Theoretical Study On Noncompatible Symplectic Finite Element Of Three Elastomers

The displacement finite element method has low computing resource consumption and high displacement accuracy,which is convenient to accomplish analysis of large-scale problem.However,due to the differentiation of displacements,the precision of stresses is low.In composite laminated structure of general engineering,out-plane stress is usually nonlinear along the thickness direction,and its gradient is large.Supposing the displacement method is adopted,in order to obtain the reliable numerical results,the mesh must be divided more finely,and the stress results should be modified by grinding scheme.The mixed finite element method can obtain displacement and stress directly.It has high precision of stress.Nonetheless,the number of field variables increases and more computing resources are needed.In view of the above several problems,the main work of this paper includes the following parts:(1)Based on the principle of minimum potential energy combined with the modified Hellinger-Reissner(H-R)variational principle or the modified generalized H-R variational principle respectively,two noncompatible symplectic isoparametric equations can be established.The same shape function is used to discretize displacement and out-plane stress,which lays a foundation for the introduction of out-plane stress boundary condition.Since the control equation contains only two kinds of variables,displacement and out-plane stress,the space occupied by linear equations is less than mixed method.In-plane stress can be obtained by displacement and out-plane stress.The reliability of two methods is proved by analyzing the eigenvalues and modes of element matrix.Patch test proves the convergence and stability of results.(2)Numerical examples of composite laminate show that noncompatible symplectic elements guarantee the continuity of interlaminar out-plane stress,and the accuracy is high.Therefore,the noncompatible symplectic finite element method has advantages of low computational resource consumption,high precision and good stability.(3)The form of noncompatible symplectic element is relatively simple.It can be easily extended to engineering materials containing multi-physical fields or related smart structures,such as laminated structure of piezoelectric material or electromagnetic material.The theoretical derivation and numerical examples of these aspects further show that the force,electricity and magnetic results have high precision and fast convergence speed,which are calculated by using noncompatible symplectic finite element method contains generalized displacement and generalized out-plane stress.The work provides a simple and reliable way to analyze statics problems of related engineering structure.