Finite difference time domain calculation of three-dimensional phononic band structures using a postprocessing method based on the filter diagonalization

Abstract

If the band structure of a three-dimensional (3D) phononic crystal (PNC) is calculated by using the finite difference time domain (FDTD) method combined with the fast Fourier transform (FFT)-based postprocessing method, good results can only be ensured by a sufficiently large number of FDTD iterations. On a common computer platform, the total computation time will be very long. To overcome this difficulty, an excellent harmonic inversion algorithm called the filter diagonalization method (FDM) can be used in the postprocessing to reduce the number of FDTD iterations. However, the low efficiency of the FDM, which occurs when a relatively long time series is given, does not necessarily ensure an effective reduction of the total computation time. In this paper, a postprocessing method based on the FDM is proposed. The main procedure of the method is designed considering the aim to make the time spent on the method itself far less than the corresponding time spent on the FDTD iterations. To this end, the FDTD time series is preprocessed to be shortened significantly before the FDM frequency extraction. The preprocessing procedure is performed with the filter and decimation operations, which are widely used in narrow-band signal processing. Numerical results for a typical 3D solid PNC system show that the proposed postprocessing method can be used to effectively reduce the total computation time of the FDTD calculation of 3D phononic band structures.

Topology optimization of two-dimensional phononic crystals using FDTD and genetic algorithm

“…For a given Bloch wave vector, the obtained difference equations are solved by iterations in the time domain after a certain initial condition is set. The data series generated during the iterations are generally post-processed by the fast Fourier transform (FFT) or some high-resolution spectral estimation method [9,10] to obtain the eigenfrequencies of the PNC. For the details of the FDTD method for PNCs, see Refs.…”

A matrix‐exponential decomposition‐based time‐domain method for band structure calculation of one‐dimensional periodic structures

“…The band structure calculation is an important aspect in the studies of periodic structures. The numerical methods developed for photonic/phononic band structure calculations include the plane wave expansion method [3,4], multiple scattering theory [5,6], transfer matrix method [7,8], wavelet method [9,10], finite-difference time-domain (FDTD) method [11][12][13][14][15][16][17][18], finite element method (FEM) [19,20], and boundary element method (BEM) [21,22], etc. Among them, the FDTD method has some noticeable advantages.…”

“…Therefore, the overall calculation efficiency of the FDTD may be drastically reduced just because of the presence of a small domain with a very high wave speed. Considering the large number of FDTD time-steps, a sophisticated spectral analysis method rather than the classical fast Fourier transform (FFT) may often be used to serve as the post-processing method following the FDTD time-stepping process [11,17,18]. In this way, the calculated results can be improved significantly when the FDTD is run with relatively few steps.…”

An improvement of the filter diagonalization-based post-processing method applied to finite difference time domain calculations of three-dimensional phononic band structures

“…the frequency intervals where the propagation of elastic waves (EWs) is fully forbidden in any direction. The finite difference time domain (FDTD) [2][3][4][5][6][7][8][9][10][11] method is a widely used method for the band structure calculation of PNCs. Although it is well known that the computational cost is high, relatively less attention has been paid to the improvement in the FDTD calculation of three-dimensional (3D) phononic band structures.…”

“…The FDTD-TS obtained from the phononic band structure calculation is a deterministic time signal formed by a number of harmonic components. For this kind of time signal, the filter diagonalization method (FDM) [10,[13][14][15][16][17][18][19][20][21], originally developed in the nuclear magnetic resonance signal processing community [13], has significantly higher frequency resolution than not only the FFT, but also some high-resolution spectral estimation methods [22] in modern signal processing. Instead of the classical FFT, the FDM has been proposed and used in post-processing for the FDTD calculation of 3D photonic or phononic band structures [10,16,17,20].…”