以第二類保群算法研究地震行為下結構反應訊號

工程問題多能以常微分方程式的形式描述，而以數值分析的形式求取常微分方程式的近似解也發展了一段時間。保群算法為基於群論所發展而來的方法，在方程式的求解過程中皆保有勞倫茲群的架構，具有良好的穩定性以及準確度。本文中主要引入了喬登結構的概念，改良原本保群算法的增廣動態系統，衍生出更加準確的第二類保群算法，並以其他方法像是保群算法、四階Runge-Kutta法予之比較。而第二類保群算法在求解的過程中能提供反映系統解特性的訊號條碼，於是將此方法應用於地震下單自由度的動力系統，並且將訊號做分類以便於詳細分析訊號的切換和物理意義，進一步的建立起訊號條碼與結構反應的關係，就可以藉由掃瞄條碼來獲取我們想要的資訊。在本文中除了介紹有關於第二類保群算法的基本理論外，將以MATLAB撰寫程式建構出第二類保群算法及其訊號，在不同的地震下予以模擬分析，並且歸納其應用，以及未來研究方向。

關鍵字

李群；李代數；保群算法；喬登結構；第二類保群算法；符號函數；單自由度運動方程式

並列摘要

Most of engineering problems can be formulated in the form of ordinary differential equations (ODEs). Nowadays, people usually use numerical analysis to find an approximate solution when the exact solution of ODEs is hard to derive. The group preserving scheme (GPS) developed by Liu is a numerical method based on group theorem. It is a stable and accurate method for solving ODE because in every step of GPS, it can retain the group structure in a Lorentz group form. The present thesis mainly introduces the concept of Jordan structure to modify the original augmented dynamic system of GPS, hence a more accurate numerical method, named the second type of GPS (GPS2) comes up, and in the example there will show the comparison between GPS2 and other methods, like the GPS and the famous RK4 method. Moreover, it is interesting that we can find signal barcodes reflected to the characteristics of the system, and the barcodes are generated by signum function through the process of using GPS2 to solve ODEs. Consequently, by applying the GPS2 to solve SDOF motion equation under earthquakes and classifying the sign in order to realize both physical meaning and the switch of the sign in detail can construct a bridge between the signal barcode and the structural response, as a result, we can abstract the information by scanning barcodes. In this thesis, we will not only describe the basic theory of GPS2 but also write a program to simulate the response of SDOF system and its signal barcode under different earthquakes by MATLAB. After that, there are some conclusions and future works at last.

並列關鍵字

Lie-group ； Lie-algebra ； group preserving schemes(GPS) ； the second augmented Lie-symmetry SOo(n,1) ； Jordan structure ； signum function ； SDOF motion equations.

參考文獻

[1] Liu C.-S., Cone of non-linear dynamical system and group preserving schemes. International Journal of Non-linear Mechanics, 36(7): pp. 1047-1068 (2001).

[3] Hairer E., Lubich C., Wanner C., Geometric Numerical Integration: Structure-

Preserving Algorithms for Ordinary Differential Equations. 31, Springer (2002).

[4] Iserles A., Solving linear ordinary differential equations by exponentials of iterated commutators. Numerische Mathematik, 45: pp. 183-199 (1984).

[5] Iserles A, Norsett SP., On the solution of linear dfferential equations in Lie group.

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以第二類保群算法研究地震行為下結構反應訊號

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