Ullmann, Sebastian (2015)
POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions.
Buch, Erstveröffentlichung, Postprint
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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: | Buch |
---|---|
Erschienen: | 2015 |
Autor(en): | Ullmann, Sebastian |
Art des Eintrags: | Erstveröffentlichung |
Titel: | POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions |
Sprache: | Englisch |
Publikationsjahr: | 14 Januar 2015 |
Ort: | Darmstadt |
Publikationsdatum der Erstveröffentlichung: | 2014 |
Verlag: | Dr. Hut |
URL / URN: | https://tuprints.ulb.tu-darmstadt.de/4296 |
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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. |
Status: | Postprint |
URN: | urn:nbn:de:tuda-tuprints-42964 |
Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
Hinterlegungsdatum: | 19 Jun 2024 14:55 |
Letzte Änderung: | 09 Aug 2024 09:06 |
PPN: | 35330123X |
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- POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions. (deposited 19 Jun 2024 14:55) [Gegenwärtig angezeigt]
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