臺灣博碩士論文加值系統

本文應用移動Trefftz近似法模擬Mindlin一階剪變形平板之受力行為。為了使位移與接觸力之邊界條件納入變分原理中，本法採用了Hellinger-Reissner變分原理，並將原勢能泛函式中之物理量以局部近似函數替代以建立數值模擬計算程序，且為使局部近似位移場與全域位移場在解析架構中保持關聯，將兩者之誤差以平方加權置入勢能泛函式中成為修飾之H-R變分原理。
本法之另一特點為引入了Trefftz Method之概念，選擇滿足微分方程式之函數做為局部近似函數之基底，並於修飾之H-R變分原理中使用了無網格法之移動近似，藉由佈設節點並建立其之間之變量關係形成聯立方程組，進而求解以獲得位移場與合應力場。此外，有鑒於使用多項式之表達形式於數值分析中之便利性，本文利用多項式之運算特性，建立了一套系統化之方法，以求出於移動近似法中所需之特解與基底函數。
數值範例中，模擬了在不同載重與邊界條件下平板之變形與受力行為，並與解析解比較，藉由改變佈點數與基底函數之階數，以評估本法之精度與誤差收斂程度。藉著這些分析，也驗證了本法解析解再生之特性(Reproducing Property)，當基底函數之階數與多項式解析解之階數相同時，數值解即為解析解。而近似解隨著基底函數階數之提高與佈點數之增多，誤差皆呈下降之趨勢，可達到高精度之要求，且誤差收斂性佳。

In this thesis, we derive a numerical method named Moving Trefftz Method to simulate the mechanical behavior of Mindlin Plates. In order to incorporate the essential and natural boundary conditions into the variational principle, we adopt the Hellinger-Reissner variational principle. The characteristic of this method is adopting the concept of the Trefftz Method. We choose the functions that satisfy the differential equations as the basis of the local approximation function. By using the moving approximation of the Meshless Method in the modified H-R variational principle, we set nodes to construct the relationship between variables. Then, we can use the relationship to form simultaneous equations and solve the displacement field and the resultant stress field.
In view of the convenience of using polynomial expressions in numerical analysis, we use the computational properties of polynomials to establish a systematic method to obtain the particular solutions and the bases required in the moving approximation.
In numerical examples, we simulate the responses of the plate with different loads and boundary conditions. By changing the number of nodes and the order of bases, we compare the numerical solutions with the analytical solutions to validate the accuracy and the convergence of this method.

摘要 I
Abstract II
誌謝 VIII
目錄 IX
表目錄 XI
圖目錄 XII
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 2
1.3 研究方法 4
1.4 本文架構 5
第二章 基於Hellinger-Reissner變分原理之Mindlin平板理論推導 6
2.1 Hellinger-Reissner變分原理 6
2.2 平板之平衡方程式 8
2.3 邊界條件 12
2.4 承受雙正弦載重之四邊簡支端平板解析解 13
2.5 多項式解析解 15
第三章 移動Trefftz近似法理論 20
3.1 多項式函數線性微分運算之矩陣表示法 20
3.2 滿足局部平衡方程式之基底 22
3.3 移動近似法 24
3.4 鄰近點數目與影響半徑 26
3.5 權重函數 27
3.6 邊界條件判斷參數 27
3.7 誤差計算 28
第四章 數值範例 29
4.1 四邊皆為簡支承之平板承受雙正弦載重 29
4.2 多項式解析解與數值解之比較 30
4.2.1 平板主要承受x向純彎矩 30
4.2.2 平板主要承受均布載重 30
4.2.3 平板主要承受線性載重 31
4.2.4 平板主要承受二次載重 31
第五章 結論 32
參考文獻 34
表 37
圖 48

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