臺灣博碩士論文加值系統

摘 要

本文主要是探討有限元素法中的高階元素對求解裂縫端之應力強度因子（SIF）的效能，並與奇異元素作個比較。在這之前也對超收斂點的位置和靜態縮減的方法作一些的探討。
對能量誤差而言， 階Lagrange型態的元素在n*n個高斯積分點的位置上有(n+1)的超收斂率；且其收斂率和精確度方面亦較其它兩種元素來的好。若將高階Lagrange型態之元素在求解部分搭配靜態縮減的方法，將可大量的縮短求解時間，尤其是整體自由度越來越大時。
當使用位移外插法來求SIF時，二次外插法似乎較線性外插法來的合理，且有時其精確度亦較高。針對奇異性為1/sqrt(r)的問題時，奇異元素較高階元素有著更好的近似結果，特別是在裂縫端配置三角形奇異元素時
，然而高階元素亦有不錯的結果（誤差小於1%）。
針對相同總自由度的問題，若高階元素能搭配靜態縮減的方法，其不僅較低階元素求解時間快且精度亦較高。







關鍵字：有限元素法、超收斂率、高階元素、位移外插法、應力強度因
子、靜態縮減矩陣

Abstract

The purpose of this research is to study the effect of higher order elements
in solving the crack tip stress intensity factor (SIF), and to compare the results
with those obtained by using the singular elements. We also study the positions
of the superconvergent points as well as the method of static condensation
beforehand.

In energy norm,the n-th order Lagrange elements have the superconvergent
rate with the order of n+1,and their locations are on n by n Gauss integration
points. The accuracy and convergent rate of the Lagrange elements are better
than the other two kinds of elements which are Serendipity and Enriched
Serendipity. The method which using the static condensation in solving process
will save a large amount of time with lager and lager degrees of freedom
especially.

When using the method of displacement extrapolation to solve SIF, we
find that the second order interpolation method seems to be more reasonable
than the first order one.Singular elements have the better approximate results
than higher order elements with the problems of the singularity equaling to 1/sqrt(r), especially in the case of using triangular singular elements. But the
higher order elements also have satisfying results (the error less than 1% ).

For the problem of the same total degrees of freedom,the higher order
elements spend less time and have better accuracy than lower order elements,
if these higher order ones can collocate the method of static condensation in
solving process.






keywords : finite element method,superconvergent rate, higher order
elements, displacement extrapolation method,stress intensity
factor, static condensation

目 錄
中文摘要……………………………………………………Ⅰ
英文摘要……………………………………………………Ⅱ
誌謝…………………………………………………………Ⅲ
目錄…………………………………………………………Ⅳ
表目錄………………………………………………………Ⅶ
圖目錄………………………………………………………Ⅸ
符號說明……………………………………………………XⅢ

第一章 緒論…………………………………………………1
1.1 前言…………………………………………………………1
1.2 文獻回顧……………………………………………………3
1.3 研究動機與目的……………………………………………4
1.4 論文架構……………………………………………………6
第二章 相關理論……………………………………………7
2.1 有限元素法…………………………………………………7
2.1.1 平面應力和平面應變………………………………....8
2.1.2 元素的型態…………………………………………....12
2.2 靜態縮減矩陣………………………………………………15
2.3誤差分析與收斂的定義…………………………………….16
2.4 破壞力學……………………………………………………18
2.4.1 奇異元素……………………………………………....20
2.4.2 位移外插法…………………………………………....21
第三章 程式驗證與問題描述………………………………23
3.1 程式的驗證…………………………………………………23
3.1.1 『ANSYS』的驗證………………………………….....23
3.1.2 含正確解範例的驗證………………………………....25
3.2 收斂性與靜態縮減的探討…………………………………29
3.2.1 超收斂點的位置……………………………………....29
3.2.2 求解時間的比較……………………………………....31
3.3 問題的定義與分析的目標…………………………………33
第四章 數值結果與討論……………………………………36
4.1 奇異元素對誤差的影響……………………………………36
4.1.1 程式之結果…………………………………………....36
4.1.2 與『ANSYS』的比較……………………………….....41
4.1.3 三角形元素的影響…………………………………....42
4.2 高階元素對誤差的影響……………………………………44
4.2.1 二階元素……………………………………………....45
4.2.2 三階元素……………………………………………....46
4.2.3 四階元素……………………………………………....46
4.2.4 五階元素……………………………………………....47
4.2.5 六階元素……………………………………………....48
4.2.6 七階元素……………………………………………....48
4.2.7 八階元素……………………………………………....49
4.2.8 三角形元素的影響…………………………………....50
4.3 高階元素的優勢……………………………………………60
第五章 結論與建議…………………………………………62
參考文獻…………………………………………………………64
自述………………………………………………………………67

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