A numerical scheme for rate-independent systems: analysis and realization

Abstract

Many materials in the field of continuum mechanics can be considered, at least in parts, as rate-independent. Such systems are generally driven by external forces and thereby independent of their speed (rate) but still dependent on their direction. In this dissertation, we consider rate-independent systems which can be described by means of a, in general, nonconvex energy and a positively homogeneous dissipation. Both properties in combination allow the formation of abrupt changes in state, even if the external forces evolve smoothly. Mathematically speaking, this means that temporal discontinuities (jumps) may develop. In order to be able to reflect such phenomena, suitable (weak) notions of solutions are required. The first section of this dissertation is therefore devoted to the presentation of precisely such solution concepts and their interrelationship. In addition to the energetic solutions, which are by now widely known and analyzed, we particularly focus on the so-called parametrized solutions, the main concept within this thesis. In the second section of this work, we analyze a scheme that provides a discretization in time and space based on the well-known local incremental minimization scheme. The main difference compared to the latter one is that, instead of solving a minimization problem, only stationary points have to be determined here. This scheme - referred to as local incremental stationarity scheme (LISS) - is therefore well suited for numerical methods. Beyond this practical applicability, the convergence analysis for LISS also provides the existence of parametrized solutions, even in the case of an unbounded dissipation. Therefore, the local incremental stationarity scheme can also be used to approximate unidirectional processes, such as those that occur in damage models for example. In the third section of this work, we then deal with a priori error estimates for LISS, whereby we do not incorporate the discretization in space here. A crucial assumption in this context is the uniform convexity of the energy. Without this, solutions are, in general, not unique and not continuous, so that convergence rates are not to be expected in this case. With sufficient convexity assumptions, on the other hand, we derive convergence rates of the order Ο(√τ) for a general setting and of the order Ο(τ) in case of a semilinear energy. We also extend the latter result to the so-called locally convex case, in which the solution trajectory remains in an area of uniform convexity of the energy. The last section, finally, is devoted to the presentation of a possible realization of the introduced approximation scheme LISS and to visualize the numerical results. In particular, we provide several examples that illustrate the theoretical findings of the preceding sections.