高精度、高效率及顯式剛度矩陣的八節(jié)點固體殼單元與四節(jié)點膜單元
發(fā)布時間：2018-08-06 09:05
【摘要】：固體殼單元是一種對于具有板殼類拓撲特性的工程結構進行高效有限元分析的新型三維有限元模型,被廣泛地應用于非線性板殼、復合材料層合結構以及金屬薄板沖壓成型等領域。但是目前固體殼單元的研究還遠不夠完善,容易出現(xiàn)各種自鎖現(xiàn)象,因此近年來固體殼單元的開發(fā)也成為國際計算力學界的研究熱點。四節(jié)點膜單元是在考慮面內(nèi)彈性變形問題以及四節(jié)點板殼單元的開發(fā)中應用最廣泛的一種二維有限元模型。由于最早提出的基于位移法的雙線性Q4單元在承受面內(nèi)彎曲的情況時容易出現(xiàn)剪切自鎖現(xiàn)象,幾十年來專家學者們一直致力于精確、高效、可靠的平面四節(jié)點膜單元的開發(fā),這對平面單元列式的理論基礎創(chuàng)新和更加高效地解決工程問題都具有重要的現(xiàn)實意義。鑒于上述兩種單元的研究現(xiàn)狀與應用前景,本文的研究工作主要包括以下內(nèi)容:本文采用擬協(xié)調(diào)元方法推導了一個具有顯式單元剛度矩陣的八節(jié)點固體殼單元。該單元每個節(jié)點僅具有3個位移自由度,共計24個節(jié)點位移參數(shù)。根據(jù)固體殼單元中各應力分量的特點,擬協(xié)調(diào)固體殼單元在單元內(nèi)假設了合理的應變場,從而可有效地避免固體殼單元中容易出現(xiàn)的各種自鎖現(xiàn)象。擬協(xié)調(diào)固體殼單元的另一個顯著優(yōu)點是可以得到顯式單元剛度矩陣,這極大地提高了所得單元的計算效率。此外,本文還采用了基于彈性力學平面問題解析解的位移試探函數(shù)近似單元圍面上的位移場,從而提高所得固體殼單元的計算精度。算例表明,本文所給出的八節(jié)點擬協(xié)調(diào)固體殼單元不僅有效地克服了剪切自鎖,而且擁有很高的計算效率和良好的計算精度。在笛卡爾直角坐標系內(nèi),本文利用擬協(xié)調(diào)元方法構造一個四節(jié)點四邊形平面單元,相應的每個節(jié)點具有兩個位移自由度(屬于Q4類型膜單元)。該精確、高效的四節(jié)點擬協(xié)調(diào)膜單元的假設應變場僅有五個獨立的應變參數(shù),并且考慮了泊松效應的影響;此單元假設應變場還與由平面彈性問題控制方程給出的位移解析解相一致協(xié)調(diào)。此外,本文還給出在1991年提出的另一個基于假設應變場的四節(jié)點膜單元的性能考核。以上兩個四節(jié)點膜單元均沒有任何單元內(nèi)部參數(shù),并且在確定應變參數(shù)時不涉及任何數(shù)值積分,它們的單元剛度矩陣均可在笛卡爾直角坐標系內(nèi)顯式地計算出來。因此,這兩個四節(jié)點擬協(xié)調(diào)膜單元的列式極其簡單,具有非常高的計算效率;同時它們均能夠通過分片試驗,無剪切自鎖。與其他四邊形膜單元的數(shù)值結果對比表明這兩個四節(jié)點擬協(xié)調(diào)四邊形膜單元不僅可靠穩(wěn)定,而且給出的位移和應力結果都非常精確。
[Abstract]:Solid shell element is a new kind of three-dimensional finite element model which is widely used in nonlinear plate and shell, which is used for efficient finite element analysis of engineering structures with the topological characteristics of plates and shells. Composite laminated structure and metal sheet stamping and other fields. However, the research of solid shell element is far from perfect, and it is easy to appear various self-locking phenomena. Therefore, in recent years, the development of solid shell element has become a research hotspot in the field of international computational mechanics. Four-node membrane element is the most widely used two-dimensional finite element model in the consideration of in-plane elastic deformation and the development of four-node plate-shell element. Since the first bilinear Q4 element based on displacement method is prone to shear self-locking when it is subjected to in-plane bending, experts and scholars have been devoting themselves to the development of accurate, efficient and reliable planar four-node film element for decades. This is of great practical significance to the theoretical foundation innovation of plane element formulation and to solving engineering problems more efficiently. In view of the research status and application prospect of the two kinds of elements mentioned above, the research work in this paper mainly includes the following contents: in this paper, an eight-node solid shell element with explicit element stiffness matrix is derived by using the quasi-conforming element method. Each node of the unit has only 3 degrees of freedom and a total of 24 node displacement parameters. According to the characteristics of each stress component in the solid shell element, the reasonable strain field is assumed by the quasi-conforming solid shell element in the element, which can effectively avoid all kinds of self-locking phenomena which are easy to occur in the solid shell element. Another significant advantage of the quasi-conforming solid shell element is that the explicit element stiffness matrix can be obtained, which greatly improves the computational efficiency of the resulting element. In addition, the displacement-heuristic function based on the analytical solution of the plane problem of elasticity is used to approximate the displacement field on the circumplane of the element, so as to improve the calculation accuracy of the obtained solid shell element. The numerical examples show that the proposed eight-node quasi-conforming solid shell element not only overcomes the shear self-locking effectively, but also has high calculation efficiency and good calculation accuracy. In the Cartesian Cartesian Cartesian coordinate system, a quadrilateral plane element with four nodes is constructed by using the quasi-conforming element method. Each node has two degrees of displacement (belonging to the Q4 type membrane element). The assumed strain field of the four-node quasi-conforming membrane element has only five independent strain parameters, and the Poisson effect is taken into account. This element assumes that the strain field is also consistent with the analytical solution of displacement derived from the governing equation of the plane elastic problem. In addition, the performance evaluation of another four-node membrane element based on the hypothetical strain field proposed in 1991 is also presented in this paper. Neither of the above two four-node membrane elements has any internal parameters and no numerical integration is involved in the determination of strain parameters. Their element stiffness matrices can be calculated explicitly in Cartesian Cartesian coordinate system. Therefore, the formulation of these two quasi-conforming membrane elements is extremely simple and highly efficient, and both of them can pass the shearing self-locking experiment. Compared with other quadrilateral membrane elements, the numerical results show that the two quadrilateral quasi-conforming quadrilateral membrane elements are not only reliable and stable, but also the results of displacement and stress are very accurate.
【學位授予單位】：天津大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：TB115
本文編號：2167226
[Abstract]:Solid shell element is a new kind of three-dimensional finite element model which is widely used in nonlinear plate and shell, which is used for efficient finite element analysis of engineering structures with the topological characteristics of plates and shells. Composite laminated structure and metal sheet stamping and other fields. However, the research of solid shell element is far from perfect, and it is easy to appear various self-locking phenomena. Therefore, in recent years, the development of solid shell element has become a research hotspot in the field of international computational mechanics. Four-node membrane element is the most widely used two-dimensional finite element model in the consideration of in-plane elastic deformation and the development of four-node plate-shell element. Since the first bilinear Q4 element based on displacement method is prone to shear self-locking when it is subjected to in-plane bending, experts and scholars have been devoting themselves to the development of accurate, efficient and reliable planar four-node film element for decades. This is of great practical significance to the theoretical foundation innovation of plane element formulation and to solving engineering problems more efficiently. In view of the research status and application prospect of the two kinds of elements mentioned above, the research work in this paper mainly includes the following contents: in this paper, an eight-node solid shell element with explicit element stiffness matrix is derived by using the quasi-conforming element method. Each node of the unit has only 3 degrees of freedom and a total of 24 node displacement parameters. According to the characteristics of each stress component in the solid shell element, the reasonable strain field is assumed by the quasi-conforming solid shell element in the element, which can effectively avoid all kinds of self-locking phenomena which are easy to occur in the solid shell element. Another significant advantage of the quasi-conforming solid shell element is that the explicit element stiffness matrix can be obtained, which greatly improves the computational efficiency of the resulting element. In addition, the displacement-heuristic function based on the analytical solution of the plane problem of elasticity is used to approximate the displacement field on the circumplane of the element, so as to improve the calculation accuracy of the obtained solid shell element. The numerical examples show that the proposed eight-node quasi-conforming solid shell element not only overcomes the shear self-locking effectively, but also has high calculation efficiency and good calculation accuracy. In the Cartesian Cartesian Cartesian coordinate system, a quadrilateral plane element with four nodes is constructed by using the quasi-conforming element method. Each node has two degrees of displacement (belonging to the Q4 type membrane element). The assumed strain field of the four-node quasi-conforming membrane element has only five independent strain parameters, and the Poisson effect is taken into account. This element assumes that the strain field is also consistent with the analytical solution of displacement derived from the governing equation of the plane elastic problem. In addition, the performance evaluation of another four-node membrane element based on the hypothetical strain field proposed in 1991 is also presented in this paper. Neither of the above two four-node membrane elements has any internal parameters and no numerical integration is involved in the determination of strain parameters. Their element stiffness matrices can be calculated explicitly in Cartesian Cartesian coordinate system. Therefore, the formulation of these two quasi-conforming membrane elements is extremely simple and highly efficient, and both of them can pass the shearing self-locking experiment. Compared with other quadrilateral membrane elements, the numerical results show that the two quadrilateral quasi-conforming quadrilateral membrane elements are not only reliable and stable, but also the results of displacement and stress are very accurate.
【學位授予單位】：天津大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：TB115
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