Master’s Thesis Project : Fibre-based preconditioner for granular matter simulation

Abstract: Simulating granular media at large scales is hard to do because of ill-conditioning of the associated linear systems and the ineffectiveness of available iterative methods. One common way to improve iterative methods is to use a preconditioner which involves finding a good approximation of a linear system A. A good preconditioner will improve the condition number of A. If a linear system has a set of large eigenvalues of comparable magnitude, and the rest of the eigenvalues are small, so that the gap between the set of large eigenvalues and the small ones is large, the ill-conditioning caused by the small eigenvalues will not appear in the early iterations. We investigate a new fibre-based preconditioner that involves finding chains of contacting particles along the particles of a granular medium and reordering the system, which leads to a diagonal preconditioner. We show its effects on the relative residual and error of the velocity on linear systems where the ill-conditioning is caused by a big gap between a set of large eigenvalues and small eigenvalues for three differentiterative methods: Uzawa, the Conjugate Residual (CR) and the Minimum ResidualMethod (MINRES).