本文主要研究如何將小波理論導入有限元素法，並且應用在結構振動的分析。在傳統的有限元素法中，使用多項式當作內插函數來近似結構的位移，單位元素的自由度會被多項式的階數所限制，因而對於要解結構局部高變化梯度的問題，就必需提高多項式階數或增加分析的單位元素數目，這都將使計算上更為複雜。
在小波有限元素法架構中，將小波尺度函數作為內插函數，小波係數作為單位元素自由度。因為單位元素建構在小波空間，需要建構一空間轉換矩陣將單位元素的自由度轉為節點位移場，才能處理單位元素之間的連接和邊界條件的代入。
文中選擇的 Daubechies小波函數具有正交化、緊密支撐及優秀的頻域及時域局部定位解析特性，導入有限元素法後應用在桿結構振動分析，驗證小波有限元素法的可行性和優於傳統有限元素法的收斂性。再將小波有限元素法推展到樑結構的動態特性分析，驗證小波有限元素法在不同結構的適用性。

The objective of this dissertation is to study the construction of the wavelet-finite element method. In traditional finite element methods, polynomials are used as interpolation functions to construct an element; the degrees of freedom are restricted by the order of polynomial. When the problem with local high gradient is analyzed by using traditional finite element methods, the higher order polynomial or denser mesh must be employed to ensure the accuracy.
In the wavelet-finite element method, wavelet functions are employed as interpolation functions and wavelet coefficients are employed as the degrees of freedom. We must construct the space transform matrix to transform wavelet coefficients to nodal displacements and rotations, because elements are constructed in wavelet space. By using the transform matrix, neighboring elements can be connected and processing boundary conditions can be processed directly.
Daubechies scaling functions possess elegant properties of orthonormal, compact support and time-frequency localization. The wavelet-element is introduced into the finite element procedure and the dynamic problems of a bar structure. The accuracy and the convergence rate are verified. Then same dynamic problems of a beam structure are solved by the present wavelet-element modal.

摘要I
Abstract II
誌謝III
目錄V
表目錄VII
圖目錄VIII
符號說明X
第一章緒論
1-1研究背景1
1-2文獻回顧2
1-3本文架構4
第二章Debauchies 小波函數
2-1 小波函數簡介5
2-2 Debauchies 尺度函數及其小波函數之計算8
2-3 尺度函數微分值10
2-4 尺度函數動量值11
2-5 Debauchies 小波函數之聯結係數13
第三章小波有限元素法在桿結構的應用
3-1 桿結構的運動方程式24
3-2 有限元素法25
3-3 小波有限元素法27
3-4 結果與討論30
3-5 結論31
第四章小波有限元素法在樑結構的應用
4-1 樑結構的運動方程式42
4-2 有限元素法43
4-3 小波有限元素法45
4-4 結果與討論51
4-5 結論51
第五章綜合結論與建議
5-1 綜合討論60
5-2 未來研究方向與建議61
參考文獻62

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