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Type: Report × Subject: Navier–Stokes equations ×

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    Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Dreyer, Wolfgang; Druet, Pierre-Étienne; Klein, Olaf; Sprekels, Jürgen
    This paper deals with the mathematical modeling and simulation of crystal growth processes by the so-called Czochralski method and related methods, which are important industrial processes to grow large bulk single crystals of semiconductor materials such as, e.,g., gallium arsenide (GaAs) or silicon (Si) from the melt. In particular, we investigate a recently developed technology in which traveling magnetic fields are applied in order to control the behavior of the turbulent melt flow. Since numerous different physical effects like electromagnetic fields, turbulent melt flows, high temperatures, heat transfer via radiation, etc., play an important role in the process, the corresponding mathematical model leads to an extremely difficult system of initial-boundary value problems for nonlinearly coupled partial differential equations ...
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    Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Frigeri, Sergio; Grasselli, Maurizio; Sprekels, Jürgen
    In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the NavierStokes equations, nonlinearly coupled with a convective nonlocal CahnHilliard equation. The system rules the evolution of the (volume-averaged) velocity u of the mixture and the (relative) concentration difference ' of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force v acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map v 7! [u; '], and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E. Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C. G. Gal in [14].
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    A numerical investigation of velocity-pressure reduced order models for incompressible flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Caiazzo, Alfonso; Iliescu, Traian; John, Volker; Schyschlowa, Swetlana
    This report has two main goals. First, it numerically investigates three velocity-pressure reduced order models (ROMs) for incompressible flows. The proper orthogonal decomposition (POD) is used to generate the modes. One method computes the ROM pressure solely based on the velocity POD modes, whereas the other two ROMs use pressure modes as well. To the best of the authors knowledge, one of the latter methods is novel. The second goal is to numerically investigate the impact of the snapshot accuracy on the ROMs accuracy. Numerical studies are performed on a two-dimensional laminar flow past a circular obstacle. It turns out that, both in terms of accuracy and efficiency, the two ROMs that utilize pressure modes are clearly superior to the ROM that uses only velocity modes. The numerical results also show a strong correlation of the accuracy of the snap shots with the accuracy of the ROMs.
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    Pressure-induced locking in mixed methods for time-dependent (Navier)Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Linke, Alexander; Rebholz, Leo G.
    We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and NavierStokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the L2 convergence order for high order methods, and even a complete stall of the L2 convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.