無元素分析之
無元素分析之積分權值調整法

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沈國瑞(Kou-Juei Shen) 查詢紙本館藏

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土木工程學系

論文名稱

無元素分析之無元素分析之積分權值調整法
(A Weighting Adjustment Scheme of Gauss Integration in Element Free Method )

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摘要(中)

摘要
無元素法( Element Free method，EFM)處理網格邊界與材料邊界不一致的不規則邊界問題，常以高斯積分點在材料邊界內外做為取捨標準，容易造成數值誤差。本論文提出積分權值調整觀念，由高斯積分的幾何意義著手，將積分網格中積分點與權的值拓展成積分面積的觀念，處理邊界網格積分時直接估算材料邊界內的高斯積分面積，使能量計算誤差降低，取代以往以高斯積分點點位座落於介質體內外為取捨的積分觀念。數值範例顯示積分權值調整觀念處理網格邊界與材料邊界不一致的邊界積分，不必細分網格也可以達到相當的分析精度，不論處理傾斜邊界或曲線邊界均呈現出穩定而且良好的分析結果。運用本法結合J積分計算版裂縫應力強度因子與運用完整高斯積分法情況所得相近。積分權值調整觀念使用於裂縫拓展分析中也相當便利，由彈性樑垂直邊裂縫拓展與版雙裂縫拓展分析顯示本法可以在處理移動性邊界問題時真正做到不必網格重建。此一新的積分法以簡單有效的權值調整觀念處理複雜的邊界網格積分問題，不僅符合無元素法的原創精神，也確保了無元素法能量積分的正確性。

摘要(英)

A Weighting Adjustment Scheme of Gauss Integration in Element Free Method
In this thesis, a modified numerical integration scheme is presented that improves the accuracy of the numerical integration of the Galerkin weak form, within the integration cells of the analyzed domain in the element-free methods. A geometrical interpretation of the Gaussian quadrature rule is introduced to map the effective weighting territory of each quadrature point in an integration cell. Then, the conventional quadrature rule is extended to cover the overlapping area between the weighting territory of each quadrature point and the physical domain. This modified numerical integration scheme can lessen the errors due to misalignment between the integration cell and the boundary or interface of the physical domain. Numerical examples illustrate that this proposed integration scheme for element-free methods does effectively improve the accuracy of solving solid mechanics problems, and is easy to handle the crack propagation problems of solids than it does in the finite element method.

關鍵字(中)

★ 無元素法
★ 元素釋放法
★ 高斯積分

關鍵字(英)

★ EFGM
★ element free

論文目次

目錄
摘要 I
A Weighting Adjustment Scheme of Gauss Integration in Element Free Method II
目錄 III
表目錄 VI
圖目錄 VII
第一章緒論 1
1.1 前言 1
1.2 研究動機與目的 3
1.3 論文內容 4
第二章文獻回顧 6
2.1無網格法的發展 6
2.2 斷裂損傷力學上的應用 10
2.3無元素法的邊界與介面處理 15
2.4 無元素法的積分處理 17
第三章無元素法之基本理論 19
3.1 移動式最小平方(MLS)內插法之應用 19
3.1.1 一致性(Consistency)檢驗 23
3.1.2 形狀函數(Shape Function)之性質 24
3.1.3 加權函數(Weight Function)之探討 26
3.2變分原理 28
3.3離散方程式 30
第四章無元素法的實行與積分改良 32
4.1無元素法的實行 32
4.2高斯積分的幾何新義 37
4.3不規則邊界處之積分處理 41
4.4 無元素法的積分改善 48
4.5權值調整法程式流程 51
第五章裂縫拓展分析的基礎 53
5.1 應力強度因子的求法 53
5.2 裂縫拓展的原則 59
5.3 無元素法於裂縫處的影響圓建構 64
5.3.1不連續邊界處的影響圓決定方法 64
5.3.2通視法定影響圓 66
5.3.3繞射法定影響圓 68
5.3.4透明法定影響圓 69
5.4 權值調整法在裂縫處之應用 70
第六章積分驗證與數值分析例 72
6.1積分驗證 72
6.1.1 一階多項式函數積分檢測 72
6.1.2 高階多項式函數積分檢驗 73
6.1.3指數函數積分檢驗 77
6.2一維微分方程數值例 80
6.2.1具一階線性解的一維微分方程問題 80
6.2.2 具較高階解的一維問題 91
6.3二維彈性介質數值例 96
6.3.1 拉桿 96
6.3.2 含圓孔之無限鈑 100
6.3.3 曲桿純彎曲 104
6.4 破壞力學數值例 105
6.4.1第一型應力強度因子分析 106
6.4.2複合型應力強度因子分析 108
6.4.3 含45度邊緣斜裂縫的版 110
6.4.4 樑三點彎曲試驗 112
6.4.5 含兩組圓孔與裂縫的版 116
第七章結論與建議 123
7.1 結論 123
7.2建議 125
參考文獻 127

參考文獻

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指導教授

王仲宇(Chong-Yui Wang)

審核日期

2002-5-21

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