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A symplectic method in 2D Thermo-viscoelasticity

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A Symplectic Method in 2D Thermo-viscoelasticity

Explicit symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation

The second Asia-Pacific International Conference on Computational methods in Engineering（ICOME2006）

作者： Weixiang Zhang Xinsheng Xu Key Laboratory of Structure Analysis of Industrial Equipment Dalian University of Technology Dalian 116024 China

This paper describes problems of two-dimensional thermo-viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of end effects. Thus the original problem is reduced to finding zero eigensolutions and non-zero eigensolutions. Meanwhile, an effective method for end conditions is given in the symplectic space. As its application, local effects in the ends are discussed in the numerical results. The symplectic method and numerical method provide an idea for other researching also.

关键词： thermo-viscoelasticity duality eigenvalue end effects symplectic method

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IEEE/CAA Journal of Automatica Sinica 2018年第2期5卷 479-488页

作者： Hong-Yan Zhang Zi-Hao Wang Lu-Sha Zhou Qian-Nan Xue Long Ma Yi-Fan Niu School of Information Science and Technology Hainan Normal University Haikou 571158 China the Department of Engineering Sino-European Institute of Aviation Engineering Civil Aviation University of China Tianjin 300300 China Civil Aviation University of China Tianjin 300300 China

Solving quaternion kinematical differential equations(QKDE) is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly,a generalized Euler's formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.

关键词： Linear time-varying system navigation system quaternion kinematical differential equation(QKDE) real-time computation symplectic method

Multi-revolution low-thrust trajectory optimization using symplectic methods

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Science China(Technological Sciences) 2020年第3期63卷 506-519页

作者： E ZhiBo GUZZETTI Davide School of Aerospace Engineering Tsinghua UniversityBeijing 100084China Department of Aerospace Engineering Auburn UniversityAuburn AL 36849USA

Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered as a challenging problem.This paper describes a high-precision symplectic method and optimization techniques to solve the minimum-energy low-thrust multi-revolution orbit transfer problem. First, the optimal orbit transfer problem is posed as a constrained nonlinear optimal control problem. Then, the constrained nonlinear optimal control problem is converted into an equivalent linear quadratic form near a reference solution. The reference solution is updated iteratively by solving a sequence of linear-quadratic optimal control sub-problems, until convergence. Each sub-problem is solved via a symplectic method in discrete form. To facilitate the convergence of the algorithm, the spacecraft dynamics are expressed via modified equinoctial elements. Interpolating the non-singular equinoctial orbital elements and the spacecraft mass between the initial point and end point is proven beneficial to accelerate the convergence process. Numerical examples reveal that the proposed method displays high accuracy and efficiency.

关键词： low-thrust trajectory optimization symplectic method multi-revolution transfers

A symplectic moving horizon estimation algorithm with its application to the Earth-Moon L2 libration point navigation

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Astrodynamics 2019年第2期3卷 137-153页

作者： Xinwei Wang Haijun Peng Department of Engineering Mechanics State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalian 116024China

Accurate state estimations are perquisites of autonomous navigation and orbit maintenance missions.The extended Kalman lter(EKF)and the unscented Kalman lter(UKF),are the most commonly used method.However,the EKF results in poor estimation performance for systems are with high nonlinearity.As for the UKF,irregular sampling instants are required.In addition,both the EKF and the UKF cannot treat constraints.In this paper,a symplectic moving horizon estimation algorithm,where constraints can be considered,for nonlinear systems are developed.The estimation problem to be solved at each sampling instant is seen as a nonlinear constrained optimal control problem.The original nonlinear problem is transferred into a series of linear-quadratic problems and solved iteratively.A symplectic method based on the variational principle is proposed to solve such linear-quadratic problems,where the solution domain is divided into sub-intervals,and state,costate,and parametric variables are locally interpolated with linear approximation.The optimality conditions result in a linear complementarity problem which can be solved by the Lemke's method easily.The developed symplectic moving horizon estimation method is applied to the Earth-Moon L2 libration point navigation.And numerical simulations demonstrate that though more time-consuming,the proposed method results in better estimation performance than the EKF and the UKF.

关键词： moving horizon estimation symplectic method quasilinearization variational principle L2 libration point navigation

Nonlinear resonance phenomenon of one-dimensional Bose Einstein condensate under periodic modulation

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Chinese Physics B 2014年第2期23卷 112-116页

作者：花巍刘世兴 College of Physics and Technology Shenyang Normal University College of Physics Liaoning University

We investigate the effect of an external periodic modulation on the one-dimensional （1D） Bose-Einstein conden- sate with harmonic trapping potential. By numerically solving the Gross-Pitaevskii equation with symplectic algorithm, the nonlinear resonance phenomenon is shown and the corresponding Fourier spectrum is given. The autoresonance phe- nomenon is also presented under almost periodic external modulation, and it shows that the condensate eventually evolves into quasi-periodic oscillation.

关键词： Bose-Einstein condensate resonance periodic modulation symplectic method

symplectic STRUCTURE OF POISSON SYSTEM

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Applied Mathematics and Mechanics(English Edition) 2005年第11期26卷 1484-1490页

作者：孙建强马中骐田益民秦孟兆 GU Yuan-xian Institute of High Energy Physics Chinese Academy of Sciences Beijing 100039 P. R. China Institute of Applied Physics and Computational Mathematics Beijing 100088 P. R. China Institute of Software Chinese Academy of Sciences Beijing 100080 P. R. China Institute of Computational Mathematics Chinese Academy of Sciences Beijing 100080 P. R. China 不详

When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform. Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the mid-point scheme. Numerical results show the effectiveness of the nonlinear transform.

关键词： Poisson system nonlinear transformation symplectic method rigid body problem

symplectic SCHEMES FOR TELEGRAPH EQUATIONS

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Journal of Computational Mathematics 2016年第3期34卷 285-299页

作者： Yi Lu Yaolin Jiang School of Mathematics and Statistics Xi'an Jiaotong University Shaanxi 710049 China

A new numerical algorithm for telegraph equations with homogeneous boundary con- ditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether＇s theorem. The algorithm origins from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Mean- while, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.

关键词： Telegraph equation Klein-Gordon equation symplectic method Explicitmethod.

VOLUME PRESERVING RK methodS FOR LINEAR SYSTEMS

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Acta Mathematicae Applicatae Sinica 2000年第4期16卷 430-434页

作者：秦孟兆李洪伟 LSEC Institute of Computational Mathematics and Scientific/Engineering Computing the Academy of Mathematics and Systems Sciences the Chinese Academy of Sciences Beijing China

In this article, we analyze and study under what conditions a source-free system has volumepreserving RK schemes. For linear systems, we give a comparatively thorough discussion about RK methods to be phase volume preserving integrators. We also analyze the relationship between volume-preserving integrators and symplectic integrators.

关键词： Runge-Kutta method volume preserving method symplectic method

Dynamics of cubic–quintic nonlinear Schr?dinger equation with different parameters

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Chinese Physics B 2016年第5期25卷 23-30页

作者：花巍刘学深刘世兴 College of Physics Science and Technology Shenyang Normal University Shenyang 110034 China Institute of Atomic and Molecular Physics Jilin University Changchun 130012 China College of Physics Liaoning University Shenyang 110036 China

We study the dynamics of the cubic–quintic nonlinear Schrodinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter.

关键词： nonlinear Schrodinger equation phase space symplectic method