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**Ullmann, Sebastian** (2014):

*POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions.*

München, Verlag Dr. Hut, TU Darmstadt, ISBN 978-3-8439-1568-7,

[Online-Edition: http://tuprints.ulb.tu-darmstadt.de/4296/],

[Ph.D. Thesis]

## 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.

Item Type: | Ph.D. Thesis |
---|---|

Erschienen: | 2014 |

Creators: | Ullmann, Sebastian |

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: | München |

Publisher: | Verlag Dr. Hut |

ISBN: | 978-3-8439-1568-7 |

Divisions: | Exzellenzinitiative Exzellenzinitiative > Graduate Schools Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE) 04 Department of Mathematics 04 Department of Mathematics > Numerical Analysis and Scientific Computing Zentrale Einrichtungen |

Date Deposited: | 18 Jan 2015 20:55 |

Official URL: | http://tuprints.ulb.tu-darmstadt.de/4296/ |

URN: | urn:nbn:de:tuda-tuprints-42964 |

Additional Information: | Zugl.: Darmstadt, Techn. Univ., Diss. 2014 |

Referees: | Lang, Prof. Dr. Jens |

Refereed / Verteidigung / mdl. Prüfung: | 2014 |

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