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.

Just as photonic crystals can be used to manipulate light, phononic crystals (PCs) with inclusions in a lattice with single, double, or triple periodicity can be used to manipulate sound [

Several computational methods have been adapted and developed to study wave propagation through PCs. Two such methods are the purely numerical finite-difference time domain (FDTD) method and the semianalytical layer multiple-scattering (LMS) method, which is developed from Korringa-Kohn-Rostoker theory [

The LMS method has also been applied to the 2D case with infinite cylindrical scatterers. It is mainly the in-plane propagation case that has been considered [

In the present paper, basic LMS equations for propagation of plane waves of any direction through a 2D PC slab are first provided in Section

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

As in Figure

Horizontal

Sound waves with time dependence

As detailed in [

Explicit expressions for the

For scatterers at

The vector

The R/T matrices are obtained, finally, by transforming the expansion (

Explicit expressions for the

An incident plane compressional wave

An incident plane shear wave of SV or SH type can be expanded by a superposition of two cases. The first case,

The second case,

The lattice translation matrix

For a homogeneous cylindrical scatterer, the interior field and the exterior field can be expanded in cylindrical waves

The LMS method is commonly applied for lattices with identical cylindrical scatterers within the same layer. As shown below, however, different types of scatterers within the same layer can also be accommodated. A similar extension for the restriction to in-plane wave propagation is made in Ivansson [

An illustration is given in Figure

The configuration from Figure

The generalization of the expression (

With

In order to form the R/T matrices, incident plane waves with different horizontal wavenumber vectors

The transformation of the expansion (

It is well known that band gaps can appear when scatterers with a large density are arranged periodically in a host with a small density. A particular 2D PC slab example with lead cylinders in epoxy was considered in Mei et al. [^{3} for the density. Corresponding lead material parameters were 2160 and 860.568 m/s, and 11.4 kg/dm^{3}. The slab was formed by sixteen layers of lead cylinders, each with a radius of 3.584 mm, with a spacing between cylinder centers of 11.598 mm in the

Figure

Contour plot of transmittance for a PC slab with lead cylinders in an epoxy host. Parameters of the slab are given in the text. The direction vector of the incident compressional wave is (sin

Global optimization methods can be used to design PC slabs with desirable properties. Simulated annealing, genetic algorithms and differential evolution (DE) are three kinds of such methods, that have become popular during the last fifteen years. DE, to be applied here, is related to genetic algorithms, but the parameters are not encoded in bit strings, and genetic operators such as crossover and mutation are replaced by algebraic operators [

As a very simple example, to try to position and widen the gap in Figure ^{3 }^{3}], layer thickness

Table

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

Scatterer compressional-wave velocity | 2995.6 m/s |

Scatterer shear-wave velocity | 1200.0 m/s |

Scatterer density | 14.000 kg/dm^{3} |

Layer thickness | 13.292 mm |

Largest scatterer radius | 4.2702 mm |

Smallest scatterer radius | 4.2300 mm |

Lattice period | 22.670 mm |

Compared to Figure

Contour plot of transmittance as in Figure

It turns out that the transmittance for the optimized PC slab remains small within the 45–65 kHz band for all out-of-plane incidence angles (an “absolute” band gap). Figure

Contour plot of transmittance for the optimized PC slab specified in Table

Apparently, the optimized PC slab has large transmittance below about 30 kHz and there are regions with large transmittance above 80 kHz as well. The band gap with small transmittance extends to higher frequencies, than those in the band 45–65 kHz, for plane-wave incidence directions with either large

The efficacy of the DE technique can be illustrated by showing the decrease of the maximum transmittance within the specified frequency/angle region as the number of tested parameter settings for the PC slab evolves. Figure

Evolution of the maximum transmittance with the number of tested parameter combinations for the DE optimization leading to the PC slab specified in Table

The difference between

The layer multiple-scattering (LMS) method is a fast semianalytical technique for computing scattering from layers including periodic scatterer lattices. For the 2D case with cylindrical scatterers and any solid angle direction of an incident plane wave, an extension has been made to scatterer lattices with cylindrical scatterers of two different sizes in the same horizontal plane.

Global optimization methods from inverse theory are useful for designing PC slabs with desirable properties. A differential evolution algorithm has been applied here to position and widen an “absolute” band gap for a certain 2D PC slab. Although only limited angle intervals for the direction of incidence were included for the optimization, small transmittance was achieved for all solid angles specifying the direction of the incident wave.

The possibility to include cylindrical scatterers of two different sizes in the same horizontal plane provides an additional degree of freedom that can be useful for PC design purposes. In the presented example, with its specification of objective function and search intervals for the parameters, however, the difference between the optimal cylinder radii was rather small and the improvement only marginal. The additional flexibility is expected to be more important in more complicated filtering applications. For example, specified regions in frequency-angle space with large transmittance could be desired in addition to specified regions with small transmittance.

Alexander Moroz kindly provided his Fortran routine for calculation of lattice sums.