Numerical Methods for Large Thermo-Mechanical Systems

Numerical methods are investigated for solving large-scale sparse linear systems of equations, that can be applied to thermo-mechanical models and wafer-slip models. This thesis examines efficient numerical methods, in terms of memory, number of iterations required for convergence, and computation time. To be more specific, algebraic multigrid (AMG) methods and deflation methods are considered as preconditioners for the conjugate gradient method. We investigate if smoothed aggregation AMG or adaptive smoothing and prolongation based AMG improve upon the classical Ruge-Stüben AMG. It is shown that Ruge-Stüben AMG needs fewer iterations for the test problems. However, smoothed aggregation AMG has a smaller data-size, which is of interest for situations with limited memory or large systems of equations. Moreover, the mechanical problems considered have a coefficient matrix with a block structure, which can be exploited by preconditioners like block Jacobi or the incomplete block Cholesky decomposition; but also the smoothed aggregation AMG can take the block structure into account when creating coarser grids. Further, we examine if the results of the conjugate gradient method can be improved by adding a deflation preconditioner based on the proper orthogonal decomposition or rigid body modes. They are combined with a direct or stationary iterative preconditioner, resulting in two-level preconditioned conjugate gradient methods. The various implementations of such methods are discussed, and the deflation preconditioner is shown to generally reduce the number of iterations compared to the single preconditioner.Applied Mathematic