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研究生:黃柏彰
研究生(外文):Po-Chang Huang
論文名稱:尤拉梁外力的閉合係數展開識別法
論文名稱(外文):A closed-form coefficients expansion method for recovering spatial-dependent load on the Euler-Bernoulli beam equation
指導教授:劉進賢鍾立來鍾立來引用關係
指導教授(外文):Chein-Shan,LiuLap-Loi,Chung
口試日期:2017-07-19
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:中文
論文頁數:66
中文關鍵詞:邊界積分方程法尤拉梁伴隨Trefftz測試函數廣義格林第二恆等式自我伴隨運算子源識別問題閉合係數展開識別法
外文關鍵詞:Boundary Integral Equation Method (BIEM)Euler-Bernoulli BeamAdjoint Trefftz Test FunctionClosed-Form Expansion Coefficients Method
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在土木工程議題中,橋梁的振動是非常重要的一塊領域,而橋梁又可以理想地簡化為尤拉梁進行分析。在正算尤拉梁問題時處理手段有許多種,而當處理欲反求外力的尤拉梁問題(源識別)時,計算量與困難度將會增加許多。本文使用邊界積分方程法(BIEM),搭配伴隨Trefftz測試函數之概念,引入振態的概念做為基底,並發展出新的閉合係數展開識別法。由於過程之中沒有重積分及大尺度的反矩陣運算,且求出之解為閉合解,因此能夠得到誤差小、精度高的結果。論文中將以數值算例,實際求解尤拉梁的外力反算問題,其中包含各種不同邊界條件的梁以及其對應之振態做為基底,另外並加入噪音進行分析,最後與精確解進行比對,分析數值結果。
In the field of Civil engineering, the forced vibration of a bridge is a big issue. When discussing the behavior of a bridge, we can apply the Euler-Bernoulli beam theory to analyze it simply. There are several methods to analyze the Euler-Bernoulli beam equation under an external force; however, without knowing the external force, it becomes an inverse source problem which is much more complicated.In this thesis, we adopt the boundary integral equation method (BIEM) with the mode shapes as adjoint test functions. Then, we can develop a non-iterative method to recover a space-dependent load on the Euler-Bernoulli beam named closed-form expansion coefficients method. Finally ,we give some numerical examples to demonstrate the efficiency and accuracy of the proposed new method.
目錄
誌謝 I
摘要 III
ABSTRACT IV
圖目錄 VII
表目錄 IX
第一章 緒論 1
  1.1 前言 1
  1.2 文獻回顧 1
  1.3 研究動機與目的 2
  1.4 論文架構 2
第二章 理論基礎 4
  2.1 尤拉梁理論 4
  2.2 自我伴隨運算子(Self-Adjoint Operators) 6
  2.3 廣義格林第二恆等式(Generalized Green’s Second Identity) 7
  2.4 伴隨Trefftz測試函數(Adjoint Trefftz Test Functions) 9
  2.5 擬時間積分法 11
  2.6 高斯-克朗羅德法(Gauss-Krontod Quadrature Formula) 12
  2.7 閉合係數展開識別法(Closed-Form Expansion Coefficients Method) 13
第三章 邊界積分方程法(BIEM) 18
3.1 等斷面尤拉梁 18
  3.2 各邊界條件之尤拉梁振態推導 20
第四章 數值算例 29
  4.1 數值算例一 30
  4.2 數值算例二 32
  4.3 數值算例三 34
  4.4 數值算例四 35
  4.5 數值算例五 36
  4.6 數值算例六 38
  4.7 算例表 40
  4.8 算例圖 46
第五章 結論與未來工作 62
參考文獻 64
參考文獻
[1]C. S. Liu, A non-iterative method for recovering a space-dependent load on the Euler-Bernoulli beam equation,in press(2017)
[2] Artur Maciag and Anna Pawinska, Solution of the direct and inverse problems for beam, Computational and Applied Mathematics, Volume 35, Issue 1, pp 187-201 (2014).
[3]C. S. Liu, A BIEM using the Trefftz test functions for solving the inverse Cauchy and source recovery problems, Engineering Analysis with Boundary Elements 62:pp. 177-185 (2016).
[4]C. S. Liu, A global domain/boundary integral equation method for the inverse wave source and backward wave problems, Inverse Problems in Science and Engineering(2016).
[5]C. S. Liu, A global boundary integral equation method for recovering space-time dependent heat source, International Journal of Heat and Mass Transfer 92:1034-1040 (2016).
[6]C. S. Liu, Atluri, S. N., A novel time integration method for solvinga large system of non-linear algebraic equations.CMES: Computer Modeling in Engineering & Sciences, vol. 31,71-83 (2008).
[7]C. S. Liu, A Lie-group adaptive differential quadrature method to identify an unknown force in an Euler–Bernoulli beam equation, Acta Mech 223,2207-2223 (2012).
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[12]George Lindfield and John Penny, Numerical Methods: Using MATLAB, Academic Press, 3rd edition, 233-276 (2012).
[13]L.F. Shampine, Vectorized adaptive quadrature in MATLAB, Journal of Computational and Applied MathematicsVolume 211, Issue 2, 131-140 (2008).
[14]M. A. Jawson , Integral equation methods in potential theory, I, Proc. Roy. Soc., Ser. A 275, 23-32 (1963).
[15]Prem K. Kytbe, An Introduction to Boundary Element Methods, CRC Press, 15-18 (1995).
[16]Raymond W. Clough and Joseph Penzien, Dynamics of Structures, Mcgraw-Hill College, 3rd edition, 389-390 (1995).
[17]Serge Nicaise and Ouahiba Zair, Determination of Point Sources in Vibrating Beams by Boundary Measurements: Identifianility,Stability, And Reconstruction Results, Electronic Journal of Differential Equations, Vol. 2004, No. 20, pp. 1–17.ISSN: 1072-669 (2004).
[18]T . A. Cruise and F. J. Rizzo, A direct formulation and numerical solution of the general transient elasto-dynamic problem, I, J. Math. Anal. Appl. 22, 244-259 (1968).
[19]Trefftz,E., Ein Gegenstuck zum Ritzschen Verfahren, in Proceedings 2nd International Congress of Applied Mechanics, Zurich, pp.131-137 (1926).
[20]Zill, Differential Equations with Boundary-Value Problems, Brooks Cole, 7th edition (2008).
[21] Huan-Cheng Hsu,By Using Boundary Integral Equation Method to Solve the I
Inverse Problems of Forces of Euler-Bernoulli Beams,Master thesis(2016)
[22] Bo-Jun Chang, By Using Bounday Integral Equation Method to Solve The Direct Euler-Bernoulli Beam Problem,Master thesis,pp.5-43(2016)
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