A matrix-exponential decomposition based time-domain method for calculating the defect states of scalar waves in two-dimensional periodic structures

作者：Su, Xiao-Xing^*; Wang, Yue-Sheng; Zhang, Chuanzeng

来源：Journal of Computational Physics, 2017, 337: 403-420.

DOI：10.1016/j.jcp.2017.02.042

• Periodic structures
• Scalar waves
• Defect states
• Time-domain method
• Matrix-exponential decomposition

A time-domain method for calculating the defect states of scalar waves in two-dimensional (2D) periodic structures is proposed. In the time-stepping process of the proposed method, the column vector containing the spatially sampled field values is updated by multiplying it with an iteration matrix, which is written in a matrix-exponential form. The matrix exponential is first computed by using the Suzuki's decomposition based technique of the fourth order, in which the Floquet-Bloch boundary conditions are incorporated. The obtained iteration matrix is then squared to enlarge the time-step that can be used in the time-stepping process (namely, the squaring technique), and the small nonzero elements in the iteration matrix is finally pruned to improve the sparse structure of the matrix (namely, the pruning technique). The numerical examples of the super-cell calculations for 2D defect-containing phononic crystal structures show that, the fourth order decomposition based technique for the matrix-exponential computation is much more efficient than the frequently used precise integration technique (PIT) if the PIT is of an order greater than 2. Although it is not unconditionally stable, the proposed time domain method is particularly efficient for the super-cell calculations of the defect states in a 2D periodic structure containing a defect with a wave speed much higher than those of the background materials. For this kind of defect-containing structures, the time-stepping process can run stably for a sufficiently large number of the time -steps with a time step much larger than the Courant-Friedrichs-Lewy (CFL) upper limit, and consequently the overall efficiency of the proposed time-domain method can be significantly higher than that of the conventional finite-difference time-domain (FDTD) method. Some physical interpretations on the properties of the band structures and the defect states of the calculated periodic structures are also presented.

• 出版日期2017-5-15
• 单位中国地震局工程力学研究所; 北京交通大学