| 刘诗源,陈飙松,丁海宽.线性常微分方程灵敏度分析的精细积分法[J].计算力学学报,2009,26(4):453~459 |
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| 线性常微分方程灵敏度分析的精细积分法 |
| Precise integration method for sensitivity analysis of linear ODEs |
| 投稿时间：2007-07-16 |
| DOI：10.7511/jslx20094001 |
| 中文关键词: 精细积分 指数矩阵 指数矩阵导数 灵敏度分析 |
| 英文关键词:Precise time integration matrix exponential derivatives of matrix exponential sensitivity analysis |
| 基金项目:国家自然科学基金(10721062,90715037);国家基础性发展规划(2005CB321704)资助项目. |
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| 中文摘要: |
| 利用指数矩阵的导数计算来求解一类一阶线性常系数微分方程组对某一设计变量的灵敏度计算问题。对于初值问题,利用指数矩阵的导数,递推得到状态向量的灵敏度;对于线性两点边值问题,通过两点之间的状态向量的导数关系,得到全部初始条件,进而转化为初值问题计算。指数矩阵及其导数阵的高精度计算基于2N类算法,在此基础上可实施灵敏度分析的计算。本文给出了初值和两点边值常微分方程的高精度灵敏度计算方法,计算结果可认为是计算机上的精确解,算例验证了算法的有效性。 |
| 英文摘要: |
| In this paper, the derivatives of matrix exponential is employed to solve the sensitivity analysis of linear ordinary differential equations(ODEs) with respect to a given design variable. For the initial value problems, the derivatives of state vector with respect to design variables can be obtained by using the derivatives of matrix exponential. For the two-point boundary value problem, the matrix exponential and its derivatives are employed to link the boundary derivative conditions between the two points. With full conditions at initial point, the liner two-point boundary value problems can be transformed into initial value problem and then solved by time marching scheme. The computation of matrix exponential and its derivatives are performed based on 2N algorithm. And then sensitivity analysis can be carried out. Sensitivity analysis method with high numerical precision is developed. Numerical examples demonstrate the extremely high numerical precision of the method. |
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