# Karlstad Applied Analysis Seminar (KAAS)

## Next seminars:

**When: **April 8th, Wednesday, 11:00.**What: **A look behind the curtain: Modelling of elastomers**Who: **Nils Hendrik Kröger, Deutsches Institut für Kautschuktechnologie e.V., Hannover**Where: ***Zoom*: https://kau-se.zoom.us/j/172202538 (Meeting ID: 172 202 538)

**Abstract: **In mechanical and automotive engineering, elastomer materials are required for components that are subjected to extreme mechanical stress and are resistant to temperature and chemicals. Typical examples are tires, toothed belts and V-belts, engine and unit mounts, air springs and chassis bushings, as well as statically and dynamically loaded seals. The requirements in terms of reliability and durability are particularly high, especially in connection with the highly sensitive, safety-relevant functions of the components. Simulation models are becoming increasingly important, particularly in the context of digitization in production. Many processing steps in the manufacture of elastomer components, beginning with mixing, rolling and extrusion or injection molding, through vulcanization, influence the final mechanical properties. In the course of their service life, these properties change due to thermo-oxidative aging, so that even lifetime predictions become a challenge. The reliable generation of "digital twins" for elastomer components thus requires a description of many effects, some of which are coupled with each other. This lecture offers insights into different modelling approaches of individual stages of the life of elastomers.

**When: **April 15th, Wednesday, 11:00.**What: **Two-scale model for moisture transport in concrete carbonation process**Who: **Toyohiko Aiki, Japan Women’s University, Tokyo, Japan**Where: ***Zoom*: https://kau-se.zoom.us/j/853785478 (Meeting ID: 853 785 478)

**Abstract: **In this talk we consider a two-scale model describing moisture transport appearing in concrete carbonation process. The model consists of the diffusion equation for the relative humidity in macro domain and the free boundary problem in each micro domain. Here, we note that the boundary condition of the free boundary problem contains the relative humidity given on the macro domain and the coefficient of one term of the diffusion depends on the free boundary. Accordingly, it is necessary to solve infinite number of the free boundary problems at the same time. The aims of this talk is to introduce our modeling of the two-scale model and show recent mathematical results on the model. This is a joint work with Kumazaki (Nagasaki University, Japan)

## Future seminars:

**When: **T.B.A.**What: **Topological and Interfacial Effects on the Glass Transition in Confined Polymers**Who: **Alexey V. Lyulin, Department of Applied Physics, Technische Universiteit Eindhoven, The Netherlands**Where:** T.B.A.

**Abstract: **Glasses in general, and polymer glasses in particular, are not, perhaps surprisingly, technically solid in a crystalized form, but are substances frozen in a liquidlike structure. Many fundamental questions remain as to exactly how glasses form, transitioning from flowing liquid-like state to solid polymer glass. A central factor materials scientists study is the temperature where this occurs, the glass-transition temperature Tg. After some introduction, I will discuss in more detail the recent results of the molecular-dynamics computer simulations of atactic polystyrene (PS), for the bulk and free-standing films, and for both linear and cyclic polymers. Simulated volumetric glass-transition temperatures ([1,2] show a strong dependence on the film thickness below 10 nm [3]. Our studies reveal that the fraction of the chain-end groups is larger in the interfacial layer with its outermost region approx. 1 nm below the surface than it is in the bulk. The enhanced population of the end groups is expected to result in a more mobile interfacial layer and the consequent dependence of Tg on the film thickness. In addition, the simulations show an enrichment of backbone aliphatic carbons and concomitant deficit of phenyl aromatic carbons in the interfacial film layer. This deficit would weaken the strong phenyl-phenyl aromatic interactions and, hence, lead to a lower film-averaged Tg in thin films, as compared to the bulk sample. To investigate the relative importance of the two possible mechanisms (increased chain ends at the surface or weakened p-p interactions in the interfacial region), the data for linear PS are compared with those for cyclic PS. For the cyclic PS the reduction of the glass-transition temperature is also significant in thin films, albeit not as much as for linear PS. Moreover, the deficit of phenyl carbons in the film interface is comparable to that observed for linear PS. Therefore, chain-end effects alone cannot explain the observed pronounced Tg dependence on the thickness of thin PS films; the weakened phenyl-phenyl interactions in the interfacial region seems to be an important cause as well [4]. I will also discuss the interface characteristics of polystyrene in free-standing thin films and on a graphite surface simulated employing an explicit all-atom force field [5]. References [1] Ediger, M.D.; Forrest, J.A. Macromolecules 2014, 47, 471-478. [2] Barrat, J.-L.; Baschnagel, J.; Lyulin, A.V. Soft Matter 2010, 6, 3430-3446. [3] Hudzinskyy, D.; Lyulin, A.V.; Baljon, A.R.C.; Balabaev, N.K.; Michels, M.A.J. Macromolecules 2011, 44, 2299-2310. [4] A. V. Lyulin, N. K. Balabaev, A. R.C. Baljon, G. Mendoza, C. W. Frank and Do Y. Yoon, J. Chem. Phys. 2017, 146, 203314. [5] S. Lee, A. V. Lyulin, C. W. Frank, Do Y. Yoon, Polymer, 2017, 116, 540-548.

**When: **June 2, Tuesday 15:15**What: **A generalized Monge-Ampère equation to compute freeform lens surfaces**Who: **Martijn Anthonissen, Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands**Where: **T.B.A.

**Abstract: **The key problem in computational illumination optics is to find the shape of an optical surface given a source and desired target light distribution. Our mathematical model is based on the principles of geometrical optics, formulated in terms of the optical map connecting source and target domain, and energy conservation. This leads to a fully nonlinear partial differential equation of generalized Monge-Ampère type subject to the transport boundary condition. We show the derivation of this PDE for several model problems. The numerical solution method is a two-stage procedure. In the first stage it computes the optical map. In the second stage it constructs the shape of the optical surface from the optical map. Both stages are solved using a least squares method. The computation is an iterative process with fast convergence. The resulting algorithm gives accurate results and is very efficient. This is illustrated by numerical experiments. We consider e.g. laser beam shaping, where an incident Gaussian source distribution is transformed into a uniform top hat illuminance

**When:** T.B.A.**What**: On robotics, falling cat, parallel parking and stock trading ** ****Who**: Maria Ulan, Linnaeus University**Where:** T.B.A.

**Abstract**: Configuration spaces of many real mechanical systems appear to be manifolds with singularity. A singularity often indicates that geometry of motion may change at the singular point of configuration space. During the talk, I will present one possible solution by considering a certain algebra instead of the configuration space, with a structure completely determined by the geometry of the singularity. We will discussed how one could describe different types of complex motion. We will start with simple examples like linkages, manifolds with corners. Then we will discussed examples with given motion of subsystem. Finally, we will consider some classical reachability problems. At the end of the talk, I will show some applications in industry.

**When:** T.B.A.**What**: Some methods of approximation by spline functions ** ****Who**: Adrian Branga, Lucian Blaga University, Sibiu, Romania**Where:** T.B.A.

**Abstract**: We investigate some classes of spline functions, which can be used to approximate linear functionals defined on unidimensional Sobolev spaces, as well as to find the approximate solutions of some classes of ordinary differential equation and partial derivative equations. The existence and uniqueness of the approximate solution, together with convergence criteria for these methods and estimates of the approximation error are studied and some computational results are given.

**When:** T.B.A.**What**: Ideals generated by quadratic forms in the exterior algebra** ****Who**: Veronica Crispin Quinonez, Uppsala University**Where:** T.B.A.

**Abstract**: (joint work with S. LUNDQVIST and G. NENASHEV) There is a longstanding conjecture due to Fröberg about the minimal Hilbert series of C[x_1, ..., x_n]/(f_1,...,f_r), where f_i are homogeneous forms. We introduce the problem and give some results.

Now let E_n denote the Exterior algebra on n generators over C. It is natural to believe that the Hilbert series of E_n/(f_1,...,f_r) should be equal to the conjectured series in the commutative case, if the f_i's are generic forms of even degree. In 2002, Moreno and Snellman showed it to be true for only one generic form f. However, the same year Fröberg and Löfwall gave a counterexample for the case of two generic forms.

We use the structure theory of pairs of skew-symmetric matrices to study the Hilbert function of two generic quadratic forms f and g in E_n. Further, we use combinatorial methods to describe the Hilbert series of E_n/(f,g). Among our results, we have a conjecture for the minimal Hilbert series.