Finite Element Methods Of Linearly Elastic Shallow Shell And Plate Models

As a part of the elastic shell,plates and shallow shells have wide application prospects in civil,aviation and mechanical engineering fields.Based on the search of the existing literature,there is no numerical method on the linearly elastic clamped plate and shallow shell model proposed by Ciarlet in the 1990s.Therefore,effective numerical methods are proposed for clamped plate and shallow shell model in this paper,which provides feasible optimization algorithms for the research of shell structure.The main contents of the research work are as follows:(1)For the two-dimensional linearly elastic clamped plate model,the conforming finite element method is proposed for the first time.Based on the different regularities of the three displacement components,linear Lagrange element is used to discretize the first two components of the displacement,Hsieh-Clough-Tocher(HCT)element is used to discrete the third component.Furthermore,the existence and uniqueness of the numerical solution and the convergence independent of the mesh step size are proved.Finally,numerical experiment is performed to circular plate,and the stability of the numerical method as well as the effectiveness of the model are further validated by the experimental results.(2)For the two-dimensional linearly elastic clamped plate model,the nonconforming finite element method is proposed for the first time.The first two components of displacement are discretized by linear Lagrange elements,and the third component is discretized by Morley elements.After that,the existence and uniqueness of numerical solution under nonconforming elements are analyzed,and the prior error estimation of the the numerical solution is studied according to Aubin-Nitsche lemma.Finally,the rectangular plate and circular plate are simulated to verify the accuracy of theoretical analysis.(3)For the two-dimensional linearly elastic shallow shell model,conforming finite element is firstly proposed to approximate the three components of displacement.The existence,uniqueness and convergence of numerical solution are analyzed.Finally,the deformation process of displacement after applying load is simulated by the paraboloids,which verifies the numerical algorithm is stable and effective.(4)For the two-dimensional linearly elastic shallow shell model,a general nonconforming finite element is proposed to discretize displacement component.Specifically,linear P₁ triangle and bilinear P₁ elelment are used to discretize the first two components of displacement,Morley triangle,Zienkiewicz triangle,Fraeijs de Veubeke triangle,Specht triangle,rectangle Morley and ACM element to discretize the third component.Then,the existence and uniqueness of numerical solution are also analyzed,and it is proved that they have the same convergence rate under the energy norm.Finally,numerical experiments are carried out with paraboloid,spherical dome and cylindrical surface.The results describe the deformation under force well.In addition,numerical experiment also show that the computational complexity of the discrete shallow shell model is apparently lower than that of the discrete Koiter shell.