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Bandgap design of three-phase phononic crystal by topological optimization X.K. Han, Z. Zhang

∗

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China

article

info

Article history: Received 25 June 2019 Received in revised form 13 December 2019 Accepted 13 December 2019 Available online 17 December 2019 Keywords: Phononic crystal Topological optimization Plane wave expansion Genetic algorithm

a b s t r a c t Material topologies in phononic crystals take the key role in wave controlling. A genetic algorithm with a plane wave expansion method was adopted to optimize the threephase phononic crystal for a larger relative band gap in both out-of-plane and in-plane wave modes. Fourier displacement property was used to calculate the structure function in the plane wave expansion. The mutation and crossover rates were calculated based on the adaptive GA method. Results indicate that the volume fraction and the symmetry are two key factors for the design of topological configurations. The relations between the key factors and the different topologies are correlated. © 2019 Elsevier B.V. All rights reserved.

1. Introduction As a kind of periodic artificial structure, phononic crystals (PnCs) have been widely studied with regard to the unique frequency characteristics related to acoustic and elastic wave propagations [1–9]. One of the remarkable features is the bandgap (BG), in which the elastic wave cannot propagate through the PnC. The bandgaps produced by PnC are valuable for both sound insulations and environmental noise controls [10–13]. According to the formation of BGs, PnCs can be divided into two categories, the Bragg scattering PnC or the local resonance PnC [14–16]. The BG width and center frequency can be controlled by material configurations and parameters [17–23]. To obtain the best structural performance, topology optimization is used to obtain material distributions within a prescribed design domain [24,25]. Initial topological optimization design of PnC is introduced by Sigmund et al. [26]. The method of moving asymptotes with the finite element method are adopted to optimize the two-dimensional PnC with the square lattice. The other optimization methods, like the genetic algorithm (GA) [27–30] and the bi-directional evolutionary structural optimization algorithm [31–33], are developed for the optimization of PnCs due to better convergence and higher efficiency. An adaptive GA with an improved fast plane wave expansion method [34] is used to optimize the PnC for the in-plane elastic wave attenuation. This method combined with GA and sparse point sampling-based Chebyshev polynomial expansion [35] is proposed for topology optimization of 2D PnCs with unknown-but-bounded parameters. GA combined with the classical homogenization results [36] in the case of the Helmholtz equation, which can be used to realize the geometrical transformation for anisotropic inhomogeneous media like the acoustic cloaking. The optimization of the one-dimensional PnC is mainly focused on the design of material thickness [37], while the material distribution becomes the main factor for the optimization of the two-dimensional PnC [38]. The optimization method mentioned above can also be used to design the PnC for unidirectional acoustic transmission [39] by maximizing the minimum imaginary ∗ Corresponding author. E-mail address: [email protected] (Z. Zhang). https://doi.org/10.1016/j.wavemoti.2019.102496 0165-2125/© 2019 Elsevier B.V. All rights reserved.

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X.K. Han and Z. Zhang / Wave Motion 93 (2020) 102496

Fig. 1. The unit cell. (a) 10 × 10 cell structure (b) displacement characteristic.

part of the wave vector in a specific direction. Combined with PDE-constrained optimization with suitable modifications, a systematic approach [40] is proposed to obtain a target bandgap by casting the metamaterial design problem as an inverse medium problem. Significant achievements have been made by combining topological optimization with PnC design. However, most previous studies are concentrated on designing PnC unit cell composed of two types of materials. Three-phase PnCs have rarely been investigated. However, many studies [41–44] revealed that the three-phase PnCs can show BGs in lower frequency domains. It is interesting to study the internal relations between the topologies of three-phase PnCs and their properties. This is the motivation for our current research. The plane wave expansion method is a simple and convenient method for the calculation of BGs. This method is successful in convergence when the materials are in all solid or all-liquid states [45–47]. Finite element software like Comsol and Ansys are usually used for BG calculations [48–51]. When GA is used for optimization, the plane wave expansion method is much faster and more convenient for data processing. The optimization design of the three-phase PnCs is studied to obtain the maximum relative BG for in-plane and out-ofplane wave modes by the plane wave expansion method and GA. The optimized topological configuration of the PnCs plate can then be obtained. The key factors, including volume fraction and symmetry, are correlated with different optimized topologies. 2. Model and method 2.1. Analysis of the three-phase PnC The two dimensional three-phase PnC is discussed. The lattice constant a is 0.02 m. The square lattice unit cell is divided into 10 × 10 elements in Fig. 1(a). The materials of the PnC unit cell are gold, rubber and epoxy resin (black represents gold, gray represents rubber, white represents epoxy resin). The Young’s modulus, the density and the shear modulus of gold are E1 = 8.5 × 1010 Pa, ρ1 = 19 500 kg/m3 and ν1 = 2.99 × 1010 Pa. The Young’s modulus, the density and the shear modulus of rubber are E2 = 1.175 × 105 Pa, ρ2 = 1300 kg/m3 and ν2 = 4 ×104 Pa. The Young’s modulus, the density and the shear modulus of epoxy resin are E0 = 4.35 × 109 Pa, ρ0 = 1180 kg/m3 and ν0 = 1.59 × 109 Pa. For each element in unit cell, the material parameters are defined as:

ρ (xe ) = xe (xe − 1)ρ2 + xe (2 − xe )ρ1 +

E(xe ) = xe (xe − 1)E2 + xe (2 − xe )E1 +

(1 − xe )(2 − xe )

ρ0 ,

(1a)

E0 ,

(1b)

2 (1 − xe )(2 − xe )

µ(xe ) = xe (xe − 1)µ2 + xe (2 − xe )µ1 +

2

(1 − xe )(2 − xe ) 2

µ0 ,

(1c)

where e ∈ [1,2,3, . . . , 100], xe = 0 or 1 or 2. There is no external force when the eigen field is studied. For the two-dimensional PnC, the wave equations [52] are shown in Eq. (2).

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For the in-plane mode:

∂ 2 ux ∂ ∂ ux ∂ uy ∂ 2 ux = [λ (r)( + ) ] + 2 µ (r) ∂t2 ∂x ∂x ∂y ∂ x2 ∂ ∂ ux ∂ uy + [µ(r)( + )], ∂y ∂y ∂x

(2a)

∂ 2 uy ∂ ∂ ux ∂ uy ∂ 2 uy = [λ(r)( + )] + 2µ(r) 2 ∂t ∂y ∂x ∂y ∂ y2 ∂ ∂ ux ∂ uy + [µ(r)( + )]. ∂x ∂y ∂x

(2b)

ρ (r)

ρ (r)

For the out-of-plane mode:

ρ (r)

∂ 2 uz ∂ uz ∂ ∂ uz ∂ = [µ(r)( )] + [µ(r)( )], ∂t2 ∂x ∂x ∂y ∂y

(2c)

where ux,y,z represents the displacements along x, y, and z direction; r is the space vector; the relation between the two µ(E −2µ) lame constants is λ = 3µ−E . The characteristic equations are obtained by the plane wave expansion [53]:

ω2

∑

ρ (G ′′ − G ′ )uxk +G

G′

⎤ ⎡ ′′ ′ ′ ′′ ∑ λ(G − G )(k + G )x (k + G )x ⎣+µ(G ′′ − G ′ )(k + G ′ )y (k + G ′′ )y ⎦ uxk +G = G′ +2(k + G ′ )x (k + G ′′ )x [ ∑ λ(G ′′ − G ′ )(k + G ′ ) (k + G ′′ ) ] y y x u , + +µ(G ′′ − G ′ )(k + G ′ )x (k + G ′′ )y k +G ′

(3a)

G

ω2

∑

ρ (G ′′ − G ′ )uyk +G

G′

⎤ ⎡ ′′ ′ ′ ′′ ∑ λ(G − G )(k + G )y (k + G )y ⎣+µ(G ′′ − G ′ )(k + G ′ )x (k + G ′′ )x ⎦ uyk +G = G′ +2(k + G ′ )y (k + G ′′ )y [ ∑ λ(G ′′ − G ′ )(k + G ′ ) (k + G ′′ ) ] x y ux , + +µ(G ′′ − G ′ )(k + G ′ )y (k + G ′′ )x k +G ′

(3b)

G

ω2

∑

ρ (G ′′ − G ′ )µzk (G ′ ) =

G′

∑

µ(G ′′ − G ′ )(k + G ′ ) · (k + G ′′ )uzk (G ′ ),

(3c)

G′

where ω represents the frequency, k is the wave vector in first Brillouin zone; G ′ is the reciprocal lattice vector. This method combined with GA and plane wave expansion is adopted to investigate the optimization of twocomponent PnC Ref. [2]. When the method is extended to the optimization of three-component material, the material distribution expression and the Fourier expansion coefficients of the parameters, which are important for the calculation of Eq.(3), should be changed. The Fourier expansion coefficient of the center element P0 for parameters is 0 when it is filled with the epoxy resin in Fig. 1(b). When P0 is filled with the gold or rubber, the Fourier expansion coefficient of the parameters can be expressed as:

⎧ 1 sin(Gx a/2π ) sin(Gy a/2π ) ⎪ ⎪ (g1,2 − g0 ) , G ̸= 0 ⎨ 100 Gx a/2π Gy a/2π go1,2 (G) = ⎪ 1 ⎪ ⎩ (g − g ) + g ,G = 0 100

1 ,2

0

(4)

0

where g represents the parameters ρ , λ, µ. According to the Fourier displacement property, the Fourier expansion coefficient of the parameters of the element Pr (Pr ∈P1 or P2 : the set of the elements including gold or rubber) in Fig. 1(b): gr1,2 (G) = go1,2 (G)eiGr ,

(5)

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For the unit cell: g(G) =

∑

gr1 (G) +

r ∈P1

=

∑

∑

gr2 (G)

(6)

r ∈P2

gr1 (G)δ1 (G) +

r

∑

gr2 (G)δ2 (G)

r

= go1 (G)

∑

eiGr δ1 (G) + go2 (G)

∑

eiGr δ2 (G)

r

r

= go1 (G)e(G) · δ1 + go2 e(G) · δ2 , where e(G) = [eiGr 1 , eiGr 2 , . . . , eiGr L ], δ = δ1 + δ2 = [δ (r 1 ), δ (r 2 ), . . . , δ (r 100 )],

⎧ ⎨0, δ (r) = 1, ⎩2,

⎧

else ⎨0, r ∈ P1 , δ1 (r) = 1, ⎩0, r ∈ P2

⎧

else ⎨0, r ∈ P1 and δ2 (r) = 0, ⎩2, r ∈ P2

else r ∈ P1 . r ∈ P2

2.2. The genetic algorithm The GA begins with initializing a population of Np = 20 chromosomes randomly. The chromosomes are then evaluated by the plane wave expansion method to obtain the RBG. The objective function is the RBG, and the optimization can be simplified as: To find ρ (xi )

( Max

2

mink ωj+1 (k , X ) − maxk ωj (k , X )

)

mink ωj+1 (k , X ) + maxk ωj (k , X )

(7)

st. st. ρ (xi ) = 0 or 1 or 2; sum X 1 = f1 · 100, 0 ≤ f1 ≤ 1; sum X 2 = f2 · 200, 0 ≤ f2 ≤ 1; X = X 1 + X 2 , 0 ≤ f1 + f2 ≤ 1 ; where X is the chromosome of the topological configuration, X 1 is a matrix, which only includes 0 and 1, X 2 a matrix, which only includes 0 and 2, k represents the wave vector, j is the band number, and f1 and f2 are the volume fraction of the gold and rubber, respectively. GA operation includes selection, crossover and mutation after population initiation. The fitness-based roulette wheel selection is adopted to filter the individuals with larger fitness to be the parents of the next generation in the selection operation. The multi-point crossover is used to avoid trapping the local maximum and to keep high diversity for the population. By adjusting the values of some genes of the chromosome in the mutation operation, generations with new individuals are obtained. The crossover rate Pc and the mutation rate Pm are defined by the adaptive GA from Srinivas [54]:

⎧ (Pc1 − Pc2 )(f ′ − favg ) ⎪ ⎪ , f ′ ≥ favg ⎨Pc1 − fmax − favg Pc = ⎪ ⎪ f ′ < favg ⎩Pc1 ,

(8)

⎧ (Pm1 − Pm2 )(fmax − f ) ⎪ , f ≥ favg ⎨Pm1 − fmax − favg Pm = , ⎪ ⎩Pm1 , f < favg

(9)

where f is the individual fitness. f ′ represents larger fitness of the two individuals. favg represents the average fitness. fmax is the maximum fitness. Pc1 is 0.9, Pc2 is 0.6, Pm1 is 0.1, and Pm2 is 0.001. The evolution flow chart is shown in Fig. 2. 3. Results and discussions 3.1. The optimization results for the in-plane mode and out-of-plane mode To validate the current algorithm, the band structures of a PnC are recalculated by the finite element software Comsol when the gold volume fraction is 0.16 and the rubber volume fraction is 0.2, as shown in Fig. 3. Single unit cell is used for

X.K. Han and Z. Zhang / Wave Motion 93 (2020) 102496

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Fig. 2. The genetic algorithm evolution flow chart.

Fig. 3. The verification PnC model.

calculation due to the periodicity. Due to the Bloch theorem, periodic boundary conditions are adopted on the interfaces between the adjacent unit cells. When wave vectors are swept on Brillouin zone, eigenvectors and the corresponding eigenfrequencies can be obtained. Then the band structures are obtained. As shown in Fig. 4(a), for in-plane mode, the BG calculated by algorithm of the verification PnC model ranges from 329 Hz to 691 Hz between the third and fourth bands. The RBG is 0.71. For the band structure calculated by Comsol, the BG ranges from 327 Hz to 693 Hz in Fig. 4(b). The RBG is 0.72. The error between the RBG calculated by algorithm and Comsol is the ratio of the difference between two RBGs and the RBG calculated by Comsol. The error is 0.05%. For out-of-plane mode, the bandgap is calculated by the current model. The bandgap calculated by the algorithm ranges from

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Fig. 4. Band structures of the verification model in-plane mode: (a) band structure by algorithm (b) band structure by Comsol out-of-plane mode: (c) band structure by algorithm (d) band structure by Comsol.

121 Hz to 264 Hz between the first two order bands in Fig. 4(c). The RBG is 0.74. For the band structure calculated by Comsol, the bandgap ranges from 123 Hz to 265. The RGB is 0.73. The error between the current model and Comsol is only 0.02%. The comparison shows the validity of the proposed model. This serves as the basis for further optimization of PnC structures. For optimization, the square lattice unit cell is divided into 10 × 10 elements. When the unit cell is symmetric along four axes, the number of design variables is reduced to a quarter of the total. The optimization objective is the relative width of the first bandgap. The volume fraction, as well as the material distribution, can be optimized. The GA starts with an initial population of Np = 20 chromosomes, randomly. The convergence criteria is the continuous equality of best fitness and average fitness. For each iteration of our calculations, the generated different random state leads to the same optimal configurations. For in-plane mode, convergence curves of best fitness and average fitness are shown in Fig. 5. We have tested the calculation up to 800 generations. After 500 generations, best fitness converges at a stable value of 1.0857 in Fig. 5. The convergence speed is rapid at an earlier stage of the evolution and becomes more gradual in the following stage. The average fitness of the first generation is 0.1. This means that the BG is very small or, perhaps, that the BG does not exist for the most initial individuals produced randomly in the initial phase. But the best fitness for the first generation is 0.26. This indicates that the BG of the PnC opens. The 3 × 3 unit cell obtained from topological optimization is shown in Fig. 6(a) for in-plane mode. The corresponding gold and rubber volume fraction is 0.52 and 0.28, respectively. According to the corresponding dispersion curves shown in Fig. 6(b), the absolute BG ranges from 300 Hz to 1011 Hz between the third and the fourth bands. The optimized RBG is 1.0857. For the traditional PnC (see Fig. 3) with the same volume fraction, the bandgap is 272 Hz–785 Hz in Fig. 6(c).

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Fig. 5. Convergence curves for in-plane mode.

Fig. 6. The topologic optimization results of the in-plane mode. (a) the optimized 3 × 3 unit cell (b) corresponding band structure by GA (c) band structure of the traditional PnC.

The band gap obtained by topological optimization design is 1.4 times wider than the initial design with the same volume fraction of gold and rubber. Convergence curves of the average and best fitness are shown in Fig. 7(a) for out-of-plane mode. The best fitness converges to a stable value of 1.36 after 150 generations. The convergence speed is rapid initially and becomes slower later. The average fitness of the first generation is 0.1, which means that BG is very small or does not exist for individuals produced randomly in the initial phase of evolution. The iteration history of volume fractions of gold and rubber are given in Fig. 7(b). The corresponding gold and rubber volume fractions of the optimized unit cell are 0.32 and 0.36, respectively. For out-of-plane mode, the 3 × 3 unit cell obtained from topological optimization is shown in Fig. 8(a). As the corresponding BG shows in Fig. 8(b), the absolute BG ranges from 229 Hz to 1206 Hz between the fourth and fifth bands. The optimized RBG is 1.36. For the traditional PnC with the same volume fraction, the bandgap ranges from 90.9 Hz to 309.6 Hz in Fig. 8(c). The optimized BG is 4.5 times wider than the BG of the traditional PnC. 3.2. The influence of volume fraction When the volume fraction of materials is limited, the optimized topologies for wave modulations can be different. So, it is necessary to design adaptive configurations for PnCs with wider BGs. The corresponding PnC structure with 3 × 3 unit cells of the three-phase PnC with different volume fractions are studied to provide internal relations between volume fraction and the optimized configurations. For the in-plane mode, the optimization results are shown in Fig. 9. When f1 = 0.28 and f2 = 0.36, the optimized configuration of the scatterer is a square with corners along the diagonal, as indicated in Fig. 9(a, b). The bandgap range is from 289 Hz to 825 Hz and the RBG = 0.96 in Fig. 9(c). When f1 = 0.2 and f2 = 0.16, the optimized configuration is

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Fig. 7. Curves for the out-of-plane mode: (a) convergence curves (b) volume fractions variation.

Fig. 8. The topologic optimization results of the out-of-plane mode. (a) the optimized 3 × 3 unit cell (b) the corresponding band structure by GA (c) band structure of the traditional PnC.

shown in Fig. 9(d, e) and the bandgap range is from 442 Hz to 1022 Hz and the RBG = 0.79 in Fig. 9(f). When f1 = 0.48 and f2 = 0.16, the optimized configuration is shown in Fig. 9(g, h) and the optimized bandgap range is from 1583 Hz to 2652 Hz and the RBG = 0.29 in Fig. 9(i). When f1 = 0.32 and f2 = 0.04, the optimized unit cell is shown in Fig. 9(j, k). The bandgap range is from 2051.3 Hz to 2051.8 Hz and the RBG is 2.59 × 10−4 in Fig. 9(l). By comparing the optimized band structures in Fig. 9(c) and (i), the optimized BG is in the higher frequency domain and the RBG becomes smaller when f1 > f2 . When f2 decreases to 0.04, the BG becomes very small in Fig. 9(l). For the out-of-plane mode, the optimized configurations in a given volume fraction are shown in Fig. 10(a, b) when f1 = 0.28 and f2 = 0.36 and the BG range is from 239 Hz to 1206 Hz and the RBG = 1.34 in Fig. 10(c). When f1 = 0.2 and f2 = 0.16, the optimized unit cell is shown in Fig. 10(d, e) and the bandgap range is from 185 Hz to 434 Hz and the RBG is 0.81 in Fig. 10(f). When f1 = 0.48 and f2 = 0.16, the optimized configuration is shown in Fig. 10(g, h) and the bandgap range is from 168 Hz to 238 Hz and the RBG is 0.35 in Fig. 10(i). When f1 = 0.32, f2 = 0.04, the optimized unit cell is shown in Fig. 10(j, k). The bandgap range is from 982 Hz to 985 Hz and the RBG is 3.4 × 10−3 in Fig. 10(l). The optimized BG is between the fourth and the fifth bands when f1 is smaller than f2 in Fig. 10(c). When f1 is larger than f2 , the optimized BG is between the second and the first bands. The optimized RBG decreases when the ratio of f1 and f2 increases. 3.3. The influence of the symmetry By changing the symmetries of the unit cells, the optimization of the RBG can be affected. Three kinds of PnCs with different symmetries (four axis symmetry, biaxial symmetry, and uniaxial symmetry) are considered for the optimization. The diagrams are given in Fig. 11. The feasible sets between different symmetry cases are independent, which indicates that the solution spaces of the above three kinds of problems are mutually exclusive.

X.K. Han and Z. Zhang / Wave Motion 93 (2020) 102496

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Fig. 9. The optimization results for different volume fraction for in-plane mode. the optimized unit cell: (a) f1 = 0.28, f2 = 0.36 (d) f1 = 0.2, f2 = 0.16 (g) f1 = 0.48, f2 = 0.16 (j) f1 = 0.32, f2 = 0.04; the 3 × 3 unit cells: (b) f1 = 0.28, f2 = 0.36 (e) f1 = 0.2, f2 = 0.16 (h) f1 = 0.48, f2 = 0.16 (k) f1 = 0.32, f2 = 0.04; the corresponding band structure: (c) f1 = 0.28, f2 = 0.36 (f) f1 = 0.2, f2 = 0.16 (i) f1 = 0.48, f2 = 0.16 (l) f1 = 0.32, f2 = 0.04.

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X.K. Han and Z. Zhang / Wave Motion 93 (2020) 102496

Fig. 10. The optimization results for different volume fraction for out-of-plane mode. the optimized unit cell: (a) f1 = 0.28, f2 = 0.36 (d) f1 = 0.2, f2 = 0.16 (g) f1 = 0.48, f2 = 0.16 (j) f1 = 0.32, f2 = 0.04; the 3 × 3 unit cells: (b) f1 = 0.28, f2 = 0.36 (e) f1 = 0.2, f2 = 0.16 (h) f1 = 0.48, f2 = 0.16 (k) f1 = 0.32, f2 = 0.04; the corresponding band structure: (c) f1 = 0.28, f2 = 0.36 (f) f1 = 0.2, f2 = 0.16 (i) f1 = 0.48, f2 = 0.16 (l) f1 = 0.32, f2 = 0.04.

X.K. Han and Z. Zhang / Wave Motion 93 (2020) 102496

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Fig. 11. Three kinds of symmetries: (a) uniaxial symmetry (b) biaxial symmetry (c) four axis symmetry.

Fig. 12. The first Brillouin zone.

The first Brillouin zone is shown in Fig. 12. The calculation of band structure of the biaxial symmetry PnC unit cells needs to sweep along the boundaries M-T-X-M-Y-T in the first Brillouin zone in Fig. 12. The calculation of band structure of the uniaxial symmetry PnC unit cells needs to sweep along the boundaries Y2 -Γ -X-M2 -T-M-Y-T-X-M in the first Brillouin zone in Fig. 12. When the PnC is symmetry is along four axes, the optimized configuration is shown in Fig. 13(a), and the corresponding RBG is 1.1 for the in-plane mode. The corresponding f1 and f2 are 0.52 and 0.28, respectively. The RBG of the biaxial symmetry PnC in Fig. 13(b) is 0.52 when f1 is 0.28 and f2 is 0.36. For the uniaxial symmetry PnC in Fig. 13(c), the RBG is 0.61 when f1 is 0.4 and f2 is 0.3. When the topological configuration of the filling material is symmetric along four axes, a wider bandgap is obtained. The bandgap becomes narrow when it is symmetrical only along one or two axes. For the out-of-plane mode, the optimized configuration for the four axes symmetry PnC is shown in Fig. 14(a). The optimized RBG is 1.36. The corresponding f1 and f2 are 0.32 and 0.36, respectively. The RBG of the biaxial symmetry PnC in Fig. 14(b) is 1.38 when the volume fraction f1 and f2 are both 0.36. For the uniaxial symmetry PnC in Fig. 14(c), the optimized RBG is 1.37 when the volume fraction f1 is 0.54 and f2 is 0.22. When the unit cell is in lower symmetry, f1 of the optimized configuration is bigger than f2 . Imperfect symmetry can lead to a wider, optimized RBG. 4. Conclusions In this paper, three-phase PnC in a square lattice with maximum relative BG for in-plane and out-of-plane modes are designed by the GA in combination with a plane wave expansion method. Volume fraction and symmetry are key factors for the design of topological configurations. The optimized RBG for the in-plane mode can be obtained and was found to be 1.1 in the optimized design of volume fraction (f1 = 0.52 and f2 = 0.28). The optimized RBG for out-of-plane mode is 1.36 when f1 and f2 are 0.32 and 0.36, respectively. The optimized configurations are also different for different volume fractions. For in-plane mode, the optimized BG is in the higher frequency domain and the RBG becomes smaller when f1 > f2 . For out-of-plane mode, the optimized BG is between the fourth and the fifth bands when f1 is smaller than f2 . When f1 is larger than f2 , the optimized BG is between the first and second bands. The optimized RBG decreases when the ratio of f1 and f2 increases. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Fig. 13. Variation of the RBG with the symmetry for in-plane mode. Topological configuration: (a) four axes symmetry (b) biaxial symmetry (c) uniaxial symmetry; corresponding band structure: (d) four axes symmetry (e) biaxial symmetry (f) uniaxial symmetry.

Fig. 14. Variation of the RBG with the symmetry for out-of-plane mode. topological configuration: (a) four axes symmetry (b) biaxial symmetry (c) uniaxial symmetry; corresponding band structure: (d) four axes symmetry (e) biaxial symmetry (f) uniaxial symmetry.

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11572074) and Liaoning Provincial Natural Science Foundation (2019-KF-05-07). References [1] M. Maldovan, Sound and heat revolutions in phoninics, Nature 503 (2013) 209–217. [2] X.K. Han, Z. Zhang, Topological optimization of phononic crystal thin plate by a genetic algorithm, Sci. Rep. 9 (2019) 8331.

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