Sep 24, 2014
Wolfgang Dreyer, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany Joint work with Clemens Guhlke We study reversible storage systems serving to store electrical or chemical energy for later use. Particularly, we consider storage systems that consist of an ensemble of many interconnected storage particles as they appear in lithium-ion batteries. During charging and discharging of the battery, one observes a phase transition with two coexisting phases and hysteretic behavior. There are two regimes for fast and slow charging with different storage mechanisms. In this lecture we describe in detail the slow charging regime, where the time to approach equilibrium in a single storage particle is much smaller than the time for full charging of the ensemble. Here the observed phase transition is a many-particle effect and happens in the ensemble instead within a single particle. Its evolution is modelled by a nonlocal and nonlinear conservation law of Fokker-Planck type. There are two parameter that control if the ensemble transits the 2-phase region along a Maxwell line, along a hysteresis path, or if the ensemble shows the same behaviour as its constituents.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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