WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Abstract Journal of Sound and Vibration 301 (2007) 319–340 Topology optimization problems for reflection and dissipation of elastic waves Jakob S. Jensen JOURNAL OF SOUND AND VIBRATION Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Allé, Building 404, DK-2800 Kgs. Lyngby, Denmark Received 29 August 2006; received in revised form 11 October 2006; accepted 11 October 2006 Available online 28 November 2006 This paper is devoted to topology optimization problems for elastic wave propagation. The objective of the study is to maximize the reflection or the dissipation in a finite slab of material for pressure and shear waves in a range of frequencies. The optimized designs consist of two or three material phases: a host material and scattering and/or absorbing inclusions. The capabilities of the optimization algorithm are demonstrated with two numerical examples in which the reflection and dissipation of ground-borne wave pulses are maximized. r 2006 Elsevier Ltd. All rights reserved. 1. Introduction ARTICLE IN PRESS www.elsevier.com/locate/jsvi This work deals with two fundamental optimization problems encountered in the study of elastic wave propagation through a finite slab of material. Wave propagation can be suppressed if the wave reflection or the wave dissipation is maximized and in this work these two problems are addressed with the method of topology optimization. The method is used to find an optimized distribution of inclusions of scattering and/or absorbing material that maximizes the reflection and/or the dissipation, respectively. Recently, much work has been devoted to highly reflecting materials and structures created with a periodic distribution of scattering inclusions—the so-called bandgap materials. If inclusions are distributed periodically a large reflection of propagating waves can occur due to destructive interference. For a comprehensive overview see e.g. the recent review paper by Sigalas et al. [1]. Bandgap structures are promising candidates as optimal wave reflecting structures. In Sigmund and Jensen [2] it was demonstrated that optimized designs are typically periodic-like structures with modifications near the boundaries that compensate for edge effects. A material optimization problem was also considered in which the topology of repetitive identical inclusions was optimized. Rupp et al. [3] studied similar problems with a topology optimization algorithm. Halkjær et al. [4] considered bending waves in beams and plates and structures with limited spatial dimensions that did not allow for a repetitive periodic structure. Hussein et al. [5] analyzed one-dimensional wave propagation through a layered medium and used a genetic algorithm to generate optimized structures. E-mail address: jsj@mek.dtu.dk. 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.10.004
320 Optimization of the dissipation of elastic waves has been far less studied than the reflection/transmission problem. Planar structures that maximize dissipation of scalar waves was previously demonstrated by the author [6]. Razansky et al. [7] studied dissipation of acoustic waves and gave bounds for maximal dissipation in thin and thick dissipative layers. Work has also been done on optimal placement of dampers to reduce structural dynamic compliance [8,9]. Related work with the method topology optimization were seen in Jog [10] who studied the forced vibration problem and minimized the dissipated energy and in Wang and Chen [11] who maximized heat dissipation in cellular structures. Closely related to this present work are optimization studies related to propagation of electromagnetic waves. Cox and Dobson [12] studied optimal design of infinite periodic structures with a material distribution method and maximized the optic (photonic) bandgaps and Jensen and Sigmund [13] studied a finite photonic bandgap structure and topology optimized the material distribution in a 90 bend so that the transmitted power through the bend was maximized. This paper extends the work of Jensen [6] and considers dissipation of elastic waves (pressure and shear) for multiple frequencies instead of scalar waves at a single frequency. The work of Halkjær et al. [4] is extended to deal with the multiple frequency case. Furthermore, both optimization problems are extended to allow for the distribution of three material phases instead of two. The paper is organized as follows: first, the general model for elastic wave propagation is presented (Section 2), and the formulation of the two optimization problems are discussed in detail (Section 3). Section 4 describes the parametrization of the design domain with two continuous design fields, the material interpolation model, and the artificial damping penalization. In Section 5 the numerical implementation of the optimization problem is explained, including boundary conditions, FEM discretization, and a mathematical formulation of the problem. In Sections 6 and 7 two numerical examples are presented that demonstrate optimization of the material distribution for maximized reflection and dissipation of ground-borne wave pulses. Finally, main conclusions are given in Section 8. 2. Elastodynamic model ARTICLE IN PRESS J.S. Jensen / Journal of Sound and Vibration 301 (2007) 319–340 The computational model is based on the full 3D elastodynamic equations for an inhomogeneous medium. The 3D model is reduced to a 2D model under the assumption that the waves propagate in the plane and material properties vary in the same plane. The 3D elastodynamic equations: r €U ¼ = R, (1) govern the displacements U ¼fUðx; tÞ Vðx; tÞ Wðx; tÞg T of an elastic medium with position-dependent density r ¼ rðxÞ. The position vector is denoted x ¼fxyzg T ,andR ¼ Rðx; tÞ is the stress tensor. Eq. (1) is transformed to complex form with the complex variable transformation U ! ~U and R ! ~R, so that U ¼ Reð ~UÞ and R ¼ Reð ~RÞ. Time-harmonic motion with frequency o gives a steady-state solution to the complex version of Eq. (1): ~Uðx; tÞ ¼uðxÞe iot , (2) ~Rðx; tÞ ¼rðxÞe iot , (3) in which the displacement amplitude vector uðxÞ and the stress amplitude tensor rðxÞ are generally complex. The solution forms (2) and (3) are inserted into the complex version of Eq. (1) to give the standard timeharmonic elastic wave equation: = r þ ro 2 u ¼ 0. (4) With the appropriate boundary conditions Eq. (4) gives the displacement field used as the basis for the optimization problems introduced in the following section. With a specific solution to the complex equation (4) the instantaneous displacement and stresses are: Uðx; tÞ ¼ReðuðxÞe iot Þ¼u r cos ot u i sin ot, (5)
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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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Page 103 and 104: wave-guiding properties show promis
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Page 109 and 110: ω/κ ω/κ 1.5 1.4 1.3 1.2 1.1 1 0
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Page 113 and 114: [B 400 ] 0.25 0.2 0.15 0.1 0.05 B.S
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Page 136 and 137: Inverse design of phononic crystals
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Page 149 and 150: 322 The reflection of the wave is f
Page 151 and 152: 324 which averaged over a wave peri
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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
Page 173 and 174: 972 In the following section, we sh
Page 175 and 176: 974 Design variable, t Design varia
Page 177 and 178: 976 Eigenvalue ratio, ω n+1 2 /ωn
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Page 181 and 182: 980 ARTICLE IN PRESS J.S. Jensen, N
Page 183 and 184: 982 respect to the higher bound C1
Page 185 and 186: 984 instead vary the value of b. Th
Page 187 and 188: 986 References ARTICLE IN PRESS J.S
Page 189 and 190: a thick solid was considered. A few
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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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