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

• Published in: International journal of numerical modelling Vol. 29; no. 4; pp. 773 - 791
• Main Authors: Su, Xiao-Xing, Wang, Yue-Sheng, Zhang, Chuanzeng, Yu, Yaxin
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 01.07.2016; Wiley Subscription Services, Inc
• Subjects: A defects; band structure calculation; Band structure of solids; Decomposition; Finite difference time domain method; Mathematical analysis; Mathematical models; matrix-exponential decomposition; Periodic structures; Time domain techniques; time-domain method
• ISSN: 0894-3370; 1099-1204
• DOI: 10.1002/jnm.2141

Summary A time‐domain method for calculating the band structure of one‐dimensional periodic structures is proposed. During the time‐stepping of the method, the column vector containing the spatially sampled field data is updated by multiplying with an iteration matrix. The iteration matrix is first obtained by using the matrix‐exponential decomposition technique. Then, the small nonzero elements of the matrix are pruned to improve its sparse structure, so that the efficiency of the matrix–vector multiplication involved in each time‐step is enhanced. The numerical results show that the method is conditionally stable but is much more stable than the conventional finite‐difference time‐domain (FDTD) method. The time‐step with which the method runs stably can be much larger than the Courant–Friedrichs–Lewy (CFL) limit. And moreover, the method is found to be particularly efficient for the band structure calculation of large‐scale structures containing a defect with a very high wave speed, where the conventional FDTD method may generally lose its efficiency severely. For this kind of structures, not only the stability requirement can be significantly relaxed, but also the matrix‐pruning operation can be very effectively performed. In the numerical experiments for large‐scale quasi‐periodic phononic crystal structures containing a defect layer, significantly higher efficiency than the conventional FDTD method can be achieved by the proposed method without an evident accuracy deterioration if the wave speed of the defect layer is relatively high. Copyright © 2016 John Wiley & Sons, Ltd.