Seminar "Material Modeling"

Title: Mean-field limits for interacting particle systems.
Speaker: Angeliki Koutsimpela (University of Augsburg, Faculty of Mathematics, Natural Sciences and Technology) [link]
Time: Tuesday, 19.11.2024, 13:30
Location: WIAS-406

In this talk, we study stochastic particle systems on a complete graph and derive mean-field equations in the limit of diverging system size. We establish a connection between tagged particles and size-biased empirical processes in interacting particle systems, in analogy to classical propagation of chaos, which links the dynamics of unbiased empirical measures with that of occupation numbers on a fixed site. In a mean-field scaling limit, the evolution of the occupation number on the tagged particle site converges to a time-inhomogeneous Markov process with non-linear master equation given by the law of large numbers of size-biased empirical measures. The latter are important in recent efforts to understand the dynamics of condensing interacting particle systems with unbounded occupation numbers, such as zero-range or inclusion processes. Joint work with Stefan Grosskinsky.

Title: On the use of viscoelastic fluids flows models in hemolysis prediction.
Speaker: Tomáš Bodnár (Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Technical Mathematics)
Time: Thursday, 07.11.2024, 15:00
Location: WIAS-406/Online (link on request)

The mathematical modeling and numerical simulations of blood flows is a very challenging problem due to complex rheological properties of blood. The blood can be considered as a suspension of red blood cells in blood plasma, resulting in a shear-thinning and viscoelastic behavior of the whole blood. It is well known that high levels of stress in blood flows, typically found in ventricular assist devices for example, can lead to blood damage - the hemolysis. Under high stress the red blood cells can be damaged or ruptured, which can lead to serious medical consequences. This is why a considerable effort has been dedicated by numerous researchers to prediction of the blood damage, based on the detailed knowledge of the flow field. There exists a number of models for hemolysis, but non of them is generally accepted as a definitive, convenient and reliable tool for prediction of hemolysis. The aim of the presented talk is to point out the importance of inclusion of non-Newtonian rheology of blood in the hemolysis predictions. The similarity between some of the promising stretch based tensorial models of hemolysis and the non-linear viscoelastic rheology models offers the possibility to estimate the hemolysis directly from the local stretch tensor. This possibility will be presented and discussed in detail.

Title: Composite solutions to a liquid bilayer model.
Speaker: Georgy Kitavtsev (Middle East Technical University, Northern Cyprus Campus) [link]
Time: Thursday, 05.09.2024, 15:00
Location: WIAS-406

In this talk we will review the research initiated in Jachalski et. al SIAP 73(3), 2014. Explicit formulae for the leading order profiles of eleven types of stationary solutions to a one-dimensional two-layer thin-film liquid model considered with an intermolecular potential depending on both layer heights. The found solutions are composed of the repeated elementary blocks (bulk, contact line and ultra-thin film ones) being consistently asymptotically matched together. Once considered on a finite interval with Neumann boundary conditions these stationary solutions are either dynamically stable or weakly translationally unstable. Other composite solutions are found to be numerically unstable and rather exhibit complex coarsening dynamics.

Title: Asymptotic methods for lithium-ion battery models.
Speaker: Ferran Brosa Planella (University of Warwick) [link]
Time: Thursday, 08.08.2024, 13:30
Location: WIAS-406

Lithium-ion batteries have become ubiquitous over the past decade, and they are called to play even a more important role with the electrification of vehicles. In order to design better and safer batteries and to manage them more efficiently, we need models than can predict the battery behaviour accurately and fast. However, in many cases these models are still posed in an ad hoc way, which makes them hard to extend and may lead to inconsistencies. In this talk we will see some examples on how asymptotic methods can be applied to obtain simple models that can be used in battery control and parameterisation.

Title: A posteriori error estimates for systems of hyperbolic conservation laws modeling compressible flows.
Speaker: Jan Giesselmann (Department of Mathematics, TU Darmstadt) [link]
Time: Thursday, 18.07.2024, 15:00
Location: WIAS-406/Online (link on request from peschka{at}wias-berlin.de)

Hyperbolic conservation laws are a class of PDEs that have many applications, most notably in compressible fluid flows when viscosity is neglected. While these equations are successfully used by engineers in simulations, there are many open questions regarding their mathematical study. These are mostly driven by to the non-uniqueness of entropy solutions in multiple space dimensions but even in one space dimensions stability of solutions is challenging if one is to go beyond initial data that are small in BV.

The challenges in the mathematical analysis have lead to a situation where a variety of sophisticated numerical schemes exists and is successfully used in practice but the mathematical foundations of these methods are not as substantial as one would hope. This issue has been addressed by several authors in recent years and we plan plan to present what has been achieved in the field of a posteriori error estimates, i.e. explicit bounds for the error of the numerical solution that can be computed from the numerical solution.

A fundamental building block of a posteriori error estimates are stability theories that allow to relate certain norms of residuals, i.e. quantities measuring by how much an approximate solution fails to satisfy the PDE, to certain norms of the error, i.e. the difference between numerical and exact solution. Examples of such stability theories include the classical relative entropy framework, the theory of shifts developed by Vasseur and coworkers and recent results by Bressan and coworkers.

We will describe these stability results and discuss how they have been used to derive a posteriori error estimates for hyperbolic conservation laws.

Title: Axisymmetric capillary water waves with vorticity and swirl.
Speaker: Jörg Weber (Universität Wien) [link]
Time: Thursday, 04.07.2024, 10:00
Location: WIAS-406/Online (link on request from peschka{at}wias-berlin.de)

While the research on axisymmetric capillary water waves has a long history, only surprisingly little has been done when one wants to allow for vorticity (local spinning) and swirl (angular momentum). After introducing the problem and presenting a well-known formulation in terms of the Stokes stream function, we will explain the main analytical ingredients in order to rigorously construct travelling wave solutions with a free boundary. Then, two solution curves are constructed: First, we start at perfectly cylindrical jets, where we derive the dispersion relation and investigate it in detail in certain cases. Also, we will point out inherent connections to the famous Rayleigh instability observed in everyday life, e.g. when a tap drips. Second, we start at static configurations with constant mean curvature surfaces, so-called unduloids. There, as an interesting interplay between water waves, geometry, and elliptic integrals, we show rigorously that to any such configuration there connects a curve of non-static configurations, which confirms previous numerical observations. The talk is based on joint work with André Erhardt (WIAS), Anna-Mariya Otsetova (Aalto), and Erik Wahlén (Lund).

Title: Mixed-dimensional Coupled Finite Elements in FEniCS(x).
Speaker: Cécile Daversin-Catty (Simula Research Laboratory, (Oslo, Norway)) [link]
Time: Tuesday, 09.04.2024, 13:30
Location: Online (link on request)

Mixed-dimensional partial differential equations (PDEs) are equations coupling fields defined over distinct domains that may differ in topological dimension. Such PDEs naturally arise in a wide range of fields including geology, bio-medicine, and fracture mechanics. Mixed-dimensional models are also used to impose non-standard conditions through Lagrange multipliers. Finite element discretizations of such PDEs involve nested meshes of possibly heterogeneous topological dimension. The assembly of such systems is non-standard and non-trivial, and requires the design of both generic high level software abstractions and lower level algorithms. The FEniCS project aims at automating the numerical solution of PDE-based models using finite element methods. A core feature is a high-level domain-specific language for finite element spaces and variational forms, close to mathematical syntax. Lately, FEniCS gave way to its successor FEniCSx, including major improvements over the legacy library. An automated framework was developed in core FEniCS legacy libraries to address the challenges characterizing mixed-dimensional problems. These concepts were recently ported to FEniCSx, taking advantage of the underlying upgrades in the library features and design. This talk gives an overview of the abstractions and algorithms involved, and their implementation in the FEniCS project core libraries. The introduced features are illustrated by concrete applications in biomedicine.

Title: Microscopic and mesoscopic simulations of fluid interfaces.
Speaker: Marcello Sega (University College London, UK) [link]
Time: Tuesday, 13.02.2024, 13:30
Location: WIAS-406/Online (link on request)

I will present recent advancements in analyzing fluid interfaces from the microscopic to the mesoscopic scales. Beginning with the classical atomistic description of fluids in molecular dynamics simulations, I will discuss techniques for extracting surface information in the presence of thermal capillary waves. These modern numerical methods allow resolving the interface properties on a molecular layer basis, revealing a high degree of heterogeneity in their structure and transport coefficients. I will conclude by introducing a novel approach that bridges the microscopic and hydrodynamic scales, offering a more comprehensive understanding of droplet wetting dynamics at the mesoscopic level.