臺灣博碩士論文加值系統

無網格法為近十幾年來逐漸日趨成熟的演算法，相較於傳統的有限元素法，其具有不需使用網格建立形狀函數之特點，讓無網格法擺脫有限元素法在處理某些問題遭遇到的限制。但由於無網格法發展並未如有限元素法的歷史悠久並且成熟，在許多研究方面仍待拓展。一維梁是簡單的問題且有解析解，本研究利用無網格法分析一維問題，重點在於探討無網格法之參數對於分析誤差的影響，以便有利於日後將無網格法應用到更複雜之問題。

The meshfree method has attracted lots of attentions from the researchers in the field of numerical analysis in the recent two decades. The meshfree method has the advantage over the finite element method on constructing shape function without predefining meshes. The development of the meshfree method is still far away from maturity. Accordingly, the main purpose of the work is to study comprehensively the convergence behaviors of the solutions obtained from the element free Galerkin method. The element free Galerkin method is applied to analyze a beam under static loading and to determine the natural frequencies of a beam. The used shape functions are constructed by the moving least squares method or radial basis function approach. The accuracy of the solutions influenced by the parameters such as the type of weighting function and basis function, the number of nodes is extensively investigated.

目錄..........................................v
表目錄.....................................viii
圖目錄........................................x
第一章 緒論...................................1
1.1 前言.................................1
1.2 文獻回顧.............................2
1.3 內容概要.............................4
第二章 理論基礎...............................5
2.1 樑自然振動頻率以及靜力解之推導.......5
2.2 移動式最小平方差法建構形狀函數.......8
2.3 徑向基函數建構形狀函數..............12
2.3.1 徑向基函數..........................12
2.3.2 建構形狀函數........................14
2.3.3 徑向函數結合多項式..................15
2.3.4 邊界條件處理方式....................16
第三章 移動式最小平方差法分析結果............21
3.1 分析樑自然振動頻率問題之參數探討....21
3.1.1 權函數..............................22
3.1.2 節點數目以及佈點方式................26
3.1.3 基底多項式次方數....................27
3.1.4 高斯積分點數目以及高斯積分區間......27
3.1.5 改變基底多項式以符合邊界條件........28
3.1.6 不同邊界梁之結果....................29
3.2 分析樑靜力問題之參數探討............30
3.2.1 權函數..............................30
3.2.2 節點數目以及佈點方式................31
3.2.3 基底多項式項數......................32
3.2.4 改變基底多項式以符合邊界條件........32
3.3 分析討論............................33
第四章 徑向基函數分析結果...................35
4.1 分析樑自然振動頻率問題之參數探討....35
4.1.1 徑向基函數種類對於誤差的影響........35
4.1.2 節點數目對於誤差之影響..............36
4.1.3 添加多項式基底項次對於誤差之影響....37
4.1.4 邊界條件處理方式對於誤差之影響......38
4.1.5 不同邊界條件梁之誤差................39
4.2 分析樑靜力問題之參數探討............39
4.2.1 徑向基函數種類對於誤差的影響........39
4.2.2 節點數目對於誤差的影響..............40
4.2.3 添加多項式次方數對於誤差的影響......40
4.2.4 邊界條件處理方式對於誤差之影響......40
4.2.5 不同邊界條件梁之誤差................41
4.3 分析討論............................41
第五章 結論與建議...........................42
5.1 結論...............................42
5.2 建議...............................43
參考文獻.....................................46

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