Symplectic partitioned Runge-Kutta method based onthe eighth-order nearly analytic discrete operator and its wavefield simulations

We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.

Particle swarm optimization-based algorithm of a symplectic method for robotic dynamics and control

Multibody system dynamics provides a strong tool for the estimation of dynamic performances and the optimization of multisystem robot design. It can be described with differential algebraic equations(DAEs). In this paper, a particle swarm optimization(PSO) method is introduced to solve and control a symplectic multibody system for the first time. It is first combined with the symplectic method to solve problems in uncontrolled and controlled robotic arm systems. It is shown that the results conserve the energy and keep the constraints of the chaotic motion, which demonstrates the efficiency, accuracy, and time-saving ability of the method. To make the system move along the pre-planned path, which is a functional extremum problem, a double-PSO-based instantaneous optimal control is introduced. Examples are performed to test the effectiveness of the double-PSO-based instantaneous optimal control. The results show that the method has high accuracy, a fast convergence speed, and a wide range of applications.All the above verify the immense potential applications of the PSO method in multibody system dynamics.

SYMPLECTIC ALGORITHM IN SOLVING OPTIMAL CONTROL PROBLEMS

A symplectic algorithm is used to solve optimal control problems. Linear and nonlinear examples aregiven. Numerical analyses show that the symplectic algorithm gives satisfactory performance in that it works inlarge step and is of high speed and accuracy. This indicates that the symplectic algorithm is more effective andreasonable in solving optimal control problems.

基于辛Runge-Kutta方法的棋盘形褶皱二维薄膜-基底结构动力学特性研究

基于力学屈曲原理的褶皱薄膜-基底结构已成功应用于制备可延展无机电子器件。然而,该类电子器件在应用时需要服役于复杂动态环境中,针对棋盘形褶皱薄膜结构的动力学问题鲜有研究,此问题又是该类电子器件走向实际应用需要解决的关键问题之一。本文首先采用能量方法,分别计算了二维薄膜的弯曲能、膜弹性能和柔性基底中的弹性能以及薄膜动能;然后采用拉格朗日方程,推导出了该结构的振动控制方程;而该方程为非线性动力学方程,无法给出其解析解;因此,本文采用辛Runge-Kutta方法对其进行数值求解;数值结果表明,辛数值方法具有长期稳定的特性和系统结构特性,为高精度的可延展电子器件的动力学问题研究提供了优异的数值方法。

Difference Discrete Variational Principles,Euler—Lagrange Cohomology and Symplectic,Multisymplectic Structures Ⅲ：Application to Symplectic and Multisymplectic Algorithms

In the previous papers I and II,we have studied the difference discrete variational principle and the Euler-Lagrange cohomology in the framework of multi-parameter differential approach.We have gotten the difference discrete Euler-Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler-Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the lagrangian and Hamiltonian formalisms.In this paper,we apply the difference discrete variational principle and Euler-Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms.We will show that either Hamiltonian schemes of Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler-Lagrange cohomological conditions are satisfied.

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.

GPR Wave Propagation Model in a Complex Geoelectric Structure Using Conformal First-Order Symplectic Euler Algorithm

Possessing advantages such as high computing efficiency and ease of programming,the Symplectic Euler algorithm can be applied to construct a groundpenetrating radar(GPR)wave propagation numerical model for complex geoelectric structures.However,the Symplectic Euler algorithm is still a difference algorithm,and for a complicated boundary,ladder grids are needed to perform an approximation process,which results in a certain amount of error.Further,grids that are too dense will seriously decrease computing efficiency.This paper proposes a conformal Symplectic Euler algorithm based on the conformal grid technique,amends the electric/magnetic fieldupdating equations of the Symplectic Euler algorithm by introducing the effective dielectric constant and effective permeability coefficient,and reduces the computing error caused by the ladder approximation of rectangular grids.Moreover,three surface boundary models(the underground circular void model,the undulating stratum model,and actual measurement model)are introduced.By comparing reflection waveforms simulated by the traditional Symplectic Euler algorithm,the conformal Symplectic Euler algorithm and the conformal finite difference time domain(CFDTD),the conformal Symplectic Euler algorithm achieves almost the same level of accuracy as the CFDTD method,but the conformal Symplectic Euler algorithm improves the computational efficiency compared with the CFDTD method dramatically.When the dielectric constants of the two materials vary greatly,the conformal Symplectic Euler algorithm can reduce the pseudo-waves almost by 80% compared with the traditional Symplectic Euler algorithm on average.

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation,applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.

NUMERICAL METHOD BASED ON HAMILTON SYSTEM AND SYMPLECTIC ALGORITHM TO DIFFERENTIAL GAMES

The resolution of differential games often concerns the difficult problem of two points border value (TPBV), then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.

A Mathematical Model of Real-Time Simulation and the Convergence Analysis on Real-Time Runge-Kutta Algorithms

In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation and the convergent order of real-time algorithm is proved.

THE PROPERTIES OF A KIND OF RANDOM SYMPLECTIC MATRICES

Several important properties of a kind of random symplectic matrix used by A. Bunse_Gerstner and V. Mehrmann are studied and the following results are obtained: 1) It can be transformed to Jordan canonical form by orthogonal similar transformation; 2) Its condition number is a constant; 3) The condition number of it is about 2 618.

IRKO:An Improved Runge-Kutta Optimization Algorithm for Global Optimization Problems

Optimization is a key technique for maximizing or minimizing functions and achieving optimal cost,gains,energy,mass,and so on.In order to solve optimization problems,metaheuristic algorithms are essential.Most of these techniques are influenced by collective knowledge and natural foraging.There is no such thing as the best or worst algorithm;instead,there are more effective algorithms for certain problems.Therefore,in this paper,a new improved variant of a recently proposed metaphorless Runge-Kutta Optimization(RKO)algorithm,called Improved Runge-Kutta Optimization(IRKO)algorithm,is suggested for solving optimization problems.The IRKO is formulated using the basic RKO and local escaping operator to enhance the diversification and intensification capability of the basic RKO version.The performance of the proposed IRKO algorithm is validated on 23 standard benchmark functions and three engineering constrained optimization problems.The outcomes of IRKO are compared with seven state-of-the-art algorithms,including the basic RKO algorithm.Compared to other algorithms,the recommended IRKO algorithm is superior in discovering the optimal results for all selected optimization problems.The runtime of IRKO is less than 0.5 s for most of the 23 benchmark problems and stands first for most of the selected problems,including real-world optimization problems.

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.

一种Birkhoff形式下结构动响应问题的保辛中点格式

结构动响应预测是结构设计的基础,是结构振动控制、载荷识别的前提。本文在辛体系下针对结构动响应问题,提出了一种Birkhoff形式下的保辛中点格式。首先引入状态变量,并基于摄动方法将结构动响应方程转化为线性自治Birkhoff方程的形式,进一步利用中心差分推导出线性自治Birkhoff方程的中点格式,其证明是保辛的。该格式不要求Birkhoff方程系数矩阵非奇异,因此适用于奇数维系统。两个不同数值算例的结果充分验证了本文方法的卓越性,也凸显了相对于传统算法在计算精确度和稳定性方面的明显优势。

三维波纹型可延展结构振动特性的辛分析

基于三维组装技术的可延展结构具备优异的延展性和可调控性,使其成功应用于各类可延展电子器件的制备中。为了评估该类电子器件的稳定性,本文研究三维波纹型可延展结构的振动行为。首先,基于非线性的Euler-Bernoulli梁理论、Kelvin-Voigt粘弹性理论和考虑压电材料的表面压电效应,建立三维波纹结构的理论分析模型;其次,基于能量原理和扩展拉格朗日运动原理,推导出该结构的动力学控制方程;然后采用二级四阶辛Runge-Kutta求解该动力学方程。通过数值仿真实验验证了辛算法的优越性,同时,还发现随着三维波纹型可延展结构外界激励及其结构参数的变化,该结构的振动特性会从倍周期向分岔和混沌转化;本文结果为三维波纹型可延展结构的优化设计及应用提供理论基础。

基于多级高阶辛Runge-Kutta方法的暂态稳定性并行计算方法

将s级2s阶的辛Runnge-Kutta方法用于电力系统暂态稳定性计算,利用矩阵分裂技巧以及矩阵求逆运算的松弛方法,导出了一种新的暂态稳定性并行计算方法,具有较好的时间并行特性和超线性收敛性。利用IEEE 145节点系统,对导出的并行算法进行了仿真测试和评估。仿真测试结果表明,所提出的并行算法具有很好的收敛性,有效地解决了时间并行度与收敛性之间的矛盾,可以获得较高的加速比和很好的并行计算效率。

参数冻结精细指数积分法在非线性车桥耦合振动分析中的应用

描述车桥耦合作用的基本问题是一个时变系统问题,且很多工况下需考虑非线性特性,使得该问题难以得到解析解,甚至数值解也可能很复杂.针对该问题的求解,提出了一种参数冻结精细指数积分法,将其应用于车桥耦合动力学模型的数值分析中.该方法结合了精细积分和指数积分特点,并将时变系数矩阵在每一积分步参数冻结,用于获得系统振动响应的数值解.考虑汽车轮胎与桥面的力和位移耦合关系、桥面沥青铺装层、桥梁材料黏弹性和几何非线性特性,建立了车桥耦合动力学模型,并应用参数冻结精细指数积分法对该模型进行了求解.通过与近似解析解、辛Runge-Kutta算法以及经典的Newmark-β数值积分法计算结果进行对比,验证了所提出方法计算结果的有效性和准确性.在此基础上,制作了缩尺车桥耦合系统模型,测试了跨中挠度响应,进一步验证了理论建模和所提算法的有效性和实用性.通过数值计算分析了所提算法的数值特性,结果表明:提出的参数冻结精细指数积分法不仅可以处理时变、非线性问题,且具有良好的数值计算精度和长时间数值稳定性;由于精细积分的特点,参数冻结精细指数积分法的计算时间步长可以取的较大,可有效提高计算效率.因此,所提出的参数冻结精细指数积分法预期可成为求解车桥耦合动力学问题的一种新的高效算法.

基于梯形网格辛Runge-Kutta射线追踪的角度域层析偏移速度分析

为了与Kirchhoff叠前深度偏移速度分析进行比较,提出利用角度域共成像点道集(ADCIGs)的剩余校正和梯形网格辛Runge-Kutta射线追踪建立层析反演线性方程组。首先,通过常规叠前偏移速度分析获得矩形网格初始层速度模型,采用双平方根方程波场向下延拓偏移提取ADCIGs,并作剩余校正分析,计算旅行时残差作为方程组的右侧;采用偏移层位控制的梯形网格建立层速度模型,进行辛Runge-Kutta射线追踪,求取方程组左侧的偏导数矩阵;最后,运用LSQR并行算法、GPU-LU及递归LU分解算法求解方程组,得到速度修正量,更新初始层速度模型。此流程迭代多次直到ADCIGs同相轴拉平。该流程对青海野外地震数据的处理结果表明:该方法可行、有效。

高阶Maggi方程的Birkhoff化及其辛算法

针对非完整系统的高阶Maggi方程,在满足一定的条件时,可以对其进行Birkhoff化.通过构造生成函数,利用Birkhoff广义辛算法对其进行数值仿真.仿真结果和传统的Runge-Kutta算法结果相比较,Birkhoff广义辛算法在长期跟踪后更加准确.