对偶变量块体混合元及其位移元的收敛性和精度分析

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

弹性力学Hamilton正则方程和Hamilton混合元的等效刚度系数矩阵,均具有直观的辛特性.基于H-R变分原理和弹性力学保辛理论建立的对偶变量块体混合元,其等效刚度系数矩阵同样具有直观的辛特性.根据对偶变量块体混合元列式,可直接建立问题的控制方程,进行混合法求解.同时,通过对偶变量块体混合元列式可以导出对偶变量块体位移元列式,建立问题的控制方程后,可先求位移的解.数值实例表明:线性8结点对偶变量块体位移减缩积分元的各力学量的收敛速度均衡、收敛过程稳定、结果精度高,其应力变量的收敛速度与传统的20结点位移协调减缩积分元接近.对偶变量块体位移元具有普适性.

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.