【作者】 高强；

【导师】 钟万勰； 林家浩；

【作者基本信息】 大连理工大学 ， 固体力学， 2007， 博士

【摘要】 随着科学和技术的发展，空间、纳米、环境、能源、生命和信息等各方面都发生着深刻变化，大量新学科涌现与多学科交叉成为当代科学发展的一个突出特点。当代科学发展的另一个特点是学科统一化进程加速。科学是一个整体，其内在力量总是激励人们以尽可能统一的方式完整地认识问题。力学作为一门理论科学，其最根本的体系是经典分析力学。其中的Lagrange方程、最小作用量原理、Hamilton正则方程、对偶变量、辛变换及Hamilton-Jacobi理论等都是极为优美的数学理论体系。而且分析力学也为理论物理中的统计力学、电动力学、量子力学等学科奠定了坚实的基础。现代控制论所奠基的状态空间法的起点至少也应当回溯到哈密顿正则方程体系。将对偶变量理论体系引入到弹性力学，导向了理性的求解方法，从而可以求得许多以往半逆凑合法无法导出的结果。线性规划、二次规划以及非线性规划的基本方法也奠基于对偶体系基础上。基于以上观察，应用力学也应当自觉地、系统地将对偶体系运用于其它各个学科分支。本博士学位论文在对偶哈密顿体系的统一框架下，探讨了波的传播、鲁棒控制与保守动力系统中的辛几何求解理论和高效高精度的数值求解方法。本文的主要工作如下：（1）本文将各向同性分层半无限空间中表面波问题导向对偶体系，基于状态方程矩阵是哈密顿矩阵的性质，提出了一种求解分层半无限空间中Love和Rayleigh表面波问题的精确、高效方法。此方法联合应用了精细积分法和扩展W—W算法。精细积分法是一个求解一阶线性常微分方程组的精确数值方法，既可以应用于空间域的两点边值问题，又可以用于时间域的初值问题。它可以给出计算机精度意义下的精确解，并且保持哈密顿体系的辛结构。应用扩展的W—W算法求解从精细积分法得到的超越方程时，对任意给定的试验频率ω给出本征值计数J（ω），通过计算试验频率ω处的本征值计数可以确保不会遗漏任何本征值。（2）各向异性介质有21个独立的弹性参数，并且三个方向的位移分量都耦合在一起，这使得各向异性介质中波的传播性质变得异常复杂。本文将各向异性分层介质中的波传播问题导向对偶体系，基于状态方程是哈密顿矩阵的性质，进一步将保辛的精细积分法和扩展W—W算法推广应用于各向异性分层介质的波传播问题，该方法的有效性和高精确性得益于精细积分法对哈密顿系统的辛结构的保持和本征值计数方法的应用。应用本文方法求解各向异性分层介质的波传播问题具有与求解各向同性介质波传播问题同样简洁的公式和统一的计算步骤。（3）本文将粘弹性分层介质中的平稳和非平稳随机波传播问题导入到对偶体系，将随机性问题转换为确定性问题，通过变换将粘弹性介质中的控制方程转换为哈密顿正则方程，提出了平稳和非平稳随机波传播问题的虚拟激励—保辛精细积分法。首先，虚拟激励法将随机波传播问题转换为波数—频率域中的确定性问题进行求解。然后证明，虽然岩层是粘弹性介质，存在阻尼或者能量耗散机制，但是将控制方程导入到对偶体系后仍然是哈密顿正则方程。在对偶体系下，波数—频率域中的状态方程是线性哈密顿方程的两端边值问题，保辛的精细积分法是求解此类问题的高精度算法，可以给出计算机意义下的精确解。（4）本文提出了随机地震波作用下分层半无限空间与上部结构相互作用问题的高效、精确方法。目前，对于岩层—结构相互作用的研究都采用确定性激励，但地震激励本质上是随机的。本文中将虚拟激励方法推广应用于随机性岩层—结构相互作用问题。应用虚拟激励法后，所有控制方程都转换为确定性方程，这时可以应用精细积分法来积分波数—频率域中的控制方程，得到计算机意义下的精确解。（5）本文提出了哈密顿体系下大系统分散鲁棒控制的广义模态综合法。在大系统控制中，分散控制方法通过将整个系统划分成一系列子系统来研究系统的控制问题，这与结构力学中的子结构分析技术类似。本文表明分散控制的提法与结构力学中的子结构分析相互模拟：静力子结构分析对应于通常意义的分散控制理论，而动力子结构分析则对应于分散H_∞控制理论。本文由此建立了H_∞分散控制与子结构振动问题之间的对应关系，指出大系统分散控制的最优参数γ_cr^-2对应于整体结构的一阶本征值，并利用模态综合法计算出H_∞分散控制问题中的最优参数γ_cr^-2。给出了大系统分散鲁棒控制广义模态综合法的正交性、完备性和展开定理。（6）已有哈密顿系统的保辛算法多是基于差分近似的Runge-Kutta方法。保辛Runge-Kutta法的困难之处在于构造高精度算法。本文基于分析结构力学，利用时间有限元的概念，将广义位移用形函数离散，得到近似作用量。利用近似作用量作为生成函数，构造无约束和有约束保守系统的保辛算法。利用多个节点构造高阶形函数，能显著提高有限元保辛算法的精度。最终通过天体力学、分子动力学和统计物理学等领域的大量数值算例验证了本文方法的有效性。更多还原

【Abstract】 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.更多还原