Super-Convergence Computation And Adaptive Analysis Of 3D FEM Based On EEP Technique

The importance of super-convergence computation of finite element method(FEM)lies in two aspects.Firstly,super-convergence computation yields solutions with higher accuracy on relatively coarse finite element meshes.Secondly,super-convergent solutions could be used to construct a posteriori error estimators in adaptive finite element analysis,which is the main purpose of this study.The super-convergence technique named Element Energy Projection(EEP)method proves to be highly effective and efficient in finite element analysis of various one-dimensional(1D)and two-dimensional(2D)problems,but a severe challaged was faced in the extension to three-dimensional(3D)problems.In this dissertation,theory and implementation of EEP technique in multi-dimensional problems are re-studied before super-convergence computation of 3D problems is finally realized and primary success in adaptive analysis of 3D FEM is achieved.The main works of this dissertation are as follows.(1)The theoretical framwork of “generalized 1D problems” is established,in which a multi-dimensional problem is treated as a generalized 1D problem.Uniform formulations for different dimensions are derived based on the mature 1D-based EEP method,therefore solving an n-dimensional problem can be equivalently viewed as solving n successive steps of generalized 1D problems.(2)Super-convergence strategy for generalized 1D problem based on EEP technique is proposed,concluding from a re-study of super-convergence schemes for 2D problem.The influence of the input solutions of 1D problem on the output solutions of 2D problem is scrutized,and the associated basic requirements are summarized,based on which the super-convergence algorithm is systematically proposed.Besides,second order derivatives of 2D problems are also computed,which paves the way for the following study of 3D problems.(3)Super-convergence computation of 3D FEM based on EEP technique is realized on irregular meshes.Taking the algorithm in 2D problem as a reference,the algorithm for 3D FEM is systematically proposed,which can deal with Poisson's equation and elasticity problem,and works well on irregular meshes and with various boundary conditions,which have made a breakthrough over the limitation on regular meshes in previous studies.Firstly super-convergent displacements and derivatives in all directions are computed for 3D Poisson's equation on irregular meshes composed of 3D hexahedron elements,and then this algorithm is extended for 3D elasticity problems,where a group of dimension-by-dimension(D-by-D)discretization formulas entirely equivalent to 3D FEM,the associated D-by-D recovery formulas based on EEP technique and their alternative forms for program implementation,are derived,and finally super-convergent displacements and derivatives in all directions for 3D elasticity problem are obtained.(4)Adaptive analysis of 3D FEM is achieved for model problems in this paper.Super-convergent displacements are used to replace exact solutions in a posteriori error estimate for finite element solutions,and error-averaging method on element edges is introduced for element division and mesh refinement,until the maximum norm of errors of finite element solutions on the whole domain satisfy the tolerance prescribed by user,and the final output is the adaptive finite element mesh and solutions.Tested with examples with exact solutions,the adaptive procedures in this paper accomplish the aim of adaptivity by giving solutions point-wisely satisfying the tolerance.These procedures are applicable to 3D Poisson's equation and elasticity problems and are highly practical in engineering computation.