Research On Superconvergent Recovery Algorithm For FE Analysis Of Free Vibration Of Planar Curved Beams And Shells Of Revolution

The natural frequency and vibration mode reflect the dynamic properties of structures,which are the basis of structural dynamic analysis.It is of high theoretical significance and research value to solve them efficiently and accurately.In the recent structural free vibration analysis,the displacement based finite element method is widely used for its easy implementation and convenient programming.However,in order to obtain satisfactory accuracy,the traditional finite element method requires refining the mesh or increasing the degree of elements when faced with a complex structure or highorder modes,and the amount of computation increases rapidly.It is hard to achieve high accuracy and efficiency at the same time.The theories and a large number of numerical experiments of finite element method have proved that the stresses or displacements on some points in the element have higher accuracy and convergence rate than on other points.The superconvergence recovery based on these points becomes an efficient method to improve the accuracy of the finite element solution.In free vibration problem,the p-type superconvergent recovery method is based on the superconvergence properties on frequencies and nodal displacements in modes and sets up a linear ordinary differential boundary value problem(BVP)which approximately governs the mode on each element.This linear BVP is solved by using a higher order element from which the mode on each element is recovered.Then by substituting the recovered mode into the Rayleigh quotient,the frequency is recovered.The method has been successfully applied in the static analysis of one-dimensional variable-section beam,planar curved beam,axisymmetric shell,thin-walled curved beam,one-dimensional Galerkin problem and free vibration analysis of beams and bars.This paper will extend this method into the free vibration of planar curved beams and axisymmetric shells.Planar curved beams and axisymmetric shells are widely used in engineering structures.Due to the existence of curvature,multi-directional vibrations are coupled and the vibration modes are complex.This paper presents the finite element solution schemes for the free vibration analysis of Timoshenko planar curved beams,Euler planar curved beams,moderately thick axisymmetric shells and thin axisymmetric shells and analyses the error characteristics and the convergence of their frequencies and modes.In order to improve the accuracy of finite element solutions,this paper extends the p-type superconvergent recovery method into the finite element analysis of the free vibration of these four kinds of structures,then summarizes the convergence order of the recovered vibration modes and recovered frequencies through numerical examples.The concept of this method is clear and the thought is ingenious,and it is easy to be programmed.A large number of numerical tests prove that this method is efficient and reliable.This method can improve the accuracy and quality of the solution significantly with a small amount of finite element post computation.This thesis shows that this method effectively improves the accuracy,quality and convergence order of the frequencies and vibration modes of planar curved beams and axisymmetric shells.It has strong engineering application prospects and is worthy of further exploration and application.