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The development and extended applications of precise integration method

DOI：10.7511/jslx20231126001

 作者 单位 E-mail 姚伟岸 大连理工大学 力学与航空航天学院 工业装备结构分析优化与CAE软件全国重点实验室, 大连 116024 ywa@dlut.edu.cn 高强 大连理工大学 力学与航空航天学院 工业装备结构分析优化与CAE软件全国重点实验室, 大连 116024 谭述君 大连理工大学 力学与航空航天学院 工业装备结构分析优化与CAE软件全国重点实验室, 大连 116024 吴锋 大连理工大学 力学与航空航天学院 工业装备结构分析优化与CAE软件全国重点实验室, 大连 116024

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。

In 1991, ZHONG Wanxie first proposed the Precise Integration Method(PIM) for calculating matrix exponential, the main points of which are a 2N algorithm and increment storage.The PIM can give the precise solution of the matrix exponential in the computer sense and provide a high-precision stable algorithm for the numerical calculation of ordinary differential equations, and it has been successfully applied in many fields such as structural dynamics response, random vibration, heat conduction, optimal control.This paper first introduces the proposal, basic idea and development of PIM for matrix exponential calculation.Then various PIMs for solving time-invariant and time-varying linear differential equations, nonlinear differential equations and large-scale problems are presented, and their advantages, disadvantages and the range of application are also analyzed.Finally, the extended application of the basic idea of PIM for two-point boundary value problems, elliptic functions, ill-conditioned algebraic equations and other problems is described, which further demonstrates the characteristics of PIM.