研究生: |
陳建琦 Chien-Chi Chen |
---|---|
論文名稱: |
二維六點應變有限元素法之推導 The Derivation of the 6-node Strain Finite Element Method for the two Dimensional Problems |
指導教授: |
潘誠平
Cheng-ping Pan |
口試委員: |
施俊揚
Jun-Yang Shi 郭世榮 Shyh-Rong Kuo |
學位類別: |
碩士 Master |
系所名稱: |
工程學院 - 營建工程系 Department of Civil and Construction Engineering |
論文出版年: | 2021 |
畢業學年度: | 109 |
語文別: | 中文 |
論文頁數: | 141 |
中文關鍵詞: | 應變有限元素法 、應變諧和 、插值函數 、LST 、εFEM3元素 、εFEM6元素 |
外文關鍵詞: | strain finite element method, strain compatibility, interpolation function, LST, εFEM3 element, εFEM6 element |
相關次數: | 點閱:205 下載:0 |
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傳統有限元素法通常係以位移作為基本未知數,而其在求解應變時需透過微分降次,因此導致應變場的結果較為不精確。而本研究係採用以應變作為基本未知數來計算,目的在於將應變積分後得到的位移場較傳統位移場之函數高階,因此能獲得較精確之結果,而位移係基於應變積分而得到的結果,因此最後得到的位移場與應變場必然是諧和的。一般微分是增加數值誤差的過程,而積分則是反向使得數值誤差變小。另外,此法可使得套裝軟體最愛用的三角形元素獲得更高的精確度。
應變有限元素法以三角形元素為基礎,以往取三個位於角點的節點,在二維平面應力懸臂梁的案例上單排劃分的情況下有良好的收斂性,而雙排劃分時的收斂性反而較為不佳。本研究係以取六個節點的三角形元素為基礎,在二維平面應力懸臂梁的分析結果相當滿意,也未出現雙排劃分位移收斂性趨於不佳的情況。
The traditional finite element method usually takes displacements as the basic unknowns, and it needs to reduce the order through differentiation when solving the strains. Therefore, the result of the strain field is less accurate. In this study, strains are used as the basic unknowns. The displacement field is obtained from the integration of strain field. Therefore, the displacement function is in a higher order function compared to the traditional method. More accurate results can be obtained. The displacement is based on the result obtained by strain integration. Therefore, the final displacement field and strain field must be compatible. Generally, differentiation is the process of increasing the numerical error, while integration is the reverse to make the numerical error smaller. In addition, this method can make the favorite triangular element obtain higher accuracy.
The strain finite element method is based on triangular elements. In the past, three nodes located at the corner points were used. In the case of a two-dimensional plane stress cantilever beam, there is good convergence in the case of single-row division, but the convergence in double-row division is rather poor. This study is based on the triangular elements of six nodes. The analysis results of the two-dimensional plane stress cantilever beam are quite satisfactory, and the convergence of the double-row division is still pretty good.
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有限元素法理論推導與驗證」,國立台灣科技大學營建工程研究所碩士論文,潘誠平指導,2018。
【5】 楊孟軒,「一維及二維應變有限元素法之推導」,國立台灣科技大學營建工程研究所碩士論文,潘誠平指導,2020。
【6】 張昱賢,「二維應變有限元素法理論推導與程式實踐」,國立台灣科技大學營建工程研究所碩士論文,潘誠平指導,2020。
【7】 Bo-nan Jiang and Jie Wu. The least-squares finite element in elasticity. Part I: Plane stress or strain with drilling degrees of freedom.
【8】 Bo-nan Jiang. The least-squares finite element method in elasticity. Part II: Bending of thin plates.
【9】 歐陽,「材料力學論衡」,2016。