A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

Handle URI:
http://hdl.handle.net/10754/597348
Title:
A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics
Authors:
Schick, M.; Heuveline, V.; Le Ma, O. P.
Abstract:
The determination of stable limit-cycles plays an important role in quantifying the characteristics of dynamical systems. In practice, exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves, resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method using polynomial chaos (PC). We are able to overcome the difficulties of PC regarding its well-known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the rescaled governing equations, we aim at determining an initial condition and a period such that the trajectories close after completion of one stochastic cycle. Furthermore, we verify the numerical method by computation of a vortex shedding of a flow around a circular domain with stochastic inflow boundary conditions as a benchmark problem. The results are verified by comparison to purely deterministic reference problems and demonstrate high accuracy up to machine precision in capturing the stochastic variations of the limit-cycle.
Citation:
Schick M, Heuveline V, Le Ma OP (2014) A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics. SIAM/ASA J Uncertainty Quantification 2: 153–173. Available: http://dx.doi.org/10.1137/130908919.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM/ASA Journal on Uncertainty Quantification
Issue Date:
Jan-2014
DOI:
10.1137/130908919
Type:
Article
ISSN:
2166-2525
Sponsors:
The work of this author was supported in part by the FrenchAgence Nationale pour la Recherche (Project ANR-2010-Blan-0904), by the US Department of Energy, Office ofAdvanced Scientific Computing Research, Award DE-SC0007020, and the SRI Center for Uncertainty Quantificationat the King Abdullah University of Science and Technology.
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Full metadata record

DC FieldValue Language
dc.contributor.authorSchick, M.en
dc.contributor.authorHeuveline, V.en
dc.contributor.authorLe Ma, O. P.en
dc.date.accessioned2016-02-25T12:31:16Zen
dc.date.available2016-02-25T12:31:16Zen
dc.date.issued2014-01en
dc.identifier.citationSchick M, Heuveline V, Le Ma OP (2014) A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics. SIAM/ASA J Uncertainty Quantification 2: 153–173. Available: http://dx.doi.org/10.1137/130908919.en
dc.identifier.issn2166-2525en
dc.identifier.doi10.1137/130908919en
dc.identifier.urihttp://hdl.handle.net/10754/597348en
dc.description.abstractThe determination of stable limit-cycles plays an important role in quantifying the characteristics of dynamical systems. In practice, exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves, resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method using polynomial chaos (PC). We are able to overcome the difficulties of PC regarding its well-known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the rescaled governing equations, we aim at determining an initial condition and a period such that the trajectories close after completion of one stochastic cycle. Furthermore, we verify the numerical method by computation of a vortex shedding of a flow around a circular domain with stochastic inflow boundary conditions as a benchmark problem. The results are verified by comparison to purely deterministic reference problems and demonstrate high accuracy up to machine precision in capturing the stochastic variations of the limit-cycle.en
dc.description.sponsorshipThe work of this author was supported in part by the FrenchAgence Nationale pour la Recherche (Project ANR-2010-Blan-0904), by the US Department of Energy, Office ofAdvanced Scientific Computing Research, Award DE-SC0007020, and the SRI Center for Uncertainty Quantificationat the King Abdullah University of Science and Technology.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.titleA Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamicsen
dc.typeArticleen
dc.identifier.journalSIAM/ASA Journal on Uncertainty Quantificationen
dc.contributor.institutionEngineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology (KIT), 76133 Karlsruhe, Germanyen
dc.contributor.institutionEngineering Mathematics and Computing Lab (EMCL), Karlsruhe Institute of Technology (KIT), 76133 Karlsruhe, Germany. Current address: Interdisciplinary Center for Scientific Computing, Heidelberg University, 69115 Heidelberg, Germanen
dc.contributor.institutionLIMSI-CNRS, 91403 Orsay cedex, France, and Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708en
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