用有限元广义混合法分析不可压缩或几乎不可压缩弹性体

Application of Generalized Mixed Finite Element Method to Research on Incompressible and Nearly Incompressible Elasticity

不可压缩或几乎不可压缩问题在数学上表现为最小势能原理中的某些项趋于无穷大,使得有限元方程产生病态。本文给出了不可压缩或几乎不可压缩弹性分析的广义混合变分原理,以此为基础建立了该类问题的有限元广义混合法。该变分原理的泛函中不含有上面这种奇异项,故其有限元方程不会产生病态。算例表明该有限元法可以同时进行可压缩、不可压缩或几乎不可压缩弹性分析,且精度良好;有限元常规位移法及Herrmann法是该法的特例。

The characteristic of the incompressible or nearly incompressible problems is that some items in the functional of minimum potential energy variational principle trend to infinity, it makes the displacement finite element model ill-conditioned. In this paper, generalized mixed variational principle for incompressible or nearly incompressible elasticity is introduced, and the generalized mixed finite element method (GMFEM) is established. There are no singular items in the new variational principle, so ill-conditioned problem will not happen. The numerical examples show that the GMFEM could be used in the analysis of the incompressible and nearly incompressible elasticity, and the precision of the results is very higher. The minimum potential energy variational principle and L.R. Herrmann s variational principle are the special cases of the generalized mixed variational principle.