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8.62The State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University20.29Swinburne University of Technology+ 137.76Swinburne University of TechnologyShow more authorsThe realization of unidirectional acoustic transmission (UAT) has recently aroused great attention owing to the versatile possibility in acoustics-based applications. This paper extends the bi-directional evolutionary structural optimization (BESO) method to the design of phononic crystal (PC) for achieving UAT. The optimization objective is to enlarge the minimum imaginary part of wave vectors along G-X while keep that along G-M less than the constraint value. We systematically studied the design of symmetric and asymmetric PCs at various frequencies. Numerical examples demonstrate that the proposed optimization algorithm is effective for creating the partial band gap at the specified frequency. The UAT with high rectifying efficiency is then successfully realized by placing the optimized PC in a bend wave guide. The results also show that the asym-metric PCs are more favorable for the design of broadband UAT devices compared with symmetric ones.Discover the world's research15+ million members118+ million publications700k+ research projects(a) Schematic illustration of the 2D model of the UAT device. (b) Calculated dispersion bands of 2D PC model. (c) Illustration of the propagation direction specified by the angle q and the irreducible Brillioun zone (G-X-M-G).
Topological design of phononic crystals for unidirectionalacoustic transmissionYafeng Chena, Fei Mengb, Guangyong Suna, Guangyao Lia,Xiaodong Huanga,b,*aKey Laboratory of Advanced Technology for Vehicle Body Design &Manufacture, Hunan University, Changsha, 410082, ChinabFaculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, VIC, 3122, Australiaarticle infoArticle history:Received 5 April 2017Received in revised form 3 August 2017Accepted 12 August 2017Keywords:Unidirectional acoustic transmissionPhononic crystalsPartial band gapTopological optimizationBi-directional evolutionary structural opti-mization (BESO)abstractThe realization of unidirectional acoustic transmission (UAT) has recently aroused greatattention owing to the versatile possibility in acoustics-based applications. This paperextends the bi-directional evolutionary structural optimization (BESO) method to thedesign of phononic crystal (PC) for achieving UAT. The optimization objective is toenlarge the minimum imaginary part of wave vectors alongG-X while keep that alongG-Mless than the constraint value. We systematically studied the design of symmetric andasymmetric PCs at various frequencies. Numerical examples demonstrate that the pro-posed optimization algorithm is effective for creating the partial band gap at the spec-ified frequency. The UAT with high rectifying efficiency is then successfully realized byplacing the optimized PC in a bend wave guide. The results also show that the asym-metric PCs are more favorable for the design of broadband UAT devices compared withsymmetric ones.(C)2017 Elsevier Ltd. All rights reserved.1. IntroductionElectrical diodes, due to their capability of realizing the unidirectional propagation of electric current, have led tosignificant revolutions in science and technology [1]. Inspired by this one-way effect of electric currents, acousticdiodes (ADs) which can realize the unidirectional acoustic transmission (UAT) have widespread potential applicationsin many fields, such as noise insulation [2] and medical imaging [3]. Liang et al. [4e6] proposed the theoretical modelof an AD and demonstrated the phenomenon of UAT by combining a superlattice with a nonlinear acoustic medium.Boechler et al. [7] identified acoustic switching rectification based on bifurcations and chaos. However, the nonlinearAD changes the frequencies of acoustic waves, which generally suffers from the low efficiency of nonlinear conversion[8]. Therefore, many researchers have turned to realize UAT using linear materials [8e21], e.g. phononic crystals (PCs)[8 ,11,12 ,17e21].*Corresponding author. Key Laboratory of Advanced Technology for Vehicle Body Design &Manufacture, Hunan University, Changsha, 410082, China.E-mail address: xhuang@swin.edu.au (X. Huang).Contents lists available at ScienceDirectJournal of Sound and Vibrationjournal homepage: www.elsevier.com/locate/jsvihttp://dx.doi.org/10.1016/j.jsv.2017.08.015X/(C)2017 Elsevier Ltd. All rights reserved.Journal of Sound and Vibration 410 (2017) 103e123
PCs are composites usually made of one material periodically embedded in others with different physical properties(elasticity modulus and mass density) [22]. Li et al. [20] realized the UAT in a PC based acoustic diode that had brokenspatial inversion symmetry. Li et al. [17] proposed a scheme of realizing broadband UAT using a gradient-index structure,and demonstrated an implementation utilizing PC-based metamaterial. Cicek et al. [19] investigated the UAT across ajunction of two PCs with different orientations and lattice constants. Song et al. [8] proposed a novel type of waveform-preserved UAT device composed of an impedance-matched acoustic metasurface and a PC structure. For practical acous-tic devices, acoustic signals are generally conducted within waveguides for a variety of important applications [23,24].Toexplore the possibility of realizing the UAT within a waveguide, Yuan et al. [18] designed a two-dimensional (2D) modelconstructed by placing a PC with partial band gap (PBG) in a bent waveguide. The UAT with high efficiency was achieved inthe frequency range of the PBG. Huang et al. [11] realized the UAT with multi-bands by placing a PC inside a bent waveguideimmersed in water.Achieving an efficient PBG in the dispersion relations is the key to realize the UAT in waveguides. The previous works areempirical designs based on the observation of their band diagrams [11,18]. Therefore, it is hard to obtain a PBG at the desirablefrequency. Moreover, the spatial decay in band gaps determines how much the system becomes an insulator. It should belarge enough to guarantee the rectifying efficiency of UAT devices [4]. The spatial decay can only be revealed by the complexband diagram [25], but it was totally neglected in the previous studies. As a result, the performance of the resulting UATdevices can not be accurately controlled.The UAT behavior of PCs highly depends on the spatial distribution of the constituent materials. Therefore designingphononic crystals for UAT can be looked as a typical topology optimization problem. The design of PCs by topology opti-mization has aroused a growing attention in recent years [26]. So far, topology optimization has been successfullyimplemented in the design of various PCs and acoustic devices, such as phononic band gap crystals [27e30], waveguidesand filters [31], self-collimation [32] and metamaterials [33]. The effects of the symmetry on the optimization of PCs havealso been discussed [34,35]. However, topology optimization of PCs for achieving UAT in a waveguide has never beeninvestigated.This paper will develop a new topology optimization algorithm of PCs for UAT based on the bi-directional evolutionarystructural optimization (BESO) method. In order to achieve UAT, we need to open the PBG and maximize the spatial decay atthe specific frequency simultaneously. It is well known that the minimum imaginary of the wave vector (k)reflect the spatialdecay in the band gap [25,36]. Instead of creating the PBG directly, the optimization is converted to maximize the minimumimaginary part of the wave vector alongG-X (minimum Im (kG-X)) and keep that alongG-M (minimum Im (kG-M)) less than aconstraint value. Thus, the PBG is expected to open automatically and the UAT with high rectifying efficiency can be realizedby placing the optimized PC in a bent wave guide.The rest of the paper is organized as follows. Section 2describes the mechanism of the UAT based on PBG, in-troduces the calculation method of the complex band diagram, and then derives the mathematical formulation of theoptimization problem. Numerical implementation for the proposed BESO method is described in Section 3. In Section4, numerical examples of optimized PC with C4vsymmetry and without symmetry are presented. Meanwhile, thetransmission properties of the UAT devices based on the optimized PC are discussed. The conclusions are drawn inSection 5.2. Theory and method2.1. UAT based on PBGTo illustrate the mechanism of UAT based on PBG, an example from Ref. [18] is shown in Fig. 1(a). The UAT device iscomposed of a 2D PC, and a bent rectangular tube whose internal width is Land the bending angle is 45?. The PC is placed atthe joint part inside the tube. The device is assumed to be filled with air. The PC consists of steel rods periodically arranged inthe square lattice with lattice constant a, and the radius of steel rods is 5a/22. The band diagram of the PC is shown in Fig. 1(b),which indicates a PBG ranging from normalized frequency 0.39 to 0.56 along boundaryG-X of the first Brillouin zone (denotedin Fig. 1(c)). When the driving frequency falls within this frequency range, the incident waves from the right side fall withinthe PBG of the PC while that from the left side fall within the pass band. UAT is thus achieved. However, such a trail-and-errorprocedure is extremely tedious and challenging for realizing UAT at a desirable frequency. It can be seen that UAT essentiallyutilizes the anisotropic transmission property of PCs. This paper aims to achieve UAT through controlling the transmissionproperties in different directions by topology optimization.2.2. Analysis of evanescent waves in PBGFor acoustic waves propagate in PCs, the dynamic equilibrium equation can be expressed as follows [37]V$?1r?r?Vp?r?? 1/4 v2vt2?p?r?B?r??(1)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123104
where Bis bulk modulus,ris mass density, pis pressure and r?x;y?denotes the position vector. Due to the periodicity of PCs asshown in Fig. 1(a), the pressure p?r?can be expressed by the Bloch wave expansion asp?r;k? 1/4 pk?r?ei?ut?k$r?(2)where pk?r?is a periodic function of rwith the same periodicity to the structure. p?r? 1/4 p?r?a?and B?r? 1/4 B?r?a?, where aisthe lattice translation vector. k 1/4 ?kx;ky?is the Bloch wave vector [38]. Thus, it is sufficient to solve the governing equation (1)in the primitive unit cell under periodic boundary conditions. By substituting eqn. (2) into eqn. (1), the governing equationcan be converted to an eigenvalue problem, which can be solved by the k?u?oru?k?method. Since wave propagation isinhibited within a band gap, only evanescent Bloch waves are left, which explains the exponentially decreasing transmissionof waves. To reveal the characteristics of wave propagation within a PC, it is necessary to use the k?u?method by sweepingfrequency [25].Consider a given wave vector with wavenumber kand directionqas illustrated in Fig. 1(c). When the primitive unit cell isdiscretized with finite elements, the eigenvalue problem can thus be written in the matrix form as,?k2KI?kKII ?KIII?p 1/4 0 (3)where p 1/4 pk?r?is the nodal pressure and the matrices KI,KII,KIIIare the functions of material properties and angleq, as givenin the Appendix.Since the eigenvalue problem in eqn. (3) is quadratic in the wavenumber, it should be rewritten in a standard eigenvalueform asFig. 2. Complex band diagram alongG-X: (a) Real par (b) Imaginary part of the wave vector.Fig. 1. (a) Schematic illustration of the 2D model of the UAT device. (b) Calculated dispersion bands of 2D PC model. (c) Illustration of the propagation directionspecified by the angleqand the irreducible Brillioun zone (G-X-M-G).Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 105
?K?q;u??kM?q??P 1/4 0 (4)whereK 1/4 ?KII KIII?0?;M 1/4 ??KI00??;P 1/4 ?kpp?For the given frequencyuand angleq, eqn. (4) can be solved for eigenvalues k. It should be noted that the wavenumber kisgenerally a complex value k 1/4 k0?ik00.For the PC shown in Fig. 1(a), the complex band diagram inG-X direction ?q 1/4 0+?andG-M direction ?q 1/4 45+?are givenin Figs. 2 and 3, respectively. The real and imaginary parts of the wave vectors are shown in the left and right panelsrespectively. The imaginary part of the wave vector is only displayed within the limited range (jIm (ka/2p)j&1.4), but otherlarger values are also exist [39]. To clearly show the complex band diagrams, the wave vectors are classified into four groupsshown with different colors: (i) pure real wave vectors characterized by values of Re(k) in the Brillouin zone and Im(k) 1/4 0are shown with black circle points, these wave vectors correspond to the classical band dia (ii) pure imaginary wavevectors characterized by Im(k)&0 and Re(k) 1/4 0 are shown by
(iii) complex wave vectors characterizedby Im(k)&0 and Re(k) 1/4 1/2(pointX)or1=ffiffiffi2p(point M) are shown by green pentacle p (iv) other complex wavevectors whose Re(k)inthefirst Brillouin zone and Im(k)s0 are shown by red square points. The classical band diagram (asshown in Fig. 1(b)) resulting from theu?k?method is also given in Figs. 2(a) and 3(a) as red solid lines for comparison. Thefrequency range of PBG alongG-X is illustrated by the grey areas. It can be seen that the purely real wave vectors giveexactly the same results with classical band diagrams, which correspond to the propagation of real Bloch waves withoutspatial decay. However, there are additional complex bands, which can not be revealed by the classical band diagram, e.g.bands within the PBG alongG-X.Thesecomplexbandsreflect the evanescent Bloch waves corresponding to wave numberswith nonzero imaginary part, which denotes spatial decays of the waves [36].ItcanbeseenfromFigs. 2(b) and 3(b) that, inthe frequency range of PBG, the minimum Im (kG-X) is not 0 while the minimum Im (kG-M) is 0. Thus, the UAT device shownin Fig. 1(a) requires that the minimum Im (kG-X) be large enough to make the system an insulator for incident waves fromright side. Meanwhile, the minimum Im (kG-M) should be small enough to make the device a conductor for incident wavesfrom left side.2.3. Topology optimization formulationsAiming to achieve UAT ina PC, the optimization needs maximize the minimum Im (kG-X), meanwhile keeps them in Im (kG-M) smaller than a limit, k*. When the primitive unit cell is discretized with finite elements, the structure of the PC can berepresented by assigning artificial design variables, xi(i 1/4 1, 2, …,n), for all elements. The PC considered in this paper isconstructed by two materials. A design variables xiis assigned to 1 if element iis composed of material 1 and xi 1/4 0 formaterial 2. The optimization problem can be mathematically formulated as follows8&:max :f?xi? 1/4 k000+s:t::k0045+&k*xi 1/4 0or1;i 1/4 1;2;…;n(5)Fig. 3. Complex band diagram alongG-M: (a) Real par (b) Imaginary part of the wave vector.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123106
where k000+and k0045+denote the minimum Im (kG-X) and minimum Im (kG-M), respectively.The sensitivity of the objective function with respect to design variable xican be written as,vf?xi?vxi 1/4 vk000+vxi(6)Differentiating both sides of eqn. (3) with respect to the design variable xiyields the following equation,v??k2KI?kKII ?KIII?p?vxi 1/4 vkvxi?2kKIp?KIIp???k2vKIvxi?kvKIIvxi?vKIIIvxi?p??k2KI?kKII ?KIII?vpvxi 1/4 0(7)Multiplying both sides of eqn. (7) with a vector vTgetsvkvxi?2kvTKIp?vTKIIp??vT?k2vKIvxi?kvKIIvxi?vKIIIvxi?p?vT?k2KI?kKII ?KIII?vpvxi 1/4 0 (8)To eliminate the unknownvpvxiin the above equation, vTis chosen asvT?k2KI?kKII ?KIII? 1/4 0 (9)Eqn. (9) denotes the adjoint equation and vis the eigenvector. It should be noted that the adjoint equation is different fromthe original eigenvalue problem in eqn. (3) since the matrix KIIis non-symmetric. To this end, the sensitivity of the wavevector isvkvxi 1/4 ?vT?k2vKIvxi?kvKIIvxi?vKIIvxi?p?2kvTKIp?vTKIIp?(10)andvk00vxi 1/4 imag?vkvxi?(11)The liner material interpolation scheme has been applied successfully to maximize the spatial decay of evanescent wavesin phononic crystals [40]. Since the acoustic governing equation (1) in this paper can be obtained by substituting the densityrand shear modulusmin the governing equation of solid-solid PCs for out-of-plane mode with1Band1r, respectively [37],aninverse linear material interpolation is adopted here1r?xi? 1/4 ?1?xi?1r1?xi1r2(12)1B?xi? 1/4 ?1?xi?1B1?xi1B2(13)where subscripts ‘1’and ‘2’represent material 1 and 2, respectively. It is assumed thatr1&r2and B1&B2. Based on thematerial interpolation functions, the variations of the matrices KI,KII,KIIIwith regard to design variables can be calculated, asgiven in the Appendix.3. Numerical implementation and BESO procedure3.1. Modification of the objective functionCurrent optimization problem stated in eqn. (5) has an additional constraint on the k0045+. Like the optimization problem ofcellular phononic band gap crystals with a stiffness constraint [28], the latter one can be imposed by introducing a Lagrangianmultiplierl. The objective function can be modified asmax :f?xi? 1/4 k000+?l?k*?k0045+?(14)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 107
When k0045+is equal to k*, the above equation is equivalent to the original objective function defined in eqn. (5). Otherwise,if k0045+&k*which means the constraint is already satisfied andlis set to be 0; if k0045+&k*, which means the constraint is notsatisfied yet and we need to minimize k0045+first, solneeds to be infinity. Therefore, before the calculation of the overallsensitivity of the modified objective function, the Lagrangian multiplier should be determined first. Here an intermediateparameter wis introduced in the program and defined byFig. 4. Evolution histories of the volume fraction, the minimum imaginary part of wavenumbers atq 1/4 0+and 45?and topology of the primitive unit cell forU 1/4 0.7.Fig. 5. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.5: (a) Optimized 3 ?3 primitive unit cells (black-Air, white-PVC); (b)M (c) C (d) Imaginary part of complex band diagram (red circle and black triangle points denote wave vectors alongG-XandG-M, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123108
l 1/4 1?ww(15)where wis a constant value which varies from an extremely small value wmin, e.g., 10?20, to 1, and the resulting Lagrangianmultiplierllocates in the range of 0 to infinity.To find the proper value of w, we introduce two bound values of w,wlower 1/4 wminand wupper 1/4 1. The program starts froman initial guess w 1/4 1 and the sensitivity numbers can be determined by eqn. (14). Then, based on the ranking of thesensitivity numbers the design variables are updated to satisfy the volume fraction in the next iteration. The value of k0045+inthe next iteration, k00iter?145+, can be estimated ask00iter?145+zk00iter45+?Xidk00iter45+dxiDxi(16)Thereafter, if k00iter?145+&k*, which means the constraint is not satisfied and the Lagrangian multiplierlis too small, we willmove the upper bound wupperto wand update wwith a smaller valuew 1/4 w?wlower2(17)On the other hand, if k00iter?145+&k*, we move the lower bound wlowerto wand update wwith a larger valueFig. 6. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.7: (a) Optimized 3 ?3 primitive unit cells (black-Air, white-PVC); (b) Minimum (c) Classic band diagram (black solid line and red dotted line denote band without and with considering the acoustic-solid coupling, respectively);(d) Imaginary part of complex band diagram (red circle and black triangle points denote wave vectors alongG-X andG-M, respectively). (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 109
w 1/4 w?wupper2(18)This procedure is repeated for several iterations until wupper?wloweris less than 10?15and an appropriate Lagrangianmultiplier is obtained.3.2. BESO procedureThe basic concept of the BESO method is to remove low-efficient materials from the ground structure and add materialsnear the most efficient regions simultaneously, so that the resulting design evolves to an optimum [41e43]. BESO generallydrives the update of structural topology by elemental sensitivity numbers, which reflect the relative ranking of elementalsensitivities asai 1/4 vf?xi?vxi(19)It is well known that topology optimization of continuum structures often encounters numerical instabilities,e.g. checkerboard and mesh-dependency. These numerical instabilities can be avoided by employing a filter scheme forelemental sensitivity numbers [43]. The modified elemental sensitivity numbers after filtering can be expressed by~ai 1/4 Pw?rij?ajPw?rij?(20)where rijis the distance between the element iand element j;w?rij?is weight factor defined by,Fig. 7. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.9: (a) Optimized 3 ?3 prim itive unit cells (black-Air, white-PVC); (b)Minimum de (c ) Classic b (d) Im aginary part of complex band diagram (red circle and black tria ngle point s denote wave vectorsalongG-X andG-M, respectively ). (For interpretation of the references to colour in this figure legend, the re ader is refer red to the web version of thisarticle.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123110
Fig. 8. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized PC forU 1/4 0.5; (b) numerical contrast ratio of the UAT (c) Thepressure field for RI withU 1/4 0.5; (d) The pressure field for LI withU 1/4 0.5.Fig. 9. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized PC forU 1/4 0.7; (b) numerical contrast ratio of the UAT (c) Thepressure field for RI withU 1/4 0.7; (d) The pressure field for LI withU 1/4 0.7.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 111
w?rij? 1/4 max?rmin ?rij ;0?with rminas the radius of the filter, which can be specified by the user. Furthermore, the resulting sensitivity numbers will beaveraged with their corresponding values in the last iteration.baiteri 1/4 ?~aiteri?baiter?1i?.2 (21)where kis the iteration number.Normally, BESO gradually decreases the volume fraction of the structures until the volume constraint is satisfied[41,42]. However, there is no volume fraction constraint for the current optimization problem as expressed by Eqn. (5).The solution space for the optimization problem is essentially complicate, and the objective function may notmonotonously increase or decrease with the variation of the volume fraction of materials. Thus, it might have manylocal optimal solutions corresponding to different volume fractions and more discussion based on numerical exampleswill be given in Section 4.2. The alternative way used in this paper is seeking the optimal volume fraction near aprescribed volume fraction of material 2, V*f. BESO starts from an initial design which is almost fully composed ofmaterial 2, and then gradually decreases V2fwhen it is larger than V*f. The evolution of the volume fraction can beexpressed byV2f;iter?1 1/4 V2f;iter?1?ER?when V2f;iter?1&V*f(22)where ER is the evolutionary rate and the subscript iter denotes the iteration number. Once the volume fraction is less than theprescribed value, the volume fraction is then determined by the variation trend of the objective function f?xi?,asV2f;iter?1 1/4 V2f;iter0@1??f?xiteri??f?xiter?1i??$?V2f;iter ?V2f;iter?1????f?xiteri??f?xiter?1i????$???V2f;iter ?V2f;iter?1???ER1A(23)Once the volume fraction is vibrantly convergent to a value, it will keep constant for the remaining iterations.Fig. 10. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized PC forU 1/4 0.9; (b) numerical contrast ratio of the UAT (c) Thepressure field for RI withU 1/4 0.9; (d) The pressure field for LI withU 1/4 0.9.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123112
Based on the calculated sensitivity numbers, baiteri, the threshold of the sensitivity number, baiterth, is determined by using thebi-section method so that the volume fraction of material 2 in the next iteration is equal to V2f;iter?1. Numerical experienceshows that the objective function is very sensitive to the variation of design variables since the location of min?k00q?varies asthe topology of the primitive unit cell changes as mentioned above. Instead of using finer mesh, the design variables in thispaper are updated according toxiter?1i 1/4 (min?xiteri?Dx;1?;if ~aiteri&~aiterthmax?xiteri?Dx;1?;if ~aiteri&~aiterth(24)Fig. 11. The LI transmittance spectra of the UAT devices based on the optimized PCs under (a) and (b)U 1/4 0.5; (c) and (d)U 1/4 0.7; (e) and (f)U 1/4 0.9, respectively.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 113
whereDx 1/4 0:05 in the paper. It is different from the traditional BESO method using two discrete design values (0 and 1).Although discrete intermediate design variable are used during the optimization process, the final designs are convergent toclear 0/1 designs as demonstrated in numerical examples.The proposed BESO method is briefly outlined as followsStep 1: Define BESO parameters: volume fraction, evolution rate and filter radius.Step 2: Construct the initial design and discretize the structure into finite elements.Step 3: Conduct finite element analysis for PCs and compute complex wave vector according to eqn. (4).Step 4: Calculate sensitivity numbers according to eqn. (19).Step 5: Filter and average the sensitivity numbers according to eqns. (20) and (21).Step 6: Determine the target volume fraction of material 2 according to eqns. (22) and (23).Step 7: Update design variables according to eqn. (24).Step 8: Update Lagrangian multiplier of the constraint according to eqns. (14)e(18).Step 9: Repeat Steps 3e8 until volume fraction constraint is satisfied and the objective function and topology are stablyconvergent.4. Results and discussionTo demonstrate the capability of the proposed BESO algorithm, this section will present a number of optimization resultsfor the PCs with and without symmetry under various specific frequencies. The 2D solid-fluid crystal with the square lattice isconsidered. Materials in use are PVC as material 1 and air as material 2. The physical properties arer1 1/4 1560kg=m3andB1 1/4 7.8 GPa for PVC andr2 1/4 1:204kg=m3and B2 1/4 1.42 ?10?4GPa for air [37]. Due to the extreme mismatch of the acousticimpendence between PVC and air, the shear wave modes in PVC is ignored [37]. We will show later that this simplificationFig. 12. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.5 for primitive unit cell without symmetry: (a) Optimized 3 ?3 primitive unitcells (black-Air, white-PVC); (b) M (c) C (d) Imaginary part of complex band diagram (red circle and black trianglepoints denote wave vectors alongG-X andG-M, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123114
does not alter the physics of the system. Material 1 and material 2 are respectively denotedas white and black in the pictures.The side length of square lattice a 1/4 1 cm and the primitive unit cell is discretized into 64 ?64 linear 4-node elements. It isnoted that the governing eqn. (1) cannot be applied to the case which fluid is surrounded by solid. To guarantee the validity ofeqn. (1), the design domain is set as 60 ?60 elements locating in the center of the unit cell and surrounded by air. Forconvenience, the frequency is normalized byU 1/4 ua=2pClwhere Cl 1/4 ffiffiffiffiffiffiffiffiffiffiffiffiffiB2=r2pdenotes the longitudinal wave speed in ma-terial 2; the wavenumbers are normalized by ka=2p. BESO parameters used in most cases are ER
1/4 0.002, k* 1/4 1 (corre-sponding to the normalized value 0.0016), and rmin 1/4 ffiffiffi2pa=50. V*fis selected as 0.78 for PC with C4vsymmetry and 0.85 for PCwithout symmetry.The UAT device is constructed by placing the optimized PC at the joint part of a bent rectangular tube, as illustrated in Fig.1.The PC is composed of 8 ?8 unit cells and the internal width Lof the tube is 8a. The upper and lower boundaries of thewaveguide are defined as the sound hard boundary. The normally incident acoustic waves from the left and right sides of thedevice are represented by LI and RI, respectively. The transmission property of the device is numerically simulated by thecommercial FEA software, COMSOL Multiphysics.4.1. UAT device based on PC with C4vsymmetryThe phononic crystals used in previous studies on UAT are all lattices with 4-fold reflection symmetry and 4-foldrotational symmetry (C4v), so the unit cell of PC is restricted with C4vsymmetry in the first examples. Three operationfrequencies,U 1/4 0.5, 0.7 and 0.9, are selected. ForU 1/4 0.7, Fig. 4 shows the evolution histories of volume fraction, theminimum Im (kG-X) and minimum Im (kG-M), and topology of primitive unit cell. It is evident that BESO starts from asimple unit cell, whose volume fraction Vf2is close to 1. At the first stage of the optimization, the volume fractiongradually decreases and the minimum Im (kG-X) rapidly rises after 96 iterations. Then the volume fraction decreasesFig. 13. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.7 for primitive unit cell without symmetry: (a) Optimized 3 ?3 primitive unitcells (black-Air, white-PVC); (b) Minimum decay contour; (c) C (d) Imaginary part of complex band diagram (red circle and black trianglepoints denote wave vectors alongG-X andG-M, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 115
continually and finally achieves an optimal value, 0.720. Meanwhile, minimum Im (kG-X) gradually converges to itsmaximum value, 0.603, and topology of the primitive unit cell is also stably convergent. The minimum Im (kG-M)maintains a small value during the optimization. The whole optimization takes 214 iterations for this case. Althoughdiscrete intermediate design variables are employed in the current BESO method, the final optimized design is very closeto 0/1 design without any “grey”area.Figs. 5e7show the optimized topology, its minimum decay contour, classical band diagram and imaginary part ofcomplexband diagram forU 1/4 0.5, 0.7 and 0.9, respectively. The optimized topology shows 3 ?3 primitive unit cells and the middleone is marked out in the dash box. The volume fraction of material 2 is 0.782, 0.720 and 0.782, respectively. The minimum Im(kG-X) under the specified frequencies is respectively 0.146, 0.603, and 0.439 for these optimized PCs, while the minimum Im(kG-M) is approximate zero. As expected, the PBG covering the specified frequency opens automatically in all cases. Thecorresponding PBGs are illustrated by the grey area in the band diagrams, and their frequency ranges are (0.322e0.504),(0.699e0.884) and (0.680e0.940). It can be seen from the imaginary part of the complex band diagrams that, within thefrequency range of PBG, the minimum Im (kG-X) has nonzero value while minimum Im (kG-M) is equal to zero. It means that thebasic conditions for UAT is achieved and the rangeof working frequency can be easily controlled by the specified frequency inthe proposed optimization algorithm.As mentioned above, the shear wave modes in PVC is ignored. To further verify its effectiveness, we re-calculate theclassical band diagram of the optimized PC forU 1/4 0.7 by considering shear modes in PVC and acoustic-solid coupling. Thecorresponding results are shown in Fig. 6(c) with red dash lines. It can be seen that the classical band diagrams with andwithout considering shear modes in PVC are almost the same. Thus, for simplicity but without losing the essential physics, theshear modes are reasonably ignored for the air/PVC PCs in this paper.Numerical simulations using the commercial FEA software, COMSOL Multiphysics, are carried out to demonstrate thetransmission of energy flux and acoustic pressure field when incident waves come from two sides of the UAT device. Thetypical results are shown in Figs. 8e10. It can be observed that the transmittance spectra of energy flux exhibit a significantFig. 14. Optimized topology and its band diagrams obtained under frequencyU 1/4 0.9 for primitive unit cell without symmetry: (a) Optimized 3 ?3 primitive unitcells (black-Air, white-PVC); (b) M (c) C (d) Imaginary part of complex band diagram (red circle and black trianglepoints denote wave vectors alongG-X andG-M, respectively). (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123116
difference between LI and RI for frequencies located in the PBGs shown in Figs. 5e7, respectively. To evaluate the per-formance of the UAT device, we define a contrast ratio: Rc 1/4 jTR?TLj=?TR?TL?[20]. The absolute value of the contrast ratiorepresents the relative transmittance weight between LI and RI. It can be seen from Figs. 8(b)e10 (b) that Rcz1formostfrequencies in PBGs, indicating high performance of the UAT devices. The frequency range of which Rc&0.90 are defined asUAT range in this paper. They are respectively (0.317e0.492), (0.492e0.877) and (0.632e0.923) for the three cases, asunderlined with black area in the contrast ratio diagrams. For the optimized PC withU 1/4 0.5, the UAT range is slightlynarrower than the frequency ranges of corresponding PBG. This attributes to the fact that the minimum imaginary part ofwavenumbers close to the edge of gaps is small, thus the decay for RI is low and the relative transmittance weight betweenLI and RI is thus small. On the contrary, the UAT range are wider than the corresponding PBG for the sample obtained underU 1/4 0.7 and 0.9. In those cases, the minimum Im (kG-X) is larger enough than the minimum Im (kG-M) although full band gaps(0.492e0.699) and (0.632e0.680) appear. Therefore the decay for RI is bigger enough than that for LI and the relativetransmittance weight between LI and RI is still high. To clearly illustrate the UAT behavior, acoustic pressure fields for thethree cases are also presented in Figs. 8e10 and the frequencies of the incident plane waves areU 1/4 0.5, 0.7 and 0.9,respectively. It can be seen that the acoustic wave could transmit through the PC for LI, but negligible transmission isobserved for RI.For practical acoustic devices, the angleabetween the two tubes needs to be small. Here, based on the optimized PC,three angles, 45?,30?and 15?, are selected to investigate the influence ofaon the UAT behavior. Changingawill notsignificantly affect the transmittance for RI, so we only plot the transmittance spectrums of LI and the contrast ratioTable 1Optimized topology, objective function and PBG range forU 1/4 0.5 uning various prescribed volume fractions.V*fV2f3?3 unit cells Min [Im (kG-X)] PBG range0.850 0.658 0.270 0.241e0.5000.800 0.671 0.269 0.240e0.5000.750 0.671 0.269 0.240e0.5000.700 0.666 0.270 0.241e0.5000.650 0.652 0.270 0.242e0.500Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 117
diagrams within the original UAT ranges of the three samples, as shown in Fig. 11. It can be seen that the decay for LI withinthe original UAT range generally increase with the decrease ofa. However, the contrast ratios for most frequencies in theoriginal UAT range are still higher than 0.90 even whenais 15?. This is due to that the minimum Im (kG-X) is maximized inour algorithm and the transmittance for RI is thus extremely low, although the decay for LI increases and the transmittancebecomes lower, the contrast between RI and LI is still high. The mechanism is similar to the acoustic diode constructed withnonlinear structures, in which both the transmittance for RI and LI are small but the transmittance for RI is several orderslower [4,5].4.2. UAT device based on PC without symmetryThe optimization algorithm is now modified for PCs without symmetry constraints. Figs. 12e14 present the optimi-zation results forU 1/4 0.5, 0.7 and 0.9, respectively. The volume fraction of material 2 is 0.658, 0.843 and 0.856, respectively.The minimum Im (kG-X)underthespecified frequencies is respectively 0.270, 0.414 and 0.528 for these optimized designs,and the minimum Im (kG-M) is equal to zero. Similar to the optimization of PC with C4vsymmetry, the PBGs for asymmetricdesigns also occur automatically around the operation frequency. The frequency ranges are (0.241e0.500), (0.609e0.888)and (0.636e0.932) respectively, which are wider than the counterpart obtained under PCs with symmetry. The imaginarypart of complex band diagrams shows that, within the frequency range of PBG, minimum Im (kG-X)&0 and minimum Im(kG-M) 1/4 0.Table 2Optimized topology, objective function and PBG range forU 1/4 0.9 uning various prescribed volume fractions, V*f.V*fV2f3?3 unit cells Min [Im (kG-X)] PBG range0.850 0.856 0.528 0.636e0.9320.800 0.800 0.614 0.599e0.9530.750 0.770 0.640 0.633e0.9420.700 0.704 0.664 0.613e0.9230.650 0.652 0.758 0.600e0.945Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123118
Then we selectU 1/4 0.5 and 0.9 as illustrative examples to investigate the effects of the prescribed volume fraction V*fas shown in Tables 1 and 2.WhenU 1/4 0.5, the optimized topologies, values of minimum Im (kG-X) and partial band gapsare almost the same although various prescribed volume fractions are used. The optimal volume fraction of material 2 isaround 0.66 and increasing or decreasing the volume fraction leads to the decrease of the objective function. However, fora higher frequencyU 1/4 0.9, the value of the prescribed volume fraction has significant effect on the optimized solution.This indicates that the variation of objective function against the volume fraction is much more complicate than that ofU 1/4 0.5. As shown in Table 2 , a different prescribed volume fraction results in different solutions and the final volumefraction are close to the prescribed value. Even so, the partial band gap coveringU 1/4 0.9 is successfully obtained for allcases.Figs. 15e17 demonstrate the transmission of energy flux and acoustic pressure field when the acoustic incident wavescome from two sides of the UAT devices constructed by the optimized asymmetric PCs. The transmittance spectra of energyflux exhibit a significant difference between LI and RI within frequencies located in the PBGs shown in Figs. 12e14. The broadUAT ranges are 0.236e0.644, 0.504e0.865 and 0.585e0.911 for these optimized designs, respectively. The relative bandwidthdefined by the ratio between the bandwidth and center frequency achieves 0.927, 0.475 and 0.436, respectively. The UATranges are all wider than the corresponding frequency ranges of PBGs due to the same reason. Here we take the optimizeddesign underU 1/4 0.7 as an example. The UAT range (0.504e0.865) covers two PBGs ((0.504e0.560 and 0.609e0.865)) and afull band gap (0.560e0.609), as evident from the classical band diagram, Fig. 13(c). It can be found from the imaginary part ofcomplex band diagram, Fig. 13(d), that minimum Im (kG-X)&minimum Im (kG-M) within the full band gap, (0.560e0.609).Therefore, the decay for RI is still bigger enough than that for LI and the relative transmittance weight between LI and RIremains high. It is also noted that the UAT ranges for optimized asymmetric PCs underU 1/4 0.5 and 0.9 are much wider thantheir symmetric counterparts. Although the UAT range forU 1/4 0.7 is slightly narrower than its symmetric counterparts, thetransmittance of symmetric design within frequency range (0.492e0.690) is extremely small compared with that of theasymmetric design. Therefore removing symmetry constraints is favor for the design of broadband UAT devices based on PBG.The acoustic pressure fields in Figs. 12e14 show that acoustic wave could transmit through the PC in LI, but is blocked in RIunder the operation frequencies.The effect of the angleaof the two tubes on the UAT behavior is also investigated for the three asymmetric PCs. The resultis shown in Fig. 18. It can be seen that, within the originalUAT range, the decay for LI generally increase with the decrease ofa,which is consistent to the UAT devices based on symmetric PCs. For the casesU 1/4 0.5 and 0.7, the contrast ratios within theoriginal UAT range are higher than 0.90 even whena 1/4 15?, which has been explained in Section 4.1.Fig. 15. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized asymmetric PC forU 1/4 0.5; (b) numerical contrast ratio of the UAT (c) The pressure field for RI withU 1/4 0.5; (d) The pressure field for LI withU 1/4 0.5.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 119
Fig. 17. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized asymmetric PC forU 1/4 0.9; (b) numerical contrast ratio of the UAT (c) The pressure field for RI withU 1/4 0.9; (d) The pressure field for LI withU 1/4 0.9.Fig. 16. (a) Numerical LI and RI transmittance spectra of the UAT device using the optimized asymmetric PC forU 1/4 0.7; (b) numerical contrast ratio of the UAT (c) The pressure field for RI withU 1/4 0.7; (d) The pressure field for LI withU 1/4 0.7.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123120
5. ConclusionsThis paper has systematically studied the topology optimization of2D PC with or without symmetry for UAT based on PBG.The complex wave vector at a specified frequency is calculated through finite element analysis. Based on the sensitivityanalysis, the BESO method is established by gradually enlarging the minimum Im (kG-X) while keeping the minimum Im (kG-M)below a constraint. BESO starts from an initial design almost fully consist of air and gradually evolves the topology of theprimitive unit cell to an optimal one. Gradually, the minimum Im (kG-X) is maximized and the minimum Im (kG-M) becomesFig. 18. The LI transmittance spectra of the UAT devices based on the optimized asymmetric PCs under (a) and (b)U 1/4 0.5; (c) and (d)U 1/4 0.7; (e) and (f)U 1/4 0.9,respectively.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123 121
zero. The PBG at the specific frequency open automatically. Then the UAT devices are constructed by placing the optimizedPCs in the joint of a bend waveguide. Numerical simulations show that they have high rectifying efficiency. It is alsodemonstrated that optimized PCs without symmetry constraints are more favor for the design of broadband UAT behaviorcompared to that with C4vsymmetry. The effect of the bend angle of the waveguide on the efficiency of UAT for differentdesigns is explored. Surprisingly, some designs exhibit high rectifying efficiency even when the bend angle decreases to15?.Itis noted that the band width of UAT may not be maximum since the current optimization objective is to maximize spatialdecay at the specific frequency. Further work on fabrication and experiment of the resulting optimized designs isrecommended.AcknowledgementsThe authors wish to acknowledge the financial support from the Australian Research Council (FT130101094).AppendixThe matrices KI,KII,KIIIin eqn. (3) are calculated asKI 1/4 ?cos2q?sin2q?XK1(A1)KII
1/4 i?cosqXK2?sinqXK3?(A2)KIII
1/4 XK4?u2XM(A3)where Smeans the assembly of elemental matrices andK1 1/4 1rZU?NTN?dUK2 1/4 1rZU?vNTvxN?NTvNvx?dUK3 1/4 1rZU?vNTvyN?NTvNvy?dUK4 1/4 1rZU?vNTvxvNvx?vNTvyvNvy?dUM 1/4 1BZU?NTN?dUwhereUdenotes the area of an element, and Nis the shape function matrix of an element. For a 4-node plane strain element,Ncan be expressed by:N 1/4
1/2 N1N2N3N4?The variations of KI,KII,KIIIwith respect to design variables, xiare given as belowvKIvxi 1/4 ?cos2q?sin2q?vK1vxi(A4)vKIIvxi 1/4 i?cosqvK2vxi?sinqvK3vxi?(A5)vKIIIvxi 1/4 vK4vxi?u2vMvxi(A6)wherevMvxi 1/4 ?1B2?1B1?RU?NTN?dUandvKjvxi?j 1/4 1;2;3;4?is equal to Kjby replacing1rwith1r2?1r1.Y. Chen et al. / Journal of Sound and Vibration 410 (2017) 103e123122
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ArticleMay 2018ProjectPrivate Profile[...]ProjectPrivate Profile[...]Project[...]Project[...]Phononic crystals (PnCs) are artificial periodic materials, which offer exceptional control over phonons.
In PnCs, elastic or acoustic waves within a specific frequency range cannot propagate, whi…& [more]ArticleApril 2017 · The propagation of evanescent waves inside phononic band gaps is important for the design of phononic crystals with desirable functionalities. This paper extends the bi-directional evolutionary structure optimization (BESO) method to the design of phononic crystals for maximizing spatial decay of evanescent waves. The optimization objective is to enlarge the minimum imaginary part of wave... [Show full abstract]ArticleMay 2018 · The viscoelasticity of constituent materials has a significant effect on the dispersion relation of waves in viscoelastic phononic crystals (PCs). This paper extends the bi-directional evolutionary structure optimization (BESO) method to the design of viscoelastic PCs with the maximum attenuation and stiffness. The attenuation factor is calculated by the k(ω)-method, and the effective... [Show full abstract]ArticleAugust 2017 · ArticleMarch 2016 · Phononic band gap crystals are made of periodic inclusions embedded in a base material, which can forbid the propagation of elastic and acoustic waves within certain range of frequencies. In the past two decades, the systematic design of phononic band gap crystals has attracted increasing attention due to their wide practical applications such as sound insulation, waveguides, or acoustic wave... [Show full abstract]Last Updated: 01 Aug 18