Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system

Abstract

Abstract-This paper aims at studying the band gap phenomena of three-dimensional phononic crystals using the finite difference time domain (FDTD) method and a PC cluster system. In the paper, Bloch's theorem is applied to the wave equation and to the boundary conditions of the periodic structure. We calculate the variations of displacements and take discrete Fourier transform to acquire the resonances of the structures. Then, the dispersion relations of the bulk acoustic wave can be obtained and the band gaps are predicted accordingly. On the other hand, because of larger data calculation in three-dimensional phononic crystals, we introduce the PC cluster system and parallel FDTD programs written with respect to the architecture of a PC cluster system. Finally, we discuss the numerical calculation of two-dimensional and three-dimensional phononic crystals consisting of steel/epoxy and draw conclusions regarding the band gap phenomena between these phononic crystals.

Multidimensional Phononic Bandgaps in Three-Dimensional Lattices for Additive Manufacturing

“…A relative gap to mid-gap ratio (relative BG width) can be used to compare the calculated BGs of the BCC xyz and res-BCC xyz lattices to those of the BCC-inspired multimaterial lattices of Lu et al [13] and Husieh et al [45]. A relative BG width is the quotient of the bandwidth and the BG mean frequency (BGMF) and is independent of the unit cell size; a high relative BG width is more desirable as it indicates a wide BG with low frequency.…”

“…Syam et al [42] determined the natural frequencies of BCC lattices with additional reinforcement struts in the x, y and z directions (designated as BCC xyz ) for vibration isolation purposes, but did not model the dispersion curves (DCs) of the lattice, and did not report on the effect of the lattice volume fraction on achieving vibration isolation. Lu et al [13] and Hsieh et al [45] independently predicted the DCs of multimaterial BCC lattice designs, but to date there have been no reports on the manufacturability or performance of these designs. In comparison to single-material AM, which is well-established, multimaterial AM currently requires manual assembly (such as in the lattice work of Matlack et al [15]), requires support structures that constrain the design of the part, necessitates post-processing (such as in the work of Choi et al [46]), and is limited to a small range of materials.…”

Three-dimensional phononic nanocrystal composed of ordered quantum dots

“…The first step to investigate the consequent phononic band gaps is based on a 3D FDTD method. [7][8][9] Here, the dynamics of the longitudinal phonons with wavevectors along the surface normal z, i.e., one of the principal axes of the studied lattice, was specially addressed as a demonstration. Figure 1͑b͒ shows the calculated square of laterallyaveraged displacement in the QD crystal under the excitation of an impulsive longitudinal stress T zz , exhibiting two folded dispersion curves.…”

Acoustic mirage in two-dimensional gradient-index phononic crystals

“…To investigate the propagation of acoustic waves in the GRIN PC, we simulated the SV-mode BAW by a FDTD method. 15,28 The FDTD program was developed per the theory of elasticity: The equation of motion and the constitutive law were discretized to simulate wave propagation in linear elastic materials. A tilted line source with a width of 4a was placed in the epoxy region ͑centered at y =−5͒ and was defined by setting an initial value of body force in the equation of motion along the z axis ͑parallel to the cylinder axis͒ to generate a SV-mode BAW.…”