Mixed Finite Element Method Based On Two Variational Principles

Finite Element Method(FEM)is a very effective numerical method in computational mechanics.It has a very important role and status in the field of modern engineering applications.However,there are still some problems in the existing conventional finite element method.In view of problem that the low stress results of displacement method,The research work of this article mainly includes the following three parts:First,the basic equations of elastic mechanics,the minimum potential energy variational principle,the H-R variational principle and the finite element stiffness equation of the displacement method based on the principle of minimum potential energy are introduced.Meanwhile,the three-dimensional modified H-R variational principle is derived.Secondly,through the analysis of the minimum potential energy principle and the the H-R variational principle,the basic equations of the combined finite element are obtained by the combined application of the two groups.So the essence of the joint finite element is no longer the finite element method of the single variational principle,but the displacement method based on the minimum potential energy principle is first to find the result of the displacement variable,and then the stress is solved by the finite element equation between the displacement and the stress variables based on the H-R variational principle.The numerical examples verify the correctness of the method.Further,a nonconforming finite element formulation for two-dimensional and three-dimensional problems is established by the theory of incompatible displacement finite element method.Numerical examples further improve the accuracy of the calculation and verify the correctness of the joint finite element method.The results of each calculation in this paper can be used as a reference solution for other numerical methods.