利用擬時間積分法與李群方法識別Euler-Bernoulli梁的外力

Abstract

本文主要討論Euler-Bernoulli梁的正算與反算問題，正算問題方面使用Euler法、RK4與保群算法(group preserving scheme)，而對於求解非線性動態系統，保群算法(group preserving scheme)是一個全新的求解形式，它可以保有求解的微分方程系統中的內部對稱群。 而反算問題方面使用擬時間積分法(fictitious time integration method)、李群打靶法(Lie-group shooting method)與李群調整法(Lie-group adaptive method)；其中保群算法(group preserving scheme)也同時是李群方法中非常重要的理論基礎。本文結合擬時間積分法(fictitious time integration method)與李群打靶法(Lie-group shooting method)去推導出其問題的代數方程，並在一個封閉形式下求出其解。比較於其他求解方法，李群打靶法(Lie-group shooting method)的優點有以下幾點：(1)不需要事先知道外力方程式的形式。(2)不需要疊代計算。(3)擁有封閉形式的解。而李群調整法(Lie-group adaptive method)則是利用層層計算疊代的技巧，此方法的最大特色即是不需要量測的變位數據，只需要邊界條件與初始條件即可求出正確的解。 本文將會詳盡介紹使用的所有方法，利用其方法求解Euler-Bernoulli梁動態方程式，並且使用程式語言FORTRAN進行數值模擬分析。同時，針對反算病態問題，本文也將會驗證李群方法在對抗噪音方面有著非常良好的結果。

The present paper mainly discusses the direct problem and the inverse problem of the Euler-Bernoulli beam dynamic equation. For the direct problem, we use the Euler method, the fourth-order Runge-Kutta method (RK4) and the group preserving scheme (GPS). The group preserving scheme (GPS) is a new form for solving the non-linear dynamical system, and it can preserve the internal symmetry group of the considered ordinary differential equations (ODEs) system. For the inverse problem, we use the fictitious time integration method (FTIM), the Lie-group shooting method (LGSM) and the Lie-group adaptive method (LGAM). Particularly, the group preserving scheme (GPS) also is a very important basic theory for the Lie-group methods. We have applied the fictitious time integration method (FTIM) and the Lie-group shooting method (LGSM) to deriving the algebraic equations and solved them in a closed-form. In contrast to other estimation methods, the advantages of the Lie-group shooting method (LGSM) are that it does not need any prior information on the functional form of the external force, no iterations are required and the closed-form solution is available. The other Lie-group method we used is the Lie-group adaptive method (LGAM) which is using the layer-stripping technique which can be used to find the unknown force layer by layer via iterations. The layer-stripping technique together with the Lie-group adaptive method (LGAM) leads to that solving the inverse Euler-Bernoulli beam equation does not require the extra measurement of data, in addition to the usual boundary conditions and initial conditions for the direct problem. In this paper, we introduce all of the methods which are used, deeply and thoroughly, and via using them to calculate the Euler-Bernoulli beam dynamic equation. Moreover, we use the programming language, FORTRAN, to analyze the numerical identifications. For the ill-posed behavior of inverse problems, we have tested and verified that the Lie-group methods are very useful to be directed against them, namely using the Lie-group methods we can obtain good results even for the ill-posed problems.