I³MS - Sander Seminar
Prof. Dr. Oliver Sander - Multigrid Methods for Nonsmooth Problems in Mechanics
Numerical Mathematics, RWTH Aachen University
Most partial differential equations are either linear or nonlinear. Of the latter, some contain operators that are not even differentiable; these are called nonsmooth.
Several interesting problems in solid mechanics are of this type. Examples are contact problems, various friction problems, and linear plasticity. Standard Newton-type solvers cannot be used here, because they rely on differentiability of the operator. Usually, some form of regularization is then used, which leads to a difficult trade-off between approximation quality and the condition number of the linear problems.
On the other hand, many of these problems can be written as minimization problems for strictly convex functionals. This point of view allows to construct new multigrid algorithms that are globally convergent, work without regularization parameters, and are as fast as standard linear multigrid. We show how one such method, the Truncated Nonsmooth Newton Multigrid (TNNMG) method, is built, and how it performs in practice.