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Das digitale Repositorium erfasst, speichert, erhält, erschließt und verbreitet digitale Forschungsergebnisse.2024-08-11T19:17:52ZSimulation of micron-scale drop impact
https://hdl.handle.net/20.500.11811/11813
Simulation of micron-scale drop impact
Klitz, Margrit; Griebel, Michael
The numerical simulation of droplet impact is of interest for a vast variety of industrial processes, where practical experiments are costly and time-consuming. In these simulations, the dynamic contact angle is a key parameter, but the modeling of its behavior is poorly understood so far. One of the few models which considers the overall physical context of the involved ‘moving contact line problem’ is Shikhmurzaev’s interface formation model [1]. In addition to keeping the problem well-posed, all surface and bulk parameters, such as the contact angle, are determined as part of the solution rather than being prescribed functions of contact line speed. In this article, we couple an asymptotic version of the interface formation model with our three-dimensional incompressible two-phase Navier-Stokes solver. Additionally, we employ a simple constant contact angle approach. We compare the results of these two numerical models with those from laboratory experiments for the micron-scale droplet impact on substrates with a variety of wetting characteristics. With our sophisticated asymptotic interface formation model, the droplet shapes, heights and diameters compare very well with practical experiments.
2018-10-01T00:00:00ZHaar system as Schauder basis in Besov spaces: the limiting cases for 0 < p ≤ 1
https://hdl.handle.net/20.500.11811/11812
Haar system as Schauder basis in Besov spaces: the limiting cases for 0 < p ≤ 1
Peter Oswald
We show that the <em>d</em>-dimensional Haar system <em>H<sup>d</sup></em> on the unit cube <em>I<sup>d</sup></em> is a Schauder basis in the classical Besov space B<em><sup>s</sup><sub>p,q,1</sub></em>(<em>I<sup>d</sup></em>), 0 < <em>p</em> < 1, defined by first order differences in the limiting case <em>s</em> = <em>d</em>(1/<em>p</em> − 1), if and only if 0 < <em>q</em> ≤ <em>p</em>. For <em>d</em> = 1 and <em>p</em> < <em>q</em> < ∞, this settles the only open case in our 1979 paper [4], where the Schauder basis property of <em>H</em> in B<em><sup>s</sup><sub>p,q,1</sub></em>(<em>I</em>) for 0 < <em>p</em> < 1 was left undecided. We also consider the Schauder basis property of <em>H<sup>d</sup></em> for the standard Besov spaces B<em><sup>s</sup><sub>p,q</sub></em>(<em>I<sup>d</sup></em>) defined by Fourier-analytic methods in the limiting cases <em>s</em> = <em>d</em>(1/<em>p</em>−1) and <em>s</em> = 1, complementing results by Triebel [7].
2018-09-01T00:00:00ZStochastic subspace correction methods and fault tolerance
https://hdl.handle.net/20.500.11811/11811
Stochastic subspace correction methods and fault tolerance
Griebel, Michael; Oswald, Peter
We present convergence results in expectation for stochastic subspace correction schemes and their accelerated versions to solve symmetric positive-definite variational problems, and discuss their potential for achieving fault tolerance in an unreliable compute network. We employ the standard overlapping domain decomposition algorithm for PDE discretizations to discuss the latter aspect.
2018-07-01T00:00:00ZIncremental kernel based approximations for Bayesian inverse problems
https://hdl.handle.net/20.500.11811/11810
Incremental kernel based approximations for Bayesian inverse problems
Rieger, Christian
We provide an interpretation for the covariance of the predictive process of Bayesian Gaussian process regression as reproducing kernel of a subset of the Cameron Martin space of the prior. We demonstrate that this deterministic viewpoint enables us to relate particular greedy methods using that subset kernel to instances of powerful low-rank matrix approximation techniques such as <em>adaptive cross approximation</em> or <em>pivoted Cholesky decomposition</em>. In particular, we can show convergence results for such algorithms which appear to be novel in the case of finitely smooth kernels. <br> Moreover, we consider the inverse problem to reconstruct a parametrized diffusion coefficient from point evaluations of the solution to a diffusion equation with that parametrized coefficient. To this end, we present a Gaussian process regression based approach to approximate the observati- on operator. The error estimates for this approximation methods are capable to take deterministic model errors explicitly into account. Finally, we show how the findings about incremental low rank approximations can be applied to these reconstruction problems.
2018-05-01T00:00:00Z