap structure with layers of stiff inclusions moving with the propagating wave. The dynamic structure was demonstrated to reduce the transmission of a wave pulse by about a factor of three compared to an optimized static structure. The present paper extends the described work by allowing materials that have not only time-varying stiffness but also a timevarying mass density. This extension requires special attention to the choice of time-integration scheme since many standard schemes fail. However, it allows for extended manipulation of the wave propagation as illustrated in the example in the present paper, in which a single gaussian wave pulse is compressed when propagating through the optimized structure. Preliminary results for this design problem in the case of time-varying stiffness were presented in Jensen (2008). The basic setting for obtaining optimal space and time distributions of materials for problems governed by the wave equation was first presented in Maestre, Münch and Pedregal (2007), Maestre and Pedregal (2009). These papers analyze 1D and 2D problems with a strong focus on the mathematical aspects of the optimization problem. Both the present paper and the aforementioned works root in the fundamental concept of dynamic materials. This concept was introduced by Lurie and Blekhman (Lurie, 1997; Blekhman and Lurie, 2000; Blekhman, 2008) who unfolded the rich and complex behavior of materials with properties that vary in space and time. The dynamics of structures with space and time varying properties was also studied in the work by Krylov and Sorokin (Krylov and Sorokin, 1997) and later in Sorokin, Ershova and Grishina (2000), Sorokin and Grishina (2004). The basis for the presented optimization problem is time-integration of the transient model equation coupled with adjoint sensitivity analysis. Thus, the problem closely resembles previous studies that have been carried out for topology optimization of static structures using a transient formulation, e.g. Min et al., (1999), Turteltaub (2005), Dahl, Jensen and Sigmund (2008). The outline of the paper is as follows. In Section 2 the governing equation is presented and the basic setup defined. In Section 3 the design parametrization is defined and design sensitivities are derived. Section 4 is devoted to numerical analysis of the transient direct and adjoint equations and numerical simulation results are presented. In Section 5 an optimization problem is defined and examples of optimized designs are presented. Section 6 summarizes and gives conclusions.
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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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