17.10.2017

24.10.2017

Einladender: Dr. O. Kamps

07.11.2017

Einladender: Dr. O. Kamps

A Phase Field Model for Thin Elastic Structures with Topological Constraint

Prof. Patrick Dondl
Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg

With applications in the area of biological membranes in mind, we consider the problem of minimizing Willmore’s energy among the class of closed, connected surfaces with given surface area that are confined to a fixed container. Based on a phase field model for Willmore’s energy originally introduced by de Giorgi, we develop a technique to incorporate the connectedness constraint into a diffuse interface model of elastic membranes. Our approach uses a geodesic distance function associated to the phase field to discern different connected components of the support of the limiting mass measure. We obtain both a suitable compactness property for finite energy sequences as well as a Gamma-convergence result. Furthermore, we present computational evidence for the effectiveness of our technique. The main argument in our proof is based on a new, natural notion of convergence to describe phase fields in three dimensions.

Einladender: Prof. B. Wirth

Nonlinear dynamics and time delays in engineering applications

Dr. Andreas Otto
Institut für Physik, TU Chemnitz

Time delay effects appear in many dynamical systems. Often the delay effect is a result of a transport phenomena and feedback and the systems are nonlinear. In this talk we discuss general aspects of such systems, which can be often found in engineering. The relevant applications ranging from manufacturing processes, such as rolling and metal cutting over gasoline engines to traffic flow dynamics. After an introduction to the field of time delay systems, we will first focus on systems with state-dependent delays. We show that in many situations equivalent representations with constant delays exist, which are much easier to analyze. In a second part systems with multiple and distributed delays are studied and we discuss our recent results on systems with dissipative delays. Dissipative delays are a specific class of time-varying delays and may lead to a hitherto unknown type of chaotic behavior in nonlinear delay systems.

Einladender: Dr. O. Kamps

Einladender: Dr. O. Kamps

Thin film modelling of surfactant-driven biofilm spreading
Sarah Trinschek (AG Thiele)

The spreading of bacterial colonies at solid air interfaces hinges on physical processes connected to the properties of the involved interfaces. The production of surfactant molecules by the bacteria is one strategy that allows the bacterial colony to efficiently expand over a substrate. These surfactant molecules affect the surface tension which results in an increased wettability as well as in
outward-pointing Marangoni fluxes that promote spreading. These fluxes may cause an instability of the circular colony shape and the subsequent formation of fingers. In this work, we study the front instability of bacterial colonies at solid-air interfaces induced by surfactant production in the framework of a passive hydrodynamic thin film model which is extended by bioactive terms. We show that the interplay between wettability and Marangoni fluxes determines the spreading dynamics and decides whether the colony can expand over the substrate. We observe four different types of spreading behaviour, namely, arrested and continuous spreading of circular colonies, slightly modulated front lines and the formation of pronounced fingers.

Collective Cell Migration in Embryogenesis Follows the Laws of Wetting
Bernhard Wallermeyer (AG Betz)

Collective cell migration is a fundamental process during embryogenesis and its initial occurrence, called epiboly, is an excellent in vivo model to study the physical processes involved in collective cell movements that are key to understand organ formation, cancer invasion and wound healing. In zebrafish, epiboly starts with a cluster of cells at one pole of the spherical embryo. These cells are actively spreading in a continuous movement towards its other pole until they fully cover the yolk. Inspired by the physics of wetting we determine the contact angle between the cells and the yolk during epiboly. Similar to the case of a liquid drop on a surface one observes three interfaces that carry mechanical tension. Assuming that interfacial force balance holds during the quasi-static spreading process, we employ the physics of wetting to predict the temporal change of the contact angle. While the experimental
values vary dramatically, the model allows us to rescale all measured contact angle dynamics onto a single master curve explaining the collective cell movement. Thus, we describe the fundamental and complex developmental mechanism at the onset of embryogenesis by only three main parameters: the offset tension strength 𝛼, the tension ratio 𝛿 and the rate of tension variation 𝜆.

Prof. Dr. Konrad Polthier
Freie Universität Berlin, AG Mathematical Geometry Processing

Multivalued functions and differential forms naturally lead to the concept of branched covering surfaces and more generally of branched covering manifolds in the spirit of Hermann Weyl's book "Die Idee der Riemannschen Fläche" from 1913. This talk will illustrate and discretize basic concepts of branched (simplicial) covering surfaces starting from complex analysis and surface theory up to their recent appearance in geometry processing algorithms and artistic mathematical designs.
Applications will touch differential based surface modeling, image and geometry retargeting, global surface and volume remeshing, and novel weaved geometry representations with recent industrial applications.

Einladender: Prof. B. Wirth

Interfacial turbulence and regularization in falling films

Dmitri Tseluiko
School of Mathematics, Loughborough University, UK

We consider a liquid film flowing down an inclined wall that may be subjected to an additional external effects, such as an electric field. We analyse the Stokes-flow regime, using both a non-local long-wave model and the full system of governing equations. For an obtuse inclination angle and strong surface tension, the evolution of the interface is chaotic in space and time. However, a sufficiently strong electric field has a regularising effect, and the time-dependent solution evolves into an array of continuously interacting pulses, each of which resembles a single-hump solitary pulse. For an acute inclination angle and a sufficiently small supercritical value of the electric field, solitary-pulse solutions do not exist, and the time-dependent solution is instead a modulated array of short-wavelength waves. When the electric field is increased, the evolution of the interface first becomes chaotic, but then is regularised so that an array of pulses is generated. A coherent-structure theory for such pulses is developed and corroborated by numerical simulations.

Einladender: Prof. U. Thiele

Turbulence and pattern formation in a model for active fluids

Dr. Michael Wilczek, MPISD Göttingen

We consider a liquid film flowing down an inclined wall that may be subjected to an additional external effects, such as an electric field. We analyse the Stokes-flow regime, using both a non-local long-wave model and the full system of governing equations. For an obtuse inclination angle and strong surface tension, the evolution of the interface is chaotic in space and time. However, a sufficiently strong electric field has a regularising effect, and the time-dependent solution evolves into an array of continuously interacting pulses, each of which resembles a single-hump solitary pulse. For an acute inclination angle and a sufficiently small supercritical value of the electric field, solitary-pulse solutions do not exist, and the time-dependent solution is instead a modulated array of short-wavelength waves. When the electric field is increased, the evolution of the interface first becomes chaotic, but then is regularised so that an array of pulses is generated. A coherent-structure theory for such pulses is developed and corroborated by numerical simulations.

Einladender: Dr. O. Kamps

Curve fitting on Riemannian manifolds

Prof. Pierre-Antoine Absil

Université Catholique de Louvain

In this talk I will discuss curve fitting problems on manifolds. Manifolds of interest include the rotation group SO(3) (generation of rigid body motions from sample points), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion
tensors) and the shape manifold (morphing). Ideally, we would like to find the curve that minimizes an energy function E defined as a weighted sum of (i) a sum-of-squares term penalizing the lack of fitting to the data points and (ii) a regularity term defined as the mean squared acceleration of the curve. The Euler-Lagrange necessary conditions for this problem are known to take the form of a fourth-order ordinary differential equation involving the curvature tensor of the manifold, which is in general hard to solve. Instead, we simplify the problem by restricting the set of admissible curves and by resorting to suboptimal Riemannian generalizations of Euclidean techniques.

Einladender: Prof. B. Wirth

Branched flows in weakly scattering random media: from electronic transport to tsunami propagation

Dr. Ragnar Fleischmann

Max-Planck-Institut für Dynamik und Selbstorganisation, Göttingen

Wave propagation in random media — this might sound abstract but is in fact very tangible and almost omnipresent in science and everyday life. Examples are wind driven ocean waves, but also light, sound, electrons, tsunamis and even earth quakes are waves that in a natural environment typically propagate through a complex medium. Due to its complexity, the medium is often best described as random with inherent correlations. Examples include the turbulent atmosphere, complex patterns of ocean currents or semiconductor crystals sprinkled with impurities. In recent years it has become clear that even very small fluctuations in a random medium, if they are correlated, lead to focussing of the waves in pronounced branch-like spatial structures and to heavy-tailed intensity distributions. These branches are closely connected with the occurrences of random caustics, i.e. singularities in the corresponding ray fields.
I will give an overview over the phenomenon of branching and the statistical characteristics of branched flows, discussing examples from ballistic electron transport in semiconductors to the random focusing of tsunamis waves.

_{Example of a branched flow emitted from a point ray-source propagating in a slightly anisotropic random medium}