Algorithm of Super-Convergent in Two-Dimensional Finite Element of Lines Based on Improved Displacement Mode
-
摘要: 提出了基于改进位移模式的二维有限元线法超收敛算法.利用单元内部需满足平衡方程的条件,推导了超收敛计算的解析公式的显式,即将高阶有限元线法解的位移模式用常规有限元线法解的位移模式表示.用常规有限元线法解的位移模式与高阶有限元线法解的位移模式之和构造新的位移模式,基于线性形函数,采用变分形式推导了有限元线法求解的修正的常微分方程组.该算法在前和后处理同时使用超收敛计算公式,在原有试函数的基础上,增加了高阶试函数.使得单元内平衡方程的残差减少,从而达到提高精度的目标.对于二维Poisson方程问题,给出了有代表性的算例,结点和单元内的位移、导数的收敛精度得到了极大的提高.Abstract: Algorithm of super-convergent in two-dimensional finite element of lines (FEMOL) based on improved displacement mode is presented. An explicit analytical formula of super-convergent calculating was derived with the conditions of equilibrium equations stuictly met within the element, of which the displacement mode of high-order finite element of lines solution was expressed with that of a conventional finite element of lines solution. The new displacement mode was constructed with the sum of the displacement mode of conventional finite element of lines solution and that of high-order finite element of lines solution. Based on the linear shape function, the improved ordinary differential equations for FEMOL solution were derived in the rariation form. The super-convergent formula was used for this algorithm in both the pre-processing and post-processing to improve the accuracy of the solution and reduce the residual of balance equation, with the higher-order trial function added the original trial function. A calculation example is presented for Poisson’s equation of a two-dimensional problem, the convergence accuracy of the displacement and derivative at nodes and in elements is greatly improved.
-
Key words:
- FEMOL /
- two-dimensional problem /
- pre-processing /
- displacement mode /
- Poisson’s equation /
- super-convergence
-
[1] Zienkiewicz O C, Zhu J Z. The super-convergent patch recovery and a posteriori error estimates—partⅠ: the recovery technique[J].International Journal for Numerical Methods in Engineering,1992, 33(7):1331-1364. [2] 陈传淼. 有限元超收敛构造理论[M]. 长沙: 湖南科学技术出版社, 2002.(CHEN Chuan-miao. Structure Theory of Super Convergence of Finite Elements [M]. Changsha: Hunan Science and Technology Press, 2002. (in Chinese)) [3] 袁驷. 从矩阵位移法看有限元应力精度的损失与恢复[J].力学与实践, 1998, 20(4):1-6.(YUAN Si. The loss and recovery of stress accuracy in FEM as seen from matrix displacement method[J].Mechanics and Practice,1998, 20(4): 1-6. (in Chinese)) [4] 袁驷, 王枚. 一维有限元后处理超收敛解答计算的EEP法[J].工程力学, 2004, 21(2):1-9.(YUAN Si, WANG Mei. An element-energy-projection method for post-computation of super-convergent solutions in one-dimensional FEM[J].Engineering Mechanics, 2004, 21(2): 1-9. (in Chinese)) [5] 袁驷, 王枚, 和雪峰. 一维C-1有限元超收敛解答计算的EEP法[J].工程力学, 2006 , 23(2):1-9.(YUAN Si, WANG Mei, HE Xue-feng. Computation of super-convergent solutions in one-dimensional C-1 FEM by EEP method[J].Engineering Mechanics,2006, 23(2):1-9. (in Chinese)) [6] 王玫, 袁驷. Timoshenko梁单元超收敛结点应力的EEP法计算[J]. 应用数学和力学, 2004 , 25(11):1124-1134.(WANG Mei, YUAN Si. Computation of super-convergent nodal sresses of Timoshenko beam elements by EEP method[J].Applied Mathematics and Mechanics,2004, 25(11):1124-1134.(in Chinese)) [7] 袁驷, 林永静. 二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法[J]. 计算力学学报, 2007, 24(2):142-147.(YUAN Si, LIN Yong-jing. An EEP method for post-computation of super-convergent solutions in one-dimensional Galerkin FEM for second order non-self-adjoint boundary value problem[J].Chinese Journal of Computational Mechanics,2007, 24(2): 142-147. (in Chinese)) [8] 袁驷, 王枚, 王旭. 二维有限元线法超收敛解答计算的EEP法[J]. 工程力学, 2007, 24(1): 1-10.(YUAN Si, WANG Mei, WANG Xu. An element-energy-projection method for super-convergence solutions in two-dimensional finite element method of lines[J].Engineering Mechanics,2007, 24(1): 1-10. (in Chinese)) [9] 唐义军, 罗建辉.基于改进位移模式的一维C-1有限元超收敛算法[J].计算力学学报, 2012, 29(5):721-725.(TANG Yi-jun, LUO Jian-hui. Algorithm of super-convergent in one-dimensional C-1 FEM based on improved displacement mode[J].Chinese Journal of Computational Mechanics,2012, 29(5):721-725. (in Chinese)) [10] 袁驷. 有限元线法[J]. 数值计算与计算机应用, 1992, 13(4): 252-260.(YUAN Si. The finite element method of lines[J].Journal on Numerical Methods and Computer Applications,1992, 13(4): 252-260. (in Chinese)) [11] 唐义军, 罗建辉.基于改进位移模式的一维有限元超收敛算法[J].计算力学学报, 2012, 29(6):954-959.(TANG Yi-jun, LUO Jian-hui. Algorithm of super-convergent in one-dimensional FEM based on improved displacement mode[J].Chinese Journal of Computational Mechanics,2012, 29(6):954-959. (in Chinese)) [12] LUO Jian-hui, LONG Yu-qiu, LIU Guang-dong. A new orthogonality relationship for orthotropic thin plate theory and its variational principle[J].Science in China Series G-Physics and Astronomy,2005, 38(3): 371-380.
点击查看大图
计量
- 文章访问数: 1613
- HTML全文浏览量: 133
- PDF下载量: 1216
- 被引次数: 0