Applications Of Symplectic Algorithm In Wave Propagation And Vibration Problems

With the development of science and technology, the engineering problems becomemore and more complicated, and more and more researchers attach importance to thenonlinear region. The computational science, which benefits from the development ofcomputer technology, got a booming development. The computational science, together withthe theories and experiments, become the three major pillars of the contemporary scientificresearch. Applications of symplectic geometry algorithm based on Hamilton system in wavepropagation and vibration problems are presented in this doctoral thesis. Comparing with thetraditional algorithms, the symplectic geometry algorithm has unique advantages. Aconservative system can be described with the Hamilton system methodology, and itscharacteristic is the conservation of symplecticness, which is the most important feature ofconservative systems. By translating into the conservative system, some non-conservativesystems can also be dealt with symplectic algorithms. A series of algorithm for wavepropagation and vibration problems is presented in the framework of the Hamilton systemmethodology, for which the precise integration method (PIM), the extendedWittrick-Williams (W-W) algorithm, the sub-structure analysis and the inter-belt theory areintegrated to solve the concerned problems. Plenty of numerical results indicated that thesymplectic geometry algorithm used in this thesis has advantages in efficiency and precision.The main research work covers the following topics:(1) This thesis presents the precise integration methods of second order elliptic functions.After validating the algorithm by numerical examples, the improvement of the algorithm onsingular points is also presented. By comparing with other existent algorithm and software, itis found that the PIM of elliptic functions are better in precision, efficiency and applicablerange. The following wave guide problems in layered media prove the advantages of PIMagain.(2) The optical wave guide problems in layered media are solved by using the symplectictheories in electromagnetism. The wave guide problems in linear layered media are analyzedfirst, which are complicated eigenvalue problems. Then a new algorithm, which combineswith the PIM and the extended W-W algorithm, is presented with high precision. The newalgorithm can calculate the eigenvalues in any specified range without missing anyeigenvalues, which is also the main characteristic of the new algorithm. After solving the linear problems, the wave guide problems in nonlinear layered media are also analyzedbriefly. As an example, the wave guide problem in layered nonlinear Kerr material is alsonumericaly solved the PIM.(3) Based on the displacement method, a new type of shallow water wave equation isdeduced. Different from the traditional shallow water wave equation in Euler coordinatesystem, the new shallow water wave equation is deduced in Lagrange coordinate system.Hence the shallow water wave can be analyzed by using analytic structural mechanicstheories, such as corresponding Lagrange function, the variational principle etc. Consequentlythe shallow water wave equation based on displacement method has the merit of symplecticconservation. The solitary wave phenomena, which is well know in shallow water wave, alsocan be obtained by the equation based on displacement method. Under the basic assumptionof shallow water, plenty of numerical results are calculated by the equation based ondisplacement method, which are found very similar to the corresponding results given buy thetraditional KdV equation. The characteristics and differences of two kinds of equations arealso discussed in this thesis, the shallow water wave equation based on displacement methodbreaks a new way to study the shallow water wave in the framwork of the analytic structuralmechanics theories.(4) The nonlinear Duffing equation is solved by the symplectic numerical method, inwhich numerical methods with symplectic conservation, such as time domain finite elementmethod (FEM) etc, are used. According to the variational principle, the space domain FEM isautomatic symplectic conservation. Hence the matrices of time domain FEM, which are alsobased on the variational principle, are symmetrical, which proves the symplectic conservationof the time domain FEM. The small parameter perturbation approximation is applied quiteoften in solving nonlinear vibration problems. Addition perturbation is used in traditionalmethod of small parameter perturbation, but it is not symplectic conservation. Themultiplicative perturbation is used in this thesis, therefore the transfer matrices keeping theirsymplecticness, which insures the perturbation method is symplectic conservation. Comparedwith the traditional Runge-Kutt method, the validity and stability of the symplecticperturbation method in this thesis are proved by plenty of numerical examples.(5) The cross domain influence is widely taken into consideration in recently engineeringproblems. By expanding the region (interface) between two sub-structures to some width, theregion present multi-layer behavior, named inter-belt. The symplectic theories and algorithmof sub-structure inter-belt analysis and dispersion analysis are presented in this thesis. Thenthe phonon dispersion relation problem in carbon nanotubes is analyzed with the inter-belttheory. In the Hamilton system methodology, the phonon dispersion relations of carbonnanotubes calculated by the traditional structural mechanics models are dissatisfactory. After analyzing the numerical results obtained by traditional models, a new inter-belt structuralmodel is presented in this thesis. The corresponding symplectic geometry algorithm, which iscomposed of the sub-structure method, the W-W algorithm and the inter-belt analysis, isestablished. Plentiful numerical examples and comparisons are given, which show theparticular advantages of inter-belt model and symplectic theories again.