Accepted Manuscript Numerical study and topology optimization of 1D periodic bimaterial phononic crystal plates for bandgaps of low order Lamb waves Saeid Hedayatrasa, Kazem Abhary, Mohammad Uddin PII: DOI: Reference: S0041-624X(14)00331-X http://dx.doi.org/10.1016/j.ultras.2014.11.001 ULTRAS 4955 To appear in: Ultrasonics Received Date: Revised Date: Accepted Date: 21 May 2014 14 October 2014 4 November 2014 Please cite this article as: S. Hedayatrasa, K. Abhary, M. Uddin, Numerical study and topology optimization of 1D periodic bimaterial phononic crystal plates for bandgaps of low order Lamb waves, Ultrasonics (2014), doi: http:// dx.doi.org/10.1016/j.ultras.2014.11.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Numerical study and topology optimization of 1D periodic bimaterial phononic crystal plates for bandgaps of low order Lamb waves Saeid Hedayatrasa *, Kazem Abhary and Mohammad Uddin School of Engineering, University of South Australia, Mawson Lakes, SA 5095, Australia Abstract The optimum topology of bimaterial phononic crystal (PhCr) plates with one-dimensional (1D) periodicity to attain maximum relative bandgap width of low order Lamb waves is computationally investigated. The evolution of optimized topology with respect to filling fraction of constituents, alternatively stiff scattering inclusion, is explored. The underlying idea is to develop PhCr plate structures with high specific bandgap efficiency at particular filling fraction, or further with multiscale functionality through gradient of optimized PhCr unitcell all over the lattice array. Multiobjective genetic algorithm (GA) is employed in this research in conjunction with finite element method (FEM) for topology optimization of silicon-tungsten PhCr plate unitcells. A specialized FEM model is developed and verified for dispersion analysis of plate waves and calculation of modal response. Modal band structure of regular PhCr plate unitcells with centric scattering layer is studied as a function of aspect ratio and filling fraction. Topology optimization is then carried out for a few aspect ratios, with and without prescribed symmetry, over various filling fractions. The efficiency of obtained solutions is verified as compared to corresponding regular centric PhCr plate unitcells. Moreover, being inspired by the obtained optimum topologies, definite and easy to produce topologies are proposed with enhanced bandgap efficiency as compared to centric unitcells. Finally a few cases are introduced to evaluate the frequency response of finite PhCr plate structures produced by achieved topologies and also to confirm the reliability of calculated modal band structures. Cases made by consecutive unitcells of different filling fraction are examined in order to attest the bandgap efficiency and multiscale functionality of such graded PhCr plate structures. Keywords: Phononic Crystal; Topology Optimization; Lamb Wave; Filling Fraction * Corresponding author. Tel. +61883025374 Email address: saeid.hedayatrasa@mymail.unisa.edu.au (S. Hedayatrasa) 1 1 Introduction Artificial metamaterials with periodically modulated acoustic properties using two or more base materials forming a crystal lattice, called Phononic (or sonic) Crystal (PhCr), have attracted great attention during the past decade [1]. PhCrs possess promising capability to manipulate propagation of elastodynamic waves having wave length in the order of their lattice spacing, resembling photonic crystals affecting electromagnetic waves. The main extraordinary feature of PhCrs is existence of phononic bandgaps referred to as frequency ranges over which propagation of elastic waves is prohibited due to superposition of Bragg and Mie resonant scattering [2]. Usually scattering inclusions are embedded into a matrix material periodically, causing subsequent reflections of waves at the interfaces. Phononic bandgaps are applicable in producing vibrationless structure, e.g. solid attachment of a vibrating device to a substrate while destructing transmission of vibroacoustic waves. Besides, creating point and line defects in a PhCr structure produces resonance modes within bandgaps which can trap and guide acoustic energy at relevant frequencies. This capability is used in production of acoustic energy harvesting, resonators and wave guiding devices [3]. Another attractive characteristic of PhCrs is negative refraction index at the edge of phononic bandgaps [4]; an extraordinary phenomenon making them ideal for acoustic waves focusing. PhCrs are also capable of altering thermal heat conductivity in semiconductor materials in which thermal energy is dominantly transferred by phonons (vibration of atomic lattices) [3, 5] . Many types of structures have been developed in different applications, benefiting from features of phononic bandgaps from Nano to Macro scale. Design of PhCr composite pipes for flexural vibration attenuation in fluid piping systems [6, 7], layered PhCr foundation to mitigate seismic damage [8], micro fabricated PhCr wave guides [9] and Nano PhCr plate resonator with 200 nm lattice constant and 2.25 GHz central frequency [3], all are examples of these applications at different scales. Considerable research has been conducted on modal band structure of bulk acoustic waves in infinite mediums and surface acoustic waves in semi-infinite mediums. Very recently there has been growing interest in bandgaps of acoustic waves in periodic plate structures. Vibroacoustic analysis and optimization of thin-walled platelike elements are highly important as they have been widely used in various structural applications. Moreover, special characteristics of guided waves confined by finite thickness of such structures, make them ideal for non-destructive evaluation purposes [10] as well as production of low loss resonators, filters and waveguides [11]. Theoretically a plate structure satisfying two stress free surfaces, supports 3 well-known wave modes: two in-plane symmetric and asymmetric modes (Lamb waves) which are the resultant of interacting longitudinal and shear vertical modes, and 2 an anti-plane shear horizontal mode [12]. Wave dispersion in PhCr plate is governed by its planar anisotropy measured by lattice periodicity and unitcell features as well as its transversal anisotropy defined by plate’s thickness. The arrangement, orientation, filling fraction and acoustic impedance of constitutive materials are basic features of a PhCr unitcell determining the location, width (frequency range) and depth (reflection strength) of bandgaps. So for specific material contrast, the topology and aspect ratio of unitcell (plate’s thickness over lattice periodic constant) are two key factors defining the band structure of a PhCr plate. Therefor investigations have been dedicated towards theoretical analysis and experimental verification of band structure of PhCr plates. The most popular type of PhCr plates is the one made by periodic insertion of a centric scattering layer or circular scattering inclusions in a base plate for 1D and 2D PhCrs, respectively. The modal band structure of centric PhCr plates and influence of unitcell’s filling fraction and aspect ratio have been widely studied [13-17]. Li, Wang [18] studied propagation of elastic waves in phononic crystal slabs with Archimedean-like tilings through FEM modelling and defined relevant band structure. However, other types of PhCr plates have been introduced and studied [19-21] e.g. those produced by periodical attachment of studs or cutting circular cavities in homogeneous plates. Although considering centric scattering inclusions in the base plate produces a PhCr plate with acoustic bandgaps, the optimum shape and arrangement of scattering phase can be defined by inverse design of PhCr unitcell in order to obtain wide bandgaps of plate wave modes of interest. Wider bandgap leads to vibration isolation over wider frequency range. Besides, the width of bandgap determines the frequency range over which acoustic waves can be manipulated for design of wave guides and resonators. Hence, topology optimization has been efficiently applied to design of PhCr unitcells using either gradient based or evolutionary based optimization methods. Essentially the widest bandgap at lowest frequency range is desired in order to obtain the smallest feature size (unitcell) capable to manipulate frequency range of interest. Halkjær, Sigmund [22] implemented gradient based topology optimization to design basic bi-material unitcells including Aluminium and Polymethyl methacrylate (PMMA) for maximized bandgap of in-plane and bending modes in infinite 1D PhCr beams and infinite 1D PhCr slabs. 2D square unitcells were optimized which in a periodic transversal layup represent thick PhCr slabs. Finite PhCr slabs were also optimized for maximum in-plane shear and longitudinal bandgaps. Further Halkjær, Sigmund [23] optimized single-material unitcell of 2D periodic Mindlin’s plate for maximizing bandgap width between first and second bending modes. Then the frequency response of a finite structure was studied to define dependence of 3 bandgap quality to distance from excitation point as well as size of finite structure. Also Larsen, Laksafoss [24] defined optimized topology of finite square bi-material Mindlin plates for either suppression of vibration or control transport of vibration energy. Based on delivered literature, the band structure of PhCr plates with strip inclusions in 1D PhCrs and regular (e.g. circular) inclusions in 2D PhCrs have been well studied. However, to the best of the Author’s knowledge, there is no thorough study on optimum topology of PhCr plates for maximization of in-plane Lamb wave bandgaps. Moreover it is of great value to study the optimum topology of PhCr plate unitcells for maximum specific bandgap width as a function of filling fraction of inclusions. High contrast of constitutive materials has essential role in opening bandgaps and subsequently adjusting fraction of soft or stiff inclusions is applicable e.g. to satisfy weight, stiffness or production requirements. Plus, multiscale modulation of material properties by gradient of PhCr unitcell throughout the lattice array can produce a multifunctional structure which has additional functionality at scales larger than PhCr unitcell size. Hence, it is worthwhile to: (i) investigate the optimum topology of PhCr plate unitcell for maximum Lamb bandgap functionality with respect to filling fraction of constituents, and (ii) explore relative bandgap efficiency of graded PhCr plate comprised of consecutive unitcells of different filling fractions with optimized topology. To start with, 1D bi-material silicon-tungsten PhCr plates are investigated in this study. Multiobjective GA is employed in conjunction with FEM to explore the optimum topology of PhCr plate unitcell. Plate wave expansion (PWE) is the most commonly used method for this purpose by explaining the displacement field as an infinite sum of plate waves through Fourier series. FEM, however, is a discretization based approach for efficient calculation of modal response and band structure of PhCrs specially for inverse design of unitcell that makes using of PWE impractical. The governing equations and implemented FEM-GA basic formulation are firstly described in Section 2. The results of developed FEM code for dispersion analysis of single material plate along with band structure analysis of bi-material plate unitcells are delivered and discussed in Sections 3. Then topology optimization results are presented in Section 4 and definite prescribed topologies are introduced and evaluated premised on achieved optimum topologies. Finally, the frequency response of sample finite PhCr plate structures comprised of a variety of unitcell topologies is investigated in Section 5. 4 2 2.1 Theory and constitutive formulation Wave propagation in 1D periodic PhCr plate The focus of this research is to study the dispersion and modal band structure of elastic waves in longitudinally periodic PhCr plate with no material gradient through the thickness. A schematic of such a 1D periodic plate with thickness h and its basic unitcell with periodicity length a (along x direction) is shown in Figure 1. Figure 1. Schematic of a 1D PhCr plate along x direction with lattice constant a and thickness h Relatively large width (along z direction) is considered in which the effect of free edges (at z + and z − ) can be neglected and so by assumption of uniform displacement field in z direction ( ∂ / ∂z = 0 ) the problem is confined to x − y plane. Based on equation of motion in solids [12], two sets of independent equations can be developed for plate wave propagation in such heterogeneous plate, a couple of equations for in-plane (Lamb) waves: ∂u ∂ 2u ∂ ∂v ∂ 2u ∂ 2 v C = C + C + + ρ 2 11 12 44 ∂x ∂x ∂y ∂t ∂y 2 ∂x∂y (1) ∂ ∂v ∂u ∂ 2v ∂ 2v ∂ 2u + + C 44 = C + C 11 12 2 2 ∂x ∂y ∂x∂y ∂x ∂t ∂y (2) ρ and an equation for anti-plane shear horizontal (SH) waves: ρ ∂2w ∂ ∂w ∂2w = C + C 44 44 ∂x ∂x ∂t 2 ∂y 2 (3) where C11 , C12 and C44 are elastic material properties, ρ material density, t is time and U = {u, v, w} is the displacement vector. In order to theoretically solve the Eqs. (1) to (3) for periodic plate of interest the Bloch-Floquet theory is implemented [25]. Accordingly, the general modal solution for displacement U of 1D periodic structure having lattice constant a is defined by harmonic modulation of an a periodic filed U p , as follows: 5 U(x,y,t) = U p(x,y)e i(kx− t) (4) U p (x,y) = U p(x + a,y) (5) where ω is temporal angular frequency and k is spacial angular frequency (wave number) in x direction. So Bloch-Floquet complex strain matrices for in-plane and anti-plate waves respectively are: [ε x εy γ xy [γ ] T = ∂u p ∂x γ zy ] = T zx + iku p ∂w p ∂x ∂v p ∂u p ∂y ∂y + ikw p + ∂w p ∂v p ∂x T + ikv p e i ( kx −ωt ) (6) T ∂y e i ( kx −ωt ) (7) where U p = {u p , v p , w p }. Periodic plates made by silicon (Si) and tungsten (W) strips are modelled and studied in this research with material properties listed in Table 1. Silicon and tungsten are of potential matrix and scattering materials, respectively, for fabrication of phononic bandgaps and their micro fabricated samples have been produced by Sandia National Lab, USA [2, 9]. Table 1. Material parameters used for FEM modelling Material 2.2 Silicon (Si) Tungsten (W) C11 (GPa) 165.7 502 C12 (GPa) 63.9 199 C44 (GPa) 79.6 152 (kgm-3) 2330 19200 Specialized FEM model A specialized FEM model using first order quadrilateral elements is developed in MATLAB (MATLAB 8.0, The MathWorks Inc.) for dispersion analysis of guided waves and calculation of the band structure of hypothesized periodic plate. Taking into account Bloch-Floquet solution (Eq. (4)), the FEM notation of displacement vector and equation of motion for a loss free elastic model with stiffness matrix K and mass matrix M become: U = Nq p e i ( kx −ωt ) 2 (K (k ) − Mω )q p e i ( kx −ωt ) (8) =0 (9) Where N is matrix of shape function and q p is the vector of nodal displacements corresponding to periodic term of displacement filed U p (Eq. (4)). With regard to Eq. (6) and Eq. (7) the strain6 displacement matrix and so stiffness matrix K are complex and dependent on longitudinal wave number k . Due to periodicity of structure just a representative unitcell with length equal to lattice constant a is modelled. Periodic boundary condition is therefore applied by equalizing corresponding periodic degrees of freedom ( q p ) i at two left and right edges of the model. The top and bottom traction free edges are left with natural free boundary condition. Finally, the modal response of unitcell is obtained by dropping e i ( kx −ωt ) in Eq. (9) and Eigen value analysis for non-trivial solutions over: K r (k ) − M r =0 (10) where K r and M r are reduced mass and stiffness matrices obtained after applying periodic boundary condition through master-sleeve transform [26], and eigenvalues i = 2 . So the gradient of versus wave number k is determined, giving the band structure of periodic unitcell. Principally k can have any value, but due to periodicity in Bloch-Floquet condition it can be limited to the first Brillouin zone − π / a < k < π / a [27, 28] and further to irreducible Brillouin zone 0 < k < π / a due to symmetry. Appropriate mesh resolution should be defined in FEM simulation so that the tiny wave oscillations of highest frequency of interest are approximated adequately. According to the common practice, 10~20 nodes have to be considered over the wave length. Hence mesh density with at least 15 nodes per shortest wave length is ensured based on lower elastodynamic wave speed which pertains to the tungsten inclusion in this study. 2.3 Multiobjective optimization using Genetic Algorithm Evolutionary based optimization through genetic algorithm (GA) is employed in this study for topology optimization of PhCr plate unitcell. GA searches over a population of potential solutions, applies the principle of survival of the fittest using crossover and mutation functions and iteratively produces better and better approximations to a solution [29]. Crossover operation explores design space for better individuals by retaining good parts of mating parents. Mutation, on the other hand, exploits the design space by maintaining the diversity of population and reduces the likelihood of premature convergence. GA is a desirable strategy for intelligent design of discrete and strongly multimodal problems, with the possibility of sub‐optimal solutions. Ideally, topology optimization provides the best material layout to achieve the desired objective when a continuous design space is considered for design variables i.e. material properties. Although continuous GA has been adopted for such problems, it is computationally intensive and gradient based optimization techniques e.g. method of moving asymptotes (MMA) 7 are more appropriate for this purpose. Consequently, the optimality can be fulfilled by pseudo domains with intermediate artificial material properties which may be achieved by considering a multiscale design with equivalent homogenized properties, or faded via appropriate penalization approach. However, in the case of PhCrs, the optimum topology providing widest bandgap naturally converges to distinct material layout because highest material contrast is required for maximization of interfacial reflections. So GA which implements a coded string of design variables is an appropriate tool for optimization of PhCrs with favored discrete design space. Moreover, GA searches for optimum topologies within a population of designs in a single run and performs stochastic search over the entire search space. Hence it is unlikely to stick to a local optimum, specially when a broad band gap PhCr is desired involving gaps of several modal branches. The application of GA for optimization of PhCrs is adequately described in [30]. Some other works have also taken the advantage of GA for optimization of 1D and 2D bimaterial phononic bandgaps of longitudinal, flexural and bulk waves [30-33]. This study is concerned with finding optimum topology of 1D PhCr plate unitcell with respect to filling fraction of stiff inclusion for maximizing the bandgap width of Lamb waves. In order to get optimized topology with prescribed filling fraction, two approaches were examined: (i) setting filling fraction as a constraint to the optimization problem and (ii) introducing the deviation of individuals from reference filling fraction as another objective to be minimized. The former approach was firstly implemented where the average fitness of population dropped after a number of generations and the problem did not converge to the best solution. It is possibly due to crossover between individuals of population that interrupts the filling fraction of individuals and disturbs constrained convergence. But the later one, as a multiobjective optimization, capably produced a set of optimal solutions for the desired filling fraction and adjacent values. As for the first objective, the fundamental requirement is to open widest bandgaps at lowest potential frequency range for specific plate’s thickness at prescribed lattice constant [28, 31]. In other words the smallest feature size able to manipulate frequency range of interest is defined in this way. Thus relative gap width of Eigen values is sought as the objective function: Rj = min in=k1 Ω j +1 ( k i ) − max ni=k1 Ω j ( k i ) 0.5(min in=k1 Ω j +1 ( k i ) + max ni=k1 Ω j (k i )) (11) Indeed R j is the Eigen value gap width over mean gap value of two subsequent modes jth and (j+1)th over the n k search points located in irreducible Brillouin zone. So the first objective function F1 to be maximized is formulated as the sum of relative Eigen value gap widths of desired modes j1 to j2: 8 j2 F1 = Rj (12) j = j1 The second objective F2 to be minimized is then defined as absolute deviation of filling fraction of individual solution v f from its desired reference value v r : F2 = v f − vr (13) MATLAB multiobjective optimization toolbox (MATLAB 8.0, The MathWorks Inc.) is employed for optimization purpose which uses a controlled elitist nondominated sorting GA as a variant of NSGA-II [34, 35]. GA with controlled elitist evaluates objective functions to sort individual designs based on their non-dominated fitness and crowding distance. In this way the designs with the most dominated fitness are placed at the first rank (Pareto front) and among them those with highest crowding distance are preferred. The crowding distance enforces even distribution and diversity of optimized solutions within Pareto front [34]. The basic steps of multiobjective optimization algorithm implemented in this research are briefly as follows: 1. Create a random bit-string initial population for the bi-material problem defining the material type at each element of FEM model as a design variable (gene). Verify the fitness of individuals, define their rank and crowding distance and sort them accordingly. 2. Select individuals to be considered as parents of next generation through tournament selection. Two individuals are selected at random and the one with highest rank is selected as parent. If both have the same rank, the one with highest crowding distance is selected. 3. Create offspring population by applying single point crossover with certain probability on couples of randomly selected parents providing two Childs per couple. 4. Mutate gens of offspring population with specified probability 5. Produce intermediate population by accumulating offspring and current population and sort them based on ranking and crowding distance 6. Select individuals for next generation first based on their ranking and second, if the first rank is bigger than required population, based on their crowding distance. 7. Repeat from step-2 until the desired maximum number of generations is reached. For more information regarding implementation of GA for structural topology optimization see e.g. [30, 36-38]. 9 3 Plate waves dispersion and bandgaps The modal band structure of plate waves in 1D silicon-tungsten PhCr plate unitcells with regular centric topology is studied in this section for various filling fractions at different aspect ratios. Initially, the accuracy and competency of developed FEM model in determining the plate wave dispersion and frequency response of structures is evaluated. Multilayer FEM model of a pure silicon square cell with unit length as shown in Figure 2 is employed for this purpose. 50 element divisions through thickness are considered to determine the unfolded dispersion curves of plate waves including first 10 Lamb wave modes. Whereas linear quadratic elements are implemented, applying Bloch periodic boundary condition on two edges of single longitudinal element leads to x independent U p over the entire length. Consequently, pure harmonic oscillations of plate waves along a single material plate are simulated through this multilayer model (Eq. (4) and (5)). Figure 2. Multilayer FEM model of a plate unitcell for Lamb wave dispersion analysis, having periodic boundary at left and right edges and 50 element divisions evenly through thickness Figure 3(a) shows the plate wave dispersion in terms of phase velocity versus product of frequency and plate’s thickness fh comparable to the reference analytical results by Degertekin, Honein [33]. Excellent agreement was observed by comparison of these two. Moreover, the nondimensional modal band structure, as displayed in Figure 3(b), is determined based on dimensionless wave number k d = ka / π and dimensionless angular frequency ω d = ωh / cm , where cm = E m / ρ m is the longitudinal elastodynamic wave speed in silicon matrix. Solid lines stand for in-plane (Lamb) modes and dash lines for anti-plane SH modes. The colors differentiate various frequency modes, however Lamb modes intersect between adjacent in-plane modes as highlighted by bold dots in Figure 3 for symmetric modes S1 and S 2 . This is investigated by identifying symmetry of modal shapes at different wave lengths. 10 (a) (b) Figure 3. Plate wave dispersion curves obtained for a pure silicon plate; (a) Phase velocity versus product of frequency and plate’s thickness fh , (b) non-dimensional modal band structure. 11 In order to study the band structure of plate waves in 1D periodic plate structures, primarily a regular bi-material PhCr plate unitcell with centrally concentrated scattering inclusion shown in Figure 4(a) is modelled. The aspect ratio of unitcell r = a / h is 2 and filling fraction of tungsten scattering phase v f is 0.5. The non-dimensional modal band structure of this unitcell is then calculated as shown in Figure 4(b). (a) (b) Figure 4. (a) FEM model of Bi-material plate unitcell with centrally concentrated scattering inclusion, filling fraction v f = 0.5 and aspect ratio r = 2 , and (b) related plate wave band structure and bandgaps for few first modes Solid lines define the in-plane (Lamb) modes and dash lines the anti-plane SH modes. Dispersion of different modal frequencies is highlighted by colors, and intersections of symmetric (S) and asymmetric (A) Lamb modes are defined by appropriate markings on first 6 in-plane modes. By comparing the dispersion curves of this silicon-tungsten periodic plate unitcell (Figure 4(b)) with that of pure silicon plate (Figure 3(b)), two major effects are obvious 12 arising from tungsten scattering inclusion and structure’s periodicity. First, the dispersion curves are periodic with respect to wave number and symmetric about k d = 1 in the range 0 ≤ k d ≤ 2 . That is why only first irreducible Brillouin zone ( 0 ≤ k d ≤ 1) is usually searched for dispersion analysis of plate wave modes. This happens due to appearance of folded branches of modes resonated by scattering. Second, frequency bandgaps open in the band structure of periodic plate unitcell in which some or all plane modes are not feasible. The Gap 1 is the first bandgap exclusively for principal asymmetric Lamb mode (A0) since principal symmetric mode (S0) exists in this range. Gap 2 is the first complete bandgap between in-plane modal frequencies (modes 3rd and 4th) and on the other hand the second gap of A0. Although no in-plane mode appears in the frequency range of Gap 2, two anti-plane SH modes interrupt it. So the Gap 4 is the only absolute bandgap in which no plate wave is feasible. Gap 3 is also the first exclusive bandgap of principal symmetric mode S0. The second complete bandgap observed among displayed modes is Gap 5 between in-plane modes 7th and 8th which is interrupted by one antiplane mode. However, the focus of this research is to study complete bandgaps of in-plane modes which stop propagation of low order Lamb waves regardless of anti-plane modes, e.g. Gap 2 and Gap 5 in Figure 4(b). The results of Figure 4 show only the band structure of a PhCr unitcell with specific aspect ratio and filling fraction. So the existence and variation of bandgaps between subsequent frequency modes are further investigated for the first few in-plane modes at different filling fractions and aspect ratios. First 10 Lamb bandgaps which are likely to open between subsequent pairs of first eleven in-plane modes are studied, orderly named 1 to 10 (e.g. the Gap 2 in Figure 4(b) that is 3rd likely gap opened between 3th and 4th in-plane modes). Figure 5 presents the variation of relative bandgap width of two major gaps 3rd and 7th (Figure 4(b)) versus aspect ratio of plate unitcell. Relative bandgap width is defined as frequency bandwidth over midgap frequency ( ∆ω / ω ). High relative bandgap width is desired in design of PhCrs to manipulate lowest frequencies achievable by specified unitcell size. The filling fraction is 0.5 and the length of unitcell with fixed unit thickness varies to change its aspect ratio. Total relative bandgap of all 10 likely gaps opened within this range is also delivered showing the contribution of other gaps. Accordingly, maximum total relative gap occurs at r = 1.1 while for 3 rd and 7th gaps it happens at r = 1.9 . The two dash lines plot the gradient of midgap frequency of 3rd and 7th gaps versus aspect ratio, confirming lower frequency gap at higher aspect ratios. This is justified by bigger feature size of unitcells at higher aspect ratios (for specified thickness) that can affect larger wave lengths carrying lower frequencies. At high ratios the 3rd gap diminishes, however 7th gap remains and rises a bit. 13 1 Sum (10 gaps) Relative bandgap width 0.9 3rd Gap 0.8 7th Gap 0.7 Normalised midgap freq. 3rd Gap 0.6 Normalised midgap freq. 7th Gap 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 r=a/h Figure 5. Variation of relative bandgap width of 3rd , 7th Lamb gaps and sum of first 10 likely gaps as well as gradient of normalized midgap frequencies versus aspect ratio of regular PhCr unitcell with centric scattering inclusion, constant thickness and v f = 0.5 . The total relative bandgap width climbs after around r = 7 where 3 rd band vanishes, explaining significant contribution of other gaps in this range. For more thorough understanding, the opening and variation of all 10 likely bandgaps at a set of aspect ratios 1, 2, 4 and 10 versus filling fraction are scrutinized and presented in Figure 6. The left side figures show gradient of bandgap widths for thickness h = 1mm . The right side ones show gradient of relative bandgap width of first major gap as well as total relative bandwidth of all gaps. At aspect ratio r = 1 the 3 rd and 6 th gaps are noticeable. At aspect ratio r = 2 the 3 rd gap is outstanding over the range of filling fractions and therefore the total relative bandwidth follows the 3rd gap and have low difference with it. As the aspect ratio rises, more gaps open and grow leading to considerably higher total bandgap compared to 3 rd gap. Eventually at r = 10 the 3 rd gap is absent and 4th gap stands instead as the first major gap. The 3 rd gap has maximum bandwidths at filling fraction of around v f = 0.4 ~ 0.6 , however the total bandwidth and other gaps have multiple extremes at different filling fractions. 1 st gap doesn’t open at all and 2nd gap is minor at all cases. 14 800 600 400 200 0 0 _0.5 Gap No. 2 3 4 5 6 7 8 9 1 10 Relative bandgap width Bandgap width (kHz) 1000 0.8 Sum (10 gaps) 0.6 0.4 0.2 0 0 0.2 0 _0.5 Gap No. 2 3 4 5 6 7 8 9 1 10 0.8 _0.5 80 60 40 20 0 0 _0.5 1 3rd Gap 0.2 0 0.2 0.4 _ 0.6 0.8 1 r=2 Gap No. 2 3 4 5 6 7 8 9 1 10 0.8 Sum (10 gaps) 3rd Gap 0.6 0.4 0.2 0 0 0.2 0.4 _ 0.6 0.8 1 r=4 Gap No. 2 3 4 5 6 7 8 9 1 10 Relative bandgap width Bandgap width (kHz) 100 0.8 0.4 r=4 120 0.6 0.6 0 Relative bandgap width Bandgap width (kHz) 0 _ Sum (10 gaps) r=2 350 300 250 200 150 100 50 0 0.4 r =1 Relative bandgap width Bandgap width (kHz) r =1 700 600 500 400 300 200 100 0 3rd Gap Sum (10 gaps) 0.6 4th Gap 0.4 0.2 0 0 0.2 0.4 _ 0.6 r = 10 r = 10 (a) (b) 0.8 1 Figure 6. (a) Bandwidth of first 10 likely Lamb gaps for thickness 1mm and (b) relative bandwidth of first major Lamb gap and total relative bandwidth of first 10 likely Lamb gaps, versus filling fraction of tungsten inclusion v f at different aspect ratios r 15 4 Topology optimization studies Following the results presented in Section 3 regarding modal band structure analysis of few first in-plane modes, two bandgap objectives are defined for topology optimization of 1D PhCr plate unitcells as follows: (i) Bandgap Objective 1: Maximizing relative bandwidth of the 3rd Lamb gap (between in-plane modes 3rd and 4th) as the major low frequency gap stopping propagation of principal symmetric and asymmetric Lamb wave modes (Figure 4(b)) (ii) Bandgap Objective 2: Maximizing the total relative bandwidth of 10 Lamb gaps (if any) within first 11 in-plane modes prohibiting propagation of low order lamb waves. Bandgap Objective 2 is defined so that the feasibility of maximizing Lamb gap width over a wide frequency range is investigated. Existence and significance of Lamb gaps opening in this range, of course for centric PhCr topology, were interrogated in previous section. The length of unitcell is divided into 50 vertical layers evenly which the material type of each layer represents a design variable. So the optimum topology is sought by defining the best arrangement of stiff (tungsten) and soft (silicon) strips within these layers. Two definitions of design space are introduced: one by prescribing symmetric topology for the unitcell with respect to its center line leading to a reduced design space with 25 independent variables; and the other one with no symmetric constraint which has 50 independent design variables. Hence each independent variable stands for two percent of filling fraction for unconstrained topology and four percent for symmetric one. Execution of multiobjective optimization for each value of filling fraction results in a number of Pareto front topologies at those adjacent values which provide higher bandgap width. However, the Pareto front doesn’t include all possible filling fractions and a set of filling fractions from 0.06 to 0.88 are searched to define relevant optimized topologies. The optimum solution obtained for specific filling fraction is not necessarily the best and search for an adjacent value may lead to slightly improved solution. Hence, the Pareto front solutions obtained for various filling fractions are collected and the best solutions are defined as optimized topology. GA operation parameters introduced in Section 2.3 including population size, mutation and crossover probability as well as termination strategy are important factors in ensuring the efficiency and reliability of optimization. It is usually difficult to assign definite operation probabilities and population size for GA since they are highly problem dependent. Relatively larger population size, with respect to the number of design variables, provides wider search space for the GA but imposes high computational cost, though; larger population may cause 16 faster convergence after fewer generations and vice versa. For present study concerning optimization of bimaterial 1D periodic PhCrs, all randomly generated individuals are practically feasible, in contrary to e.g. single material (solid-void) topology optimization in which disconnected topologies are produced frequently. So, entire population unconditionally contributes to evaluation and optimization procedure that leads to faster convergence for specified population size. High crossover probability is desired as new designs produced via this GA operation generally make better generations. However, high crossover probability near to 1 provides a narrow gap for old generation to be reproduced just by mutation. Mutation has great role in introducing new designs by randomly altering design variables. However the mutation probability should be defined with much care and generally small enough to marginally exploit new alternatives while avoiding damage to the original design. As for the stopping criteria, often maximum number of generations is prescribed in which no improvement is observed for a number of generations before reaching the maximum limit. The GA parameters chosen for optimization execution of present work are listed in Table 2. Their reliability is examined by convergence performance and repeatability of results through several runs for specified objective. Table 2. GA parameters used for multiobjective optimization of PhCr plate unitcell Design Population Crossover Mutation Design space Generations variables size probability probability Symmetric 25 75 40 0.95 0.02 Unconstrained 50 120 80 0.95 0.02 In order to verify each individual topology in GA population, 15 points along first irreducible Brillouin zone are searched. However for calculating the frequency band structure and relative bandwidth of achieved topologies, 50 points are searched for higher accuracy. 4.1 Bandgap Objective 1 Initially unitcell with aspect ratio r = 2 is studied showing high relative bandwidth in regular centric topology for specific plate’s thickness (Figure 5). GA multiobjective optimization is so carried out to maximize the relative bandwidth of 3rd Lamb gap (Eq. (12)) as well as minimizing the deviation from the desired filling fraction of scattering inclusion (Eq. (13)). The Pareto front and relevant convergence history for the case with reference filling fraction v f = 0.12 are shown in Figure 7. Figures 7-a & b illustrate optimization of symmetric topology while Fig.7-b & c depict optimization of unconstrained topology. The number of solutions achieved from 17 optimization with unconstrained topology is more than symmetric one. Actually, the symmetric constraint decreases the diversity of design space and provides solutions in 0.04 steps of filling fraction. Also the number of Pareto front solutions varies based on the individuals sorted in the first rank, and the non-dominated results pertaining to other filling fractions may be located in higher ranks. The algorithm converged to the topology with desired filling fraction v f = 0.12 (i.e. F2 = 0 ) after 7 and 8 generations for symmetric and unconstrained optimizations, respectively. Afterwards, the Pareto solution with lowest bandgap objective F1 relates to this filling fraction which converges at early stages of optimization (Figure 7.b and d). (a) (b) (c) (d) Figure 7. Pareto front and convergence history of GA optimization for PhCr plate unitcell with r = 2 and maximized Bandgap Objective 1 at v f = 0.12 , (a) and (b) symmetric topology, (c) and (d) unconstrained topology The relative frequency bandwidth ( ∆ω / ω ) of 3rd gap for all obtained Pareto front solutions for various filling fractions is then calculated and shown in Figure 8. Moreover, the relative bandwidth of regular centric topologies versus filling fraction is inserted in this figure to somehow verify the efficiency of obtained optimized topologies. 18 Relative band width (3rd gap) 0.4 0.3 0.2 Regular (Centric) Pareto front solutions-Symmetric 0.1 Pareto front solutionsUnconstrained 0 0 0.2 0.4 _ 0.6 0.8 1 Figure 8. Pareto front solutions obtained from multiobjective topology optimization of silicon-tungsten PhCr plate unitcell with aspect ratio r = 2 versus filling fraction of scattering tungsten inclusion v f , for maximized Bandgap Objective 1 By comparing the results of topology optimization with regular centric topology, obviously the relative bandwidth is increased in the range 0.06 < v f < 0.4 and converges to the centric topology for the remaining. Moreover, the bandgap widths of best solutions for both symmetric and unconstrained topologies are close, although unconstrained solution is much better at v f = 0.08 . The best optimized topologies which have increased bandgap width compared to regular centric topologies are collected from Pareto front solutions and presented in Figure 9(a) and (b) for symmetric and unconstrained cases, respectively. The unconstrained topologies are rearranged so that the scattering part is concentrated at the right hand of unitcell. Regarding these unconstrained topologies, despite having no symmetric constraint, the solutions are perfectly symmetric at v f = 0.28 and 0.32 (with regard to periodicity of unitcell). However, getting asymmetric topology in some other cases could be due to the limitation of introduced design space that includes only 50 layers in unitcell model with no possible symmetric topology at certain filling fractions e.g. v f = 0.26 . Furthermore, the plate wave band structures of unitcells with regular centric topology as well as optimized ones are presented in Figure 10 for two arbitrary filling fractions 0.20 and 0.28. Accordingly, the relative bandwidth of 3rd gap, highlighted in grey, is widened and the midgap frequency is lowered at both cases. For v f = 0.20 the upper edge of bandgap is increased a bit, and the bandgap is mostly extended by its lower edge. However, for v f = 0.20 the upper edge of 19 the gap is also lowered a bit. For centric topology, the 3rd gap is actually limited by the first asymmetric Lamb gap at k d = 0 while the first Symmetric Lamb gap at k d = 1 is much wider. But for optimized topology with maximized width of 3 rd gap, the Symmetric Lamb gap is compromised; however it is still a bit more than Asymmetric Lamb gap. Also it should be noted that 3 rd gap of centric topology is an almost absolute plate wave gap while for optimized topologies it is solely a Lamb gap and is interrupted by an anti-plane mode (dash lines). 20 v f = 0.08 v f = 0.12 v f = 0.16 v f = 0.20 v f = 0.24 v f = 0.28 v f = 0.32 v f = 0.36 v f = 0.40 (a) v f = 0.06 v f = 0.08 v f = 0.10 v f = 0.12 v f = 0.16 v f = 0.18 v f = 0.20 v f = 0.22 v f = 0.24 v f = 0.26 v f = 0.28 v f = 0.30 v f = 0.32 v f = 0.34 v f = 0.36 (b) Figure 9. Optimized topologies for silicon-tungsten PhCr plate unitcell with aspect ratio r = 2 versus filling fraction of scattering tungsten inclusion v f , for maximized Bandgap Objective 1, (a) Symmetric, (b) unconstrained topology 21 Regular (Centric) Optimized-Symmetric Optimized-Unconstrained (a) v f = 0.20 (b) v f = 0.28 Figure 10. Plate wave band structure and 3rd Lamb wave bandgap of regular PhCr with centric scattering inclusion as well as optimized symmetric-unconstrained topologies for (a) v f = 0.20 and (b) v f = 0.28 As discussed in previous section, aspect ratio of PhCr plate unitcell is an important parameter affecting its band structure. Thus, other alternate aspect ratios r = 3 and r = 4 are studied further. Since no significant difference was observed in obtained symmetric/unconstrained topologies for aspect ratio r = 2 , just optimum symmetric topologies are defined for other aspect ratios. The efficiency of achieved Pareto front solutions and optimum topologies in terms of relative bandwidth of 3 rd gap are depicted in Figure 11(a) and b for r = 3 and r = 4 , respectively. 22 Relative band width (3rd gap) 0.4 0.3 0.2 Regular (Centric) 0.1 Optimised-Symmetric Pareto front solutionsSymmetric 0 0 0.2 0.4 vf 0.6 0.8 1 (a) Relative band width (3rd gap) 0.4 0.3 0.2 Regular (Centric) 0.1 Optimised-Symmetric Pareto front solutionsSymmetric 0 0 0.2 0.4 _ 0.6 0.8 1 (b) Figure 11. Pareto front solutions obtained from multiobjective topology optimization of silicon-tungsten PhCr plate unitcell with aspect ratio (a) r = 3 and (b) r = 4 , versus filling fraction of scattering tungsten inclusion v f , for maximized Bandgap Objective 1 The optimized results show increased bandgap width in the range 0.08 < v f < 0.4 and 0.16 < v f < 0.4 for aspect ratios 3 and 4, respectively. But for aspect ratio 3 the improved range of filling fractions is narrower and the raise in bandwidth is considerably less. The Pareto front solutions with improved bandgap properties, as compared to the relative bandgap width of centric topologies, are only included in Figure 10. Similar to the results of aspect ratio 2, the optimized results again converge to the regular centric ones for the remaining portion of filling fractions. This is observable in the relevant topologies delivered in Figure 12 in which those out of ranges mentioned above are centric. 23 v f = 0.08 v f = 0.12 v f = 0.16 v f = 0.20 v f = 0.24 v f = 0.28 v f = 0.32 v f = 0.36 v f = 0.40 (a) v f = 0.08 v f = 0.12 v f = 0.16 v f = 0.20 v f = 0.24 v f = 0.28 v f = 0.32 v f = 0.36 v f = 0.40 (b) Figure 12. Optimized symmetric topologies of silicon-tungsten PhCr plate unitcell with aspect ratio (a) r = 3 and (b) r = 4 versus filling fraction of scattering tungsten inclusion, for maximized Bandgap Objective 1 4.2 Bandgap Objective 2 As another objective, in this section the unitcell topology with highest total relative bandwidth of first 10 Lamb gaps (if any) is explored for aspect ratio r = 2 , in order to maximize attenuation of low order Lamb waves in this range. GA multiobjective optimization is therefore performed for both symmetric and unconstrained topologies and resultant Pareto front solutions are collected as shown in Figure 14. The Pareto front and relevant convergence history for the case with reference filling fraction v f = 0.08 is shown in Figure 13. For example, Figures 13-a&b show optimization of symmetric topology while Figs. 13-c&d optimization of unconstrained topology. The algorithm converged to the topology with desired filling fraction v f = 0.08 (i.e. F2 = 0 ) after 4 and 13 generations for symmetric and unconstrained optimizations, respectively. Afterwards, the Pareto solution with lowest bandgap objective F1 relates to this filling fraction which is maximized and remained unchanged for considerable number of generations (Figure 13.b & d). 24 (a) (b) (c) (d) Figure 13. Pareto front and convergence history of GA optimization for PhCr plate unitcell with r = 2 and maximized Bandgap Objective 2 at v f = 0.08 , (a) and (b) symmetric topology, (c) and (d) unconstrained topology According to the Figure 14, the optimized solutions have expanded total relative bandgap width in the filling fraction range 0.08 < v f < 0.6 compared to regular centric topologies. This is wider than the corresponding range of 3rd bandgap (Figure 8). More importantly, unconstrained results obtained for 0.08 < v f < 0.20 are considerably better than symmetric results with a local peak at v f = 0.26 . For other filling fractions the symmetric and unconstrained results are close and differ negligibly. The results again converge to the centric topology for a portion of filling fractions ( v f ≥ 0.6 ). However, the optimized solutions obtained for 0.6 < v f < 0.8 have slightly lower total bandgap width as compared to known centric topologies. This shortcoming is arisen from the fact that the number of points searched on the Brillouin zone for any verifying iteration of GA optimization is chosen to be 15 for the sake of computational cost. But the total bandgap width of optimized topologies and centric ones (Figure 14) are calculated using 50 points which is more accurate. As shown later on, in Figure 16, the first eleven modal frequencies of interest have several sharp approaching points along the Brillouin zone, varying for different topologies. 25 That is why the calculated total bandgap width of first 10 likely gaps is sensitive to the chosen number of search points. 0.7 Relative band width (10 gaps) 0.6 0.5 0.4 0.3 Regular (Centric) Pareto front solutions-Symmetric Pareto front solutions-Unconstrained Best Pareto solutions-Symmetric Best Pareto solutions-Unconstrained 0.2 0.1 0 0 0.2 0.4 _ 0.6 0.8 1 Figure 14. Pareto front solutions obtained from multiobjective topology optimization of silicon-tungsten PhCr plate unitcell with aspect ratio r = 2 versus filling fraction of scattering tungsten inclusion v f , for maximized Bandgap Objective 2 A selection of optimized topologies corresponding to the best Pareto front solutions of Figure 14 are presented in Figure 15(a) and b for symmetric and unconstrained design spaces, respectively. Bearing in mind the periodicity of unitcell, the symmetric and unconstrained topologies are exactly the same for filling fractions 0.24 and 0.48. 26 v f = 0.08 v f = 0.12 v f = 0.16 v f = 0.20 v f = 0.24 v f = 0.28 v f = 0.32 v f = 0.40 v f = 0.44 v f = 0.48 v f = 0.52 v f = 0.56 (a) v f = 0.08 v f = 0.12 v f = 0.16 v f = 0.20 v f = 0.24 v f = 0.28 v f = 0.32 v f = 0.40 v f = 0.44 v f = 0.48 v f = 0.52 v f = 0.56 (b) Figure 15. Obtained optimized topologies of silicon-tungsten PhCr plate unitcell with aspect ratio r = 2 versus filling fraction of scattering tungsten inclusion v f , for maximized Bandgap Objective 2 27 Regular (Centric) Optimized-Symmetric Optimized-Unconstrained (a) (b) Figure 16. Plate wave band structure and Lamb wave bandgaps of regular PhCr with centric scattering inclusion as well as optimized symmetric and unconstrained topologies for (a) v f = 0.20 and (b) v f = 0.48 ; Bandgaps opened within first eleven in-plane modes of interest are highlighted by grey area. For other filling fractions over 0.24, the symmetric and unconstrained solutions have converged to slightly different topologies due to negligible difference in the modal band structure of corresponding ones or impossible symmetric topologies at certain frequencies. Besides, the modal band structures of regular centric unitcells as well as optimized symmetric and unconstrained topologies are shown in Figure 14(a) and (b) for two arbitrarily chosen filling fractions 0.12 and 0.20, respectively. Lamb bandgaps opened within first eleven modes, highlighted by grey area, represent the increased total bandgap width of achieved optimum topologies as compared to regular centric ones. For v f = 0.12 , the band structure associated with unconstrained topology has an additional 10th gap opened between modes 10 and 11 with respect 28 to the symmetric one, while other 3 lower gaps have changed a bit. As for other filling fraction v f = 0.20 , a remarkable point is that the frequency range of first eleven modes of optimized topologies is contracted compared to the centric topology, with higher contraction in symmetric topology. This is expected as the highest total relative bandgap width (maximum bandgap widths at lowest midgap frequencies) is sought as the main objective of optimization. In fact the dimensionless frequency range of first eleven modes is pulled down from ωd = 3.38 for centric topology to 2.65 and 2.87 for optimized symmetric and unconstrained topologies, respectively. So in the same frequency range in which the eleven modal responses of centric unitcell topology appear ( 0 ≤ ω d ≤ 0.34 ), two more modes 12 and 13 exist for optimized topologies. Although the total bandgap width opened within first eleven modes of unconstrained topology 0.53 is considerably higher than symmetric one 0.43, it is actually effective in a wider frequency range. Nonetheless, the total bandwidth of gaps opened in the frequency range 0 ≤ ω d ≤ 0.34 , taking into account extra gaps 11th and 12th, is 0.66 for unconstrained topology which is still considerably higher than that of symmetric topology 0.54. Moreover, unlike the optimized topologies with widest 3rd gap, the width of first Symmetric Lamb gap is narrower than or equal to the Asymmetric one. 4.3 Prescribing definite symmetric topology Considering all optimised topologies obtained for Bandgap Objectives 1 and 2 described in Sections 4.1 and 4.2, definite symmetric topology with enhanced bandgap efficiency is prescribed and examined herein. To serve this purpose symmetric topology as shown in Figure 17(a) is prescribed in which the scattering inclusion is comprised of two identical counterparts located symmetrically in the unitcell. The only variable of this prescribed topology for specified filling fraction and aspect ratio is the relative distance of two counterparts D or their distance to the unitcell’s edge b . Assumption of unitcell with 50 strip layers in the optimization procedure, for the sake of computational efficiency, limited the study so that symmetric topology was not feasible at certain values. A more versatile unitcell including 100 layers is therefore modelled and studied in this section. Let’s start with a unitcell of aspect ratio r = 2 . The sensitivity of Bandgap Objectives 1 and 2 to the normalized topology variable b / a is defined at different filling fractions 0.2, 0.5 and 0.8 as depicted in Figure 17(b). Apparently, the variable b / a can vary from zero to a maximum value depending on the filling fraction. In Figure 17(b) the bandgap width has two peaks at two values of b / a corresponding to two unitcell topology modes forming the same PhCr plate in a periodic structure. Hence, b / a = 0 and b / a = Max. both stand for regular centric topology in 29 which two counterparts stick together. For v f = 0.2 the relative bandwidth of 3rd gap is maximum at b/a = 0.09 & 0.31 and the total relative bandwidth of first 10 likely gaps is maximum at b/a = 0.12, 0.14, 0.26 & 0.28. Regarding v f = 0.5 the maximum bandwidth of 3rd gap corresponds to the centric topology but the total relative bandgap width appears at b / a = 0.07 & 0.18 . Finally the best prescribed topology for filling fraction v f = 0.8 to gain both Bandgap Objectives is the centric one. (a) 3rd Gap, _ =0.2 3rd gap, _ _ 3rd gap, =0.5 =0.8 Relative bandgap width 0.7 Sum (All gaps), _ _ Sum (All gaps), =0.2 =0.5 Sum (All gaps), _ =0.8 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 b/a 0.3 0.4 (b) Figure 17. (a) Schematic of prescribed symmetric topology for PhCr plate unitcell with r = 2 , (b) gradient of relative bandgap width of 3rd Lamb gap and total relative bandwidth of first 10 likely gaps versus the topology ratio b / a . For simplicity the focus of study is narrowed further to the best unitcell’s topology mode with higher b / a (Right side peaks in Figure 17(b)) or least D/ a corresponding to unitcell with concentrated scattering inclusion at the center. Then all feasible symmetric filling fractions from 30 0.02 to 0.9 are analyzed to define the topology variable b / a or D / a associated with the unitcell having maximum relative bandgap width as shown in Figure 18. According to these results, the best topology variable D/ a has steeper variation versus filling fraction as compared to b / a . It means that the relative bandgap width is essentially determined by the external edges of scattering inclusion rather than its internal edges. The efficiency of best prescribed topologies versus filling fraction can now be investigated relative to the centric topology and previously optimized topologies. So, the relevant results are synthesized for this purpose in Figure 19(a) and (b) showing relative bandwidth of 3rd gap and total relative bandwidth of first 10 likely gaps, respectively. 0.5 0.5 3rd Gap 3rd Gap 0.4 0.4 Sum (10 gaps) Sum (10 gaps) b/a D/a 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 _ (a) _ 0.6 0.8 1 (b) Figure 18.Variation of topology ratios (a) b / a and (b) D / a (Figure 17(a)) versus filling fraction of scattering tungsten inclusion v f for maximized Bandgap Objectives 1 and 2 at aspect ratio r = 2 31 Relative band width (3rd gap) 0.5 0.4 0.3 Regular (Centric) 0.2 Optimised-Symmetric Optimised-Unconstrained Best prescribed topologies 0.1 Prescribed topology (b/a=0.30) Prescribed topology (b/a=0.31) 0 0 0.2 0.4 (a) _ 0.6 0.8 1 0.7 Relative band width (10 gaps) 0.6 0.5 0.4 0.3 0.2 Regular (Centric) Best prescribed topologies Optimised-Symmetric Optimised-Unconstrained 0.1 0 0 0.2 0.4 _ 0.6 0.8 1 (b) Figure 19. relative bandgap width of best prescribed topologies as well as centric and optimized topologies of silicon-tungsten PhCr plate unitcell with aspect ratio r = 2 versus filling fraction of scattering tungsten inclusion v f , for maximized (a) Bandgap Objective 1 and (b) Bandgap Objective 2 From Figure 19(a) it is evident that prescribed topologies are in excellent agreement with optimized ones in terms of relative bandwidth of 3rd gap except of unconstrained topologies which have slightly wider gap width. With regard to the Figure 18(a) the variable b / a varies from 0.28 to 0.33 for the filling fraction range 0.06 < v f < 0.4 in which the bandwidth of 3rd gap 32 is raised relative to the centric topology. Thus, over this filling fraction range, the relative bandwidth of third gap corresponding to unitcell topologies with constant variable b / a = 0.30 as well as b / a = 0.31are also included in Figure 17(a). Accordingly, the topologies with constant b / a have negligible deviation from the best topologies. Better agreement is observed at higher filling fractions for b / a = 0.30 and at lower filling fractions for b / a = 0.31. Figure 17(b) also shows good agreement of best prescribed topology and optimized symmetric topology in terms of total relative bandwidth of first 10 likely gaps. However, optimized symmetric topologies at v f = 0.24 & 0.28 and optimized unconstrained topologies in 0.08 < v f < 0.32 are considerably better than prescribed topologies. Also, the prescribed topologies show a minor improved total bandwidth in the range 0.64 < v f < 0.80 . Furthermore, unitcells with r = 1, 3 & 4 are studied in order to investigate the efficiency of prescribed symmetric topologies at other aspect ratios. The relevant results delivered in Figure 20 show that for r = 1 the best prescribed topology for widest 3rd gap is the centric one ( D / a = 0 ) at all filling fractions. But the total relative bandwidth is widened only in 0.18 < v f < 0.36 relative to the centric topology. For the cases r = 3 & 4 again the best prescribed topologies exactly coincide with the optimized symmetric topologies. The total bandwidth of first 10 likely gaps is also enlarged at both aspect ratios over a significant range of filling fractions 0.10 < v f < 0.72 and 0.04 < v f < 0.50 for aspect ratios 3 and 4, respectively. Consequently, the prescribed topologies introduced in this section are definite reliable substitutes for optimized topologies in a wide range of filling fractions specially in terms of relative bandwidth of 3rd gap. Getting optimized topologies of any arbitrary filling fraction through genetic algorithm demands implementation of unitcell having huge number of strip layers leading to an enormous design space. Hence the simple definite topologies prescribed herein can be reliably used for example in a continuous variation of filling fraction through a graded structure with nearly maximized bandgap efficiency. 33 D/a 0.3 Relative bandgap width 3rd gap-Centric 3rd gap-Optimised topology 3rd gap-Best prescribed topology Sum(10 gaps)-Centric Sum(10 gaps)-Best prescribed topology 3rd Gap Sum (10 gaps) 0.2 0.1 0 0.2 0.4 r=4 0.6 Sum (10 gaps) 0.1 0 0.4 0.6 0.8 0.8 1 0.6 0.8 1 0.6 0.8 1 0.6 0.4 0.2 0 0.2 0.4 _ 0.8 D/a 0 r =1 0.6 r =3 Sum (10 gaps) 0.4 0.4 _ 0 _ 0.1 0.2 0.2 0.8 1 3rd Gap 0 0 Relative bandgap width r =3 0 r=4 0.2 0.2 0.2 1 3rd Gap 0 0.4 _ 0.3 D/a 0.8 Relative bandgap width 0 0.6 0.6 _ 0.8 1 0.6 0.4 0.2 0 0 0.2 0.4 r =1 _ (a) (b) Figure 20. Left: Variation of topology ratio D / a (Figure 17(a)) and Right: relative bandgap width of corresponding prescribed topologies as well as centric and optimized topologies versus filling fraction of scattering tungsten inclusion v f for maximized Bandgap Objectives 1 and 2 at different aspect ratios, (a) r = 1 , (b) r = 3 and (c) r = 4 5 Frequency response analysis of finite PhCr plate structures 34 The modal band structure of 1D PhCr plate unitcell with infinite periodicity was defined in previous sections by applying Bloch-Floquet periodic boundary condition, and topologies with optimized bandgap properties were investigated for various filling fractions. Now, this section is dedicated to study the frequency response and elastic wave transmittance of finite plate structures comprised of consecutive silicon-tungsten PhCr unitcells. Different arrangements of topologies are studied to evaluate the depth and width of Lamb bandgaps associated with uniform and graded PhCr plate structures with regard to the band structure of perfectly periodic unitcells. In other words, these cases are selected so that comparing their transmission spectrums demonstrates the effect of unitcell optimization and unitcell gradient throughout the structure on its bandgap properties. Besides, the credibility of achieved modal band structure of associated PhCr plate unitcells is assessed. 20 mm (10 unitcells , h = 1mm and r = 2 ) Case 1 v f = 0.20 Case 2 v f = 0.20 (b / a = 0.31) Case 3 v f = 0.28 / 0.26 / 0.24 / 0.22 / 0.20 / 0.18 / 0.16 / 0.14 / 0.12 / 0.10 v f = 0.19 v f = 0.28 / 0.26 / 0.24 / 0.22 / 0.20 / 0.18 / 0.16 / 0.14 / 0.12 / 0.10 v f = 0.19 Case 4 b / a = 29 / 29 / 30 / 30 / 31 / 31 / 31 / 32 / 32 / 33 Case 5 v f = 0.28 / 0.26 / 0.24 / 0.22 / 0.20 / 0.18 / 0.16 / 0.14 / 0.12 / 0.10 v f = 0.19 (b / a = 0.31) v f = 0.12 / 0.16 / 0.20 / 0.24 / 0.28 / 0.28 / 0.24 / 0.20 / 0.16 / 0.12 v f = 0.20 (b / a = 0.31) Case 6 Figure 21. Finite PhCr plate structure cases including 10 unitcells with aspect ratio r = 2 and thickness h = 1mm (Optimized and prescribed topologies correspond to maximized Bandgap Objective 1) For this purpose, initially 6 cases are introduced as displayed in Figure 21 to mainly focus on their 3rd Lamb gap (Bandgap Objective 1). Then cases 7 to 9 presented in Figure 26 are defined to investigate their total bandgap width within first few modes (Bandgap Objective 2). All cases are plate structures including 10 unitcells of aspect ratio r = 2 and thickness h = 1mm leading to a total length of 20 mm. The average filling fraction of cases is set to 0.2 or a near value 0.19, from the filling fraction range in which significant improvement was observed in bandgap efficiency of optimized unitcells. 35 Cases 1 and 2 are uniform PhCr plate structures with centric and best prescribed topology, respectively. Prior to frequency response analysis of these cases, in order to get vision about their expected bandgap properties, the modal band structure of relevant unitcells are defined for filling fraction 0.2 and thickness h = 1mm , as shown in Figure 22. The frequency range 0 < f < 4.2MHz is chosen comprising of first eleven in-plane modes for both topologies. The Lamb gaps opened within this frequency range are marked by grey area. Also, the symmetric Lamb modes are indicated by bold dots to identify the exclusive bandgaps of Symmetric and Asymmetric Lamb modes. (a) (b) Figure 22. Plate wave band structure and bandgaps of first 11 Lamb modes along the first irreducible Brillouin zone for unitcell with h = 1mm and v f = 0.2 , (a) centric topology and (b) best prescribed topology for Bandgap Objective 1 (bold dots signify Symmetric Lamb mode) For frequency response analysis of PhCr plate structure, its FEM model is produced by standard FEM solver ANSYS APDL (ANSYS® Academic Research, Release 14.5) and the reliability of results obtained from developed model are validated. Any unitcell is modelled with 100 longitudinal and 25 transversal elements using ANSYS PLANE183 quadratic 8 node elements. A harmonic probing load of amplitude 0.1 N is applied to the left edge over the frequency range 0 < f < 4.2MHz and the relative transmission spectrum of induced oscillation to the top-right corner is determined in logarithmic scale dB. The strength of symmetric and asymmetric Lamb wave modes excited in a plate structure depends on the nature of probing load [39]. Since, in addition to complete Lamb gaps, there are partial bandgaps exclusively attenuating symmetric or asymmetric lamb modes (Figure 22), the transmission spectrum depends on the constitution of Lamb wave carried by the plate structure. 36 Thus, in order to define the effect of quality and polarization of incident wave on bandgap properties of PhCr structure, the transmittance of case structure 1 is defined for 4 different loading cases as listed in Table 3. Table 3. Loading cases applied for frequency response analysis No. Polarization Amp. (N) Description 1 Transversal 0.1 Point load on top left-corner, no bounded silicon unitcell 2 Transversal 0.1 Point load on top left-corner, with bounded silicon unitcell 3 Longitudinal 0.1 Point load on top left-corner, with bounded silicon unitcell 4 Longitudinal 0.1 Distributed load on left edge, with bounded silicon unitcell Loading case 1 corresponds to the model with a point load of transversal polarization applied to the top-left corner of structure. Of course, point load in this 2D cross sectional model is equivalent to a uniform load per unit width of plate structure along z axis (Figure 1). The load’s amplitude is selected to be small enough in order to avoid large displacements and so geometrical nonlinearity. Loading case 2 is similar to the loading case 1 with the only difference that a pure silicon unitcell is added at both ends between the probing point and structure as well as the receiving point and structure (Figure 23). Probing Receiving Figure 23. Bounding a pure silicon unitcell two both ends of structure for improved excitation and receiving of Lamb waves The transmission spectrums of case 1 subjected to all 4 loading cases is presented in Figure 22(a). The dips marked by numbered arrows stand for attenuation of frequency response due to existence of bandgaps. The deeper a gap is the higher attenuation occurs. The results show improved bandgap properties for loading case 2 compared to loading case 1, as deeper and wider gaps appear in this case, showing better agreement with obtained modal band structure. This is evident specially in dip 1 concerning the 3 rd Lamb gap. Although the loading case 1 can reliably provide the frequency response of the finite PhCr plate structure of interest, the loading case 2 shows better agreement with obtained modal band structure of unitcell with infinite periodicity. In fact the bounded pure silicon unitcells make better simulation of plate waves for the incident oscillation approaching the PhCr plate structure and transmitted oscillation leaving it. So in all other loading cases this condition is applied where a pure silicon unitcell is added to both ends of structure. 37 Regarding the spectrum of loading case 2 dips 1, 3 and 4 correspond to the 3rd, 8th and 9th Lamb gaps around 1.5, 2.9 and 3.6 MHz as shown in relevant band structure Figure 22(a). Dips 5 and 6 are related to the exclusive bandgaps of asymmetric Lamb waves. Moreover the lower band of dip 3 is extended due to existence of an asymmetric Lamb gap beside 8th Lamb gap. The symmetric Lamb modes are indicated by bold dots in modal band structure Figure 22 to highlight the exclusive bandgaps of symmetric and asymmetric Lamb modes. Although dip 2 corresponding to the 5th Lamb gap is expected around 2.4 MHz, it has been faded by strong adjacent asymmetric Lamb gap 6. It is well defined that applying a transversal load on just top side of plate structure like loading case 2 causes dominant excitation of asymmetric Lamb waves with leading transversal oscillations. That is why strong attenuation of asymmetric Lamb gaps is observed for this loading case. In loading case 3 a probing load of longitudinal polarization is applied to the top-left corner and transmission of longitudinal oscillation is defined. Loading case 3 still have asymmetric nature; nonetheless its horizontal polarization boosts symmetric Lamb modes. In the spectrum of loading case 3, as compared to the loading case 2, the asymmetric Lamb gaps 5 and 6 disappear and Lamb gap 2 becomes recognisable at around 2.4 MHz. Regarding other loading case 4 the left edge of plate structure is subjected to a uniformly distributed longitudinal load in order to excite pure symmetric Lamb modes. The relevant spectrum clearly shows appearance of robust symmetric Lamb gaps 7, 8, 9 and 10 around 1.5, 2.6, 3.5 and 4 MHz. The symmetric Lamb gap 7 is developed from just below 1.0 MHz to around 2.1 MHz, and the symmetric Lamb gap 9 from just below 3.0 MHz to around 3.9 MHz as predicted by related modal band structure Figure 22(a). Plus, the results show that the depth of wave attenuation changes significantly for different loading cases as they lead to Lamb waves with various constructions. For instance, the Lamb gap 1 (loading case 2) and symmetric Lamb gap 7 (loading case 4) show very different attenuation depths of in order 37.3 dB and 105.0 dB at 1.5 MHz which is the midgap frequency of 3rd Lamb gap. The transmission spectrums of all 6 cases introduced in Figure 21 subjected to loading case 2 are then collected in Figure 24(b). Comparing the frequency response of introduced structures enables us to assess their bandgap efficiency in terms of 3rd Lamb gap (Bandgap Objective 1). 38 20 Transmission (dB) -20 -60 5 1 -100 2 Loading case 1 -140 Loading case 2 -180 7 6 3 8 10 4 Loading case 3 9 Loading case 4 -220 0 0.5 1 1.5 2 2.5 Frequency (MHz) 3 3.5 4 (a) Case 1 Case 3 Case 5 50 Case 2 Case 4 Case 6 Transmission (dB) 0 -50 -100 3rd gap -150 -200 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (MHz) (b) Figure 24. Frequency response of PhCr plate structure cases (Figure 21) over the frequency range 0~4.2 MHz covering first 11 in-plane modes, (a) case 1 subjected to various loading cases (Table 3) , and (b) cases 1 to 6 subjected to loading case 2 The case 2 is actually a uniform PhCr plate structure with filling fraction v f = 0.2 like case 1 but with unitcells of best prescribed topology for maximum bandwidth of 3rd gap. As shown in previous section, the best prescribed topology of unitcell with v f = 0.2 and r = 2 is in excellent 39 agreement with optimized topology. The Cases 3 to 5 are alternative graded PhCr plate structures (type A) in which the filling fraction of unitcell evolves gradually from 0.1 to 0.28 providing average filling fraction of 0.19 close to introduced uniform structures with v f = 0.2 . But different topologies are considered for unitcells of these cases as illustrated in Figure 21. Finally the case 6 with another alternative graded pattern (type B) of average filling fraction v f = 0.2 is introduced by unitcells having best prescribed topology. From Figure 24(b) it is easily understood that using unitcell with best prescribed topology makes 3rd Lamb gap significantly wider and deeper. However, the first exclusively asymmetric Lamb gap around 0.5 MHz is degraded. The 3rd bandgap of case 2 with best prescribed topology is extended from 1.28 to 1.66 MHz as shown in Figure 22(b). This is considerably wider than that of case 1 with centric topology extended from 1.4 to 1.59 MHz. For clarity two sections of transmission spectrum Figure 24(b) in the low frequency range as well as the frequency range corresponding to the 3rd gap are delivered in Figure 25(a) and (b), respectively. Accordingly, the attenuation depth of 3rd gap is increased from 37.3 dB (for centric topology) to 66.4 dB (for best prescribed topology). Moreover, the results confirm that applying different arrangements of unitcells with various filling fractions to introduce graded PhCr plate, with almost the same average filling fraction, nearly maintains its bandgap efficiency. Using a graded pattern of centric unitcells (case 3) decreases the depth of 3rd gap from 37.3 to 33.3 dB bearing in mind that it has a bit lower filling fraction of 0.19 compared to case 1 with uniform structure. Likewise, the graded cases 4, 5 and 6 have 3rd gap depth of order 62.8, 63.2 and 65.5 dB with slight variation with respect to that of corresponding uniform structure (case 2) 66.4 dB. Case structure 5 with an arrangement of best prescribed topologies has slightly less gap depth compared to the case 6 with uniform prescribed topology of b / a = 0.31. But, the case 5 shows higher attenuation in low frequency band of 3rd gap as expected from Figure 19(a). Although the graded structure case 6 has nearest gap depth to the corresponding uniform structure case 2, it has filling fraction of 0.2 exactly the same as case 2. However the case 6 with gradient pattern type B shows wider 3rd bandgap as compared to cases 4 and 5 with gradient pattern type A and even corresponding uniform case 2. Of course, gradient of filling fraction of unitcells throughout the PhCr structure changes its overall stiffness and consequently principal modal response. Scrutinizing the few first modal responses of graded structures with corresponding uniform ones as shown in Figure 25(b) confirms this argument. 40 30 Transmission (dB) 10 -10 -30 -50 Case 1 Case 3 Case 5 Case 2 Case 4 Case 6 -70 1 1.1 1.2 1.3 1.4 1.5 Frequency (MHz) 1.6 1.7 1.8 (a) 60 50 Transmission (dB) 40 30 20 10 0 -10 Case 1 Case 2 -20 Case 3 Case 4 Case 5 Case 6 -30 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency (MHz) 0.7 0.8 0.9 1 (b) Figure 25. Frequency response of PhCr plate structure cases (Figure 21) over the frequency range (a) 1~1.8 MHz highlighting the differences in 3rd Lamb bandgap width and depth, and (b) 0~1.0 MHz highlighting the differences in low frequency response. 41 For example, the frequency response at modal frequencies 5th to 8th of PhCr structures is highly affected by the gradient of unitcell. The transmission at modal frequency 5th is highly attenuated by 23 dB form case 2 to case 4 or 5, by 18 dB from case 2 to case 6 and by 16 dB from case 1 to case 3. The same trend is observed for 6th frequency mode with even higher reduction rate. In contrast, the frequency response of 7th mode is escalated by 45 dB from case 2 to case 4, by 24 dB from case 2 to case 5 and by 27 dB from case 1 to case 3. A transmission rise is also present for 8th modal response. But, gradient pattern type B (case 6) shows negligible influence on transmission spectrum at modal frequencies 7th and 8th. As for second set of cases aimed at maximizing the Bandgap Objective 2, three structures are assumed. The cases 7 and 8 are uniform PhCr plate structures of filling fraction 0.2 with best prescribed topology and optimized unconstrained topologies, respectively. As shown earlier in Figure 19(b) these two topologies have considerable different efficiencies in total relative bandwidth of first 10 Lamb gaps. Last case 9 is a graded PhCr plate of type A with best prescribed topology and average filling fraction of 0.19. 42 20 mm (10 unitcells , h = 1mm and r = 2 ) Case 7 v f = 0.20 (b / a = 0.26) Case 8 v f = 0.20 Case 9 v f = 0.28 / 0.26 / 0.24 / 0.22 / 0.20 / 0.18 / 0.16 / 0.14 / 0.12 / 0.10 v f = 0.19 b / a = 0.23 / 0.23 / 0.23 / 0.24 / 0.26 / 0.27 / 0.26 / 0.26 / 0.26 / 0.26 Figure 26. Finite PhCr plate structure cases including 10 unitcells with aspect ratio r = 2 and thickness h = 1mm (Optimized and prescribed topologies correspond to maximized Bandgap Objective 2) (a) (b) (c) Figure 27. Plate wave band structure and bandgaps of first 11 Lamb modes along the first irreducible Brillouin zone for unitcell with h = 1mm and v f = 0.2 , (a) centric topology , (b) best prescribed topology and (c) optimized unconstrained topology for Bandgap Objective 2. The modal band structure of unitcells included in cases 7 and 8 as well as centric one are determined for r = 2 and h = 1mm as shown in Figure 27. Again the frequency range 0 < f < 4.2 MHz, including first eleven Lamb modes of interest for all 3 cases, is selected for analysis. Then the transmission spectrums of these cases subjected to loading case 2 are calculated as collected together in Figure 28. The spectrums of cases 1 and 2 are also included in Figure 28(a) as a reference to assess the effect of optimization on total band width of plate structures. 43 50 Transmission (dB) 0 -50 -100 -150 -200 Case 1 Case 2 Case 7 Case 8 -250 0 0.5 1 1.5Frequency 2 (MHz) 2.5 3 3.5 4 (a) 50 Case 7 Case 8 Case 9 Transmission (dB) 40 30 20 10 0 -10 -20 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency (MHz) 0.7 0.8 0.9 1 (b) Figure 28. Frequency response of PhCr plate structure cases (Figure 21 and Figure 26) over the frequency range (a) 0~4.2 MHz covering first 11 in-plane modes, and (b) 0~1.0 MHz highlighting the differences in low frequency response. From Figure 28(a) the improved wide bandgap efficiency of case 7 with prescribed symmetric topology is obvious as compared to case 1 with centric topology. Case 8 with optimized unconstrained topology shows even higher relative bandgap efficiency as compared to case 7. A narrow dip is present just below 2 MHz related to partial Lamb gap existing between two subsequent asymmetric Lamb modes in this range (Figure 27(c)). Deeper low frequency attenuation is observed in the range 2 to 3.5 MHz leading to higher relative band width of case 8 with respect to case 7. As predicted by the relevant band structure (Figure 27) a set of bandgaps 44 are present in the ranges 2.1~3.5 MHz and 2.2~3.3 MHz for cases 7 and 8, respectively, however with lower attenuation in the range 2.6~3.2 MHz for case 7. By comparing the transmission spectrum of case 7 and 2 one question arises. Why case 7 provides wider and deeper attenuation for 3rd Lamb gap around 1.5 MHz while the case 2 is supposed to show superior 3rd Lamb gap? This observation is justified by the fact that the dip shown around 1.5 MHz for case 7 is concerned with asymmetric Lamb gap. According to the Figure 27(b) the asymmetric Lamb gap in the left side of 3rd Lamb gap from 1.2 to 1.65 MHz is larger than symmetric one in the right side from 1.16 to 1.44 MHz (unlike Figure 22(b)). So, as mentioned earlier, the applied loading case 2 excites asymmetric Lamb waves dominantly leading to a wide and deep asymmetric Lamb gap in frequency response of case 7. The graded structure of case 9 with average filling fraction 0.19 exhibits good bandgap efficiency with respect to its equivalent uniform structure case 7 with filling fraction 0.2 . As for case 9 a bit lower attenuation is observed for 3rd Lamb gap around 1.5 MHz and high frequency range 3.0~3.5 MHz, while showing somewhat deeper attenuation in the range 1.7~3.0 MHz. Of course, this relative inconsistency could be explained by their different average filling fractions. Again the low frequency section delivered in Figure 28(b) shows how the principal modal responses of PhCr plate are altered by introducing gradient structure with varying unitcells. For example, the transmission is highly diminished at modal frequencies 5th and 6th by in order 32 and 27 dB, and highly escalated at subsequent modal frequencies 7th and 8th by in order 22 and 18 dB. Hence, the low frequency response of this graded PhCr plate is considerably manipulated besides maintaining its high frequency bandgap properties. In fact multiscale functionality is observed for this structure: in unitcell scale (with length 2 mm) the bandgap properties are altered from around 1.5 MHz, while in structure scale (with length 20 mm) the basic modal responses are influenced after around 0.15 MHz. 45 6 Conclusion Basic study was performed on optimum topology of 1D silicon-tungsten PhCr plate unitcell for highest specific Lamb bandgap width at prescribed filling fraction of tungsten scattering inclusion. Moreover the bandgap efficiency and multiscale functionality of gradient PhCr plate structures comprised of unitcells with various filling fractions and optimized topologies were investigated. Specialized FEM model was developed for this purpose capably calculating the dispersion curves and modal band structure of plate waves. The filling fraction dependency of low order Lamb gaps for regular topology of unitcell with centric scattering layer was studied considering various aspect ratios. Topology optimization was then performed successfully through multiobjective genetic algorithm (GA) in order to maximize the relative bandgap width of low order Lamb waves around particular filling fraction of scattering tungsten inclusion. Optimized topology was defined for two different bandgap objectives separately. Bandgap Objective 1 was defined as the widest relative bandwidth of lowest Lamb gap. Bandgap Objective 2 was also proposed as widest total relative bandwidth of 10 likely gaps within first 11 in-plane modes of interest. Besides, two types of design space with symmetric topology as well as constraint free topology were examined. The bandgap properties of optimized topologies were assessed as compared to regular centric topology. Consequently, the optimum topology proved to provide better efficiency in specific filling fraction range for both bandgap objectives. Of course, studying the modal band structure of few filling fractions in this range revealed that the existence of first absolute plane gap (including anti-plane modes) is penalized by this enhancement. The optimized topology converged to centric topology for the remaining portion of filling fractions. The Bandgap Objective 2 was increased over considerably wider range of filling fractions as compared to that of Bandgap Objective 1. The symmetric constraint on topology of unitcell had no considerable effect on efficiency of obtained topologies for Bandgap Objective 1. But major increase was observed in bandgap efficiency of unconstrained topologies, with respect to symmetric ones, achieved for Bandgap Objective 2 and for lower filling fraction range. Simplified symmetric topology was then prescribed inspired by obtained optimum ones to somehow introduce definite filling fraction based topology with improved bandgap efficiency and easy to fabricate. The prescribed topologies introduced in this study proved to be definite reliable substitutes for optimized topologies in a wide range of filling fractions specially for Bandgap Objective 1. 46 Finally, the frequency response analysis of a few cases confirmed the enhanced bandgap efficiency of finite PhCr plate structures comprised of consecutive unitcells of achieved topologies. Vibration attenuation occurred at the same frequency ranges predicted by modal band analysis of unitcell. Depending on the excited Lamb wave’s constitution high attenuation was observed coinciding with symmetric, asymmetric and complete Lamb gaps. Moreover, the results of this study confirmed that introducing graded PhCr plate, with almost the same average filling fraction as uniform plate, closely maintains its bandgap efficiency while altering the basic modal response of PhCr structure. Such a multiscale lattice structure has multi-functionality at two scales. On the one hand, the micromechanical topology is optimized in unitcell scale for maximum bandgap efficiency. On the other hand, the filling fraction of unitcell varies to create macromechanically graded structure and serve macroscale objectives, e.g. subwavelength wave manipulating or satisfying desired structural weight and/or stiffness efficiency. Further complementary study is being carried out by the authors on topology optimization of 2D PhCr plate unitcells to control plate waves’ propagation. 47 7 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 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Hedayatrasa, S., et al., Numerical modeling of wave propagation in functionally graded materials using time-domain spectral Chebyshev elements. Journal of Computational Physics, 2014. 258: p. 381-404. 49 Fig. 21 50 a b Fig. 22 51 Fig. 23 52 a b Fig. 24 53 a b Fig. 25 54 Fig. 26 55 a b c Fig. 27 56 a b Fig. 28 57 Highlights: • Optimum topology of 1D phononic crystal plate is studied versus filling fraction • Specialised FEM model is developed for modal band analysis of plate unitcell • Optimised topologies are assessed as compared to regular centric topologies • Definite symmetric topology is prescribed, inspired by obtained optimum topologies • Bandgap and multiscale functionality of graded phononic crystal plate are evaluated 58