The objective of this paper is to present a peer-review of the literature and trends surrounding the design of phononic crystals (PnCs) using topology optimization methods. After first providing a brief review of new developments, improvements, and applications of PnCs, this paper investigates the techniques and applications of topology optimization methods for phononic band gaps and functional structures. Both gradient-based and non-gradient-based topology optimization methods have been employed in the design of PnCs. The advantages and drawbacks of the methods used in this area are discussed in detail in this paper. Based on observations of the current state, we highlight suggestions and ideas for future research in the field of phononic crystal design optimization. The paper examines how applying topology optimization techniques into the design of PnCs yields promising performance and functions. This literature survey is designed to provide an overview of the recent advances and novel applications of popular topology optimization methods for the design of PnCs.

A comprehensive survey on topology optimization of phononic crystals

The paper analyses the wave propagation through a piezoelectric semiconductor slab sandwiched by two piezoelectric half-spaces. The two piezoelectric half-spaces (left and right) are both AlN materials, the middle piezoelectric semiconductor slab is ZnO material and is assumed to be transversely isotropic. The semiconductor effect is emphasized by considering the coupling mechanical displacement, electric potential and the carrier in the slab. The state transition differential equation is derived based on the reduction of order of the governing equations and the transfer matrix of state is obtained by solving the state transition equation. The present method can deal with not only the homogeneous slab but also the heterogeneous slab. Two cases (incident QP wave and incident QSV wave) are considered, and the energy reflection and transmission coefficients varying with the incident angle are calculated. The results show that the steady carrier density and the exterior biasing electric field have obvious influences on the reflection and transmission coefficients. This investigation provides a new thought for the adjustment and controlling of elastic wave propagation in the laminated structures.

Wave propagation through a piezoelectric semiconductor slab sandwiched by two piezoelectric half-spaces

Evanescent Bloch waves and complex band structure in magnetoelectroelastic phononic crystals

Surface effect on the propagation of flexural waves in periodic nano-beam and the size-dependent topological properties

A new model for determining band gaps for wave propagation in two-dimensional (2-D) periodic composite structures is developed using a modified couple stress theory. The general equation of motion and boundary conditions in the elasto-dynamics of the modified couple stress theory are first derived by a variational formulation based on Hamilton’s principle. The in-plane and anti-plane wave equations incorporating microstructure effects are then obtained explicitly from the general equation of motion. The plane wave expansion method and the Bloch theorem for periodic media are used to solve the in-plane and anti-plane wave equations, which are reduced to an eigenvalue problem in each case. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new model recovers the classical elasticity-based model when microstructure effects are not considered. To quantitatively illustrate the newly developed model, a parametric study is conducted for 2-D periodic composite structures containing circular and square inclusions. The numerical results reveal that the microstructure effects on the band gaps are significant only when the unit cell size is small for both the composite structures. In addition, it is found that the volume fraction has a significant effect on the band gap size, and the inclusion shape has a large influence on the band gaps.

Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects

The band gaps of a laminated piezoelectric/piezomagnetic phononic crystal with graded interlayer are studied in this paper. First, the transfer matrix method and the Bloch theorem are used to derive the dispersion equation. Next, the graded interlayer with different gradient profiles between the piezoelectric and the piezomagnetic materials is considered. The graded interlayer is modeled as a system of homogenous sublayers with both piezoelectric and piezomagnetic effects simultaneously. The effect of the graded interlayer on the band gap is introduced by inserting an additional interlayer transfer matrix in the calculation of the total transfer matrix. Finally, the dispersion equation is solved numerically, and the dispersive curves are shown in the Brillouin zone. The band gaps of the phononic crystal with graded interlayer are compared with that without graded interlayer. The influences of the graded interlayer with different gradient profiles on the band gap of a laminated piezoelectric/piezomagnetic phononic crystal are discussed based on the numerical results.

Band gap of piezoelectric/piezomagnetic phononic crystal with graded interlayer

Owing to the increasing ratio of surface to bulk volume, surface effects emerge in the mechanical performance of devices and materials when the characteristic size reduces to nanoscale. In this paper, the dispersion relations and the band gap properties of 2D phononic crystals with periodically arranged nanoholes or nanofibers are studied. The scattering matrix of a single scatterer is first derived from a set of nontraditional boundary conditions based on the surface elasticity theory. Then, the addition theorem of cylindrical waves and Bloch’s theorem of periodical structure are used to obtain the dispersion relations of an elastic wave polarized in the plane perpendicular to the axis of the nanoholes or nanofibers. It is found that the surface effect has remarkable influences on the dispersion relations and the band gaps for the phononic crystal with nanoholes. For the 2D phononic crystal with periodical arranged nanofibers, the interface effect is more evident for the softer fibers than for the stiffer fibers.

Surface/interface effects on dispersion relations of 2D phononic crystals with parallel nanoholes or nanofibers

The present work mainly focuses on the influence of interface material parameters, namely the interface mass density, the interface elastic rigidity, and the interface inertial interaction constants, on the reflection and transmission behavior of elastic wave propagating through dipolar gradient elastic solids. First, the interface kinetic energy density and interface deformation energy density are taken into account. By application of Hamilton’s variation principle, the governing equations and the boundary conditions of the dipolar gradient elastic solid are obtained. Due to the consideration of microstructure effects of the material, the interface conditions can be proposed in different forms. These interfacial conditions which include the microstructure effects and interface energy effects are then used to determine the amplitude ratio of reflection and transmission waves. A numerical example is provided for the generalized internal roller interface. The influence of the interface material parameters upon the reflection and transmission coefficients in terms of energy fluxes ratio is discussed based on the numerical results. It is revealed that the reflection and transmission behavior can be manipulated by the elaborated design of the interface at both the macroscale and the microscale.

Influences of interface properties on the wave propagation in the dipolar gradient elastic solid

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Band gap of piezoelectric/piezomagnetic phononic crystal with graded interlayer

Surface/interface effects on dispersion relations of 2D phononic crystals with parallel nanoholes or nanofibers

Influences of interface properties on the wave propagation in the dipolar gradient elastic solid