The P-Version Adaptive Finite Element Methods And The Corresponding Fast Solving Algorithms For Modeling Weak Discontinuous Problems

There exist many weak discontinuity problems such as inclusion problems in the practical engineering computations. For the commonly used finite element(FEM) methods, mesh refinement or the increase of element order throughout the domain is usually used in order to ensure that each point on the interface can satisfy the given high degree of accuracy. But this will lead to rapid growth of the computer’s physical memory and the CPU time which will greatly affect the computational efficiency of finite element analysis. For some of the existing numerical methods which only involves linear element and the calculation accuracy is low, especially the error of each point near the interface is too large. The related research and application have not been seen yet for the case of higher-order elements and the case of containing the complex internal interfaces. When it comes to the weak discontinuity problems, it will generate local singularity since the coefficient of the material at the interface discontinues, thus it will lead the loss of the overall accuracy of the finite element numerical solution. One effective remedy is to use the-version adaptive finite element method which can increase the order of elements automatically according to the strength of singularities, so that the error is uniformly distributed over the entire grid areas. In this thesis, we take a deep study on the-version adaptive finite element method and the robust fast algorithm of discrete systems for weak discontinuity problems, which aim to improve the overall efficiency of the finite element analysis under the premise of ensuring the accuracy of the numerical solution. The main contents include two parts as follow: p pIn the first part, we have designed the corresponding-version adaptive FEM method for modeling the weak discontinuity problems and emphatically discussed the influence of different error control standards on the computational results of each point on the interface. Moreover, we have made the numerical computation and simulation for some typical weak discontinuity problems such as the circular inclusion distributed randomly. The numerical results are shown that the-version adaptive FEM method is very efficient for the solution of the weak discontinuity problems. The method can greatly improve the efficiency of its finite element analysis and provide an efficient method for the calculation of the effective elastic constants of composites such as the self-consistent FEM method. p pIn the second part, we use the idea of hierarchical method and turn the high-order FEM discrete systems into the solution of low-level(such as a linear element) FEM discrete systems. We have designed the corresponding geometric-based algebraic multigrid methods(GAMG) based on the information of part geometry and analysis for the weak discontinuities problems of high-order FEM discrete systems. Compared to the usual AMG method, the new method of iterations does not substantially depend on the size of the problem, the order of element and the Young’s modulus, and it has better computational efficiency and robustness which will greatly improve the overall efficiency of FEM analysis for weak discontinuity problems.