Three Kinds Of Thermal Elastomers Based On Generalized Mixed Method And Symplectic Finite Element Method

The current finite element method mainly includes displacement method,stress method and hybrid stress method,as well as the mixed method with stress and displacement as basic variables.For the displacement method,the accuracy of the stress result cannot reach the requirement without special treatment.The stress of mixed method is more accurate than the displacement method,but because the control equation is non-positive,the finite element result is oscillate and its stability is poor.Therefore,it is meaningful to study a method that satisfies the requirements of high precision,stability and fast convergence.In this paper,based on H-R variational principle and minimum potential energy principle,combined with dual theory and heat transfer theory,two finite element methods that named generalized mixed finite element method and symplectic finite element method,for three types of thermal elastomers were established,and mathematica program was used to realize them.Moreover,two classes of the finite element method is applied to practical engineering,and the different models of three kinds of elastomers(thermal elastomer,piezoelectric thermal elastomer and magnetoelectronic thermal elastomer)are analyzed,and the finite element solutions under different conditions are compared with the exact solution and the numerical solutions of commercial software to verify the accuracy of the results and the advanced and applicable of the method.By analyzing different examples,compared the exact solution of laminated plate and shell theory and numerical solution of the literature,and the stress results obtained by the 8-node uncoordinated generalized mixed method and the 8-node uncoordinated symplectic method and the 20-node symplectic method are verified to be higher precision,better stability and the results is converge to the exact solution when the mesh is thinner.The innovation of this paper is that compared with other finite element methods,the numerical oscillation of mixed finite element method is solved under the requirement of high precision of finite element results.Combining the advantages of displacement method and mixed method,the two variational principles are cleverly used to eliminate zero elements on the main diagonal of coefficient matrix.Under the premise of ensuring the accuracy of the numerical results,the stress results are stable and the convergence is fast.As the standard displacement finite element method,it also has wide applicability and can be easily extended to nonlinear problems.On the other hand,it provides good theoretical support for solving complex engineering problems,such as creep,crack propagation,stress concentration of hole and damage.