Modified precise time step integration method of structural dynamic analysis

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method （e.g., an algorithm with an arbitrary order of accuracy） is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix H. If the matrix H is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.

A new simple method of implicit time integration for dynamic problems of engineering structures

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai＇s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.

RETHINKING TO FINITE DIFFERENCE TIME-STEP INTEGRATIONS

The numerical time step integrations of PDEs are mainly carried out by the finitedifference method to date. However,when the time step becomes longer, it causes theproblem of numerical instability,. The explicit integration schemes derived by the singlepoint precise integration method given in this paper are proved unconditionally stable.Comparisons between the schemes derived by the finite difference method and theschemes by the method employed in the present paper are made for diffusion andconvective-diffusion equations. Nunierical examples show the superiority of the singlepoint integration method.

A modified highly precise direct integration method for a class of linear time-varying systems

It is always a bottleneck to design an effective algorithm for linear time-varying systems in engineering applications.For a class of systems,whose coefficients matrix is based on time-varying polynomial,a modified highly precise direct integration(VHPD-T method)was presented.Through introducing new variables and expanding dimensions,the system can be transformed into a timeinvariant system,in which the transfer matrix can be computed for once and used forever with a highly precise direct integration method.The method attains higher precision than the common methods(e.g.RK4 and power series)and high efficiency in computation.Some numerical examples demonstrate the validity and efficiency of the method proposed.

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.

Applying a Step Approach Method in Solving the Multi-Frequency Radiation From a Complex Obstacle

In this paper,a step approach method in the time domain is developed to calculate the radiated waves from an arbitrary obstacle pulsating with multiple frequencies.The computing scheme is based on the Boundary Integral Equation and derived in the time domain;thus,the time-harmonic Neumann boundary condition can be imposed.By the present method,the values of the initial conditions are set to zero,and the approach process is carried forward in a loop from the first time step to the last.At each time step,the radiated pressure on each element is updated.After several loops,the correct radiated pressures can be obtained.A sphere pulsating with a monopole frequency in an infinite acoustic domain is calculated first.This result is compared with the analytical solution,and both of them are in good agreement.Then,a complex-shaped radiator is taken as the studied case.The pulsating frequency of this case is multiple,and the waves propagate in half space.It is shown that the present method can treat multiple-frequency pulsation well,even when the radiator is a complex shape,and a robust convergence can be attained quickly.

精细积分方法的发展与扩展应用

钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。

传输线方程高精度直接积分的数值求解方法

提出一种基于精细积分法与时域微分求积法相结合的传输线方程的数值求解方法。首先将传输线方程采用基于紧致有限差分法的四阶差分格式进行空间离散,得到关于时间的一阶线性常微分方程组,四阶差分格式对于空间微分有很好的近似精度。然后利用精细积分法与微分求积法对一阶线性常微分方程组进行数值求解。通过理论分析可知,与传统的传输线方程数值求解方法——时域有限差分法(Finite difference time domain,FDTD)相比,所提方法不涉及到状态矩阵求逆运算,保证了数值求解精度,并且其数值稳定性与计算时间、空间步长无关,可采用大步长进行数值计算,能够有效提高计算效率。最后利用仿真实例进行算法验证,结果显示,相比于时域有限差分法,所提方法能够抑制数值振荡,提高了计算精度。

结构动力方程的一种自适应步长增维精细积分法

增维精细积分法是一种求解结构动力方程的高精度逐步积分算法,其步长的选取会对计算精度产生极大的影响,在实际应用中存在难以确定合适步长的问题。为满足实际工程中对计算精度和效率的要求,提出了一种计算误差的估计方法,并以估计误差和迭代收敛速度为基准,建立了一种自适应步长增维精细积分法。针对三种结构动力方程的算例结果表明,在计算各类线性及非线性振动问题时,该方法均可以在保证计算精度的前提下快速有效地控制算法的计算步长,并且仅需较少的额外计算消耗,显著提高了增维精细积分法的计算效率,使该方法在求解结构动力方程时更具计算优势和实用价值。

基于多主导极点法的时滞积分系统PID控制器设计

伺服与扰动抑制是时滞积分系统最基本的控制问题,对其进行控制难度较大。文中提出一种基于直接综合法和多主导极点配置法的微分先行PID(Proportional-Integral-Derivative)整定方法,这种方法通过比较串联滤波器与时滞积分被控对象组成的特征方程与实际期望的特征方程的系数,将三阶主导极点置于-1/λ处,并将二阶非主导极点置于-5/λ处(λ为调整参数),从而获得期望的特征方程。以实现期望的鲁棒性方式获得设计的控制器参数,通过选择不同的调优参数获取相应的Ms(Maximum sensitivity)值,在参数具有标称性的限定条件下拟合出关于Ms和调优参数的关系曲线,给出整定规则的解析形式。PIPTD(Pure Integral Plus Time Delay system)、DIPTD(Double Integral Plus Time Delay system)和FOPTDI(First-Order Plus Time Delay Integral System)系统的仿真结果表明,IAE(Integral Absolute Error)指标平均可降低35.79%,TV(Total Variation)指标平均可降低18.97%。

单集中阻尼张紧弦动力时程分析方法对比研究

集中阻尼张紧弦动力时程分析方法对比研究,采用精细时程积分法与纽马克(Newmark)法两种方法进行算例分析,主要探讨精细时程积分法与纽马克法对积分步长的敏感度。研究发现,精细时程积分法和纽马克法,可以无条件稳定,当积分步长逐渐增大时,精细时程积分法依旧可以精确计算,得到准确结果,纽马克法计算精度受限,导致出现误差,故在进行单集中阻尼张紧弦的动力时程分析时,精细时程积分法优于纽马克法。

含非线性时间项的无偏灰色模型构建与应用

针对含二次多项式时间项灰色模型存在着从差分到微分的跳跃性误差,提出了其无偏模型,此模型无需已知白化方程解析解。借助新变量将原白化方程转化为矩阵微分方程,根据方程的解直接建立离散时间响应函数的递推公式,利用直接建模法给出无偏模型的参数估计,采用精细积分法直接对数据进行高精度的模拟(预测),并以误差平方和最小为目标对初始点进行优化。从理论上证明了该模型具有含线性时间项非齐次指数规律重合性和伸缩变换一致性,从非齐次指数序列和能源消费实例上也验证了所构建模型的无偏性和有效性,具有更好的拟合(预测)精度。

一种广义精细积分法

提出了求解非齐次动力方程特解的一种精细数值积分法,该方法与通解精细积分法具有相同精度.首先选取一个积分形式的非齐次方程特解,将积分区域划分为2^N份,并对之进行精细的数值积分;然后针对载荷为多项式、指数函数及三角函数的情况,将积分求和转化为一个递推过程,按此只需n次矩阵乘法就能计算出积分和,从而得到非齐次方程的特解.该方法的优点是能与通解的精细积分过程有机地结合起来,具有极高的精度和效率,同时还具有较广泛的适用范围.算例结果证明了该方法的有效性.

Duhamel项的精细积分方法在非线性微分方程数值求解中的应用

基于Duhamel项的精细积分方法,构造了几种求解非线性微分方程的数值算法。首先将非线性微分方程在形式上划分为线性部分和非线性部分,对非线性部分进行多项式近似,利用Duhamel积分矩阵,导出了非线性方程求解的一般格式。然后结合传统的数值积分技术,例如Adams线性多步法等,构造了基于精细积分方法的相应算法。本文算法利用了精细积分方法对线性部分求解高度精确的优点,大大提高了传统算法的数值精度和稳定性,尤其是对于刚性问题。本文构造的算法不需要对线性系统矩阵求逆,可以方便的考察不同的线性系统矩阵对算法性能的影响。数值算例验证了本文算法的有效性,并表明非线性系统的线性化矩阵作为线性部分是比较合理的选择。

移动荷载过桥的精细计算

对于作用点位置随时间连续变化的移动荷载,本文采用精细积分法求解桥梁结构动力平衡方程,用Her-mite插值函数模拟节点等效荷载,并推导出常量移动荷载在任意时间步长内的Duhamel积分格式,包括时间步长的开始和结束时刻集中荷载分别在同一个单元、相邻单元和不相邻单元三种情况。数值算例表明,即使采用最大时间步长,本文提出的方法仍可求得高精度数值解,与其它数值方法比较具有明显优势。

基于精细积分法的伸展悬臂结构动态特征的计算

本文针对时变结构 ,建立了一套高精度计算方法 ,其舍弃了传统的差分形式 ,具有计算精度高及数值稳定性好等特点 ,进而将上述方法应用于一种典型结构——伸展悬臂梁 ,给出了悬臂结构在不同伸展规律下体现出来的动态特征。计算结果表明 ,精细积分法能有效地解决时变结构问题 ,并具有很高的计算精度。

轴向激励下斜拉索大幅振动分析

针对大跨度斜拉空间结构的斜拉索大幅振动问题,考虑塔柱-拉索-空间结构协同作用,导出了拉索在轴向激励作用下的非线性振动方程。采用多尺度法研究拉索大幅振动的各种可能性,运用精细时程积分法数值求解振动方程。研究表明斜拉索自由振动时的振动特性与塔柱、拉索、空间结构三者之间的频率比,以及塔柱、空间结构对拉索所提供的初始扰动特性密切相关;塔柱振动对拉索的参数振动影响明显。若塔柱与空间结构的固有频率相近且远离拉索固有频率,拉索振动发散。若塔柱和空间结构的固有频率相近、且与拉索固有频率存在一定差别,拉索振动呈现“拍”的特点。若塔柱、空间结构与拉索三者的固有频率相互紧密靠近,拉索振动的位移幅值接近拉索的初始扰动值。

一种改进的精细-龙格库塔法

提出了求解非线性动力学方程的一种改进的精细龙格库塔法。首先对于线性问题,利用等步长的New-ton-Cotes积分公式计算非齐次方程Duhamel积分形式的特解。由于在此过程中提出了一种简便的算法,与常规的同精度数值积分法相比,能较大程度地降低计算量和存储量。然后将上述方法推广到非线性问题,对于各积分点上未知的状态参量,参照龙格库塔法的几何解释进行一次预估。与已有的精细龙格库塔法相比,在精度和效率上均有较大程度的改善。算例结果充分证明了该方法的有效性。

Burgers方程的小波精细积分算法

求解偏微分方程的常用方法包括有限差分法、有限元法等。近年来,小波分析在偏微分方程数值求解中的应用已引起很多学者的关注,例如采用Daubechies小波或shannon小波构造的小波配置方法已经取得较好的结果。钟万勰院士提出的偏微分方程的子域精细积分方法是一种半解析方法,方法简单,精度高。将小波方法和精细积分方法相结合应用于偏微分方程的数值求解中将有利于提高算法的精度和稳定性,为此本文以Burgers方程为例,提出了一种求解一维非线性抛物型偏微分方程的小波精细积分方法。该方法用拟小波配点法对空间域进行离散,建立起对时间的常微分方程组,然后采用精细时程积分方法对该方程组求解。数值计算结果表明,该方法同其它方法相比,具有计算格式简单,数值稳定性和精度较高的优点。

基于Markov参数精细积分法的载荷识别研究

对于结构受多点分布动态荷载识别问题提出了精细正则化算法。基于状态空间描述建立了离散动力系统滑动平均模型,并应用2N算法精细计算了系统模型的马尔科夫(Markov)参数矩阵,给出了全局时间域内多点分布动态载荷识别问题的精细识别模型。针对载荷辨识模型求解过程中遇到的方程病态问题,采用正则化截断奇异值分解技术进行处理。通过有限元数值模型仿真,验证了所提出方法的正确性和有效性。