Researches On Time Domain Integral Methods And Scattering Analysis Methods For Periodic Structures

Periodic structures are applied in many different kinds of engineering fields. Today, the dynamical analyses of periodic structures have been one of the important topics in the computational mechanics. In this PhD thesis, some researches on time domain integral methods and scattering analysis methods for periodic structures are presented. Focusing on the periodic structure, the main problems discussed are:(1) The wave scattering in the periodic structure with scatters and the phenomenon ofclose eigenvalues.(2) Computing the defect states of periodic structures with defects.(3) The analysis of large-scale periodic structures in time domain.(4) The analysis of the nonlinear periodic FPU model in time domain.Aiming at the above four problems, the PhD thesis gives deep discusses, which are based on some computational mechanics methodology, such as the precise integration method, symplectic mathematical method, symplectic perturbation method, energy-band theory, sub-structure technology, Fourier transform, and so on. The main work in this PhD thesis can be summarized as:(1) Based on the symplectic theory in applied mechanics, some characteristic theories, such as the symplectic Gram-Schmidt orthogonal algorithm, the independence between symplectic eigensoulutions, precise integration method for the problem with two end boundary conditions, and so on, are employed to analyze the wave scattering and the phenomenon of close eigenvalues. A symplectic mathematical method are presented for the wave scattering in the periodic structures with scatterers and the energy-band theory is used to study the phenomenon of close eigenvalues.(2) To compute the defect states in the discreted periodic structures with defects, a numerical method based on the extended finite Fourier transform, which is a member of the Fourier transform family, are proposed. Compared with the super-cell method, the proposed method do not need to choice the number of the cells and is truely suitable to the infinite periodic structures with defects. The accuracy of the results of the proposed method is better than that of the super-cell method under the condition that the computational scale of both methods are the same. (3) For the time analysis of the large-scale periodic structure, the sub-domain precise integration method is developed. The precise integration method and sub-structure technology are combined in the presented method, which makes use of the geometric characteristic of the periodic structure fully. From the point of the computational efficiency and the required memory space, the presented method is quite suitable to the time analysis of the large-scale periodic structure. For the time analyses of some simple discreted periodic structures, the extended finite Fourier transform are used and the exact solutions are given. The exact solutions can be combined with the sub-structure technique to further improve the time analysis of certain large-scale periodic structures.(4) For the nonlinear periodic FPU model problem with multiple time scales, a method which combines the symplectic perturbation method and average method is proposed. Compared with the classical difference method, the proposed method can calculate precisely in a long time, overcome the stiff problem, compute with a big time step, display the details of the high-frequency vibration in the multiple-scale way, and overcome the problem of numerical resonance.For the problems discussed in this PhD thesis, some numerical tests are given. The numerical results verify the validity of the presented methods and the correctness of the given conclusions.