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。     

一种基于Hamel形式的无条件稳定动力学积分算法

Time integration algorithm is a key issue in solving dynamical system. An unconditionally stable Hamel generalized α method is proposed to solve the instability issue arising in the time integration of dynamic equations and to eliminate the pseudo high order harmonics incurred by the spatial discretization of finite element simultaneously. Therefore, the development of numerical integration algorithm to solve the above-mentioned problems has important theoretical and application value. The algorithm proposed in this paper is developed based on the moving frame method and Hamel’s field variational integrators along with the strategy to construct an unconditionally stable Hamel generalized α method. It is shown that a new numerical formalism with higher accuracy can be derived under the same framework of the unconditional stable algorithm established through a special variational formalism and variational integrators. The above-mentioned formalism can be extended from general linear space to Lie group by utilizing the moving frame method and the Lie group formalism of the Hamel generalized α method has been obtained. Both the convergence and stability of the algorithm are discussed, and some numerical examples are presented to verify the conclusion. It is demonstrated by the theoretical analysis that the Hamel generalized α method proposed in the paper is unconditionally stable, second-order accurate and can quickly filter out pseudo high-frequency harmonics. Both conventional and proposed methods have been applied to numerical examples respectively. Comparisons between results of numerical examples show that the aforementioned advantages of the proposed method in terms of accuracy, dissipation and stability are tested and verified. At the same time, it can be developed that new numerical integration algorithms with even higher order accuracy. The scheme can also be proposed, which is suitable for both general linear space and Lie group space. A new way for constructing variational integrators is also obtained in this paper.      

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.     

Hamel 框架下几何精确梁的离散动量守恒律

Hamel场变分积分子是一种研究场论的数值方法, 可以通过使用活动标架规避几何非线性带来的计算复杂度, 同时数值上具有良好的长时间数值表现和保能动量性质. 本文在一维场论框架下, 以几何精确梁为例, 从理论上探究Hamel场变分积分子的保动量性质. 具体内容包括: 利用活动标架法对几何精确梁建立动力学模型, 通过变分原理得到其动力学方程, 利用其动力学方程及Noether定理得到系统动量守恒律; 将几何精确梁模型离散化, 通过变分原理得到其Hamel场变分积分子, 利用Hamel场变分积分子和离散Noether定理得到离散动量守恒律, 并给出离散动量的一阶近似表达式; Hamel场变分积分子可在计算中利用系统对称性消除系统运动带来的非线性问题, 但此框架中离散对流速度、离散对流 应变及位形均不共点, 而这种错位导致离散动量中出现级数项, 本文对几何精确梁的离散动量与连续形式的关系及其应 用进行了讨论, 并通过算例验证了结论. 上述证明方法也同样适用一般经典场论场景下的Hamel场变分积分子. Hamel场变分积分子的动量守恒为进一步研究其保结构性质提供了参考依据.      

基于对偶变量变分原理的完整约束系统保辛算法

基于对偶变量变分原理，选择积分区间两端位移为独立变量，构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先，利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似；然后，对作用量中不包含约束的积分项采用Gauss积分近似，对作用量中包含约束的积分项采用Lobatto积分近似，从而得到近似作用量；最后，在此近似作用量的基础上，利用对偶变量变分原理，将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性，能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。     

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。     

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.     

Gurtin变分原理在矩形板动力初值问题中的应用

结构动力分析是工程设计中的重要组成部分,传统动力分析方法并不能全面反映动力初值特征,而Ｇｕｒｔｉｎ变分原理则被认为是目前唯一能全面反映动力初值特征的变分原理。本文基于位移型Ｇｕｒｔｉｎ变分原理,对空间和时间同时离散,建立了一种求解板的动力初值问题的时空有限元法,并对两种边界情况板的振动问题进行了编程计算,计算结果表明时空元法精确度很高且稳定收敛。     

Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure (SWE-DP) is developed. A hybrid numerical method combining the finite element method for spatial discretization and the Zu-class method for time integration is created for the SWEDP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations (SWEs). The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent performance with respect to simulating the long time evolution of the shallow water.     

MESHLESS ANALYSIS FOR THREE-DIMENSIONAL ELASTICITY WITH SINGULAR HYBRID BOUNDARY NODE METHOD

The singular hybrid boundary node method (SHBNM) is proposed for solving three-dimensional problems in linear elasticity. The SHBNM represents a coupling between the hybrid displacement variational formulations and moving least squares (MLS) approximation. The main idea is to reduce the dimensionality of the former and keep the meshless advantage of the later. The rigid movement method was employed to solve the hyper-singular integrations. The 'boundary layer effect', which is the main drawback of the original Hybrid BNM, was overcome by an adaptive integration scheme. The source points of the fundamental solution were arranged directly on the boundary. Thus the uncertain scale factor taken in the regular hybrid boundary node method (RHBNM) can be avoided. Numerical examples for some 3D elastic problems were given to show the characteristics. The computation results obtained by the present method are in excellent agreement with the analytical solution. The parameters that influence the performance of this method were studied through the numerical examples.     

THE APPLICATION OF VARIATIONAL MULTISCALE METHOD ON A ONE-DIMENSIONAL MECHANICAL MODEL

将变分多尺度方法应用于一维缆索模型,导出受力缆索的宏观有限元模型并求得细观位移解析解,总结出变分多尺度方法应用于具体模型的关键点和缺陷. 假定刚度为常值,数值模拟一定边界和受力下的缆索,得到宏观和细观位移. 将细观与宏观位移叠加,相比于精确位移得出：细观位移可视为常规有限元模型的后验误差. 变分多尺度方法在一维力学模型中的成功应用,推进了其实用性,为其在更多力学及工程问题中的运用和发展提供了参考.     

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。     

近似不可压软材料动力分析的完全拉格朗日物质点法

物质点法(MPM)在模拟非线性动力问题时具有很好的效果, 其已被广泛应用于许多大变形动力问题的分析中. 然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁, 极大地影响模拟精度和收敛性. 本文针对近似不可压软材料的大变形动力学行为, 提出一种混合格式的显式完全拉格朗日物质点法(TLMPM). 首先基于近似不可压软材料的体积部分应变能密度, 引入关于静水压力的方程; 之后将该方程与动量方程基于显式物质点法框架进行离散, 并采用完全拉格朗日格式消除物质点跨网格产生的误差, 提升大变形问题的模拟精度; 对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法, 提升算法的准确性. 此外, 算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解. 最后, 给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性, 计算结果表明该方法可以有效处理体积自锁, 准确地模拟近似不可压软材料的大变形动力学行为.      

多体动系统动力学方程在流形上的辛算法

多体系统动力学方程的数值方法一直是数学与力学家们的热门研 究课题.特别是多体系统动力学微分/代数方程组形式的数学模型，是所 谓的指标-3问题，它的求解是一个难题.目前流行的关于它的数值方法都 有不尽人意的地方，主要是对出现的所谓的违约问题和刚性问题未很好地 解决.多体系统动力学方程在流形上的辛算法是近几年出现的新的数值方 法，它将闭环型多体系统动力学的方程的约束部分和常微分方程部分利 用所谓的辛方法同时进行处理，其中的一些方法已证明是有效的，所以， 用它求解多体系统动力学方程前景看好.本文介绍了这些新的理论，并提 出了一些有待解决的问题.     

Calibrating a J2 plasticity material model using a 2D inverse finite element procedure

Material models are the key ingredients to accurately capture the global mechanical response of structural systems. The use of finite element analysis has proven to be effective in simulating nonlinear engineering applications. However, the choice of the appropriate material model plays a big role in the value of the numerical predictions. Such models are not expected to exactly reproduce global experimental response in all cases. Alternatively, the measured global response at specific domain or surface points can be used to guide the nonlinear analysis to successively extract a representative material model. By selecting an initial set of stress–strain data points, the load–displacement response at the monitoring points is computed in a forward incremental analysis without iterations. This analysis retains the stresses at the integration points. The corresponding strains are not accurate since the computed displacements are not anticipated to match the measured displacements at the monitoring points. Therefore, a corrective incremental displacement analysis is performed at the same load steps to adjust for displacements and strains everywhere by matching the measured displacements at the monitoring points. The stress–strain vectors at the most highly stressed integration point are found to establish an improved material model. This model is used within a multi-pass incremental nonlinear finite element analysis until the discrepancy between the measured and the predicted structural response at the monitoring points vanishes. The J2 flow theory of plasticity is used as a constitutive framework to build the tangent elastic–plastic matrices. The applicability of the proposed approach is demonstrated by solving 2D inverse continuum problems. The comparisons presented support the effectiveness of the proposed approach in accurately calibrating the J2 plasticity material model for such problems.     

A NEW VARIATIONAL INEQUALITY FORMULATION FOR SEEPAGE PROBLEMS WITH FREE SURFACES

IntroductionTherearetwoclassesofsolutionsforseepageproblemswithfreesurfaces ,i.e .,theadaptivemeshmethodsandthefixedmeshmethods.Theadaptivemeshmethodsinvolvetoolargeamountofcomputationforinhomogeneoussoilsandoftenleadtodivergentcalculations,andhence,arenowbeingsupercededbythefixedmeshmethods.Thefixedmeshmethodsfallintotwocategories,theintuitivemethodsandthevariationalinequalitymethods.Theintuitivemethods[1- 3]establishusuallytheiterativeproceduresbaseduponthefactthatthereisnodischargebetweenth…     

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

Numerical solutions of linear quadratic control for time-varying systems via symplectic par conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented.It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation.The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

非完整力学

简要地概述了非完整力学的基本概念、变分原理、运动方程、专门问题、积分方法、代数结构与几何方法,论述了非完整力学未来发展的趋势,包括90篇参考文献。