Adaptive Finite Element Analysis Of 2D Eigenvalue Problems Based On EEP Technique

Mathematically converted from various physical problems such as structural free vibration and elastic stability,the eigenvalue problems are important in engineering calculation,and are usually solved by numerical methods such as finite element method(FEM).When adopting FEM,the efficiency and precision of the solution will be greatly determined by the meshes used and designed by the user.Based on the traditional FEM,the adaptive finite element method(AFEM),via the error estimating and mesh refining techniques,repeatedly improves the solution by successive mesh refinements in a selfadaptive way,and finally provides an optimized mesh and solutions that meet the accuracy requirements.The core of AFEM is the effective and computable error estimation.Element energy projection(EEP)method can provide superconvergent solution,which can then serve as reliable error estimation for adaptive analysis.The adaptive analysis based on EEP method has been successfully applied to a wide range of linear and nonlinear problems.In this work,a set of adaptive finite element strategy for 2D eigenvalue problems based on EEP method is proposed and is proven to be well applicable to various eigenvalue problems.The main research works in this dissertation are as follows:1.The basic strategy of finite element adaptive analysis for 2D eigenvalue problems based on EEP method is presented.By intrinsically combining the W-W algorithm,the linearization technique,the EEP superconvergence calculation method and the mesh update techniques based on local or global refinement,a set of solution strategy is established,which can continuously solve several orders with the eigenvalues satisfying required accuracy and the errors in eigenfunctions satisfying the error tolerance measured by the maximum norm.2.A unified superconvergence algorithm for eigenvalue problems is proposed.For various eigenvalue problems,a uniform equivalent load term is formed based on the linearization and iteration schemes,and a generalized EEP superconvergence formula is derived.The mesh pattern of adaptive analysis for eigenvalue problems is extended.The introduction of local refinement algorithm can significantly reduce the degrees of freedom while maintaining the accuracy and improve the flexibility of mesh and the efficiency of analysis.3.The adaptive analysis strategy which only controls errors of eigenvalues is presented.A superconvergent eigenvalue calculation strategy based on EEP method is proposed,and the transformation relationship between global eigenvalue error and local eigenfunction error is established.The eigenvalue error estimation is used to control the shutdown of the algorithm overall,and eigenfunction error estimation is locally used to drive mesh refinement.4.An adaptive analysis algorithm based on local calculation is proposed.Based on the analysis of the solution error distribution in the adaptive iteration,it is proposed and verified by numerical experiments that under the given error tolerance,the local refinement has almost no significant accuracy improvement on the out-of-region finite element solution,and then an improved second derivative recovery strategy of the sidenode solutions is proposed.A set of adaptive analysis algorithm based on local calculation is subsequently formed.