Ullmann, Sebastian (2014)
POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions.
Technische Universität Darmstadt
Dissertation, Bibliographie
Dies ist die neueste Version dieses Eintrags.
Kurzbeschreibung (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.
| Typ des Eintrags: | Dissertation |
|---|---|
| Erschienen: | 2014 |
| Autor(en): | Ullmann, Sebastian |
| Art des Eintrags: | Bibliographie |
| Titel: | POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions |
| Sprache: | Englisch |
| Referenten: | Lang, Prof. Dr. Jens |
| Publikationsjahr: | 2014 |
| Ort: | München |
| Verlag: | Verlag Dr. Hut |
| Datum der mündlichen Prüfung: | 2014 |
| Zugehörige Links: | |
| Kurzbeschreibung (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. |
| Zusätzliche Informationen: | Zugl.: Darmstadt, Techn. Univ., Diss. 2014 |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | Exzellenzinitiative Exzellenzinitiative > Graduiertenschulen Exzellenzinitiative > Graduiertenschulen > Graduate School of Computational Engineering (CE) 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen Zentrale Einrichtungen |
| Hinterlegungsdatum: | 18 Jan 2015 20:55 |
| Letzte Änderung: | 02 Jul 2024 14:52 |
| PPN: | |
| Referenten: | Lang, Prof. Dr. Jens |
| Datum der mündlichen Prüfung / Verteidigung / mdl. Prüfung: | 2014 |
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| Suche nach Titel in: | TUfind oder in Google |
Verfügbare Versionen dieses Eintrags
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POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions. (deposited 19 Jun 2024 14:55)
- POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions. (deposited 18 Jan 2015 20:55) [Gegenwärtig angezeigt]
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