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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。
Incompressible or almost incompressible problems mathematically behave as infinitesimals of some of the principles of minimal potential energy, making finite element equations ill-posed. In this paper, the generalized mixed variational principle for incompressible or incompressible elastic analysis is given. Based on this, the generalized mixed finite element method for this kind of problem is established. The variational principle of functional does not contain the above singular terms, so the finite element equation does not produce pathological. The examples show that the finite element method can perform compressible, incompressible or incompressible elastic analysis at the same time with good accuracy. The conventional finite element method and Herrmann method are special cases.