Preprints der Fakultät für Mathematik
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Item Numerical simulation and mixing characterization of Taylor bubble flows in coiled flow inverters(2024-08) Mierka, Otto; Münster, Raphael; Surkamp, Julia; Kockmann, Norbert; Turek, StefanThe here presented work is dedicated to the development of a software analysis tool specialized for Taylor bubble flows in a wide range of applications and covering a variety of geometrical realizations. Accordingly, a higher order FEM based interface tracking simulation software has been designed which due to the underlying isoparametric discretization and interface aligned mesh construction allows a semi-implicit treatment of the surface tension force term. The such designed numerical framework guarantees only a negligible amount of mass loss rate resulting in suitability of the tool for long time-scale simulations. Exploiting these numerical advantages the software has been applied for the system of Coiled Flow Inverter (CFI) capillaries characterized by a range of coil diameters and Reynolds numbers for which the pseudo-periodic flowfield has been obtained and extracted for further mixing quantification studies by the help of particle tracking based analysis. In the framework of this postprocessing analysis such a suitable transformation of the results has been applied which reveals the most important features of the characteristic flow patterns and makes it possible to qualitatively but even quantitatively characterize the behavior of the established flow patterns. Therefore, such a combination of robust CFD techniques together with the respective process performance quantification makes this approach suitable for tailored process design.Item Direct numerical simulation of dispersion and mixing in gas-liquid Dean-Taylor flow with influence of a 90° bend(2024-08) Mierka, Otto; Münster, Raphael; Surkamp, Julia; Turek, Stefan; Kockmann, NorbertGas-liquid capillary flow finds widespread applications in reaction engineering, owing to its ability to facilitate precise control and efficient mixing. Incorporating compact and regular design with Coiled Flow Inverter (CFI) enhances process efficiency due to improved mixing as well as heat and mass transfer leading to a narrow residence time distribution. The impact of Dean and Taylor flow phenomena on mixing and dispersion within these systems underscores their significance, but is still not yet fully understood. Direct numerical simulation based on finite element method enables full 3D resolution of the flow field and detailed examination of laminar flow profiles, providing valuable insights into flow dynamics. Notably, the deflection of flow velocity from the center axis contributes is followed by tracking of particle with defined starting positions, aiding in flow visualization and dispersion characterization. In this CFD study, the helical flow with the influence of the centrifugal force and pitch (Dean flow) as well as the capillary two-phase flow (Taylor bubble) is described and characterized by particle dispersion and related histograms. Future prospects in this field include advancements in imaging techniques to capture intricate flow paterns, as well as refined particle tracking methods to beter understand complex flow behavior.Item Inf-sup theory for the quasi-static Biot's equations in poroelasticity(2024-07) Zanotti, PietroWe analyze the two-field formulation of the quasi-static Biot's equations by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.Item Effective sound absorbing boundary conditions for complex geometries(2024-05) Schweizer, BenWe analyze a system of equations that describes the propagation of sound waves. We are interested in complex constructions along a part of the boundary of the domain, for example constructions with small chambers that are connected to the domain. We also allow that different flow equations are used in the chambers, e.g., modelling a damping material. In addition to the complex geometry, we assume that the viscosity vanishes in the limit. The limiting system is given by wave equations, we derive these equations and determine the effective boundary conditions. The effective boundary conditions replace the large number of small chambers. We provide examples for sound absorbing constructions and their Dirichlet-to-Neumann boundary conditions.Item The harmonic Maxwell's equations in periodic waveguides(2024-01) Kirsch, Andreas; Schweizer, BenWe study Maxwell’s equations with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and to solve the radiation problem. We furthermore characterize the set of all bounded solutions to the homogeneous problem. The case of a compact perturbation of the medium is included, the scattering problem and the limiting absorption principle are discussed.Item The time horizon for stochastic homogenization of the one-dimensional wave equation(2023-07) Schäffner, Mathias; Schweizer, BenThe wave equation with stochastic coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε−2. We consider the one-dimensional wave equation and are interested in the critical time scale ε−β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε−1Item Periodic wave-guides revisited: Radiation conditions, limiting absorption principles, and the space of boundes solutions(2023-07) Kirsch, Andreas; Schweizer, BenWe study the Helmholtz equation with periodic coefficients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.Item A radiation box domain truncation scheme for the wave equation(2022-03) Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, YohanesWe consider the wave equation in an unbounded domain and are interested in domain truncation methods. Our aim is to develop a numerical scheme that allows calculations for truncated waveguide geometries with periodic coefficient functions. The scheme is constructed with radiation boxes that are attached to the artificially introduced boundaries. A Dirichlet-to-Neumann operator N is calculated in these radiation boxes. Efficiency of the scheme is obtained by calculating N not with an iteration, but with a single run through the time interval. We observe speed-up factors of up to 20 in comparison to calculations without domain truncation.Item A data driven framework for evolutionary problems in solid mechanics(2021-11) Poelstra, Klaas; Bartel, Thorsten; Schweizer, BenData driven schemes introduced a new perspective in elasticity: While certain physical principles are regarded as invariable, material models for the relation between strain and stress are replaced by data clouds of admissible pairs of these variables. A data driven approach is of particular interest for plasticity problems, since the material modelling is even more unclear in this field. Unfortunately, so far, data driven approaches to evolutionary problems are much less understood. We try to contribute in this area and propose an evolutionary data driven scheme. We presenta first analysis of the scheme regarding existence and data convergence. Encouraging numerical tests are also included.Item Domain truncation methods for the wave equation in a homogenization limit(2021-09) Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, YohanesItem Travelling wave solutions for gravity fingering in porous media flows(2020-12) Mitra, Koondanibha; Schweizer, Ben; Rätz, AndreasWe study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.Item Concentration inequalities in random Schrödinger operators(2019-01-21) Schuhmacher, ChristophItem Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials(2020-09-29) Andraus, Sergio; Hermann, Kilian; Voit, MichaelItem Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions(2020-09-28) Voit, Michael; Woerner, Jeannette H. C.Item Sound absorption by perforated walls along boundaries(2020-06-03) Donato, Patrizia; Lamacz, Agnes; Schweizer, BenWe analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity ε > 0. A resonator volume of thickness ε is connected with thin channels (opening ε^3) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit ε → 0 and find that the effective system can describe sound absorption.Item Representation of solutions to wave equations with profile functions(2019-05-17) Lamacz, Agnes; Schweizer, BenSolutions to the wave equation with constant coefficients in $\mathbb{R}^d$ ca be represented explicitly in Fourier space. We investigate a reconstruction formula, which provides an approximation of solutions $u(., t)$ to initial data $u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given $u_0$, initial data for a profile equation are extracted. 2) A profile evolution equation determines the shape of the profile at time $\tau = \varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile to a function on $\mathbb{R}^d$. The sketched construction simplifies the wave equation, since only a one-dimensional problem in an $O(1)$ time span has to be solved. We prove that the construction provides a good approximation to the wave evolution operator for times $t$ of order $\varepsilon^{-2}$.Item Some central limit theorems for random walks associated with hypergeometric functions of type BC(2018-02) Artykov, Merdan; Voit, MichaelThe spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ are Heckman-Opdam hypergeometric functions of type BC, when the double coset spaces $G//K$ are identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The associated double coset hypergroups on $ C_q^B$ can be embedded into a continuous family of commutative hypergroups $(C_q^B,*_p)$ with $p\in[2q-1,\infty[$ associated with these hypergeometric functions by Rösler. Several limit theorems for random walks on these hypergroups were recently derived by Voit (2017). We here present further limit theorems when the time as well as $p$ tend to $\infty$. For integers $p$, this admits interpretations for group-invariant random walks on the Grassmannians $G/K$.Item Continuous Association Schemes and Hypergroups(2018-02) Voit, MichaelClassical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated to discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this notion and present several classes of examples. It turns out that for a given commutative hypergroup the existence of an associated continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We in particular show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.Item Existence results for the Helmholtz equation in periodic wave-guides with energy methods(2019-05-10) Schweizer, BenThe Helmholtz equation $ - \nabla \cdot (a \nabla u) - \omega^2 u = f$ is considered in an unbounded wave-guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$, where $S \subset \mathbb{R}^{d-1}$ is a bounded domain. The coefficient $a$ is strictly elliptic and (locally) periodic in the unbounded direction $x_1\in \mathbb{R}$. For non-singular frequencies $\omega$, we show the existence of a solution $u$. While previous proofs of such results were based on operator theory, our proof uses only energy methods.Item Functional central limit theorems for multivariate Bessel processes in the freezing regime(2019-01) Voit, Michael; Woerner, Jeannette H.C.Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers.Here we provide associated functional central limit theorems which are locally uniform in $t$.The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.