Hamiltonian systems in elasticity and their variational principles

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基于对偶变量变分原理和两端位移独立变量的保辛方法

将广义位移和动量同时用拉格朗日多项式近似,并选择积分区间两端位移为独立变量,然后基于对偶变量变分原理导出了哈密顿系统的离散正则变换和对应的数值积分保辛算法。当位移和动量的拉格朗日多项式近似阶数满足一定条件时,可以自然导出保辛算法的不动点格式。通过数值算例分析了位移和动量采用不同阶次插值所需最少Gauss积分点个数,并讨论了位移插值阶数、动量插值阶数以及Gauss积分点个数对保辛算法精度的影响,说明了上述不动点格式恰好是一种最优格式。     

Hamiltonian formulation of nonlinear water waves in a two-fluid system

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基于哈密顿动力系统新变分原理的保辛算法之一：变分原理和算法构造

提出了哈密顿动力系统的一个新变分原理,并基于此变分原理构造了四类保辛算法。通过新的变分原理定义修正作用量,然后将位移和动量采用拉格朗日多项式近似,并采用高斯积分对时间近似积分得到近似的修正作用量。在修正作用量的基础上,通过选择时间步两端不同的位移或动量作为独立变量,可构造四种不同类型的保辛算法。     

基于哈密顿动力系统新变分原理的保辛算法之三：数值算例

文献[1,2]给出了四种不同类型的求解哈密顿动力系统的数值方法,并证明了它们的保辛特性。本文将讨论这四类算法的具体数值性能,包括算法的线性稳定性,精度和效率等。     

基于哈密顿动力系统新变分原理的保辛算法之二：算法保辛性质证明

文献[1]给出了哈密顿系统的一个新的变分原理,并基于此变分原理,通过选择一个时间步长两端不同广义位移或广义动量为独立变量,给出了四种不同类型的求解哈密顿动力系统的数值方法。本文将分别证明这四类数值方法都是保辛的数值方法。     

UNCONVENTIONAL HAMILTON-TYPE VARIATIONAL PRINCIPLES FOR DYNAMICS OF REISSNER SANDWICH PLATE

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo(1987),some uncon- ventional Hamilton-type variational principles for dynamics of Reissner sandwich plate can be established systematically.The unconventional Hamilton-type variation principle can fully characterize the initial-boundary-value problem of this dynamics.In this pa- per,an important integral relation is given,which can be considered as the generalized principle of virtual work in mechanics.Based on this relation,it is possible not only to obtain the principle of virtual work in dynamics of Reissner sandwich plate,but also to derive systematically the complementary functionals for five-field,two-field and one-field unconventional Hamilton-type variational principles by the generalized Legender trans- formations.Furthermore,with this approach,the intrinsic relationship among the various principles can be explained clearly.     

Symplectic multi-level method for solving nonlinear optimal control problem

By converting an optimal control problem for nonlinear systems to a Hamiltonian system,a symplecitc-preserving method is proposed.The state and costate variables are approximated by the Lagrange polynomial.The state variables at two ends of the time interval are taken as independent variables.Based on the dual variable principle,nonlinear optimal control problems are replaced with nonlinear equations.Furthermore,in the implementation of the symplectic algorithm,based on the 2N algorithm,a multilevel method is proposed.When the time grid is refined from low level to high level,the initial state and costate variables of the nonlinear equations can be obtained from the Lagrange interpolation at the low level grid to improve efficiency.Numerical simulations show the precision and the efficiency of the proposed algorithm in this paper.     

Approximate solution to a class of multivalued nonlinear mixed variational inclusions

IntroductionIn recent years,the theory of variational inclusion has appeared as an elegant andfascinating branch of pure and applied mathematics.This theory provides us with a convenientmathematical apparatus for uniformly studying a wide range of problem…     

On the nonlinear stability of a truncated shallow spherical shell under a concentrated load

In this paper, the axisymmetric nonlinear stability of a clamped truncated shallow spherical shell with a nondeformable rigid body at the center under a concentrated load is investigated by use of the modified iteration method. The analytic formulas of second approximation for determining the upper and lower critical buckling loads are obtained. This paper was read at The Third East China Symposium on Solid Mechanics, Jiuhuashan, October, 1986.     

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.     

薄板动力学相空间非传统Hamilton变分原理与辛算法

将弹性薄板动力分析从Lagrange体系改换为Hamilton体系.通过罗恩提出的一条简单而统一的途径,建立了弹性薄板动力学的相空间非传统Hamilton变分原理,并从该原理推导出相应的Hamilton正则方程、边界条件与初始条件.然后基于这种相空间非传统Hamilton变分原理,提出弹性薄板动力响应分析的辛空间有限元-时间子域法,文中数值结果表明,这种方法的计算精度与效率都明显高于常用的Wilson-θ法和Newmark-β法.