Routh Order Reduction Method of Relativistic Birkhoffian Systems

Routh order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic Birkhoffian system, the cyclic integrals can be found by using the perfect differential method. Through these cyclic integrals, the order of the system can be reduced. If the relativistic Birkhoffian system has a cyclic integral, then the Birkhoffian equations can be reduced at least by two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics, and the relativistic Lagrangian mechanics are discussed, and the Routh order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.

State space solution to 3D multilayered elastic soils based on order reduction method

Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.

Methods in Model Order Reduction(MOR) field

Nowadays,the modeling of systems may be quite large,even up to tens of thousands orders.In spite of the increasing computational powers,direct simulation of these large-scale systems may be impractical.Thus,to industry requirements,analytically tractable and computationally cheap models must be designed.This is the essence task of Model Order Reduction(MOR).This article describes the basics of MOR optimization,various way of designing MOR,and gives the conclusion about existing methods.In addition,it proposed some heuristic footpath.

基于POD-αATS的油浸变压器瞬态温升降阶自适应变步长计算方法

针对油浸式电力变压器瞬态温升计算效率过低的问题,该文提出本征正交分解-αATS(proper orthogonal decomposition-adaptive time stepping based onαfactor,POD-αATS)降阶自适应变步长瞬态计算方法。首先,推导变压器绕组瞬态温升计算的有限元离散方程;其次,采用POD降阶算法改善传统瞬态计算中存在的条件数过大及方程阶数过高的问题;同时对于瞬态计算中的时间步长选择问题,提出适用于非线性问题的αATS变步长策略;然后,为验证方法的有效性,基于110 kV油浸式电力变压器绕组的基本结构建立二维八分区数值计算模型,同时将计算结果与基于110 kV绕组的温升实验结果进行对比。数值计算及实验结果表明,所提算法与全阶定步长算法在流场和温度场中的精度几乎相同,且流场计算效率提升约45倍,温度场计算效率提升约38倍,计算速度得到显著提高。这一点在温升实验中同样得到验证,说明该文所提算法的准确性、高效性及一定的工程实用性。

Analytical and Traveling Wave Solutions to the Fifth Order Standard Sawada-Kotera Equation via the Generalized exp(-Φ(ξ))-Expansion Method

In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.

Variance Reduction Technique for Estimating Value-at-Risk based on the Cross - Entropy

Value-at-Risk （VaR） estimation via Monte Carlo （MC） simulation is studied here. The variance reduction technique is proposed in order to speed up MC algorithm. The algorithm for estimating the probability of high portfolio losses （more general risk measure） based on the Cross - Entropy importance sampling is developed. This algorithm can easily be applied in any light- or heavy-tailed case without an extra adaptation. Besides, it does not loose in the performance in comparison to other known methods. A numerical study in both cases is performed and the variance reduction rate is compared with other known methods. The problem of VaR estimation using procedures for estimating the probability of high portfolio losses is also discussed.

Second-order difference scheme for a nonlinear model of wood drying process

A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Exact Solutions to Nonlinear Schr(?)dinger Equation and Higher-Order Nonlinear Schr(?)dinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.

轴向流作用下均布非线性弹簧支承二维壁板的复杂响应

考虑刚性导流段和尾流段对流场的影响,建立轴向流作用下二维板的非线性流固耦合运动控制方程,用有限差分法对控制方程进行离散。为了克服差分网格较多时带来的计算规模较大的问题,对控制方程用主模态缩减法缩减自由度,然后对离散方程进行数值积分,得到系统的复杂响应,分析其分岔和混沌特性。计算结果表明,以来流流速幅值和阻尼参数为可变参数时,系统具有极其复杂的动态响应,通过分岔图、相图和庞加莱截面图等方法判断了系统多种形式的周期、拟周期和混沌运动,在以来流流速幅值为可变参数时,系统一开始经由周期倍化分岔的方式进入混沌;在以阻尼系数为可变参数时,经由倒周期倍化分岔的方式从混沌运动退回到周期振动。

一个解特定二阶驻定方程的新方法——特征伴随方程法

本文对文献[1]中提出的一类二阶驻定方程G″/G=∑^(m)_(j=0)P_(j)(G′/G)^(j)的求法进行探讨,提出了全新的求解方法——特征伴随方程法,通过该求法得到这类方程当m取不同非负整数(本文仅讨论m=2)时的通解,并相应给出该方程作为求解非线性偏微分方程的辅助方程时所需要的G′/G的解析表达式;同时给出一个作为常微分方程中驻定方程的应用实例,以及该方程作为非线性偏微分方程的辅助方程时利用扩展G′/G展开法的求解例子,通过该实例给出方程的精确行波解。

不确定转子系统动力学降阶模型构建与模型散度参数辨识

在航空、航天、船舶等领域的实际工程转子系统中,广泛存在高维复杂的非线性系统。在航空发动机转子系统、燃气轮机转子系统等重点研究领域,通常还难以对高维复杂非线性系统进行直接的数据处理和分析统计。针对不确定性转子系统的模型维度较高等问题,提出了一种模型不确定性动力学降阶计算模型构建和模型散度参数辨识方法。首先,根据确定性动力学模型和静态矩阵降阶方法,完善了确定性动力学降阶模型;然后,基于随机矩阵理论和非参数动力学建模方法,提出了不确定性动力学降阶模型;最后,利用系统确定性模型的一阶临界转速、振型和实验数据,对不确定性动力学模型的散度参数进行了辨识;为了验证散度参数辨识方法的有效性,笔者又在转子实验平台上进行了实验验证。研究结果表明:实验结果与降阶之后振动响应均值的差异性较小,且与不确定性动力学模型相差不超过10%,表明所采用的理论模型在描述转子系统行为方面具备了较高的准确性和可靠性,该模型可以为深入研究模型不确定性转子系统提供参考。

实时周跳探测与修复的MGF周跳解算方法的改进

周跳的探测和修复是基于载波相位高精度定位的关键技术之一。基于改进的几何无关相位组合MGF方法具有直接分离不同频率上的周跳,对高频数据中较小的周跳实现有效探测,探测速度远快于TurboEdit法等优点。但MGF法无法探测4周以上的周跳,并对低采样率和低卫星高度角下的数据进行周跳探测时,会选取错误的周跳备选值作为周跳检验量从而造成周跳的误判。针对MGF法所存在的问题,通过联合相位减伪距法PRP法,构建新的周跳备选值判断方程,并对MGF法的周跳检验量进行二次历元间差分STMGF,组成PRP-STMGF周跳探测组合。PRP-STMGF周跳探测组合能够探测任意周的周跳,与TurboEdit法相比,修复时无需分离周跳,直接对PRP-STMGF法的周跳检验量进行四舍五入即可确定出正确的周跳值,且适用于低采样率和低卫星高度角下的实时周跳探测与修复。最后通过北斗双频实测数据进行实验验证,结果表明,PRP-STMGF周跳探测组合,避免了MGF法选取错误周跳备选值的问题,相较于TurboEdit法不仅低延时,准确率高,且适用于不同采样率和低卫星高度角下周跳探测与修复。

基于Krylov-Schur重启技术的Arnoldi模型降阶方法

Krylov子空间模型降阶方法是模型降阶中的典型方法之一,Arnoldi模型降阶方法是这类方法中的一类基本方法。运用重正交化的Arnoldi算法得到r步Arnoldi分解;执行Krylov-Schur重启过程,导出基于Krylov-Schur重启技术的Arnoldi模型降阶方法。运用此方法对大规模线性时不变系统进行降阶,得到具有较高近似精度的稳定的降阶系统,从而改善了Krylov子空间降阶方法不能保持降阶系统稳定性的不足。数值算例验证了此方法是行之有效的。

多步块格式求解微分-代数方程

多步块格式是一类新的一般线性方法,在求解微分-代数方程的过程中不会出现精度降低现象。研究了多步块格式的构造方法,精度条件及具有Runge-Kutta稳定性的多步块格式,多步块格式具有刚性精确的优点,且级精度与格式精度相等。构造了具有Runge-Kutta稳定性的2级和3级多步块格式,具有L-稳定性。数值算例证实多步块格式在求解微分-代数方程不会精度降低。

基于MO-RAMP插值模型的阻尼板GCMOC法拓扑减振优化

为了有效实现板件的抗振性动力学设计,研究约束阻尼板拓扑动力学优化方法。建立约束阻尼板有限元动力学分析模型,推导出模态损耗因子计算公式;建立了基于模态损耗因子最大化目标,以阻尼层单元相对密度为拓扑变量,以阻尼材料使用量及结构频率作为控制的阻尼板优化数学模型;利用序列凸规划理论而对传统优化准则法进行改进,采用改进准则法GCMOC(global extreme point converged by method of optimization criterion)解算优化模型以求取全域性优化解,推导出面向GCMOC的拓扑变量迭代式;考虑到多阶次RAMP(rational approxination of material properties)函数的形状具有较理想的可控下凹几何特征,提出在优化迭代中采用多阶次RAMP材料插值模型(MO-RAMP)对拓扑变量集合进行惩罚以实现其快速的0,1二值化,并尽量减少处于0.3~0.7的中间拓扑变量值出现;编制了面向约束阻尼板的拓扑动力学优化程序,实现了基于MO-RAMP的约束阻尼板GCMOC法变密度式减振拓扑动力学优化过程。算例分析表明,MO-RAMP与GCMOC复合的算法用于阻尼板拓扑迭代时,可将阻尼单元密度值快速地推向逼近0或1的值。它能得到清晰的阻尼单元优化密度云并有利于优化构型的实现;能在大幅减少阻尼材料用量条件下充分发挥其黏弹耗能效应,能在保证阻尼板动力学特性基本稳定的前提下使结构获得更好的减振效果。

一种基于快速BEM-FEM的声振耦合问题快速扫频方法

结构的声振耦合问题普遍存在于航海、航天等工程领域,快速边界元-有限元耦合方法是求解此类问题的有效手段。分析结构的声振耦合问题,往往需要对其进行扫频计算,完全法扫频的计算量太大,而由于快速BEM矩阵的引入,已有的快速扫频方法并不能很好地适用此类问题。本文基于以上事实,采用作者所在研究组提出的响应采样法,构造一种新型的投影空间,通过该空间对基于快速边界元-有限元的声振耦合系统进行模型降阶,发展一种适用于大规模声振耦合问题的快速扫频方法。最后通过两个典型算例验证了本文方法的正确性和性能,发现与完全法相比,快速扫频方法的计算量不足其十分之一,精度也能够稳定在1.0×10-5附近。

Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approximate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These approximate results are compared with the available analytical results and are found to be in good agreement.

热-动力学耦合多体系统建模与降阶

在轨运行的卫星天线受到太阳辐射热冲击后容易出现热致振动或指向不准确等问题,严重时会导致航天器失效.本文提出了一种基于改进模态综合法的刚柔热耦合多体系统建模与降阶方法.采用绝对节点坐标法单元形函数对柔性天线的位移场与温度场进行统一离散插值,避免了两种物理场网格不匹配带来的映射误差与效率问题,并使用绝对节点坐标参考节点描述中心刚体.在系统方程中考虑了热流输入和表面自热辐射.针对绝对节点坐标法切线刚度阵高度非线性的特点,利用一阶泰勒展开对系统动力学和传热学方程进行了分段线性化,在线性化区间内切线刚度矩阵为常数矩阵,避免了每个时间步上的弹性力及其雅克比矩阵的迭代计算,并使得基于模态的降阶手段得以应用.利用改进的模态综合方法划分子结构并缩减系统自由度.相邻子结构之间通过约束方程保证连接精度和连续性.通过纯导热半圆形薄板、薄板的热膨胀、柔性太阳能电池板和刚柔热耦合抛物线天线四个数值算例验证本文所提出方法的有效性.结果表明,本文提出的方法在保证仿真精度的前提下缩减了系统规模,提高了仿真计算效率.

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.