臺灣博碩士論文加值系統

本文分別以基於狀態變數與Hermite型近似之移動最小二乘法分析古典板問題。針對狀態變數之數值方法，將古典板之四階偏微分控制方程分解成以狀態變數表示的八個偶合在一起之一階偏微分方程，並建立各變量之近似函數，且各近似函數間相互獨立。而對於Hermite型近似方法，除了將變位與其一階導數之殘値納入加權殘値二次式計算之外，同時將高階導數如彎矩、剪力之殘値一併考慮，所建立的各變量近似函數彼此相依。兩種數值方法在節點上皆具有八個自由度，建立各主變數之近似函數，並透過移動最小二乘法一次求解出所有變量。針對在不同形狀、不同邊界條件且不同載重作用下之算例，其數值結果顯示本文所採用之兩種數值方法對各變量皆可達到良好之精度與收斂性。

The moving least square methods (MLS) based on state variables and Hermite type approximation are proposed to analyze classical plate problems. For the method based on state variables, the fourth order governing partial differential equation for a plate is decomposed into eight coupled partial differential equations of first order. The approximate functions of state variables are constructed. For the method based on the Hermite type, the residuals of the approximation at each node is considered not only the primary variable and its first-order derivatives, but also the higher order variables, such as the bending moment and shear force. Both of the present methods possess eight degrees of freedom at each node. We construct the approximate functions of all the state variables, and the moving least square technique is employed to solve all of the variables once. Several numerical examples of a plate under different loads with different geometric shapes and various boundary conditions are calculated. It is shown that the present methods have excellent accuracy and high convergence rate.

摘要 I
Abstract II
致謝 IX
目錄 XI
表目錄 XIII
圖目錄 XV
第一章 緒論 1
1.1 前言 1
1.2 文獻回顧 2
1.3 本文架構 5
第二章 古典板控制方程式推導 6
2.1 古典板控制方程式 6
2.2 數值算例解析解 9
2.2.1 外圈簡支圓形板受均佈載重 9
2.2.2 外圈固定圓形板受線性載重 10
2.2.3 外圈固定橢圓形板受均佈載重 11
2.2.4 四邊簡支承矩形板受雙正弦載重 12
2.2.5 四邊簡支承矩形板受均佈載重 13
2.2.6 四邊簡支承矩形板受線性載重 15
2.2.7 兩邊簡支兩邊固定矩形板受正弦載重 16
2.2.8 外圈簡支內圈自由端環形板受均佈載重 17
第三章 移動最小二乘法理論 19
3.1 移動最小二乘法 19
3.2 基於狀態變數之移動最小二乘法在古典板之應用 23
3.3 考慮高階導數之Hermite型近似在古典板之應用 28
3.4 鄰近點與權函數之選取 33
第四章 古典板數值算例 34
4.1 外圈簡支圓形板承受均佈載重 35
4.2外圈固定圓形板承受線性載重 36
4.3 外圈固定橢圓形板受均佈載重 37
4.4 四邊簡支矩形板受雙正弦載重 37
4.5 四邊簡支矩形板受均佈載重 38
4.6 四邊簡支矩形板受線性載重 39
4.7 兩邊簡支兩邊固定矩形板受正弦載重 40
4.8 外圈簡支內圈自由端環形板受均佈載重 41
第五章 結論 42
參考文獻 44
表 48
圖 64

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