精细积分方法研究综述
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摘要
对于线性常微分方程初值和两点边值问题,精细积分方法可给出计算机上的精确解.本文总结了精细积分方法的基本思想和算法的进一步发展.在初值问题精细积分方法方面,详细综述了精细积分方法的基本思想、对非齐次项的处理技术、大规模问题求解技术以及时变、非线性微分方程的求解.在两点边值问题精细积分方法方面,介绍了处理边值问题的基本思想和求解过程,总结了两点边值问题精细积分方法在各个领域的应用.最后,讨论了初值和边值问题精细积分方法的联系和区别,从而为精细积分方法的理解和应用提供了新的视角.
For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method(PIM) gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.
For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method(PIM) gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.
引文
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2 Dokainish M A,Subbaraj K.A survey of direct time-integration methods in computational structural dynamics-I.Explicit methods;II.Implicit methods.Comput Struct,1989,32:1371-1386
3 Lambert J D.The Initial Value Problem for Ordinary Differential Equations.New York:Academic Press,1993
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5钟万勰,杨再石.连续时间LQ控制主要本征对的算法.应用数学和力学,1991,12:45-50
6钟万勰.结构动力方程的精细时程积分法.大连理工大学学报,1994,34:131-136
7钟万勰.暂态历程的精细计算方法.计算结构力学及其应用,1995,12:1-6
8 Zhong W X,Williams F W.A precise time step integration method.P I Mech Eng C-J Mech,1994,208:427-430
9钟万勰.计算结构力学与最优控制.大连:大连理工大学出版社,1993
10 Zhong W X,Cheung Y K,Li Y.The precise finite strip method.Comput Struct,1998,69:773-783
11钟万勰.矩阵黎卡提方程的精细积分法.计算结构力学及其应用,1994,11:113-119
12林家浩,沈为平.结构非平稳随机响应的混合型精细时程积分.振动工程学报,1995,2:127-135
13谭述君,钟万勰.基于精细积分的(最优)控制系统程序库.见:技术科学论坛第二十三次学术报告会议论文集.上海:中国科学院,中国机械工程学会,2006.75-94
14吴志刚,谭述君,彭海军.现代控制系统设计与仿真-使用PIMCSD工具箱.北京:科学出版社,2012
15 Wei C Z,Guo J F,Park S Y,et al.IFF optimal control for missile formation reconfiguration in cooperative engagement.J Aerospace Eng,2013,28:04014087
16 Wei C Z,Shen Y,Ma X X,et al.Optimal formation keeping control in missile cooperative engagement.Aircr Eng Aerosp Tec,2012,84:376-389
17苗昊春,马清华,董国才,等.反坦克导弹最优一体化制导与控制.系统仿真技术,2013,9:9-13
18 Moler C,van Loan C.Nineteen dubious ways to compute the exponential of a matrix.SIAM Rev,1978,20:801-836
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20钟万勰,姚征.椭圆函数的精细积分算法.见:祝贺郑哲敏先生八十华诞应用力学报告会,应用力学进展论文集.北京:中国力学学会,2004.106-111
21 Fung T C.Computation of the matrix exponential and its derivatives by scaling and squaring.Int J Numer Methods Eng,2004,59:1273-1286
22陆静,韦笑梅,齐辉.正余弦矩阵函数的精细积分算法.广西工学院学报,2006,17:89-91
23富明慧,张文志.病态代数方程的精细积分解法.计算力学学报,2011,28:530-534
24刘勇,沈为平.精细时程积分中状态转换矩阵的自适应算法.振动与冲击,1995,14:82-85
25向宇,黄玉盈,曾革委.精细时程积分法的误差分析与精度设计.计算力学学报,2002,19:276-280
26陈奎孚,张森文.精细时程积分法的参数选择.计算力学学报,1998,15:301-305
27张洪武,钟万勰.矩阵指数计算算法讨论.大连理工大学学报,2000,40:522-525
28徐明毅,张勇传.精细辛算法的高效格式和简化计算.力学与实践,2005,27:55-57
29徐明毅,张勇传.精细辛几何算法的误差估计.数学物理学报,2006,26:314-320
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