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Designing 2D Phononic Crystal Slabs with Transmission Gaps for Solid Angle as well as Frequency Variation

Designing 2D Phononic Crystal Slabs with Transmission Gaps for Solid Angle as well as Frequency... Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 317890, 7 pages doi:10.1155/2009/317890 Research Article Designing 2D Phononic Crystal Slabs with Transmission Gaps for Solid Angle as well as Frequency Variation Sven M. Ivansson Department of Underwater Research, Swedish Defence Research Agency, FOI Kista, 16490 Stockholm, Sweden Correspondence should be addressed to Sven M. Ivansson, sven.ivansson@foi.se Received 29 January 2009; Accepted 15 June 2009 Recommended by Mohammad Tawfik Phononic crystals (PCs) can be used as acoustic frequency selective insulators and filters. In a two-dimensional (2D) PC, cylindrical scatterers with a common axis direction are located periodically in a host medium. In the present paper, the layer multiple- scattering (LMS) computational method for wave propagation through 2D PC slabs is formulated and implemented for general 3D incident-wave directions and polarizations. Extensions are made to slabs with cylindrical scatterers of different types within each layer. As an application, the problem is considered to design such a slab with small sound transmittance within a given frequency band and solid angle region for the direction of the incident plane wave. The design problem, with variable parameters characterizing the scatterer geometry and material, is solved by differential evolution, a global optimization algorithm for efficiently navigating parameter landscapes. The efficacy of the procedure is illustrated by comparison to a direct Monte Carlo method. Copyright © 2009 Sven M. Ivansson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction are its computational speed, which makes it useful for forward modeling in connection with extensive optimization Just as photonic crystals can be used to manipulate light, computations, and the physical insight it provides. It appears phononic crystals (PCs) with inclusions in a lattice with that the LMS method was first developed for the 3D case with single, double, or triple periodicity can be used to manipulate spherical scatterers [3, 4], and recent review papers include sound [1]. When a sound wave of a certain frequency Sigalas et al. [5] and Sainidou et al. [6]. penetrates the PC, the energy is scattered by the inclu- The LMS method has also been applied to the 2D sions. According to Bragg’s law, constructive and destructive case with infinite cylindrical scatterers. It is mainly the in- interference appears in certain directions. It follows that plane propagation case that has been considered [7–11], for transmission and/or reflection for certain frequencies can which one space dimension can be eliminated in the wave be absent (band gaps), even for all angles of incidence equations. Out-of-plane propagation has been treated by Mei (absolute band gaps). The band gaps are ideal and fully et al. [12] and, for band structure calculations, somewhat developed only in space-filling PCs. In a PC slab with earlier by Wilm et al. [13] (using the plane-wave method) finite thickness, the “band-gap wavefields” are reduced and by Guenneau et al. [14]. significantly but they do not vanish throughout the gap. In the present paper, basic LMS equations for propaga- Acoustic frequency selective insulators and filters are possible tion of plane waves of any direction through a 2D PC slab are applications. first provided in Section 2. The Poisson summation formula Several computational methods have been adapted and and the Graf addition theorem are utilized. As shown in Section 3,different types of scatterers at the same interface developed to study wave propagation through PCs. Two such methods are the purely numerical finite-difference can be allowed, which represents an extension as compared time domain (FDTD) method and the semianalytical layer to the treatment in [12]. Coupled equation systems are multiple-scattering (LMS) method, which is developed derived for the different scatterer types. For 3D PC slabs, from Korringa-Kohn-Rostoker theory [2]. These methods a corresponding extension has recently proved useful for complement each other. Advantages with the LMS method applications to design of anechoic coatings [15]. 2 Advances in Acoustics and Vibration Incident plane wave sound energy by (2), with a possibly complex cos θ. The vectors e = e (K )are defined by e = (sin θ cos ϕ,sin θ sin ϕ,cos θ), j 1 g j e = (cos θ cos ϕ,cos θ sin ϕ,− sin θ), e = (− sin ϕ,cos ϕ,0). 2 3 It is convenient also to introduce the compressional- and shear-wave wavenumbers k = ω/α and k = ω/β. p s y x As detailed in [17], for example, and references therein, reflection and transmission matrices R ,T and R ,T can B B A A z now be introduced, for the discrete set of waves specified by (1)–(3). The mentioned reference concerns the doubly Figure 1: Horizontal xy coordinates and a z depth coordinate periodic case for a 3D PC, with periodicity in the y direction axis are introduced. There are three scatterer interfaces in this as well, but the R/T matrix formalism is the same. There illustration. The medium is periodic with period d in the x are three scatterer interfaces in the illustration of Figure 1. direction, although only finitely many cylindrical scatterers are Individual R/T matrices can be combined recursively [18, shown at each interface, and the cylinder axes are parallel to the 19]. Layer thicknesses, as well as translations of individual y-axis. scatterer interfaces in the x direction, are conveniently accounted for by phase shifts of the complex amplitudes of the plane-wave components. Applications to design of 2D PC slabs with broad transmission gaps for incident waves of in-plane as well as out-of-plane directions are presented in Section 4.The 2.1. Interface with Periodically Distributed Scatterers of a design problem is equivalent to a nonlinear optimization Common Type. Explicit expressions for the R/Tmatrices problem, and differential evolution [16], a global optimiza- are well known for an interface between two homogeneous tion technique from inverse theory, is used. A 2D PC slab half-spaces [19]. To handle an interface with periodically example from Mei et al. [12], with lead cylinders in an epoxy distributed scatterers, as one of the three in Figure 1, the host, is revisited. Geometrical and material parameter values following cylindrical vector solutions to the wave equations are varied to position and widen its band gap. can be used [12]: 2. Basic 2D Layer Multiple-Scattering u (r) = ∇ f pr exp il η exp iκ y l ,(4) lκ Computational Method M N u (r) = ∇× u (r),(5) As in Figure 1, a right-hand Cartesian xyz coordinate lκ lκ system is introduced in a fluid-solid medium surrounded by homogeneous half-spaces. The horizontal directions are 1 ( ) u r = ∇× f qr exp il η exp iκ y e ,(6) l y lκ x and y. The medium is periodic with period d in the x direction and uniform in the y direction. where cylindrical coordinates r , η, y are used according to Sound waves with time dependence exp(−iωt), to be r = (r sin η, y, r cos η), and e = (0,1,0). The index l = suppressed in the formulas, are considered, where ω is the y 0,±1,±... ,and κ is a real number. Residual wavenumbers angular frequency. It follows that an incident plane wave with p, q are defined by horizontal wavenumber vector k = (k , κ, 0) will give rise to a linear combination of reflected and transmitted plane 1/2 1/2 2 2 2 2 waves with displacement vectors p = (k − κ ) , q = k − κ . (7) p s u(r) = exp i K · r · e . (1) g j 0L 0M 0N +L +M The notation u (r), u (r), u (r)and u (r), u (r), lκ lκ lκ lκ lκ +N u (r) is used for the two basic cases with f as the Bessel Here, r = (x, y, z), j = 1, 2, 3 for a wave of type P,SV,SH, lκ (1) respectively, s = +(−) for a wave in the positive (negative) z function J and f as the Hankel function H ,respectively. l l direction, and For scatterers at ⎡ ⎤ 1/2 ω R = x, y, z = (md,0,0),(8) ⎣ ⎦ K = k + g ± − k + g · (0, 0, 1) || || g j (2) where d is the lattice period and m runs over the integers, and for an incident plane wave as in (1), the total scattered field = · sin θ cos ϕ,sin θ sin ϕ,cos θ , j u can be written as (cf. [3]) sc ⎡ ⎤ where g belongs to the reciprocal lattice +P +P ⎣ ⎦ u (r) = b exp i k · R · u (r − R) , 2πm sc || l lκ g = k , k ,0 = ,0,0 (3) x y Pl R (9) with m running over the integers. Furthermore, c is the P = L, M, N , compressional-wave velocity α and c = c are the shear- 2 3 wave velocity β. The angular variables θ, ϕ of K are defined where κ is the y component of k . || g j Advances in Acoustics and Vibration 3 +P + −1 The vector b ={b } is determined by solving the [exp(ik e · r) e ]} with e = k (q sin η , κ, q cos η ) l s inc y inc inc inc −1 equation system and e = k (κ sin η ,−q, κ cos η ), can be expanded as inc inc inc s + 0 (I − T · Ω) · b = T · a , (10) k l 0M u (r) =− i exp −ilη u (r). (14) inc inc lκ 0P where I is the appropriate identity matrix, a ={a } gives the coefficients for expansion of the incident plane wave The second case, u (r) = exp(ik e · r) e = inc s inc inc 0P in regular cylindrical waves u (r), Ω = Ω(k d, pd, qd)is lκ i/q ∇× [exp(ik e · r)e ]with e as before and e = s inc y inc inc PP PP the lattice translation matrix {Ω },and T ={T } is (cos η ,0,− sin η ), has the expansion inc inc l;l l;l the transition matrix for an individual scatterer. Specifically, P k + + 0   s l+1 0N b = Ω· b and b = T· (a + b )where b ={b } gives the u (r) = i exp −ilη u (r). (15) l inc inc lκ 0P coefficients for expansion in regular cylindrical waves u (r) lκ of the scattered field from all scatterers except the one at the The lattice translation matrix Ω(k d, pd, qd)can be || origin. determined by applying, for k = p and k = q, the relation The R/T matrices are obtained, finally, by transforming the expansion (9) to plane waves of the type (1). Specifically, ⎡ ⎤ (9)can be rewrittenas (z + i(x − md)) ⎢ ⎥ exp imdk ⎣   ⎦ 1/2 + 2 m= / 0 (x − md) + z ( ) u r = Δ dg, j ; k d, κd, pd, qd, b sc j=1,2,3 (11) 1/2 ( ) (16) 1 2 × H k (x − md) + z × exp i K · r · e . g j ± (z + ix) The sign in K is given by the sign of z. g j = Θ k d, kd J (kr ), l−n  n 1/2 2 2 (x + z ) Explicit expressions for the Δ coefficients in (11)are readily obtained from (9)and (4)–(6) by invoking, for k = p where the last sum is taken over all integers n and and k = q, the relation (1) ⎡ ⎤ −l Θ k d, kd = i exp imdk H (mkd) ( ( )) m>0 z + i x − md ⎢ ⎥ exp imdk ⎣ ⎦ || (17) 1/2 2 (1) m l (x − md) + z + i exp −imdk H (mkd). m>0 1/2 (1) 2 × H k (x − md) + z This relation follows from the Graf addition theorem [20] PP for Bessel functions. The elements {Ω } of the lattice l;l 2πm −l −1 translation matrix Ω(k d, pd, qd) vanish unless P = P .The = 2(−k) × γ d − k + +iγ sgn(z) m || m remaining elements depend on l and l through |l − l | only. Explicitly, 2πm × exp i k + x + i |z| γ , || m LL Ω = Θ  k d, pd , l−l l;l (12) (18) MM NN Ω = Ω = Θ  k d, qd . l−l || l;l l;l 1/2 where γ = [k − (k +2πm/d) ] with Im γ ≥ 0. The m || m Moroz [21] has published a representation of lattice sums relation (12)isvalid for z= 0, and it can be verified using the in terms of exponentially convergent series, which has been Poisson summation formula. used in the present work for numerical evaluation of the Θ quantities defined in (17). 2.2. Computation of the Expansion Coefficients a ,the Matri- For a homogeneous cylindrical scatterer, the interior ces Ω,and theMatricesT. An incident plane compres- field and the exterior field can be expanded in cylindrical sional wave u (r) = exp(ik e · r)e ,where e = inc p inc inc inc 0P 0P +P waves u (r)and u (r), u (r), respectively. An equation −1 lκ lκ lκ k (p sin η , κ, p cos η ), can be expanded as inc inc PP system for the T -matrix elements T ={T },which l;l l 0L depend on κ, is then readily obtained from the standard u (r) =− i exp −ilη u (r). (13) inc inc lκ boundary conditions concerning continuity of displacement and traction at the cylinder surface. The scatterer as well as Noting that u (r) =∇[exp(ik e · r)]/ik , this fol- inc p inc p the host medium can be either fluid or solid. Because of the lows readily from the well-known Bessel function relation circular symmetry with a cylindrical scatterer, scattering only exp(iγ sin η) = J (γ)exp(ilη). l appears to the same l component (l = l). Details concerning An incident plane shear wave of SV or SH type can the case κ = 0, for which the P-SV (P, P = M)and SH be expanded by a superposition of two cases. The first (P = P = M) solutions decouple, are given by Mei et al. −1 case, u (r) = exp(ik e · r)e =−(k q) ∇×{∇ × [9]. inc s inc s inc 4 Advances in Acoustics and Vibration 0  0 + Incident plane wave sound energy With Ω = Ω(k d, pd, qd), it follows that b = Ω · b || 0 + dif and c = Ω · c .For acertain matrix Ω , to be determined, dif +  dif + b = Ω · c and c = Ω · b . The equation system for + + determination of b and c becomes 0 + dif + 0 I − T · Ω · b − T · Ω · c = T · a , y x (21) dif + 0 + 0 − U · Ω · b + I − U · Ω · c = U · a . Figure 2: The configuration from Figure 1 is extended here, In order to form the R/T matrices, incident plane waves by allowing two types of cylindrical scatterers to appear in an with different horizontal wavenumber vectors k + g = inc alternating fashion at the same interface depth. The medium is (k +g , κ, 0) have to be considered, where g belongs to the inc inc periodic with period d in the x direction, and the cylinder axes are set {2πm/d} providing the reciprocal lattice. Noting that the still parallel to the y-axis. union of the scatterer positions is a small square lattice with dif period d/2, the following expression for Ω as a difference of Ω matrices is directly obtained 3. Different Types of Scatterers at k + g d pd qd || inc theSameInterface dif 0 Ω = Ω , , − Ω . (22) 2 2 2 The LMS method is commonly applied for lattices with identical cylindrical scatterers within the same layer. As Only those g in {(2πm/d,0,0)} for which m is even are inc shown below, however, different types of scatterers within the reciprocal vectors for the small lattice with period d/2. Since same layer can also be accommodated. A similar extension a lattice translation matrix Ω is periodic in its first argument for the restriction to in-plane wave propagation is made in with period 2π , there will be two groups of g with different inc dif Ivansson [22]. Ω matrices according to (22). Specifically, An illustration is given in Figure 2. Centered at the same k d pd qd || z level, at each of the three scatterer interfaces, there are dif,even 0 Ω = Ω , , − Ω (23) two types of cylindrical scatterers. Each scatterer interface 2 2 2 is treated separately. Choosing coordinates appropriately, pertains to g = (2πm/d,0,0) with even m,and inc scatterers of the first type, with transition matrix T and scattered-field expansion coefficients denoted b ,appearat k d pd qd || dif,odd 0 R = m · (d,0,0), for integers m. Scatterers of the second Ω = Ω + π , , − Ω (24) 2 2 2 type, with transition matrix U and scattered-field expansion coefficients denoted c , appear at points S in between, that is, pertains to g = (2πm/d,0,0)withodd m. inc S = (m +1/2)· (d,0,0). The reciprocal lattice vectors become The transformation of the expansion (19) to plane waves g = (2πm/d,0,0), where m runs over the integers. of the type (1) can be done separately for each of the R and The generalization of the expression (9) for the scattered S sums. In the latter case, the translation from the origin field becomes causes a sign change for some combinations of incident (g ) inc ⎡ ⎤ and scattered (g ) reciprocal lattice vectors. Specifically, with sc +P +P g = (2πm /d,0,0) and g = (2πm /d, 0, 0), the double inc inc sc sc ⎣ ⎦ u (r) = b exp i k · R · u (r − R) sc l lκ sum corresponding to the one in (11)appears as Pl (19) ⎡ ⎤ m −m inc sc + u (r) = (−1) Δ dg , j ; k d, κd, pd, qd, c sc sc +P +P ⎣ ⎦ ( ) + c exp i k · S · u r − S . l  g lκ sc j=1,2,3 Pl S × exp iK · r · e . g j sc It follows that (25) + 0   + 0 b = T · a + b + b , c = U · a + c + c , (20) 4. Designing 2D PC Slabs with where, for a scatterer of the first type at R,exp(i k · R)b a Transmission Gap and exp(i k · R)b give the coefficients for expansion in 0P regular cylindrical waves u (r − R) of the scattered field It is well known that band gaps can appear when scatterers lκ from all other scatterers of the first and the second types, with a large density are arranged periodically in a host respectively. The vectors c and c are defined analogously. with a small density. A particular 2D PC slab example with For a scatterer of the second type at S,exp(i k · S)c and lead cylinders in epoxy was considered in Mei et al. [12]. exp(i k · S)c thus give the coefficients for expansion in The epoxy parameters were 2540 and 1159.817 m/s for the || 0P regular cylindrical waves u (r − S) of the scattered field compressional- and shear-wave velocities, respectively, and lκ from all other scatterers of the second and the first types, 1.18 kg/dm for the density. Corresponding lead material respectively. parameters were 2160 and 860.568 m/s, and 11.4 kg/dm . Advances in Acoustics and Vibration 5 (θ) (θ) ◦ ◦ 60 60 ◦ ◦ 30 30 ◦ ◦ ϕ = 0 ϕ = 0 ◦ ◦ 0 0 ◦ ◦ ϕ = 90 ϕ = 90 ◦ ◦ 30 30 ◦ ◦ 60 60 20 40 60 80 100 20 40 60 80 100 Frequency (kHz) Frequency (kHz) Figure 3: Contour plot of transmittance for a PC slab with lead Figure 4: Contour plot of transmittance as in Figure 3 but for the cylinders in an epoxy host. Parameters of the slab are given in the optimizedPCslabspecifiedin Table 1. text. The direction vector of the incident compressional wave is (sin θ cos ϕ,sin θ sin ϕ,cos θ). The black and gray contours are at −150 dB and −10 dB, respectively. Table 1: Specification of a PC slab that has been optimized by DE to produce small transmittance in the band 45–65 kHz. Corresponding transmittance results are shown in Figures 4 and 5. Optimum −163.0 dB The slab was formed by sixteen layers of lead cylinders, each 2995.6 m/s Scatterer compressional-wave velocity α with a radius of 3.584 mm, with a spacing between cylinder 1200.0 m/s Scatterer shear-wave velocity β centers of 11.598 mm in the x as well as z directions, cf. Figure 1 where there are three layers. A band gap centered 14.000 kg/dm Scatterer density ρ at about 60 kHz was found. (Dimensionless frequencies and 13.292 mm Layer thickness h distances were actually used in the paper, but a specialization 4.2702 mm Largest scatterer radius r max is made here.) 4.2300 mm Smallest scatterer radius r min Figure 3 is a contour plot of the transmittance, that is, 22.670 mm Lattice period d time- (and space-) averaged transmitted energy flux relative to the incident one, up to 120 kHz. The vertical axis is for the angle θ, where the direction vector of the incident compressional plane wave is (sin θ cos ϕ,sin θ sin ϕ,cos θ). in the yz plane). A configuration with cylinders of two In-plane and out-of-plane incidence angles are considered alternating sizes is allowed, as depicted in Figure 2. Still, the together in Figure 3, where the upper half with ϕ = 0 slab is formed by sixteen layers of cylinders. The following concerns incidence in the xz plane (in-plane propagation) seven parameters are varied within the indicated search and the lower half with ϕ = 90 concerns incidence in space: scatterer compressional-wave velocity α [1500 m/s ≤ the yz plane. Only the first Brillouin zone is involved, since α ≤ 3000 m/s], scatterer shear-wave velocity β [650 m/s k = ω/α sin θ cosϕ< 2π/d when α equals the epoxy ≤ β ≤ 1200 m/s], scatterer density ρ [7 kg/dm ≤ ρ ≤ compressional-wave velocity 2540 m/s, d = 11.598 mm, and 14 kg/dm ], layer thickness h [8 ≤ h ≤ 16 mm], the largest the frequency is less than α/d = 219 kHz. Of course, the band scatterer radius r [0.225 h ≤ r ≤ 0.4 h], the smallest max max gap from Mei et al. [12], at about 60 kHz, shows up clearly scatterer radius r [0.45 r ≤ r ≤ r ], the lattice min max min max in this kind of plot. One might wonder how the geometrical period d[8(r + r )/3 ≤ d ≤ 4(r + r )]. The layer max min max min and/or material parameters of the slab should be modified to thickness h is the z distance between subsequent scatterer achieve a prescribed desired change of the appearance of the interfaces (cf. Figure 2). gap. Table 1 shows the optimum obtained with DE, along Global optimization methods can be used to design PC with corresponding parameter values. A reduction of the slabs with desirable properties. Simulated annealing, genetic transmitted field with more than 160 dB is achieved through- algorithms and differential evolution (DE) are three kinds out the band 45–65 kHz and throughout the solid angle ◦ ◦ ◦ ◦ of such methods, that have become popular during the last intervals 0 <θ < 20 for ϕ = 0 , ϕ = 90 for the fifteen years. DE, to be applied here, is related to genetic direction of incidence. High-velocity high-density cylinders algorithms, but the parameters are not encoded in bit strings, seem preferable, since the lead velocities and density are andgenetic operatorssuchascrossoverand mutation are all increased to values close to the upper ends of the replaced by algebraic operators [16]. corresponding search intervals. In this case, as large as As a very simple example, to try to position and widen the possible values of α, β,and ρ couldinfacthavebeenfixed gap in Figure 3, an objective function for DE minimization from the start. Moreover, the cylinders are almost as densely is specified as the maximum transmittance in the frequency packed horizontally as allowed by the search interval for d. band 45–65 kHz when the incidence angle θ is varied Hence, the optimization could in fact have been simplified ◦ ◦ ◦ between 0 and 20 for the two azimuthal angles ϕ = 0 (in- considerably. With only a few free parameters, a complete plane propagation in the xz plane) and ϕ = 90 (propagation search, for example, can be a feasible alternative. 6 Advances in Acoustics and Vibration (θ) −50 ϕ = 30 −100 ϕ = 60 −150 2000 4000 6000 8000 20 40 60 80 100 Number of tested PC slabs Frequency (kHz) Figure 6: Evolution of the maximum transmittance with the Figure 5: Contour plot of transmittance for the optimized PC number of tested parameter combinations for the DE optimization slab specified in Table 1. The difference to Figure 4 is that two leading to the PC slab specified in Table 1. A corresponding curve other ϕ angles are considered for the direction of the incident for a brute force Monte Carlo method is also included (gray). compressional wave. Compared to Figure 3, the contour plot in Figure 4 of the modified values for the other parameters, was almost as transmittance for the optimized PC slab shows a band gap good as the one presented in Table 1. The advantages with around 60 kHz that has become significantly wider and also cylinders of different sizes are expected to be more significant deeper. The allowed increases of α, β, ρ,and (r + r )/d in more complicated filtering cases, involving, for example, max min are certainly essential for producing this effect. Variation more than one frequency band. of h, r ,and r is needed to position the band gap at max min the desired frequency interval. Although the optimization 5. Conclusions was performed with the restriction 0 <θ < 20 ,small transmittance is apparently achieved for all angles θ (0 < The layer multiple-scattering (LMS) method is a fast semi- θ< 90 ). analytical technique for computing scattering from layers It turns out that the transmittance for the optimized PC including periodic scatterer lattices. For the 2D case with slab remains small within the 45–65 kHz band for all out- cylindrical scatterers and any solid angle direction of an of-plane incidence angles (an “absolute” band gap). Figure 5 incident plane wave, an extension has been made to scatterer shows the transmittance in the same way as Figure 4,but for lattices with cylindrical scatterers of two different sizes in the ◦ ◦ ◦ ϕ = 0 changed to ϕ = 30 and ϕ = 90 changed to ϕ = same horizontal plane. 60 . The upper and lower halves exhibit increased symmetry, Global optimization methods from inverse theory are since the ϕ angles involved are closer to one another. useful for designing PC slabs with desirable properties. A Apparently, the optimized PC slab has large transmit- differential evolution algorithm has been applied here to tance below about 30 kHz and there are regions with large position and widen an “absolute” band gap for a certain 2D transmittance above 80 kHz as well. The band gap with small PC slab. Although only limited angle intervals for the direc- transmittance extends to higher frequencies, than those in tion of incidence were included for the optimization, small the band 45–65 kHz, for plane-wave incidence directions transmittance was achieved for all solid angles specifying the with either large θ or small ϕ. direction of the incident wave. The efficacy of the DE technique can be illustrated by The possibility to include cylindrical scatterers of two showing the decrease of the maximum transmittance within different sizes in the same horizontal plane provides an the specified frequency/angle region as the number of tested additional degree of freedom that can be useful for PC parameter settings for the PC slab evolves. Figure 6 shows design purposes. In the presented example, with its spec- this decrease for the example from Figure 4.Acomparison ification of objective function and search intervals for the to the much less efficient MonteCarlo method, with random parameters, however, the difference between the optimal selection of parameters from the search space, is included. cylinder radii was rather small and the improvement only The low efficiency of the MonteCarlo approach shows that marginal. The additional flexibility is expected to be more the desired PC slabs with small transmittance only appear in important in more complicated filtering applications. For a small portion of the search space. For example, only about example, specified regions in frequency-angle space with 0.15% of the random slab selections had a maximum trans- large transmittance could be desired in addition to specified mittance below −20 dB within the specified frequency/angle regions with small transmittance. region. 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Designing 2D Phononic Crystal Slabs with Transmission Gaps for Solid Angle as well as Frequency Variation

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Copyright © 2009 Sven M. Ivansson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 317890, 7 pages doi:10.1155/2009/317890 Research Article Designing 2D Phononic Crystal Slabs with Transmission Gaps for Solid Angle as well as Frequency Variation Sven M. Ivansson Department of Underwater Research, Swedish Defence Research Agency, FOI Kista, 16490 Stockholm, Sweden Correspondence should be addressed to Sven M. Ivansson, sven.ivansson@foi.se Received 29 January 2009; Accepted 15 June 2009 Recommended by Mohammad Tawfik Phononic crystals (PCs) can be used as acoustic frequency selective insulators and filters. In a two-dimensional (2D) PC, cylindrical scatterers with a common axis direction are located periodically in a host medium. In the present paper, the layer multiple- scattering (LMS) computational method for wave propagation through 2D PC slabs is formulated and implemented for general 3D incident-wave directions and polarizations. Extensions are made to slabs with cylindrical scatterers of different types within each layer. As an application, the problem is considered to design such a slab with small sound transmittance within a given frequency band and solid angle region for the direction of the incident plane wave. The design problem, with variable parameters characterizing the scatterer geometry and material, is solved by differential evolution, a global optimization algorithm for efficiently navigating parameter landscapes. The efficacy of the procedure is illustrated by comparison to a direct Monte Carlo method. Copyright © 2009 Sven M. Ivansson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction are its computational speed, which makes it useful for forward modeling in connection with extensive optimization Just as photonic crystals can be used to manipulate light, computations, and the physical insight it provides. It appears phononic crystals (PCs) with inclusions in a lattice with that the LMS method was first developed for the 3D case with single, double, or triple periodicity can be used to manipulate spherical scatterers [3, 4], and recent review papers include sound [1]. When a sound wave of a certain frequency Sigalas et al. [5] and Sainidou et al. [6]. penetrates the PC, the energy is scattered by the inclu- The LMS method has also been applied to the 2D sions. According to Bragg’s law, constructive and destructive case with infinite cylindrical scatterers. It is mainly the in- interference appears in certain directions. It follows that plane propagation case that has been considered [7–11], for transmission and/or reflection for certain frequencies can which one space dimension can be eliminated in the wave be absent (band gaps), even for all angles of incidence equations. Out-of-plane propagation has been treated by Mei (absolute band gaps). The band gaps are ideal and fully et al. [12] and, for band structure calculations, somewhat developed only in space-filling PCs. In a PC slab with earlier by Wilm et al. [13] (using the plane-wave method) finite thickness, the “band-gap wavefields” are reduced and by Guenneau et al. [14]. significantly but they do not vanish throughout the gap. In the present paper, basic LMS equations for propaga- Acoustic frequency selective insulators and filters are possible tion of plane waves of any direction through a 2D PC slab are applications. first provided in Section 2. The Poisson summation formula Several computational methods have been adapted and and the Graf addition theorem are utilized. As shown in Section 3,different types of scatterers at the same interface developed to study wave propagation through PCs. Two such methods are the purely numerical finite-difference can be allowed, which represents an extension as compared time domain (FDTD) method and the semianalytical layer to the treatment in [12]. Coupled equation systems are multiple-scattering (LMS) method, which is developed derived for the different scatterer types. For 3D PC slabs, from Korringa-Kohn-Rostoker theory [2]. These methods a corresponding extension has recently proved useful for complement each other. Advantages with the LMS method applications to design of anechoic coatings [15]. 2 Advances in Acoustics and Vibration Incident plane wave sound energy by (2), with a possibly complex cos θ. The vectors e = e (K )are defined by e = (sin θ cos ϕ,sin θ sin ϕ,cos θ), j 1 g j e = (cos θ cos ϕ,cos θ sin ϕ,− sin θ), e = (− sin ϕ,cos ϕ,0). 2 3 It is convenient also to introduce the compressional- and shear-wave wavenumbers k = ω/α and k = ω/β. p s y x As detailed in [17], for example, and references therein, reflection and transmission matrices R ,T and R ,T can B B A A z now be introduced, for the discrete set of waves specified by (1)–(3). The mentioned reference concerns the doubly Figure 1: Horizontal xy coordinates and a z depth coordinate periodic case for a 3D PC, with periodicity in the y direction axis are introduced. There are three scatterer interfaces in this as well, but the R/T matrix formalism is the same. There illustration. The medium is periodic with period d in the x are three scatterer interfaces in the illustration of Figure 1. direction, although only finitely many cylindrical scatterers are Individual R/T matrices can be combined recursively [18, shown at each interface, and the cylinder axes are parallel to the 19]. Layer thicknesses, as well as translations of individual y-axis. scatterer interfaces in the x direction, are conveniently accounted for by phase shifts of the complex amplitudes of the plane-wave components. Applications to design of 2D PC slabs with broad transmission gaps for incident waves of in-plane as well as out-of-plane directions are presented in Section 4.The 2.1. Interface with Periodically Distributed Scatterers of a design problem is equivalent to a nonlinear optimization Common Type. Explicit expressions for the R/Tmatrices problem, and differential evolution [16], a global optimiza- are well known for an interface between two homogeneous tion technique from inverse theory, is used. A 2D PC slab half-spaces [19]. To handle an interface with periodically example from Mei et al. [12], with lead cylinders in an epoxy distributed scatterers, as one of the three in Figure 1, the host, is revisited. Geometrical and material parameter values following cylindrical vector solutions to the wave equations are varied to position and widen its band gap. can be used [12]: 2. Basic 2D Layer Multiple-Scattering u (r) = ∇ f pr exp il η exp iκ y l ,(4) lκ Computational Method M N u (r) = ∇× u (r),(5) As in Figure 1, a right-hand Cartesian xyz coordinate lκ lκ system is introduced in a fluid-solid medium surrounded by homogeneous half-spaces. The horizontal directions are 1 ( ) u r = ∇× f qr exp il η exp iκ y e ,(6) l y lκ x and y. The medium is periodic with period d in the x direction and uniform in the y direction. where cylindrical coordinates r , η, y are used according to Sound waves with time dependence exp(−iωt), to be r = (r sin η, y, r cos η), and e = (0,1,0). The index l = suppressed in the formulas, are considered, where ω is the y 0,±1,±... ,and κ is a real number. Residual wavenumbers angular frequency. It follows that an incident plane wave with p, q are defined by horizontal wavenumber vector k = (k , κ, 0) will give rise to a linear combination of reflected and transmitted plane 1/2 1/2 2 2 2 2 waves with displacement vectors p = (k − κ ) , q = k − κ . (7) p s u(r) = exp i K · r · e . (1) g j 0L 0M 0N +L +M The notation u (r), u (r), u (r)and u (r), u (r), lκ lκ lκ lκ lκ +N u (r) is used for the two basic cases with f as the Bessel Here, r = (x, y, z), j = 1, 2, 3 for a wave of type P,SV,SH, lκ (1) respectively, s = +(−) for a wave in the positive (negative) z function J and f as the Hankel function H ,respectively. l l direction, and For scatterers at ⎡ ⎤ 1/2 ω R = x, y, z = (md,0,0),(8) ⎣ ⎦ K = k + g ± − k + g · (0, 0, 1) || || g j (2) where d is the lattice period and m runs over the integers, and for an incident plane wave as in (1), the total scattered field = · sin θ cos ϕ,sin θ sin ϕ,cos θ , j u can be written as (cf. [3]) sc ⎡ ⎤ where g belongs to the reciprocal lattice +P +P ⎣ ⎦ u (r) = b exp i k · R · u (r − R) , 2πm sc || l lκ g = k , k ,0 = ,0,0 (3) x y Pl R (9) with m running over the integers. Furthermore, c is the P = L, M, N , compressional-wave velocity α and c = c are the shear- 2 3 wave velocity β. The angular variables θ, ϕ of K are defined where κ is the y component of k . || g j Advances in Acoustics and Vibration 3 +P + −1 The vector b ={b } is determined by solving the [exp(ik e · r) e ]} with e = k (q sin η , κ, q cos η ) l s inc y inc inc inc −1 equation system and e = k (κ sin η ,−q, κ cos η ), can be expanded as inc inc inc s + 0 (I − T · Ω) · b = T · a , (10) k l 0M u (r) =− i exp −ilη u (r). (14) inc inc lκ 0P where I is the appropriate identity matrix, a ={a } gives the coefficients for expansion of the incident plane wave The second case, u (r) = exp(ik e · r) e = inc s inc inc 0P in regular cylindrical waves u (r), Ω = Ω(k d, pd, qd)is lκ i/q ∇× [exp(ik e · r)e ]with e as before and e = s inc y inc inc PP PP the lattice translation matrix {Ω },and T ={T } is (cos η ,0,− sin η ), has the expansion inc inc l;l l;l the transition matrix for an individual scatterer. Specifically, P k + + 0   s l+1 0N b = Ω· b and b = T· (a + b )where b ={b } gives the u (r) = i exp −ilη u (r). (15) l inc inc lκ 0P coefficients for expansion in regular cylindrical waves u (r) lκ of the scattered field from all scatterers except the one at the The lattice translation matrix Ω(k d, pd, qd)can be || origin. determined by applying, for k = p and k = q, the relation The R/T matrices are obtained, finally, by transforming the expansion (9) to plane waves of the type (1). Specifically, ⎡ ⎤ (9)can be rewrittenas (z + i(x − md)) ⎢ ⎥ exp imdk ⎣   ⎦ 1/2 + 2 m= / 0 (x − md) + z ( ) u r = Δ dg, j ; k d, κd, pd, qd, b sc j=1,2,3 (11) 1/2 ( ) (16) 1 2 × H k (x − md) + z × exp i K · r · e . g j ± (z + ix) The sign in K is given by the sign of z. g j = Θ k d, kd J (kr ), l−n  n 1/2 2 2 (x + z ) Explicit expressions for the Δ coefficients in (11)are readily obtained from (9)and (4)–(6) by invoking, for k = p where the last sum is taken over all integers n and and k = q, the relation (1) ⎡ ⎤ −l Θ k d, kd = i exp imdk H (mkd) ( ( )) m>0 z + i x − md ⎢ ⎥ exp imdk ⎣ ⎦ || (17) 1/2 2 (1) m l (x − md) + z + i exp −imdk H (mkd). m>0 1/2 (1) 2 × H k (x − md) + z This relation follows from the Graf addition theorem [20] PP for Bessel functions. The elements {Ω } of the lattice l;l 2πm −l −1 translation matrix Ω(k d, pd, qd) vanish unless P = P .The = 2(−k) × γ d − k + +iγ sgn(z) m || m remaining elements depend on l and l through |l − l | only. Explicitly, 2πm × exp i k + x + i |z| γ , || m LL Ω = Θ  k d, pd , l−l l;l (12) (18) MM NN Ω = Ω = Θ  k d, qd . l−l || l;l l;l 1/2 where γ = [k − (k +2πm/d) ] with Im γ ≥ 0. The m || m Moroz [21] has published a representation of lattice sums relation (12)isvalid for z= 0, and it can be verified using the in terms of exponentially convergent series, which has been Poisson summation formula. used in the present work for numerical evaluation of the Θ quantities defined in (17). 2.2. Computation of the Expansion Coefficients a ,the Matri- For a homogeneous cylindrical scatterer, the interior ces Ω,and theMatricesT. An incident plane compres- field and the exterior field can be expanded in cylindrical sional wave u (r) = exp(ik e · r)e ,where e = inc p inc inc inc 0P 0P +P waves u (r)and u (r), u (r), respectively. An equation −1 lκ lκ lκ k (p sin η , κ, p cos η ), can be expanded as inc inc PP system for the T -matrix elements T ={T },which l;l l 0L depend on κ, is then readily obtained from the standard u (r) =− i exp −ilη u (r). (13) inc inc lκ boundary conditions concerning continuity of displacement and traction at the cylinder surface. The scatterer as well as Noting that u (r) =∇[exp(ik e · r)]/ik , this fol- inc p inc p the host medium can be either fluid or solid. Because of the lows readily from the well-known Bessel function relation circular symmetry with a cylindrical scatterer, scattering only exp(iγ sin η) = J (γ)exp(ilη). l appears to the same l component (l = l). Details concerning An incident plane shear wave of SV or SH type can the case κ = 0, for which the P-SV (P, P = M)and SH be expanded by a superposition of two cases. The first (P = P = M) solutions decouple, are given by Mei et al. −1 case, u (r) = exp(ik e · r)e =−(k q) ∇×{∇ × [9]. inc s inc s inc 4 Advances in Acoustics and Vibration 0  0 + Incident plane wave sound energy With Ω = Ω(k d, pd, qd), it follows that b = Ω · b || 0 + dif and c = Ω · c .For acertain matrix Ω , to be determined, dif +  dif + b = Ω · c and c = Ω · b . The equation system for + + determination of b and c becomes 0 + dif + 0 I − T · Ω · b − T · Ω · c = T · a , y x (21) dif + 0 + 0 − U · Ω · b + I − U · Ω · c = U · a . Figure 2: The configuration from Figure 1 is extended here, In order to form the R/T matrices, incident plane waves by allowing two types of cylindrical scatterers to appear in an with different horizontal wavenumber vectors k + g = inc alternating fashion at the same interface depth. The medium is (k +g , κ, 0) have to be considered, where g belongs to the inc inc periodic with period d in the x direction, and the cylinder axes are set {2πm/d} providing the reciprocal lattice. Noting that the still parallel to the y-axis. union of the scatterer positions is a small square lattice with dif period d/2, the following expression for Ω as a difference of Ω matrices is directly obtained 3. Different Types of Scatterers at k + g d pd qd || inc theSameInterface dif 0 Ω = Ω , , − Ω . (22) 2 2 2 The LMS method is commonly applied for lattices with identical cylindrical scatterers within the same layer. As Only those g in {(2πm/d,0,0)} for which m is even are inc shown below, however, different types of scatterers within the reciprocal vectors for the small lattice with period d/2. Since same layer can also be accommodated. A similar extension a lattice translation matrix Ω is periodic in its first argument for the restriction to in-plane wave propagation is made in with period 2π , there will be two groups of g with different inc dif Ivansson [22]. Ω matrices according to (22). Specifically, An illustration is given in Figure 2. Centered at the same k d pd qd || z level, at each of the three scatterer interfaces, there are dif,even 0 Ω = Ω , , − Ω (23) two types of cylindrical scatterers. Each scatterer interface 2 2 2 is treated separately. Choosing coordinates appropriately, pertains to g = (2πm/d,0,0) with even m,and inc scatterers of the first type, with transition matrix T and scattered-field expansion coefficients denoted b ,appearat k d pd qd || dif,odd 0 R = m · (d,0,0), for integers m. Scatterers of the second Ω = Ω + π , , − Ω (24) 2 2 2 type, with transition matrix U and scattered-field expansion coefficients denoted c , appear at points S in between, that is, pertains to g = (2πm/d,0,0)withodd m. inc S = (m +1/2)· (d,0,0). The reciprocal lattice vectors become The transformation of the expansion (19) to plane waves g = (2πm/d,0,0), where m runs over the integers. of the type (1) can be done separately for each of the R and The generalization of the expression (9) for the scattered S sums. In the latter case, the translation from the origin field becomes causes a sign change for some combinations of incident (g ) inc ⎡ ⎤ and scattered (g ) reciprocal lattice vectors. Specifically, with sc +P +P g = (2πm /d,0,0) and g = (2πm /d, 0, 0), the double inc inc sc sc ⎣ ⎦ u (r) = b exp i k · R · u (r − R) sc l lκ sum corresponding to the one in (11)appears as Pl (19) ⎡ ⎤ m −m inc sc + u (r) = (−1) Δ dg , j ; k d, κd, pd, qd, c sc sc +P +P ⎣ ⎦ ( ) + c exp i k · S · u r − S . l  g lκ sc j=1,2,3 Pl S × exp iK · r · e . g j sc It follows that (25) + 0   + 0 b = T · a + b + b , c = U · a + c + c , (20) 4. Designing 2D PC Slabs with where, for a scatterer of the first type at R,exp(i k · R)b a Transmission Gap and exp(i k · R)b give the coefficients for expansion in 0P regular cylindrical waves u (r − R) of the scattered field It is well known that band gaps can appear when scatterers lκ from all other scatterers of the first and the second types, with a large density are arranged periodically in a host respectively. The vectors c and c are defined analogously. with a small density. A particular 2D PC slab example with For a scatterer of the second type at S,exp(i k · S)c and lead cylinders in epoxy was considered in Mei et al. [12]. exp(i k · S)c thus give the coefficients for expansion in The epoxy parameters were 2540 and 1159.817 m/s for the || 0P regular cylindrical waves u (r − S) of the scattered field compressional- and shear-wave velocities, respectively, and lκ from all other scatterers of the second and the first types, 1.18 kg/dm for the density. Corresponding lead material respectively. parameters were 2160 and 860.568 m/s, and 11.4 kg/dm . Advances in Acoustics and Vibration 5 (θ) (θ) ◦ ◦ 60 60 ◦ ◦ 30 30 ◦ ◦ ϕ = 0 ϕ = 0 ◦ ◦ 0 0 ◦ ◦ ϕ = 90 ϕ = 90 ◦ ◦ 30 30 ◦ ◦ 60 60 20 40 60 80 100 20 40 60 80 100 Frequency (kHz) Frequency (kHz) Figure 3: Contour plot of transmittance for a PC slab with lead Figure 4: Contour plot of transmittance as in Figure 3 but for the cylinders in an epoxy host. Parameters of the slab are given in the optimizedPCslabspecifiedin Table 1. text. The direction vector of the incident compressional wave is (sin θ cos ϕ,sin θ sin ϕ,cos θ). The black and gray contours are at −150 dB and −10 dB, respectively. Table 1: Specification of a PC slab that has been optimized by DE to produce small transmittance in the band 45–65 kHz. Corresponding transmittance results are shown in Figures 4 and 5. Optimum −163.0 dB The slab was formed by sixteen layers of lead cylinders, each 2995.6 m/s Scatterer compressional-wave velocity α with a radius of 3.584 mm, with a spacing between cylinder 1200.0 m/s Scatterer shear-wave velocity β centers of 11.598 mm in the x as well as z directions, cf. Figure 1 where there are three layers. A band gap centered 14.000 kg/dm Scatterer density ρ at about 60 kHz was found. (Dimensionless frequencies and 13.292 mm Layer thickness h distances were actually used in the paper, but a specialization 4.2702 mm Largest scatterer radius r max is made here.) 4.2300 mm Smallest scatterer radius r min Figure 3 is a contour plot of the transmittance, that is, 22.670 mm Lattice period d time- (and space-) averaged transmitted energy flux relative to the incident one, up to 120 kHz. The vertical axis is for the angle θ, where the direction vector of the incident compressional plane wave is (sin θ cos ϕ,sin θ sin ϕ,cos θ). in the yz plane). A configuration with cylinders of two In-plane and out-of-plane incidence angles are considered alternating sizes is allowed, as depicted in Figure 2. Still, the together in Figure 3, where the upper half with ϕ = 0 slab is formed by sixteen layers of cylinders. The following concerns incidence in the xz plane (in-plane propagation) seven parameters are varied within the indicated search and the lower half with ϕ = 90 concerns incidence in space: scatterer compressional-wave velocity α [1500 m/s ≤ the yz plane. Only the first Brillouin zone is involved, since α ≤ 3000 m/s], scatterer shear-wave velocity β [650 m/s k = ω/α sin θ cosϕ< 2π/d when α equals the epoxy ≤ β ≤ 1200 m/s], scatterer density ρ [7 kg/dm ≤ ρ ≤ compressional-wave velocity 2540 m/s, d = 11.598 mm, and 14 kg/dm ], layer thickness h [8 ≤ h ≤ 16 mm], the largest the frequency is less than α/d = 219 kHz. Of course, the band scatterer radius r [0.225 h ≤ r ≤ 0.4 h], the smallest max max gap from Mei et al. [12], at about 60 kHz, shows up clearly scatterer radius r [0.45 r ≤ r ≤ r ], the lattice min max min max in this kind of plot. One might wonder how the geometrical period d[8(r + r )/3 ≤ d ≤ 4(r + r )]. The layer max min max min and/or material parameters of the slab should be modified to thickness h is the z distance between subsequent scatterer achieve a prescribed desired change of the appearance of the interfaces (cf. Figure 2). gap. Table 1 shows the optimum obtained with DE, along Global optimization methods can be used to design PC with corresponding parameter values. A reduction of the slabs with desirable properties. Simulated annealing, genetic transmitted field with more than 160 dB is achieved through- algorithms and differential evolution (DE) are three kinds out the band 45–65 kHz and throughout the solid angle ◦ ◦ ◦ ◦ of such methods, that have become popular during the last intervals 0 <θ < 20 for ϕ = 0 , ϕ = 90 for the fifteen years. DE, to be applied here, is related to genetic direction of incidence. High-velocity high-density cylinders algorithms, but the parameters are not encoded in bit strings, seem preferable, since the lead velocities and density are andgenetic operatorssuchascrossoverand mutation are all increased to values close to the upper ends of the replaced by algebraic operators [16]. corresponding search intervals. In this case, as large as As a very simple example, to try to position and widen the possible values of α, β,and ρ couldinfacthavebeenfixed gap in Figure 3, an objective function for DE minimization from the start. Moreover, the cylinders are almost as densely is specified as the maximum transmittance in the frequency packed horizontally as allowed by the search interval for d. band 45–65 kHz when the incidence angle θ is varied Hence, the optimization could in fact have been simplified ◦ ◦ ◦ between 0 and 20 for the two azimuthal angles ϕ = 0 (in- considerably. With only a few free parameters, a complete plane propagation in the xz plane) and ϕ = 90 (propagation search, for example, can be a feasible alternative. 6 Advances in Acoustics and Vibration (θ) −50 ϕ = 30 −100 ϕ = 60 −150 2000 4000 6000 8000 20 40 60 80 100 Number of tested PC slabs Frequency (kHz) Figure 6: Evolution of the maximum transmittance with the Figure 5: Contour plot of transmittance for the optimized PC number of tested parameter combinations for the DE optimization slab specified in Table 1. The difference to Figure 4 is that two leading to the PC slab specified in Table 1. A corresponding curve other ϕ angles are considered for the direction of the incident for a brute force Monte Carlo method is also included (gray). compressional wave. Compared to Figure 3, the contour plot in Figure 4 of the modified values for the other parameters, was almost as transmittance for the optimized PC slab shows a band gap good as the one presented in Table 1. The advantages with around 60 kHz that has become significantly wider and also cylinders of different sizes are expected to be more significant deeper. The allowed increases of α, β, ρ,and (r + r )/d in more complicated filtering cases, involving, for example, max min are certainly essential for producing this effect. Variation more than one frequency band. of h, r ,and r is needed to position the band gap at max min the desired frequency interval. Although the optimization 5. Conclusions was performed with the restriction 0 <θ < 20 ,small transmittance is apparently achieved for all angles θ (0 < The layer multiple-scattering (LMS) method is a fast semi- θ< 90 ). analytical technique for computing scattering from layers It turns out that the transmittance for the optimized PC including periodic scatterer lattices. For the 2D case with slab remains small within the 45–65 kHz band for all out- cylindrical scatterers and any solid angle direction of an of-plane incidence angles (an “absolute” band gap). Figure 5 incident plane wave, an extension has been made to scatterer shows the transmittance in the same way as Figure 4,but for lattices with cylindrical scatterers of two different sizes in the ◦ ◦ ◦ ϕ = 0 changed to ϕ = 30 and ϕ = 90 changed to ϕ = same horizontal plane. 60 . The upper and lower halves exhibit increased symmetry, Global optimization methods from inverse theory are since the ϕ angles involved are closer to one another. useful for designing PC slabs with desirable properties. A Apparently, the optimized PC slab has large transmit- differential evolution algorithm has been applied here to tance below about 30 kHz and there are regions with large position and widen an “absolute” band gap for a certain 2D transmittance above 80 kHz as well. The band gap with small PC slab. Although only limited angle intervals for the direc- transmittance extends to higher frequencies, than those in tion of incidence were included for the optimization, small the band 45–65 kHz, for plane-wave incidence directions transmittance was achieved for all solid angles specifying the with either large θ or small ϕ. direction of the incident wave. The efficacy of the DE technique can be illustrated by The possibility to include cylindrical scatterers of two showing the decrease of the maximum transmittance within different sizes in the same horizontal plane provides an the specified frequency/angle region as the number of tested additional degree of freedom that can be useful for PC parameter settings for the PC slab evolves. Figure 6 shows design purposes. In the presented example, with its spec- this decrease for the example from Figure 4.Acomparison ification of objective function and search intervals for the to the much less efficient MonteCarlo method, with random parameters, however, the difference between the optimal selection of parameters from the search space, is included. cylinder radii was rather small and the improvement only The low efficiency of the MonteCarlo approach shows that marginal. The additional flexibility is expected to be more the desired PC slabs with small transmittance only appear in important in more complicated filtering applications. For a small portion of the search space. For example, only about example, specified regions in frequency-angle space with 0.15% of the random slab selections had a maximum trans- large transmittance could be desired in addition to specified mittance below −20 dB within the specified frequency/angle regions with small transmittance. region. About 0.01% had a maximum transmittance below −100 dB. The difference between r and r in Table 1 is max min Acknowledgment rather small. Optimization was also tried with all cylinders of exactly the same radius, as in Figure 1. The obtained Alexander Moroz kindly provided his Fortran routine for optimum, with cylinders of radius 4.2503 mm and slightly calculation of lattice sums. Maximum transmittance (dB) Advances in Acoustics and Vibration 7 References Low Energy Electron Diffraction, Academic Press, [18] J. B. Pendry, New York, NY, USA, 1974. [1] T. Miyashita, “Sonic crystals and sonic wave-guides,” Measure- [19] B. L. N. Kennett, Seismic Wave Propagation in Stratified Media, ment Science and Technology, vol. 16, no. 5, pp. R47–R63, 2005. Cambridge University Press, Cambridge, UK, 1983. [2] N. Papanikolaou, R. Zeller, and P. H. Dederichs, “Conceptual [20] M. Abramowitz and I. A. 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