Browsing by Author "Zimmermann, Dominik"

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Open Access
Justification of an approximation equation for the Benard-Marangoni problem
(2014) Zimmermann, Dominik; Schneider, Guido (Prof. Dr.)
The Benard-Marangoni problem is a mathematical model for the description of a temperature dependent fluid flow in very thin liquid layers with a free top surface. The liquid is bounded from below by a horizontal plate of a certain temperature. Above the liquid there is an atmosphere cooler than the bottom plate. There is a purely conducting steady state, where the liquid is at rest. This state is stable as long as the difference between the temperature of the bottom plate and the temperature of the atmosphere is sufficiently small. If the temperature difference surpasses a certain threshold, convection sets in, which is mainly driven by surface tension rather than buoyancy. The onset of convection can be seen as the propagation of a spatially periodic pattern, such that we interpret the Benard-Marangoni problem as a pattern forming system. In this thesis we are interested in the behaviour of the system when the purely conducting steady state becomes unstable. From the equations of the Benard-Marangoni problem we formally derive a Ginzburg-Landau like system of modulation equations, which we use to construct approximate solutions for the full problem. In this thesis we prove an approximation theorem for these modulation equations. That means, we show that the approximate solutions lie close to true solutions of the Benard-Marangoni problem, at least for a long time. The validity of the Ginzburg-Landau approximation was already shown for a number of pattern forming systems. In case of the Benard-Marangoni problem, however, we have a spectral situation that does not allow a direct application of the existing approximation proofs. Hence, we first consider a toy problem exhibiting such a kind of spectrum and develop a method for proving an approximation result in this case. Furthermore, the existing approximation proofs were restricted to semilinear problems. However, the equations of the Benard-Marangoni problem are quasilinear. Therefore, we also develop a method for proving approximation results for quasilinear problems. We then turn back to the Benard-Marangoni problem. After showing local existence and uniqueness of solutions, we apply our new methods in order to prove the desired approximation result.
Open Access
Material forces in finite inelasticity and structural dynamics : topology optimization, mesh refinement and fracture
(2008) Zimmermann, Dominik; Miehe, Christian (Prof. Dr.-Ing.)
The present work serves two major purposes. On the one hand, theoretical approaches to configurational mechanics are elaborated. For inelastic problems, the spatial and material equilibrium conditions are derived by means of a global dissipation analysis. In the dynamical framework, a variational formulation based on Hamilton's principle is established inducing the balances of physical momentum, material pseudomomentum and kinetic energy. On the other hand, configurational-force-based computational algorithms are developed. At first, configurational forces are exploited in the context of topology optimization. The theoretical basis is provided by a dual variational formulation of finite elastostatics. This scenario is applied to the r-adaptive optimization of finite element meshes and the optimization of truss structures. In the second step, a configurational-force-based strategy for h-adaptvity is presented. The discrete version of the material balance equation is exploited to formulate global and local refinement criteria controlling the overall decision on mesh refinement and the local refinement procedure. The method is specified for problems of finite elasticity and plasticity including thermal and dynamical effects as well. Finally, a configurational-force-driven procedure for the simulation of crack propagation in brittle materials is introduced. The algorithm bases on the separation of the geometry model and the finite element mesh. The process of crack propagation is carried out by a structural update of the underlying geometry model. The generation of the new triangulation incorporates a configurational-force-based adaptive refinement criterion. The capabilities of the derived algorithms are demonstrated by means of a variety of numerical examples including the comparison with benchmark analyses and experimental observations.