A Symplectic Method of Numerical Simulation on Local Buckling for Cylindrical Long Shells under Axial Pulse Loads

In this paper,the local buckling of cylindrical long shells is discussed under axial pulse loads in a Hamiltonian system.Using this system,critical loads and modes of buckling of shells are reduced to symplectic eigenvalues and eigensolutions respectively.By the symplectic method,the solution of the local buckling of shells can be employed to the expansion series of symplectic eigensolutions in this system.As a result,relationships between critical buckling loads and other factors,such as length of pulse load,thickness of shells and circumferential orders,have been achieved.At the same time,symmetric and unsymmetric buckling modes have been discuss.Moreover,numerical results show that modes of post-buckling of shells can be Bamboo node-type,bending type,concave type and so on.Research in this paper provides analytical supports for ultimate load prediction and buckling failure assessment of cylindrical long shells under local axial pulse loads.

Multi-revolution low-thrust trajectory optimization using symplectic methods

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.

Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems

Based on Feng's theory of formal vector fields and formal flows, we study the convergence problem of the formal energies of symplectic methods for Hamiltonian systems and give the clear growth of the coefficients in the formal energies. With the help of B-series and Bernoulli functions, we prove that in the formal energy of the mid-point rule, the coefficient sequence of the merging products of an arbitrarily given rooted tree and the bushy trees of height 1(whose subtrees are vertices), approaches 0 as the number of branches goes to ∞; in the opposite direction, the coefficient sequence of the bushy trees of height m(m ≥ 2), whose subtrees are all tall trees, approaches ∞ at large speed as the number of branches goes to +∞. The conclusion extends successfully to the modified differential equations of other Runge-Kutta methods. This disproves a conjecture given by Tang et al.(2002), and implies:(1) in the inequality of estimate given by Benettin and Giorgilli(1994) for the terms of the modified formal vector fields, the high order of the upper bound is reached in numerous cases;(2) the formal energies/formal vector fields are nonconvergent in general case.

Explicit Symplectic Methods for the Nonlinear Schrodinger Equation

By performing a particular spatial discretization to the nonlinear Schrodinger equation(NLSE),we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts(L-L-N splitting).We integrate each part by calculating its phase flow,and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows.A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE.The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method,and the convergence of the formal energy of this symplectic integrator is also verified.The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.

Preservation of Equilibria for Symplectic Methods Applied to Hamiltonian Systems

In this paper, the linear stability of symplectic methods for Hamiltonian systems is studied. In par- ticular, three classes of symplectic methods are considered： symplectic Runge-Kutta （SRK） methods, symplectic partitioned Runge-Kutta （SPRK） methods and the composition methods based on SRK or SPRK methods. It is shown that the SRK methods and their compositions preserve the ellipticity of equilibrium points uncondi- tionally, whereas the SPRK methods and their compositions have some restrictions on the time-step.

A symplectic eigensolution method in transversely isotropic piezoelectric cylindrical media

This paper reports establishment of a symplectic system and introduces a 3D sub-symplectic structure for transversely isotropic piezoelectric media. A complete space of eigensolutions is obtained directly. Thus all solutions of the problem are re- duced to finding eigenvalues and eigensolutions, which include zero-eigenvalue solutions and all their Jordan normal form of the corresponding Hamiltonian matrix and non-zero-eigenvalue solutions. The classical solutions are described by zero-eigen- solutions and non-zero-eigensolutions show localized solutions. Numerical results show some rules of non-zero-eigenvalue and their eigensolutions.

A relativistic canonical symplectic particlein-cell method for energetic plasma analysis

A relativistic canonical symplectic particle-in-cell(RCSPIC)method for simulating energetic plasma processes is established.By use of the Hamiltonian for the relativistic Vlasov-Maxwell system,we obtain a discrete relativistic canonical Hamiltonian dynamical system,based on which the RCSPIC method is constructed by applying the symplectic temporal discrete method.Through a 106-step numerical test,the RCSPIC method is proven to possess long-term energy stability.The ability to calculate energetic plasma processes is shown by simulations of the reflection processes of a high-energy laser(1?×?1020 W cm-2)on the plasma edge.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years.The method has yielded many new analytic solutions due to its rigorousness.In this study,the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge.The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied.The solution procedure incorporates separation of variables,symplectic eigen solution and superposition.The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems.The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use.The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system.The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation.Furthermore,they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial

An analytical method,called the symplectic mathematical method,is proposed to study the wave propagation in a spring-mass chain with gradient arranged local resonators and nonlinear ground springs.Combined with the linearized perturbation approach,the symplectic transform matrix for a unit cell of the weakly nonlinear graded metamaterial is derived,which only relies on the state vector.The results of the dispersion relation obtained with the symplectic mathematical method agree well with those achieved by the Bloch theory.It is shown that wider and lower frequency bandgaps are formed when the hardening nonlinearity and incident wave intensity increase.Subsequently,the displacement response and transmission performance of nonlinear graded metamaterials with finite length are studied.The dual tunable effects of nonlinearity and gradation on the wave propagation are explored under different excitation frequencies.For small excitation frequencies,the gradient parameter plays a dominant role compared with the nonlinearity.The reason is that the gradient tuning aims at the gradient arrangement of local resonators,which is limited by the critical value of the local resonator mass.In contrast,for larger excitation frequencies,the hardening nonlinearity is dominant and will contribute to the formation of a new bandgap.

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic con- servative perturbation method for linear time-varying Hamil- tonian system. Firstly, the dynamic equation of Hamilto- nian system is gradually changed into a high order pertur- bation equation, which is solved approximately by resolv- ing the Hamiltonian coefficient matrix into a ＂major compo- nent＂ and a ＂high order small quantity＂ and using perturba- tion transformation technique, then the solution to the orig- inal equation of Hamiltonian system is determined through a series of inverse transform. Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes, the transfer matrix is a sym- plectic matrix; furthermore, the exponential matrices can be calculated accurately by the precise time integration method, so the method presented in this paper has fine accuracy, ef- ficiency and stability. The examples show that the proposed method can also give good results even though a large time step is selected, and with the increase of the perturbation or- der, the perturbation solutions tend to exact solutions rapidly.

Multi-symplectic wavelet splitting method for the strongly coupled Schrodinger system

We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.

Explicit Multi-Symplectic Methods for Hamiltonian Wave Equations

In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model problem.Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.

Explicit Symplectic Geometric Algorithms for Quaternion Kinematical Differential Equation

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.

SYMPLECTIC STRUCTURE OF POISSON SYSTEM

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.

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.

New analytic solutions to 2D transient heat conduction problems with/without heat sources in the symplectic space

The two-dimensional(2D)transient heat conduction problems with/without heat sources in a rectangular domain under different combinations of temperature and heat flux boundary conditions are studied by a novel symplectic superposition method(SSM).The solution process is within the Hamiltonian system framework such that the mathematical procedures in the symplectic space can be implemented,which provides an exceptional direct rigorous derivation without any assumptions or predetermination of the solution forms compared with the conventional inverse/semi-inverse methods.The distinctive advantage of the SSM offers an access to new analytic heat conduction solutions.The results obtained by the SSM agree well with those obtained from the finite element method(FEM),which confirms the accuracy of the SSM.