
PDF, English
Download (12MB)  Terms of use 
Abstract
In this thesis we develop a method for the estimation of the flow behaviour of an incom pressible fluid based on observations of the brightness intensity of a transported visible substance which does not influence the flow. The observations are given in a subregion of the flow as a sequence of discrete images with in and outflow across the image boundaries. The resulting mathematical problem is illposed and has to be regularised with information of the underlying fluid flow model. We consider a constrained optimisation problem, namely the minimisation of a tracking type data term for the brightness distribution and a regularisation term subject to a system of weakly coupled partial differential equations. The system consists of the time dependent incompressible NavierStokes equations coupled by the velocity vector field to a convectiondiffusion equation, which describes the transport of brightness patterns in the image sequence. Due to the flow across the boundaries of the computational domain we solve a boundary identification problem. The usage of (strong) Dirichlet boundary controls for this purpose leads to theoretical and numerical complications, so that we will instead use Robintype controls, which allow for a more convenient theoretical and numerical framework. We will prove wellposedness and investigate the functionality of the proposed approach by means of numerical examples. Furthermore, we discuss the connection to Dirichletcontrol problems, e. g. the approximation of Dirichletcontrols by the socalled penalised Neumann method, which is based on the Robintype controls for a varying penalty parameter. We will show via numerical tests that Robintype controls are suitable for the identifi cation of the correct fluid flow. Moreover, the examples indicate that the underlying physical model used for the regularisation influences the flow reconstruction process. Thus appropriate knowledge of the model is essential, e. g. the viscosity parameter. For a time independent example we will present a heuristic, which, beside the boundary identification, automatically evaluates the viscosity in case the parameter is unknown. The developed physicsbased optical flow estimation approach is finally used for the data set of a prototypical application. The background of the application is the approximation of horizontal wind fields in sparsely populated areas like desert regions. A sequence of satellite images documenting the brightness intensity of an observable substance distributed by the wind (e. g. dust plumes) is thereby assumed to be the only available data. Wind field information is for example needed to simulate the distribution of other, not directly observ able, substances in the lower atmosphere. For the prototypical example we compute a high quality reconstruction of the underlying fluid flow by a (discrete) sequence of consecutive spatially distributed brightness intensities. Thereby, we compare three different models (heat equation, Stokes system and the original fluid flow model) in the reconstruction process and show that using as much model knowledge as possible is essential for a good reconstruction result.
Item Type:  Dissertation 

Supervisor:  Rannacher, Prof. Dr. Dr. h. c. Rolf 
Date of thesis defense:  29 April 2015 
Date Deposited:  08 May 2015 10:08 
Date:  2015 
Faculties / Institutes:  The Faculty of Mathematics and Computer Science > Department of Applied Mathematics 
Subjects:  500 Natural sciences and mathematics 510 Mathematics 