A New Semi-analytical Method And Symplectic Finite Element Method

As a numerical calculation method for solving partial differential equations,finite element has the advantages of universality,practicality,and ease of application.At present,the finite element method has become an important analytical technique in engineering design and scientific research.This paper mainly did the following work in the research of finite element:?1?The basic equations of elasticity,Hamilton canonical equation and the basic theory of semi-analytical method of Hamilton canonical equation are briefly introduced.When the Semi-analytical method of the Hamilton regular equation is employed to analyze the elastic plate shell problems,it is not affected,it can guarantees the continuity of out-of-plane stress.At the same time,this method also has some limitations,because the Hamilton mixed element is the plane element,as the mesh divisions increase,high-dimensional matrix exponential operation requires more memory and more time consuming.In addition,the semi-analytical method of Hamilton canonical equation cannot deal with some complicated boundary conditions.In order to solve this problems,based on the semi-analytical method of Hamilton canonical equation and the traditional displacement finite element method,a kind of semi-analytical method with high precision and low computational quantity is established.In this method,the displacement variable of the Hamilton regular equation is replaced with by the displacement results of displacement method,so the Hamilton equation is transformed into a state equation containing only the stress term.Numerical examples show that the method is characterized by high precision,low computational load,less computational time and low demand for computer equipment.?2?In order to guarantee the stability of numerical results,the mathematical theory of classical mixed finite element method is relative complicated.However,the generalized mixed method is automatic and stable,and the theory is simple and straightforward.Based on the generalized H-R variational principle,the compatible and noncompatible symplectic elements of two-and three-dimensional problems are proposed.The two main characteristics of symplectic elements are the same C₀ continuous interpolation polynomial functions are used to express the displacement and out-of-plane stress variables.On the other hand,the coefficient matrix of symplectic elements is symmetrical,and the displacement and stress results can be obtained directly from the finite element algebraic system without any additional stable mixing techniques.Numerical examples show that the displacement and stress variables are convergent and stable.The stress results of symplectic elements are almost the same as that of displacement ones,and its accurate is higher for coarse meshes.Compared with the semi-analytical method of the Hamilton regular equation,for the three-dimensional problems,the three coordinate directions of the symplectic element are handled by interpolation polynomial functions,therefore,it has universal applicability,and are not limited by the thickness,uniformity or complexity of the plates and shells,and the complicated boundary conditions.