Ullmann, Sebastian (2014):
POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions.
Darmstadt, [Book]
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Dissertation20140429Ullmann.pdf - Published Version Available under: only the rights of use according to UrhG. Download (3MB) | Preview |
Item Type: | Book |
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Title: | POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions |
Language: | English |
Abstract: | In the context of the numerical solution of parametrized partial differential equations, a proper orthogonal decomposition (POD) provides a basis of a subspace of the solution space. The method relies on a singular value decomposition of a snapshot matrix, which contains the numerical solutions at predefined parameter values. Often a sufficiently accurate representation of the solution can be given by a linear combination of a small number of POD basis functions. In this case, using POD basis functions as test and trial functions in a Galerkin projection leads to POD-Galerkin reduced-order models. Such models are derived and tested in this thesis for flow problems governed by the incompressible Navier-Stokes equations with stochastic Dirichlet boundary conditions. In the first part of the thesis, POD-Galerkin reduced-order models are developed for unsteady deterministic problems of increasing complexity: heat conduction, isothermal flow, and thermoconvective flow. Here, time acts as a parameter, so that the snapshot matrix consists of discrete solutions at different times. Special attention is paid to the reduced-order computation of the pressure field, which is realized by projecting a discrete pressure Poisson equation onto a pressure POD basis. It is demonstrated that the reduced-order solutions of the considered problems converge toward the underlying snapshots when the dimension of the POD basis is increased. The second part of the thesis is devoted to a steady thermally driven flow problem with a temperature Dirichlet boundary condition given by a spatially correlated random field. In order to compute statistical quantities of interest, the stochastic problem is split into separate deterministic sub-problems by means of a Karhunen-Loeve parametrization of the boundary data and subsequent stochastic collocation on a sparse grid. The sub-problems are solved with suitable POD-Galerkin models. Different methods to handle the parametrized Dirichlet conditions are introduced and compared. The use of POD-Galerkin reduced-order models leads to a significant speed-up of the overall computational process compared to a standard finite element model. |
Place of Publication: | Darmstadt |
Classification DDC: | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
Divisions: | 04 Department of Mathematics > Numerical Analysis and Scientific Computing |
Date Deposited: | 14 Jan 2015 07:36 |
Last Modified: | 23 Aug 2016 11:10 |
URL / URN: | http://www.dr.hut-verlag.de/9783843915687.html |
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URN: | urn:nbn:de:tuda-tuprints-42964 |
URI: | https://tuprints.ulb.tu-darmstadt.de/id/eprint/4296 |
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