Virtual Element Methods Of Linearly Elastic Clamped Plate Model And Shallow Shell Model

Both the clamped plate and shallow shell are common elastic structures in engineering design.Clamped plate is a special plate with fixed boundary part,while shallow shell is a special shell with large span.These two aforementioned structures possess robust load-bearing capacity and exhibit significant economic advantages,and therefore see extensive application in various engineering domains,including but not limited to bridge construction,road planning,ship design,and machinery manufacturing.Therefore,the study of numerical algorithm of plate and shell model is of great significance and value to solve practical problems in modern engineering technology.This paper is based on the two-dimensional linear elastic clamped plate model and shallow shell model proposed by Ciarlet.Conforming virtual element method is used for discretization,and theoretical analysis and numerical experiments are carried out.The main research contents include two parts:(1)The conforming virtual element method of linear elastic clamped plate model is proposed for the first time in this paper.Firstly,according to the different regularity of three unknown displacement components of displacement vector field,the local vector virtual element space satisfying the continuity requirement is constructed.Then the corresponding degree of freedom function is defined for the virtual space and the projection operator satisfies the orthogonal relation from the virtual space to the polynomial space.Then the numerical discretization of the two dimensional linear elastic clamping plate model is carried out using the projection operator and the existence,uniqueness and convergence of the discretization solutions are proved.In addition,the corresponding error estimates are given.Finally,the numerical experiments of circular plate and rectangular plate prove that the numerical method is convergent and stable.(2)The conforming virtual element method of linear elastic shallow shell model is proposed for the first time in this paper.When the neutral surface of the clamping plate model is deformed into a large span curved surface,the clamped plate is upgraded to shallow shell.Since the shallow shell model is a shell problem with variable coefficients,the computability of each operator cannot be guaranteed when the traditional H1 and H2 virtual projection operators are discretized.To ensure that the projection operators can be computed with minimal degrees of freedom,an augmented virtual element discrete space is formulated,and a corresponding virtual element discrete scheme for the model is presented.In addition,we prove the existence,uniqueness and convergence of the discrete solution,and give a general formula for error estimation.Finally,numerical experiments on cylindrical and saddle surfaces with different mesh sections are carried out to verify that the numerical method is stable and effective.