In the numerical solution of partial differential equations the basic question is how to solve the system of linear equations obtained from the discretization and linearization. Requirements for iterative linear equation solvers to be used in various solid- and structural mechanics problems are discussed in this talk. A special emphasis is given to non-linear stability problems. Determination of a critical point is the primary problem in structural stability analysis. Mathematically it means solution of an eigenvalue problem, which in general is non-linear. The non-linear stability eigenvalue problem constitutes of solving the equilibrium equations simultaneously with the criticality condition. This can double the size of the problem to be solved. If direct linear equation solver is used, the block elimination scheme is a feasible choice. However, when iterative linear solver is used, it is better to operate directly with the extended system. A similar situation is obtained in standard continuation, or path-following algorithms. Preconditioning strategies of this extended system are discussed. In addition, some comments on solving coupled multi-physical problems, like magneto-electro-mechanical problems, will be given.
31 Jan 2019
A workshop on the occasion of Miroslav Tůma's 60th birthday MTU60: Let's make it sparse