Study On Modified Parameter Of Two Kinds Of Generalized Mixed Elements

Firstly,the previous work on the generalized H-R(Hellinger-Reissner)variational principle and the relevant finite element method is reviewed.Then,through the analysis of the minimum potential energy principle and the H-R variational principle,parameters are introduced to combine them organically,and a more concise generalized H-R variational principle is obtained.No matter based on the classical generalized H-R variational principle or the non-conforming generalized mixed element or the non-conforming generalized symplectic element obtained by the generalized H-R variational principle in this paper,the parameter value has a great influence on the accuracy of the numerical results.The rarer the mesh is,the higher the sensitivity of the numerical results to the parameter value is.Therefore,the determination of the reasonable and reliable parameter value is an unavoidable problem 。In the case of the same mesh division of the model,the data results show that the accuracy of the numerical results can not reach the highest no matter the parameter value selected by the median method or the spectral condition number method.In fact,several numerical examples show that the closer the parameter is to 1.0,the more accurate the numerical results are.The main reason is that the closer the parameter value is to 1.0,the closer the strain energy density is to the residual energy density.In order to further improve the numerical results,the generalized H-R variational principle with modified parameters and the corresponding generalized mixed element and generalized symplectic element are established according to the idea of the ratio of strain energy to complementary energy.Finally,several numerical examples of different types are calculated.The results show that the displacement and stress results of the generalized mixed element and the generalized symplectic element are obviously improved when the strain energy and complementary energy are equal by adjusting the parameters.The applicability and accuracy of the criterion are verified.The main innovation of this paper is the parameter determination criterion with equal strain energy and complementary energy,and the generalized mixed element and generalized symplectic element with a modified parameters.It not only extends the theory of mixed finite element,but also provides a new theoretical basis for the application of mixed element method.