Numerical computations and thermodynamically complete models for inelastic behaviour in solids
Dienstag, 6.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some aspects of the mathematical modelling of inelastic solids, placing particular emphasis on models that are compatible with the second law of thermodynamics. In particular, I will describe a recent model from [Cichra, Pr?ša; 2020] and discuss its numerical approximation via the finite element method. One of the advantages of the approach considered here is that it is not necessary to introduce additional concepts, such as the plastic strain. Moreover, as a consequence of the thermodynamically consistent derivation, one is able to compute the evolution of the temperature field without additional complication. I will also showcase an application of this modelling approach to the Mullins effect, for which up to date there had been no simple yet fully coupled thermo-mechanical model.
The energy technique for BDF methods
Dienstag, 20.6.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract The application of the energy technique to numerical methods with very good stability properties for parabolic equations, such as algebraically stable Runge–Kutta methods or\nA-stable multistep methods, is straightforward. The extension to high order multistep methods requires some effort; the main difficulty concerns suitable choices of test functions. We\ndiscuss the energy technique for all six backward difference formula (BDF) methods. In the\ncases of the A-stable one- and two-step BDF methods, the application is trivial. The energy\ntechnique is applicable also to the three-, four- and five-step BDF methods via Nevanlinna–\nOdeh multipliers. The main new results are: i) No Nevanlinna–Odeh multipliers exist for the\nsix-step BDF method. ii) The energy technique is applicable under a relaxed condition on\nthe multipliers. iii) We present multipliers that make the energy technique applicable also to\nthe six-step BDF method. Besides its simplicity, the energy technique for BDF methods is\npowerful, it leads to several stability estimates, and flexible, it can be easily combined with\nother stability techniques.