Convergence and Precision of the Dual-Variable Brick Mixed Element and Its Displacement Element

QING Guang-hui; LIU Yan-hong

doi:10.21656/1000-0887.370089

Volume 38 Issue 2

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2017 > 38(2): 153-162

QING Guang-hui, LIU Yan-hong. Convergence and Precision of the Dual-Variable Brick Mixed Element and Its Displacement Element[J]. Applied Mathematics and Mechanics, 2017, 38(2): 153-162. doi: 10.21656/1000-0887.370089

Citation:

PDF( 826 KB)

Convergence and Precision of the Dual-Variable Brick Mixed Element and Its Displacement Element

doi: 10.21656/1000-0887.370089

College of Aeronautical Engineering, Civil Aviation University of China,Tianjin 300300, P.R.China

Funds: The National Science Fund for Young Scholars of China(11502286)

Received Date: 2016-03-28
Rev Recd Date: 2016-10-19
Publish Date: 2017-02-15

Abstract

Abstract

The symplectic characteristics of the equivalent stiffness coefficient matrix for the Hamiltonian canonical equations of elasticity and the Hamiltonian mixed element were intuitive, and the symplectic characteristics of the equivalent stiffness coefficient matrix for the dual-variable brick mixed element (DVBME), which was derived based on the Hellinger-Reissner (H-R) variational principle and the symplectic-conservative theory of elasticity, were similarly intuitive. The governing equations of elasticity were established immediately through the DVBME formulation, and the solution of the governing equations was obtained with the mixed method. Meanwhile, the dual-variable brick displacement element (DVBDE) formulation was deduced from the DVBME formulation, which was only related to displacement variables. The solution of the governing equations based on the DVBDE formulation was got with the displacement method. The numerical examples show that the convergence rates of displacement and stress variables of the 8-node DVBDE with reduced integration are balanced and stable with high precision. The convergence rate of stress of the DVBDE is almost equal to that of the translational 20-node displacement element with reduced integration. The DVBDE is universal.
- dual variable,
- Hellinger-Reissner variational principle,
- dual-variable brick mixed element,
- dual-variable brick displacement element

FullText(HTML)

References(15)

References

[1]	钟万勰. 弹性力学求解新体系[M]. 大连: 大连理工大学出版社, 1995.(ZHONG Wan-xie. A New Systematic Methodology for Theory of Elasticity [M]. Dalian: Dalian University of Technology Press, 1995.(in Chinese))
[2]	钟万勰. 应用力学对偶体系[M]. 北京: 科学出版社, 2002.(ZHONG Wan-xie. Duality System of Applied Mechanics [M]. Beijing: Science Press, 2002.(in Chinese))
[3]	钟万勰. 应用力学的辛数学方法[M]. 北京: 高等教育出版社, 2006.(ZHONG Wan-xie. Symplectic Solution Methodology in Applied Mechanics [M]. Beijing: Higher Education Press, 2006.(in Chinese))
[4]	唐立民, 褚致中, 邹贵平, 等. 混合状态Hamiltonian元的半解析解和叠层板的计算[J]. 计算力学学报, 1992,9(4): 347-360.(TANG Li-min, CHU Zhi-zhong, ZOU Gui-ping, et al. The semi-analytical solution of mixed state Hamiltonian element and the computation of laminated plates[J]. Chinese Journal of Computational Mechanics,1992,9(4): 347-360.(in Chinese))
[5]	ZOU Gui-ping, TANG Li-min. A semi-analytical solution for laminated composite plates in Hamiltonian system[J]. Computer Methods in Applied Mechanics and Engineering,1995,128(3/4): 395-404.
[6]	卿光辉, 邱家俊, 刘艳红. 磁电弹性体修正后的H-R混合变分原理和状态向量方程[J]. 应用数学和力学, 2005,26(6): 665-670.(QING Guang-hui, QIU Jia-jun, LIU Yan-hong. Modified H-R mixed variational principle for magnetoelectroelastic bolids and state-vector equation[J]. Applied Mathematics and Mechanics,2005,26(6): 665-670.(in Chinese))
[7]	QING Guang-hui, QIU Jia-jun, LIU Yan-hong. Free vibration analysis of stiffened laminated plates[J]. International Journal of Solids and Structures,2006,43(6): 1357-1371.
[8]	QING Guang-hui, QIU Jia-jun, LIU Yan-hong. A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches[J]. International Journal of Solids and Structures,2006,43(6): 1388-1403.
[9]	卿光辉, 徐建新, 邱家俊. 考虑阻尼的状态向量方程和层合板的振动分析[J]. 应用数学和力学, 2007,28(2): 231-237.(QING Guang-hui, XU Jian-xin, QIU Jia-jun. State-vector equation with damping and vibration analysis of laminates[J]. Applied Mathematics and Mechanics,2007,28(2): 231-237.(in Chinese))
[10]	Ding H J, Chen W Q, Xu R Q. New state space formulations for transversely isotropic piezoelasticity with application[J]. Mechanics Research Communicatio ns, 2003,27(3): 319-326.
[11]	田宗漱, 卞学.多变量变分原理与多变量有限元方法[M]. 北京: 科学出版社, 2011.(TIAN Zhong-shu, PIAN Theodore H H. Variational Principle With Multi-Variables and Finite Element With Multi-Variables [M]. Beijing: Science Press, 2011.(in Chinese))
[12]	钟万勰. 力、功、能量与辛数学[M]. 大连: 大连理工大学出版社, 2007.(ZHONG Wan-xie. Force, Work, Energy and Simplectic Mathematics [M]. Dalian: Dalian University of Technology Press, 2007.(in Chinese))
[13]	PIAN Theodore H H. Derivation of element stiffness matrices by assumed stress distribution[J]. AIAA Journal,1964,2(7): 1333-1336.
[14]	卞学.杂交应力有限元法的研究进展[J]. 力学进展, 2001,31(3): 344-349.(PIAN Theodore H H. Recent advances in hybrid stress finite element methods[J]. Advances in Mechanics,2001,31(3): 344-349.(in Chinese))
[15]	范家让. 强厚叠层板壳的精确理论[M]. 北京: 科学出版社, 1996.(FAN Jia-rang. Exact Theory of Laminated Thick Plates and Shells [M]. Beijing: Science Press, 1996.(in Chinese))