Multi-scale problems: an improved non-intrusive algorithm that enhances FEA platforms with the generalized finite element method; An improved preconditioned conjugate gradient solver for hierarchical generalized finite element systems of equations

The finite element method (FEM) discretizes an object of interest, say a cube, and solves for its dis placement and stress under a certain loading. The FEM is used by many commercial softwares. The generalized FEM (GFEM) adds information in the solution process that improves the displacement and stress results, and is not fully available in commercial software, but in third-party software. A GFEM method that transfers small scale information to larger scales is called GFEM global-local (GFEM gl ). The process of adding GFEM gl functionality to commercial software without modifying the commercial software is called a non-intrusive algorithm. This thesis presents a new non-intrusive algorithm, the hierarchical non-intrusive algorithm (HNA), that allows the combination of powerful FEM softwares with current and future state-of-the-art GFEM gl software, allowing the user to enjoy the capabilities of each software. The HNA is better than previous non-intrusive methods because it is faster, uses less memory, and is easy to use. In this thesis, the HNA is outlined and its accuracy is verified. It can be used to improve the simulation of vehicles flying at hyper-sonic speeds, greater than five times the speed of sound. A procedure that reduces the negative effects of machine precision (condition number) in solving GFEM gl systems of equations is called the Stable GFEM gl (SGFEM gl ). This thesis presents results on the ability of SGFEM gl to not only reduce the condition number, but improve solution accuracy over GFEM gl . The reduced conditioning from SGFEM gl systems of equations makes feasible iterative schemes that solve SGFEM gl linear system of equations. The reduced memory requirements of iterative solvers over direct solvers, combined with improved speed, make larger-scale simulations possible. An iterative solver called the preconditioned conjugate gradient method is investigated within this context, and a preconditioner is proposed for the method. This thesis shows that its proposed iterative solver is faster than previous iterative solvers. The proposed iterative solver is also shown to be faster than a sparse direct solver