Adaptive Finite Element Analysis Of Variational Inequality Problems Based On EEP Super-convergent Method

Efficient and reliable adaptive finite element analysis(AFEA)is a leading subject of modern finite element method(FEM)research,and it plays a vital role in engineering practice as well as in theoretical analysis.Aiming to solve the free and move boundary problems which are complex and difficult in engineering field,variational inequality problems and adaptive finite element method(AFEM)have been adopted respectively as the mathematical basis and method,and then this thesis has systematically carried out the AFEA of variational inequality problems based on element energy projection(EEP)method.The main work is as follows:(1)Starting with the most basic 1D C~0 variational inequality problems and taking the elastic string contact problem as a model,a new approach of AFEM techniques and algorithms has been developed,while the numerical examples prove its efficient,reliable and accurate features.Furthermore,this approach is also the foundation for the coming AFEA of 1D C~1 and 2D C~0 variational inequality problems,and serves as the cornerstone of the whole research.Specifically,the proposed bisection bounding and C-check techniques greatly accelerate the convergence rate of the conventional relaxation iteration;the proposed bisection bounding technique for higher-order element successfully deals with the issue that the inner nodes might be the ‘violating ones';and the strategy of adding and cutting ‘temporary nodes' guarantees the rationality of the finite element(FE)mesh refinement and reduces the redundancy,meanwhile allows the FE solution to approach the exact one gradually.(2)Taking the Euler-Bernoulli beam contact problem as a model,bisection bounding technique is applied in the C~1 research field,and the new C-check technique,i.e.w-check and ?-check aimed to match 1D C~1 variational inequality problems,is proposed;‘C-check subdivision' technique is developed for unique cases,ensuring a comprehensive and viable algorithm.Numerical examples show that the algorithm is efficient and stable,and the FE solution satisfies the error tolerance prescribed by users.(3)EEP-based AFEA for 2D variational inequality problems has been successfully carried out with the help of former experience.Taking the elastic membrane contact problem as an example,the present thesis proposes two novel techniques,i.e.2D bisection bounding technique and 2D C-check technique,which significantly accelerate the convergence rate of the conventional relaxation iteration in FE procedures.Once the converged FE solution is obtained,the super-convergent solution via the EEP method will be calculated to estimate the FE errors and then to guide mesh refinement.(4)For the cases of multiply connected domains,this thesis combines the classical SOR with projection method,which is an iteration method commonly used in free and moving boundary analysis,and direct method of Algebraic Equations(AE)to obtain an FE solution without ‘violating nodes' and corresponding FE mesh.Then the definitions of constraint property and periphery property are provided,with which the ‘critical nodes' could be automatically identified and then be checked by 2D C-check technique.Numerical examples presented show that the proposed algorithm is general and efficient,and the FE solution satisfies the error tolerance by maximum norm.