In acknowledgment of their scientific achievement in the area of applied mathematics and mechanics this year the Richard-von-Mises Price is awarded.
Experimental and numerical study of heterogeneous material behavior in technological processes at different scales
Modern technological processes demand for high performance materials. For such processes, the relation between microstructure, properties and the macroscopic constitutive response is crucial. To interpret and model heterogeneous metallic materials enabling us to design optimal structural properties it is essential to analyze and understand the interactions of different, even competing mechanisms at different length scales. However, the development of various microstructures depending on the chemical composition, loading path and loading rate, makes it a difficult task to elaborate directly comprehensive constitutive models. In this regard it is important to develop and identify sophisticated models for specific mechanisms at the relevant scales and additionally for scale bridging. However, an extended and valid understanding of the different mechanisms can only be obtained in a multidisciplinary context, where experimental observations and model concepts are linked strongly. In this regard, the talk will address different experimental and numerical investigations of heterogeneous materials at different length scales, focusing in particular on the modeling of microstructures, material instabilities as well as its application in technological production processes.
Multiscale Dynamics near Instability
Multiscale dynamical systems occur in an extraordinary variety of contexts in the natural sciences and engineering. From the scales of quantum mechanics, chemical reactions, and neurons, up to lasers, fluid dynamics, and celestial mechanics, many applications naturally yield widely differing space and time scales. Frequently, this situation provides a doubled-edged challenge for applied mathematics. On the one hand, having several scales available helps for analytical methods. Reducing phase space, gluing several subsystem results and deriving coarse-grained approximations are frequently possible. These strategies are all helpful for pen-and-paper approaches, which may be impossible for a completely general nonlinear dynamical system. On the other hand, several scales tend to complicate numerical simulation as standardized scientific computing methods may become unstable quickly if systems are stiff, or miss key features on subgrid scales. In this talk I shall outline three deceptively simple situations, where general multiscale aspects, and the role of small parameters in particular, are crucial for theoretical as well as practical results. (I) The first class of problems are fast-slow ordinary differential equations (ODEs) arising in chemical reactions, where substantially differing reaction rates induce multiple time scales. In this context, the main task is to analyse non-stationary reactions producing oscillations. (II) The second class of problems arises in mathematical biology as well as phase-field models, among many other disciplines. In this context, nonlocal partial differential equations (PDEs) with small nonlocality are of key interest for pattern formation. (III) The last example will be drawn from stochastic differential equations (SDEs) with small noise. In many dynamical systems with noise, one has observed in various sciences, that early-warnings extracted from time series may appear before drastic bifurcation-induced transitions. This observation is the motivation to provide a suitable mathematical analysis. The common theme among our examples is the problem of disentangling multiscale dynamics near an instability. One may simply not neglect small effects once the main driving terms are only marginally stable in a mathematical model. This causes a very challenging multi-parameter problem. In particular, we are actually facing a 'curse-of-instability', which requires higher-dimensional models and accompanying novel mathematical analysis.