Study Of Algebraic Multigrid Algorithms And Applications In Solid Mechanics

With the rapid development of computer, scientific computing, especially large-scale , high-performance computing has become more and more the impetus to the advancing of science and technology. And computational methods are the essence of scientific computing. Hence, the development of computer and computational methods decide the enhancement of scientific computing capability and capacity. In the fields of physics and mechanics, we need to solve a partial differential equations (PDEs) or a system of PDEs whose exact solutions are in general difficult to be obtained and the finite element method is the most effective way for solving such problems. The finite element solutions usually involve the mesh generation and optimization, the assembly of discrete algebraic systems using the finite element basis, and the solution of these systems by some algebraic solvers. It is well-known that the performance of finite element analysis depends critically on both the first and the third processes. Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of PDEs or a system of PDEs. Generally speaking, there are two types of multigrid methods: geometric-based approach and algebraic approach. Since the complexities for practical application problems and the requirements for the "plug and play" solvers in numerical business softwares, it is difficult to construct a sequence of nested discretizations or meshes needed for geometric multigrid method. The algebraic multigrid (AMG) method has become the hotspot due to the high performance and robustness.The key technique of AMG is to propose some algebraic ways for the coarse grid selection and the construction of prolongation operators. For many discrete systems arising from PDEs or a system of PDEs, it is difficult to reproduce the crucial properties of origin problems only by the global stiffness matrix. So it is a natural way to construct highly efficient AMG method such as AMGe and agglomeration methods, by using geometric and analytic information. This is a new direction in developing AMG algorithms currently. We call this method as geometric-based and analytic AMG method. AMG has been well developed for the scalar PDEs, namely for the case when d = 1, where d is the number of physical unknowns resides in each grid. However, the naive use of the scalar AMG does not lead to the robust and efficient solver, rather it deteriorates and oftentimes breaks down in the convergence for system cases in which d > 1 and those unknowns are coupled as well and the graph of global stiffness matrix is no longer the same as the corresponding graph of geometric meshes, especially for three dimensional problems. Therefore, it is a very significant work to develop new techniques and methodology for the coarse grid selection and the construction of prolongation operators to improve the AMG convergence rate, and so that the resulting AMG methods may be applied to more application problems.In this paper, we make some in-depth studies for AMG algorithms for some types of important application problems. We propose some types of AMG methods by using geometric-based and analytic information. A number of numerical experiments have been performed, and some crucial numerical conclusions are obtained. These researches will make the AMG algorithms richer and apply AMG methods to more researching fields. The main contents and results are listed as follows:...