微流星体碰撞对日地L2点Halo轨道的影响

空间中存在大量高速运动的微流星体,与在轨运行的航天器发生碰撞后将导致轨道偏离、性能下降、结构破坏甚至航天器失效.由于Halo轨道具有不稳定的特性,本文主要探究微流星体碰撞对日地L2点Halo轨道动力学演化规律的影响.首先,建立日地L2点附近轨道的动力学模型,通过微分修正法构造Halo轨道的初始条件,基于Grün微流星体通量模型计算微流星体与航天器碰撞的数量和碰撞引起的速度改变量.然后,采用Runge-Kutta算法求解Halo轨道的动力学方程,研究碰撞速度改变量引起的轨道偏差随时间的演化规律.此外,采用状态转移矩阵方法分析初始状态偏差的演化规律,并与数值积分方法对比.最后基于状态转移矩阵方法分析了不同碰撞速度改变量的大小和方向引起的动力学响应.研究发现,状态转移矩阵在短时间内得到的结果与数值积分方法基本一致,而且只需一次矩阵乘法即可通过初始状态偏差计算得到末时刻状态偏差,具有非常高的效率.研究结果表明,由于Halo轨道固有的不稳定特性,所以初始时刻微流星体碰撞引起的微小状态偏差会快速增长,导致消耗更多控制燃料,最终将影响航天器的寿命.此外,微流星体碰撞的速度改变量的方向对偏差传递的规律有重要的影响.     

An iterative integration scheme for solving second-order ordinary differential equations

This paper presents an iterative solution method for the numerical integration of second-order ordinary differential equations using a simple program for microcomputers (PC). The method of integration proposed is based on the geometrical considerations in the phase plane. The numerical results are compared to those obtained by the fourth-order Runge-Kutta method and to the closed form solutions when possible. Tests show good accuracy and, in some cases, computer time saving with respect to the Runge-Kutta's method for th same accuracy. The method of integration in the phase plane seems very good for treating every kind of nonlinear second-order differential equation whatever the degree of nonlinearity.     

基于RKNd算法的暂态稳定性快速数值计算方法

RKNd方法是一类新的数值积分方法。在相同级数条件下,RKNd方法可达到的最高代数阶比传统的Runge—Kutta方法以及Runge—Kutt—Nystrom方法均高,而且具有更高的计算效率。将RKNd方法引入电力系统暂态稳定性数值计算。以IEEEl45节点电力系统为例,通过数值实验将新方法与电力系统分析中常用的传统数值计算方法进行了对比分析。数值实验结果表明,RKNd方法在计算精度和计算效率等方面均具有明显的优势,N而更适合于电力系统暂态稳定性及相似问题的数值计算。     

An Implicit Multistage Integration Method Including Projection for the Numerical Simulation of Constrained Multibody Systems

To be efficient, the simulation of multibody system dynamics requires fast and robust numerical algorithms for the time integration of the motion equations usually described by Differential Algebraic Equations (DAEs). Firstly, multistep schemes especially built up for second-order differential equations are developed. Some of them exhibit superior accuracy and stability properties than standard schemes for first-order equations. However, if unconditional stability is required, one must be satisfied with second-order accurate methods, like one-step schemes from the Newmark family.Multistage methods for which high accuracy is not contradictory with stringent stability requirements are then addressed. More precisely, a two-stage, third-order accurate Implicit Runge–Kutta (IRK) method which possesses the desirable properties of unconditional stability combined with high-frequency dissipation is proposed.Projection methods which correct the integrated estimates of positions, velocities and accelerations are suggested to keep the constraint equations satisfied during the numerical integration. The resulting time integration algorithm can be easily implemented in existing incremental/iterative codes. Numerical results indicate that this approach compares favourably with classical methods.     

一种混合算法求解多体问题下行星际转移轨道

在限制性多体问题下的行星际转移轨道是一个典型的两点边值问题。求解该类问题的常用方法是采用打靶法结合Runge-Kutta积分进行求解,但是在求解行星际转移轨道时,由于方程比较复杂普通的打靶法在某些情况下不能求到解。文章提出了一种基于差分演化和打靶法的混合算法可以很好地解决这些问题。     

Explicit integration methods for constitutive equations of a mean-stress dependent elastoviscoplastic model: impact on structural finite element analyses

The strong dependent behavior of semi-crystalline polymers can lead to the use of simplified material laws in finite element structural calculations for reasons of robustness to the detriment of the quantitative response of the models. This work focuses on numerical integration methods as a solution to overcome the possible convergence and robustness limitations of mean-stress dependent elastoviscoplastic material laws, typical of the semi-crystalline polymers’ mechanical behavior. What is proposed here is a rational application of three explicit integration methods (fourth- and second-order Runge–Kutta method, a hybrid schema between Runge–Kutta, and Euler method) in engineering structural calculations, which provide a reliable solution for constitutive models of semi-crystalline polymer. These methods are examined for structure creep test and tensile test, in comparison with experimental data. The investigations have been done in terms of the stability toward convergence, the accuracy of results, the plastic consistency, and CPU time efficiency. This work, proposes an easy implementation of integration methods in any computational finite element code. It also provides a flexible modular implementation which is applicable to any different constitutive equations. An integration step sub-division technique is recommended. It is a powerful technique to improve the convergence of solution and accuracy of result by damping oscillation around stress Gauss point integration solution. The results obtained illustrate the effect of numerical integration schemas on structural analysis and provide an insight into select suitable method.

     

Optimized Forest-Ruth- and Suzuki-like algorithms for integration of motion in many-body systems

An approach is proposed to improve the efficiency of fourth-order algorithms for numerical integration of the equations of motion in molecular dynamics simulations. The approach is based on an extension of the decomposition scheme by introducing extra evolution subpropagators. The extended set of parameters of the integration is then determined by reducing the norm of truncation terms to a minimum. In such a way, we derive new explicit symplectic Forest-Ruth- and Suzuki-like integrators and present them in time-reversible velocity and position forms. It is proven that these optimized integrators lead to the best accuracy in the calculations at the same computational cost among all possible algorithms of the fourth order from a given decomposition class. It is shown also that the Forest-Ruth-like algorithms, which are based on direct decomposition of exponential propagators, provide better optimization than their Suzuki-like counterparts which represent compositions of second-order schemes. In particular, using our optimized Forest-Ruth-like algorithms allows us to increase the efficiency of the computations by more than ten times with respect to that of the original integrator by Forest and Ruth, and by approximately five times with respect to Suzuki's approach. The theoretical predictions are confirmed in molecular dynamics simulations of a Lennard-Jones fluid. A special case of the optimization of the proposed Forest-Ruth-like algorithms to celestial mechanics simulations is considered as well.     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

Simulation of supercritical flow in crossroads: Confrontation of a 2D and 3D numerical approaches to experimental results

This work deals with the modeling of a flood in an urban environment. Among the various types of urban flood events, it was decided to study specifically the severe surface flooding events, which take place in highly urbanized areas. This work concerns particularly the numerical resolution of the two-dimensional Saint Venant equations for the study of the propagation of flood through the crossroads in the city. A discontinuous finite-element space discretization with a second-order Runge-Kutta time discretization is used to solve the two-dimensional Saint Venant equations. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally more efficient than comparable second-order finite volume methods. The scheme is applied to several supercritical flows in crossroads, which are investigated by Mignot. The experimental results obtained by the author are used to verify the accuracy and the robustness of the method. The results obtained are compared to those obtained by a second-order finite volume method (Rubar20 (2D)) and by FLUENT (3D). A very good agreement between the numerical solution obtained by the Runge-Kutta discontinuous Galerkin (RKDG) method and the experimental measured data were found. The method is then able to simulate the flow patterns observed experimentally and able to predict well the water depths, the discharge distribution in the downstream branches of the crossroad and the location of the hydraulic jumps and other flow characteristics more than the other methods.     

Chaos and bifurcation in numerical computation by the Runge-Kutta method

In this paper the chaotic phenomenon and bifurcation in numerical computation using the Runge-Kutta method to discretize the nonlinear differential equation are investigated. It is shown that the bifurcation condition in the discretized equation is given by the eigenvalue of the jacobian matrix of the original differential equation. As an example, the bifurcation and chaos when a second-order nonlinear equation is discretized by the Runge-Kutta method is investigated and it is shown that the scenario from a stable fixed point to chaos when the fourth-order Runge-Kutta method is applied is quite different from those of the second-order Runge-Kutta method