An adaptive time stepping scheme for rate-independent systems with non-convex energy
Loading...
Date
2022-04
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
We investigate a local incremental stationary scheme for the numerical solution of
rate-independent systems. Such systems are characterized by a (possibly) non-convex energy
and a dissipation potential, which is positively homogeneous of degree one. Due to the non-convexity of the energy, the system does in general not admit a time-continuous solution. In
order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of
our approach in comparison to existing methods is an adaptive choice of the step size for the
update of the curve parameter depending on a prescribed tolerance for the residua in the energy-dissipation balance and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations
generated by the algorithm converge (weakly) to a so-called V-parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings and show that an adaptive
choice of the step size indeed pays off as they lead to a significant increase of the step size
during sticking and in viscous jumps.
Description
Table of contents
Keywords
rate-independent systems, parametrized balanced viscosity solutions, local incremental minimization schemes