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基于线性代数的大规模快速量纲分析算法及其在爆炸与冲击工程研究中的应用

钟巍1,2, 田宙1, 寿列枫1,3   

  1. 1. 西北核技术研究院, 西安 710024;
    2. 北京大学数学科学学院, 北京 100871;
    3. 北京理工大学 爆炸科学与技术国家重点实验室, 北京 100081
  • 收稿日期:2018-05-23 出版日期:2020-05-15 发布日期:2020-05-15

钟巍, 田宙, 寿列枫. 基于线性代数的大规模快速量纲分析算法及其在爆炸与冲击工程研究中的应用[J]. 计算数学, 2020, 42(2): 170-195.

Zhong Wei, Tian Zhou, Shou Liefeng. A LARGE-SCALE AND FAST DIMENSIONAL ANALYSIS METHOD BASED ON LINEAR ALGEBRA AND ITS APPLICATIONS IN THE FIELD OF EXPLOSION AND IMPACT ENGINEERING[J]. Mathematica Numerica Sinica, 2020, 42(2): 170-195.

A LARGE-SCALE AND FAST DIMENSIONAL ANALYSIS METHOD BASED ON LINEAR ALGEBRA AND ITS APPLICATIONS IN THE FIELD OF EXPLOSION AND IMPACT ENGINEERING

Zhong Wei1,2, Tian Zhou1, Shou Liefeng1,3   

  1. 1. Northwest Institute of Nuclear Technology, Xi'an 710024, China;
    2. School of Mathematical Sciences, Peking University, Beijing 100871, China;
    3. State Key Lab of Explosion and Safety Science, Beijing Institute of Technology, Beijing 100081, China
  • Received:2018-05-23 Online:2020-05-15 Published:2020-05-15
量纲分析是科学研究,特别是工程应用中非常重要的一个理论分析工具.从E.Buckingham提出Π定理开始算起,量纲分析已有一百多年历史,其基本理论和方法已经非常成熟,在各个领域也取得了显著的成果并且仍然有着广泛的应用.然而,随着研究的深入,面对的问题越来越复杂和细致,人们越来越关注在传统量纲分析中忽略掉的一些所谓次要因素的影响,因此涉及的物理量变得越来越多,导致按传统的量纲分析方法处理时常常显得非常繁琐甚至困难.本文从线性代数的观点出发,将量纲分析转换为线性空间问题,通过矩阵运算,完成量纲分析的关键过程.给出了量纲分析对应的线性代数问题的基本定理,并基于这些定理建立了程序化的量纲分析算法,将原本复杂的量纲分析问题转化为借助计算机代数系统能够快速方便解决的矩阵运算问题.最后,结合笔者多年的工作经历,给出了上述方法在爆炸与冲击工程研究领域中的若干应用实例,详细表述了具体操作步骤,验证了算法的优越性.
Dimensional analysis is a very important theoretical analysis tool in scientific studies and engineering applications. It has a history of more than 100 years since the Π theorem was first put forward by E. Buckingham, and has gradually formed mature basic theories and methods. Meanwhile, dimensional analysis has made remarkable achievements in various fields, and it will still be widely used in these fields. However, with in-depth study, the problems the researchers need to face have become much more complicated and detailed, and more attentions are paid to the so-called minor factors abandoned in the traditional dimensional analysis. Therefore, more and more physical quantities are involved, which make the traditional dimensional analysis very cumbersome, even extremely difficult to solve. In this paper, from the point of view of linear algebra, the dimensional analysis has been converted into linear space problem, and then, the key process of the dimensional analysis has been finished through matrix operation. The basic theorems of linear algebra problem corresponding to dimensional analysis have been proposed, and on the basis of these theorems, a programmed dimensional analysis algorithm has been built. The algorithm transforms the original complex dimesional analysis problem into a matrix operation problem, which can be solved quickly and conveniently by any computer algebra system. Finally, with years of personal experience in the field of explosion and impact engineering research, the author has provided several examples to show the detailed operation method and illustrate the advantage of the present method.

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