Adaptive Analysis Of 2D FEM With Local Mesh Refinement Based On EEP Technique

Adaptive Finite Element Method(AFEM)is a research focus in the modern computational mechanics,in which the reliable and efficient AFEM with local mesh refinement is particularly challenging.The Element Energy Projection(EEP)method is a general and powerful postprocessing method for super-convergence computation which can provide reliable error estimation for adaptive analysis.The EEP method can be directly applied to onedimension(1D)Finite Element Method(FEM)and two-dimension(2D)FEM of Lines(FEMOL).For 2D FEM,the EEP method can also be implemented using a global D-byD(Dimension by Dimension)recovery strategy with the meshes being required to be the so-called quasi-FEMOL meshes(with continuous lines throughout the domain).AFEM based on global D-by-D recovery strategy is not sufficiently effective when solving stress concentration and singularity problems because local mesh refinement is difficult to be executed.In this dissertation,adaptive FEMOL with local mesh refinement is realized firstly.Then a new super-convergent strategy is proposed and subsequent AFEM with local mesh refinement is successfully achieved with a novel combined use of the theory of EEP technique and the global D-by-D recovery strategy.The main works of this dissertation are as follows.(1)Adaptive FEMOL with local mesh refinement is realized.Boundary conditions on connected interface sides for FEMOL solutions are derived firstly.Then EEP solutions on connected interface sides are established.Local mesh refinement by inserting both nodal lines and interface sides are proposed at last.These achievements also lay the foundation for AFEM.(2)Element D-by-D recovery strategy for super-convergent calculation of 2D FEM is developed.The meshes are required to be of quasi-FEMOL type when implementing global D-by-D recovery strategy because EEP formulae for Ordinary Differential Equations of FEMOL is used.The new strategy is constructed almost on a single element with EEP formulae for FEMOL being flexibly and reasonably used twice,which reduces the requirement of mesh structures.(3)Element D-by-D recovery on local refined mesh is constructed by defining implicit elements to overcome the absence of element strips and using adjoining elements to recover displacement on hanging points which do not gain super-convergence.In this way,Element D-by-D recovery overcomes the restriction from quasi-FEMOL meshes.(4)Adaptive 2D FEM with local mesh refinement is achieved for various types of stress concentration and singularity problems of 2D Poisson Equation,plane elasticity problem,moderately thick plate bending problem and 3D axisymmetric problem.A large number of numerical examples show that the proposed adaptive analysis can produce the desired displacement solutions of FEM which satisfy the user-specified tolerances in maximum norm with an almost optimal adaptive convergence rate.Moreover,the adaptively-generated meshes can reasonably reflect the local difficulties inherent in the physical problems.