Wave Propagation, Robust Control And Symplectic Method In Hamilton Systems

With the development of science and technology, the space technology, nanotechnology,environment science, energy technology, life science and information technology all havechanged profoundly. The emergence of a great deal of new fields or intersections of differentfields is the main feature of science in the modern times. Another conspicuous feature is theacceleration of the unification process for different fields. Science is an entity and it alwaysinspires people to solve problems in a complete and uniform manner as possible as they can.The most fundamental system in mechanics is the analytical mechanics. Larange equation, thevariational principle of action, dual variable, Hamilton's equations, symplectic transformtheory and Hamilton-Jacobi theory are all very beautiful mathematical theories. And alsostatistical mechanics, electrodynamics and quantum mechanics of theoretical physics allground in the analytical mechanics. Modern control theory founded in the state space shoulddraw back at least to the Hamilton system. By introducing the dual variables into the theory ofelasticity, a rational solution method was obtained to solve many problems which could notbe solved by means of a semi-inverse method. Linear programming, quadratic programmingand nonlinear programming methods are also grounded in the fundamental basis of dualsystem. Based on the above observations, the dual system should also be applied consciouslyand systematically to other branches of mechanics. In the uniform framework of the dualHamilton system, this thesis discusses the symplectic geometric theory and accurate andefficient numerical methods associated with wave propagation, robust control andconservative dynamical systems. The main research work covers the following aspects:(1) This thesis presents theory and results for surface waves propagating in a transverselyisotropic stratified solid resting on an elastic semi-infinite space by introducing the surfacewave problem into a duality system with the system matrix of its state equation beingHamiltonian. It uses the precise integration method (PIM) and the extended Wittrick-Williams(W-W) algorithm. The PIM used here is a precise numerical method for solving sets of firstorder linear ordinary differential equations with specified two-point boundary valueconditions for space domain problems, or with specified initial value conditions for timedomain problems. It can produce numerical results up to the precision of the computer usedand preserve the symplectic structure of the Hamilton system. The extended W-W algorithmis used in the solution of the transcendental mixed variable formulations arised from the PIM, which gives the eigenvalue count for any chosen frequency. This count is just the number ofeigenvalues lying between zero and the trial value of frequency. Hence calculating it at eachtrial value of frequency will ensure that no eigenvalue will be lost.(2) The anisotropic media has 21 independent elastic parameters and its displacementcomponents are coupled together in three directions, which makes the nature of wavepropagation in the anisotropic media extremely complicated. In this thesis, the analysis ofwave propagation in the anisotropic layered media is introduced into the dual system. As thesystem matrix of the state equation is Hamiltonian, the symplectic-preserving preciseintegration method and the extended W-W algorithm are applied. The high precision andefficiency of the method are benefited from the symplectic-preserving property of the preciseintegration method for Hamilton systems and the eigenvalue counting method. The methodused in this thesis to solve the problem of wave propagation in the anisotropic layered mediahas the same simple formulas and uniform steps as for the isotropic media.(3) This thesis presents an extremely efficient and accurate solution method for thepropagation of stationary and non-stationary random waves in a viscoelastic, transverselyisotropic and stratified half space by introducing the random wave problems into the dualitysystem, transforming the random wave problems into deterministic problems andtransforming governing equations of viscoelastic materials into Hamilton equations. Theefficiency and accuracy are benefited from using the pseudo excitation method and the preciseintegration method. Firstly, PEM is used to transform the random wave equation intodeterministic ones in the frequency-wavenumber domain. Secondly, this thesis proved thatalthough the damping exists in the viscoelastic material, the governing equations is stillHamilton equation in the duality system. Finally, the symplectic preserving PIM is used tosolve the Hamilton state equation for random wave problem in the frequency-wavenumberdomain. It can produce numerical results up to the precision of the computer used andpreserve the symplectic structure of the Hamilton system.(4) This thesis presents the efficient and accurate method for soil-structure interactionexcited by stationary random waves. The soil considered is a viscoelastic, transverselyisotropic and layered half space and the structure above it is modelled by means of thestandard finite element method. So far, the studies of soil-structure interaction problems wereall refer to deterministic excitations although the seismic waves are random in nature. Thisthesis investigates soil-structure interaction due to stationary random waves. The pseudoexcitation method is used to transform this stationary random soil-structure interactionproblem into a series of deterministic harmonic response analyses and the precise integrationmethod is used to integrate the ordinary differential equations in the frequency-wavenumberdomain. (5) This thesis presents the generalized modal synthesis method of the de-centralizedcontrol for large-scale systems in the Hamiltonian system. When the de-centralizedcontrollers for large systems are designed, the whole system is divided into subsystems,which are analysed more easily by using the available control theory before they arere-combined. This strategy is analogous with the sub-structural analysis in structuralmechanics. In this thesis, it is shown that not only the static sub-structural analysis isanalogous with the de-centralized control theory, but also the dynamic sub-structural analysisis analogous with the de-centralized control theory. It is also proved that the optimalparameter of the whole large-scale system corresponds to the fundamental vibration frequencyof the whole structure. The modal synthesis methodology is transplanted from structuralmechanics to the de-centralized control theory in order to compute the optimal parameter ofthe control system. This thesis also presents the orthogonality, completeness and theexpansion theorem of eigenfunctions of the modal synthesis method for the de-centralizedcontrol.(6) The existed symplectic preserving algorithms are nearly all based on the Runge-Kuttafinite difference scheme. It is however too hard to construct a higher order symplecticpreserving Runge-Kutta method. In this thesis, by applying the concepts of analyticalstructural mechanics and using the time-domain finite element to discrete the generalizeddisplacements, an approximate action is obtained. By using it as the generating function,symplectic preserving algorithms for conserved dynamical systems with or withoutconstraints are constructed. The accuracy of the proposed method can be improvedsignificantly by using the multi-node shape function in the time-domain finite element method.Numerical examples for celestial mechanics, molecular dynamics and statistics mechanics alldemonstrate satisfactory results.