One Novel Superconvergent Scheme And Adaptive Analysis Of 1D FEM

As an approximation method,the finite element method（FEM）face with the contradiction between accuracy and computational cost,in order to solve this problem,the finite element super-convergence calculation and adaptive analysis emerged and have become a research focus of finite element analysis in recent years.It is well know that the accuracy of finite element nodal displacements are the most accurate,the accuracy of the finite element internal displacements are poorer,and the accuracy of the finite element stresses are the poorest.In order to improve the accuracy of finite element stresses and internal displacements,a novel super-convergence algorithm was proposed in this dissertation.The super-convergence displacements provided by the method have the same accuracy and convergence rate of the finite element nodal displacements,and the convergence rate of the recovered stresses provided by the method are at least one order higher than that of the finite element stresses.A similar approach:p-type super-convergence method was extended in this dissertation which can also produce super-convergent displacements and stresses.A novel meshing strategy was proposed in this dissertation,combined with error estimation in maximum norm provided by super-convergent solutions,adaptive analysis can be conducted.The main work of this dissertation is as follows:Firstly,taking one-dimensional C ⁰ self-adjoint problem as model problem,a new post-processing super-convergence algorithm:subdivided element method was proposed.Based on the super-convergence of finite element nodal displacements,the method subdivide an element into two sub-elements and then local finite element computation is carried out.The method consists of two formats:direct format and indirect format,the direct format recover displacements directly while the indirect format compute displacement errors firstly and then are added to the finite element displacements,both two formats produce super-convergent displacements and stresses.Secondly,two formats of the subdivided element method were extended to other one-dimensional finite element problems,including one-dimensional C ¹ problem,Timoshenko beam problem and Galerkin problem.The subdivided element method also show great super-convergence performance on these problems:the recovered displacements have the same convergence rate and accuracy with the nodal displacements while the convergence rate of the recovered stresses is at least one order higher than that of the finite element stresses.These reserach laid a solid foundation for application of the subdivided element method to other one-dimensional problems.Thirdly,based on the outstanding performance of the subdivided element method,a similar approach:p-type super-convergence method was extended in this dissertation.The p-type super-convergence method was proposed by Douglas and Dupont,which is also based on the super-convergence of nodal displacements and local refinement,the difference of local refinement between this method and subdivided element method is that this method improve the order of element rather than subdivide the element.The p-type super-convergence method was extended to C ¹ problem and Timoshenko beam problem in this dissertation,which also showed great super-convergence performance.Fourthly,based on pointwise super-convergence displacements and stresses provided by the above two super-convergence method,the errors of finite element displacements and stresses can be estimated in maximum norm.A new meshing strategy was proposed in this dissertation,which can get the final mesh efficiently.Based on reliable error estimation and efficient meshing strategy,taking one-dimensional C ⁰problem as model problem,displacement and stress adaptive analysis were conducted in this dissertation.A large number of numerical experiments show that the super-convergence methods in this dissertation can produce super-convergent displacements and stresses at any point of the solution domain with a small amount of calculation and the adaptive scheme in this dissertation can provide a proper nonuniform mesh with the solutions point-wisely satisfying the user specified error tolerance in maximum norm.