Applications Of Symplectic Method On One-dimensional Phononic Crystals And Perturbed Hamiltonian System

In the framework of symplectic system,this thesis explores some theoretical analysis and numerical methods for the eigenvalue computation of phononic crystals and symplectic integration of perturbed Hamilton systems.By describing the wave propagation in the phononic crystals with Hamilton system,the symplectic transformed matrix in state space is derived,and a proof of one-dimensional Bloch theorem can be obtained with the use of the character of the symplectic transformed matrix. The band structures are computed by Wittrick-Williams(W-W) algorithm which is a stable and accuracy numerical method and can ensure that no eigenvalue will be lost.The symplectic method is further applied to investigate the surface state of the semi-infinite periodical layered media.With the use of the properties of the symplecfic transformed matrix that the reciprocal of the eigenvalue is still eigenvalue of the matrix,the surface state can be calculated in one unite cell for the single freedom problem.The eigenvalues of one unit cell under different boundary conditions can be easily computed via W-W algorithm.Then the exact surface state can be obtained after taking the decay condition in the infinite direction into consideration.Due to the reproductive properties of unit cell in the semi-infinite periodical structure, one can construct a super-cell to approximate it in analogy of the idea of 2~N algorithm used in Prof.Zhong's Precise Integration Method.With this method,the semi-infinite periodical structure with defect can be approximated by a finite structure composed by the defect section and a super-cell.When the super-cell is large enough,the eigenvalues of this finite structure can be seen as good approximations of those of the real defect states.The artificial boundary condition,however,can induce some false results,which should be deleted according to the decay condition in the infinite direction.The different section combinations based on the symplectic transformed matrix and mixed energy matrix are also discussed in this thesis, respectively,and it indicates that mixed energy matrix has superiority in numerical computation.The present thesis investigates the numerical integration of the perturbed Hamilton system which is composed of an integrable system and a little perturbation.By employing the analytical solution of the integrable system as the interpolation function,the approximate integration equation of the Hamilton system can be derived on the basis of the principle of least action.Further,with the use of the analytical solution,a simple and efficient symplectic perturbed integration scheme can be obtained.Numerical examples for perturbed Kepler problem and nonlinear pendulum are used to demonstrate the validity of the scheme in this thesis.