Virtual Element Method For Generalized Membrane Shell

Elastic shell theory,a significant branch of elastic theory,is widely applied in civil construction,petrochemistry,nuclear industry and other national key engineering fields.There are many scholars and institutions all over the world dedicated to the research of elastic shell models,among which the most classic one is Koiter's model.Based on this model,ciarlet classifies the shell model as elliptic membrane shell,generalized membrane shell and flexural shell.At present,finite element method(FEM)is the most commonly numerical method for discretizing elastic shell models.As an extension of FEM on polygon or polyhedron mesh,virtual elem ent method(VEM)has the advantages of mesh-processing flexibly,high regularity and arbitrary order convergence.In summary,this work intends to study VEM of generalized membrane shell model,which is also the first time to utilize VEM to discretize elastic shell model.The main research points are as follows:(1)Determining the projection operators.In the VEM,the key is to construct operators ?? and operators ?0 related to the bilinear form of the problem.Where ?? is the Ritz projection from the virtual function space to the polynomial space,which deals with the test function and the test function in bilinear form uniformly;?0 is the L2 projection of the polynomial space,which process the non-polynomial part of the space uniformly;for the non-polynomial part of the space,the bilinear form is used to maintain the stability of the original PDE,so as to ensure the suitability of the final matrix form of the problem.The establishment and calculation process of gradient operator are showed in chapter 3.(2)Spatial discretization of generalized membrane shell model by using the VEM.This part starts with an introduction of classification of shell model.Furthermore,virtual element discretization of generalized membrane shell model is performed by mean of constructing operators ?? and no related to the bilinear form of the problem,and the existence,uniqueness and convergence of the solution of the virtual element discrete scheme are proved.(3)Numerical experiments.We use VEM to carry out numerical experiments for the special shells such as conical shell,hyperbolic shell and cylindrical shell.That is,the displacem ent distribution of conical shell,hyperbolic shell and cylindrical shell under different meshes are rovided by triangulation and rectangular subdivison.In addition,the extreme value of the three components of the displacement under different meshes are rovided by triangulation and rectangular subdivison.Our numerical results verify the effectiveness,stability and convergence of the VEM discrete scheme.Our research will provide a fast and effective numerical method for analyzing shell model,and further promote the application of shell components in engineering fields.