Design of Self-collimating Phononic Crystals Using Topology Optimization Method

Abstract

Phononic crystals (PCs), which can be viewed as an elastic version of artificial crystalline structures, have lately attracted remarkable attention in science and engineering fields due to various peculiar wave propagation phenomena found in them. Especially, the self-collimation phenomenon by using PCs, which refers to wave propagation with zero diffraction, is studied to design devices such as waveguides, lens, diodes, beam bends and splitters. This interesting wave propagation phenomenon in PCs arises from scattering mechanism originating from periodic arrangement and topology of unit cells of a PC. Previous works concerned with design of self-collimating PCs have mainly considered varying parameters controlling scattering from periodic arrangement such as lattice symmetry and periodicity. There have been also design works that treat the parameters related to localized scattering from the unit cell structure

altering the shape of inclusion in the unit cell only. However, there is no investigation to design the material distribution of the unit cell of self-collimating PCs. Furthermore, it is difficult to figure out the effects of topological layout of the unit cell on the dispersion relation of PCs due to inherent complex physics. Thus, we exploit the topology optimization method that does not necessarily require pre-determined topological material layout of the unit cells of the self-collimating PCs. In this thesis, we propose a design formulation that finds an optimal material distribution of the unit cell of self-collimating PCs in which wave propagates along the desired direction governed by the geometric features of the equi-frequency contour (EFC) of PCs. Wave behavior in self-collimating PCs can be predicted by analyzing the EFC representing a set of propagating wave vectors at the specific frequency. Because the propagating direction of waves is always perpendicular to an EFC for a given wave vector in a lossless medium, its geometric properties such as slope and curvature become important in designing self-collimating PCs. With finite element analysis for PCs and the gradient based topology optimization using the curvature and slope of EFC as the key components of objective and constraint functions, the proposed novel formulation is established. The specific design goal is to obtain the material distribution of the unit cell for PCs that collimates two-dimensional bulk elastic shear-vertical waves along a target direction. To show the feasibility of the formulation, several numerical design examples are presented.