Papers by Elfriede Friedmann
ganymed.iwr.uni-heidelberg.de
Mi:11-13 Uhr, URZ SR 215 (INF 293) E. Friedmann, Asymptotische Analysis i Es gibt zahlreiche Frag... more Mi:11-13 Uhr, URZ SR 215 (INF 293) E. Friedmann, Asymptotische Analysis i Es gibt zahlreiche Fragen asymptotischer Natur sowohl in der reinen als auch in der angewandten Mathematik. In dieser Vorlesung wird eine Einführung in die asymptotischen Methoden zur Lösung von Differentialgleichungsproblemen dargestellt. Mathematische Modelle, die in Physik, Chemie, Biologie, usw. aufgestellt werden, führen oft zu Probleme, deren exakte Lösung nur in Sonderfällen gelingt. Hier besitzen die Näherungsmethoden eine große Bedeutung. Es gibt numerische und asymptotische Näherungsmethoden. Die numerischen Methoden haben sich bewährt, werden aber instabil, wenn kleine Parameter vorkommen oder das Rechengebiet groß oder kompliziert wird. In solchen Fällen kann man viel einfachere Probleme konstruieren, indem man einen Parameter gegen Null gehen lässt oder das Gebiet vereinfacht. Die Lösung des reduzierten Problems unterscheidet sich oft wesentlich von der Lösung des ursprünglichen Problems, und Standardlösungsmethoden führen oft zu falschen Ergebnissen. Zur Behandlung solcher sogenannter singulär gestörter Probleme hilft die asymptotische Analysis weiter. Das ursprüngliche Problem wird durch immer genauere, gut zu behandelnde Probleme ersetzt. Wir werden einige spezielle analytische Methoden kennenlernen, u.a. die Methode der angepassten asymptotischen Entwicklung und die Mehrskalenmethode, die wir vor allem an Probleme aus der Strömungsmechanik (Umströmung eines Körpers oder warum schwimmt der Hai so schnell?) anwenden werden. Stichwörter: asymptotische Entwicklung, MMAE (Method of Matched Asymptotic Expansions), Mehrskalenmethoden, Navier-Stokes, Prandtl, Grenzschichten, Separation Voraussetzungen: Analysis 1,2, (nicht erforderlich aber hilfreich) Grundkenntnisse in partiellen Differentialgleichungen oder Strömungsmechanik Zielgruppe: Studierende der Mathematik, Informatik und Physik im Hauptstudium oder Promotionsstudium und alle die wissen wollen, warum der Hai so schnell schwimmt ii E. Friedmann, Asymptotische Analysis
Advances in Mathematical Fluid …, Jan 1, 2010
In this paper we will give an overview of our studies on turbulent flow over rough surfaces such ... more In this paper we will give an overview of our studies on turbulent flow over rough surfaces such as shark skin, which have a drag reducing effect known as shark-skin effect. Our mathematical model is restricted to the flow in the viscous sublayer. Here, the turbulences ...
Fundamental progress in systems biology can only be achieved if experimentalists and theoretician... more Fundamental progress in systems biology can only be achieved if experimentalists and theoreticians closely collaborate. Mathematical models cannot be formulated precisely without deep knowledge of the experiments while complex biological systems can often not be understood fully without mathematical interpretation of the dynamic processes involved. In this article, we describe how these two approaches can be combined to gain new insights on one of the most extensively studied signal transduction pathways, the Janus kinase (JAK)/ signal transducer and activator of transcription (STAT) pathway. We focus on the parameters of a model describing how STAT proteins are transported from the membrane to the nucleus where STATs regulate gene expression. We discuss which parameters can be measured experimentally in different cell types and how the unknown parameters are estimated, what the limits of these techniques and how accurate the determinations are.
archiv.ub.uni-heidelberg.de
We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving... more We consider a mathematical model of a rigid body immersed in a viscous, compressible fluid moving with a velocity prescribed on the boundary of a large channel containing the body. We show continuity of the drag functional as well as domain shape stability of solutions in the incompressible limit, with the Mach number approaching zero.
iwr.uni-heidelberg.de
Previous research has established that a smooth surface has not necessarily minimal drag: Many ex... more Previous research has established that a smooth surface has not necessarily minimal drag: Many experiments by different laboratories, e.g. NASA and DLR Berlin, indicate that an extra surface layer with tiny grooves aligned in the stream-wise direction can be used to reduce the drag. The aim of this project is to find the optimal shape of such microstructures on surfaces of submerged bodies. We assume that these microstructures remain in the viscous sublayer where the flow equations are the 3D incompressible, steady state Navier-Stokes equations with a Couette in-and outflow determinated through two boundary conditions, the no-slip condition on the lower boundary and the friction condition on the upper one. The objective function of our optimization problem is the tangential drag force, which we want to minimize. Solving this problem is difficult because of the rough boundary, which causes a big amount of data. We apply homogenization theory and replace the rough boundary by a smooth one, where the right boundary conditions have been determined. Furthermore, our optimization problem can be simplified using this approximation and we end up minimizing a scalar size, the Navier constant, which is calculated using the velocity of an auxiliary boundary layer equation. To solve the optimization problem we use sensitivity-based optimization methods. The sensitivity is calculated analytically and we use it to determine the gradient of the cost function with respect to the design variable. A minimum is sought by using the steepest descent algorithm with step size according to Armijo rule. The necessary optimality conditions are derived and a sequence of admissible domains is built which tends to the optimal solution. The state equations are solved numerically using finite elements on unstructured grids and multigrid algorithms. The results obtained with this approach give us a drag reduction of approximately 2-6% relative to the drag of the smooth configuration.
Journal of Mathematical Fluid Mechanics, Jan 1, 2011
15:10 J. Neustupa. On stability of a flow past a (rotating) body. 15:50 V. Milisic. Towards impli... more 15:10 J. Neustupa. On stability of a flow past a (rotating) body. 15:50 V. Milisic. Towards implicit multi-scale wall-laws for Poiseuille flows in rugous straight geometries. 16:10 Coffee Break 16:40 J. Naumann. Existence of weak solutions to the Oberbeck-Boussinesq equations.
Journal of Mathematical Fluid Mechanics, Jan 1, 2010
Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of s... more Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of swimming bodies in order to reduce their drag. The model describes the flow in the viscous sublayer of the boundary layer of a turbulent flow. The microscopic optimization problem is reduced applying homogenization. In the reduced so-called macroscopic optimization problem we minimize the Navier constant subject to the boundary layer equations which are solved in a very small part of the original domain. Under the assumptions that the microstructures can be represented as smooth functions the sensitivity can be determined analytically. The optimization problem is then solved by a sensitivity based method (steepest descent with optimal step size) and the state equations are solved in each iteration with an external software. Our reduced model is validated by comparing the results from the homogenized model with those obtained by simulating the whole rough channel. An improved shape is found and a drag reduction up to 10% can be shown. . Primary 35B27; secondary 76D10, 76D55, 49Q12.
Previous research has established that a smooth surface has not necessarily minimal drag: Many ex... more Previous research has established that a smooth surface has not necessarily minimal drag: Many experiments by different laboratories, e.g. NASA and DLR Berlin, indicate that an extra surface layer with tiny grooves aligned in the stream-wise direction can be used to reduce the drag. The aim of this project is to find the optimal shape of such microstructures on surfaces of submerged bodies. We assume that these microstructures remain in the viscous sublayer where the flow equations are the 3D incompressible, steady state Navier-Stokes equations with a Couette in-and outflow determinated through two boundary conditions, the no-slip condition on the lower boundary and the friction condition on the upper one. The objective function of our optimization problem is the tangential drag force, which we want to minimize. Solving this problem is difficult because of the rough boundary, which causes a big amount of data. We apply homogenization theory and replace the rough boundary by a smooth one, where the right boundary conditions have been determined. Furthermore, our optimization problem can be simplified using this approximation and we end up minimizing a scalar size, the Navier constant, which is calculated using the velocity of an auxiliary boundary layer equation. To solve the optimization problem we use sensitivity-based optimization methods. The sensitivity is calculated analytically and we use it to determine the gradient of the cost function with respect to the design variable. A minimum is sought by using the steepest descent algorithm with step size according to Armijo rule. The necessary optimality conditions are derived and a sequence of admissible domains is built which tends to the optimal solution. The state equations are solved numerically using finite elements on unstructured grids and multigrid algorithms. The results obtained with this approach give us a drag reduction of approximately 2-6% relative to the drag of the smooth configuration.
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Papers by Elfriede Friedmann