A Suzuki’s Matrix-exponential Decomposition-based Time-domain Method For Band Structure Calculation Of One-dimensional Periodic Structures

Phononic crystals(PNCs)are a kind of composite materials which have periodic structures and exhibit elastic wave band gaps where the propagation of acoustic/elastic waves is fully forbidden.Due to its band gap characteristics,it has shown broad application prospects in many areas such as new shock attenuation and noise reduction materials,waveguides,filters and transducers,etc.Therefore,calculating the band gap of a phononic crystal is an important aspect in the theoretical research of phononic crystals.The finite-difference time-domain method has become a commonly used method for calculating phononic band structures because of its outstanding advantages,for example its wide applicability,its good convergence properties,its efficient parallel and easy hardware implementation.However,a limitation of the FDTD method is its conditional numerical stability.So many improved time domain methods have been developed to overcome this deficiency.In this paper we conduct an in-depth study and development of an improved time domain method based on Suzuki Matrix Exponential Function Decomposition.The research contents are summarized as follows:(1)A new algorithm based on Suzuki ’s high-order decomposition technique for the matrix exponential computation is designed to calculate the band structures of one-dimensional phononic crystals.The 6th and 8th order algorithms are taken as examples to verify the efficiency,accuracy and stability of the method.Finally,the 4th order algorithm is found to be the best in terms of efficiency and stability based on the analysis of the numerical results.(2)A lcal high-precision algorithm based on Suzuki’s matrix exponential decomposition technique is designed.The efficiency and accuracy of the method are also verified and the stability characteristics of the algorithm are studied as well.In this algorithm,the local structure containing the defect is calculated separately from the complete structure without the defect.After completing the calculation of the complete structure,the corresponding matrix is saved,and the local structure containing the defect is calculated to obtain the corresponding matrix of the high-order algorithm.Finally,the matrix splicing and time iteration are performed.The advantage of this model of calculation is that after replacing the defect material or changing form and location,it is only necessary to recalculate the local structure containing the defect(which takes very little time)and then performs matrix stitching and iteration.