非线性动力学常微分方程组高精度数值积分方法  

ACCURATE SOLUTION OF NON-LINEAR DYNAMIC SYSTEMS BY NUMERICAL INTEGRATION METHOD

作　　者：郑兆昌[1] 沈松[1] 苏志霄[1] 

机构地区：[1]清华大学工程力学系,北京100084

出　　处：《力学学报》2003年第3期284-295,共12页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家重点基础研究发展规划资助项目(998020316);国家自然科学基金资助项目(19972029)

摘　　要：建立了一种求解非线性动力学常微分方程组初值问题的新方法.若非线性函数一阶导数存在,则给出解的积分方程表达式,计算得到按规定误差要求的高精度数值解.引入一般自治或非自治非线性系统的首次近似Jacobi矩阵,不作任何假设重构等价的非线性常微分方程组,简捷而有广泛的适应性,不改变方程的本质,但其主项构成线性化方程组,其它项则代表非线性函数高阶余项而不涉及Taylor级数展开计算,给出该方程组初值问题的Duhamel卷积分解析表达式,在时间步长内进行数值积分迭代求解,在指定误差内快速收敛,逐步递推获得非线性常微分方程的瞬态响应和全时域高精度数值解.积分解连续满足微分方程组而不是在离散的步长端点上满足代数方程组,打破了传统用增量法在离散点上建立的代数方程组迭代求解,从而使传统:Euler型逐步积分法的各种差分格式算法改变成真正的积分格式算法.数值计算中给出指数矩阵递增展开式,变矩阵乘法为乘积系数的加法,避免了大量矩阵自乘而大大提高计算效率.算法验证为无条件稳定,则保证对线性常微分方程而言,计算中舍入误差的传播不会扩散,不出现计算机字长有限而引起舍入误差导致计算不确定性问题.基于以上理论和数值方法,计算了线性非线性算例并进行了分析,验证了本方法简捷而有广泛的适应性,可以?A new method for solving the non-linear dynamic systems by numerical integration method is established. By introducing the first approach, Jacobi matrix existed for an arbitrary autonomous or non-autonomous non-linear dynamic systems, the nonlinear differential equation is reconstructed as an equivalent non-linear ordinary differential equation, in which the main terms, linear part, construct the continuous linearization equation, the remains represent the high-order remainder of the nonlinear function, it is unnecessary to expand and to calculate high-order remainder. However, the proposed method doesn't changes the nature of the systems without any assumption. The Duhamal convolution integration gives the expression of precision solution of the initial problem of the nonlinear differential equations. Using the Newton-Raphson iteration, in the time steps the accurate solution converge quickly with in given accuracy. Using step by step integration, the nonlinear instantaneous responses in inertial stage and long time steady state responses of nonlinear systems are obtained in whole time domain exactly. The Duhamel integration solutions are continuously satisfied system differential equations not in discrete form with algebra equations, so that is really integration method, which differs from all of the traditional finite difference method satisfied system equation at the end of time steps only. The truncation error is unlimited by original of method and determined by the calculating accuracy with in given accuracy. To save the computing time, the expanding of exponential function matrix gives the recurrent formula instead of multiplication of matrices itself. The unconditional stability has been easily proved. The reconstructed equivalent non-linear differential equation breakthrough the traditional iteration with linearized increment equations, it is simple and has widely acceptability. The Duhamel integration breakthrough the traditional Euler's form of finite difference methods, which the truncation error is l

关 键 词：非线性动力学 有限差分法 常微分方程组 初值问题 高精度数值积分方法 高阶余项 

分 类 号：O313[理学—一般力学与力学基础]