Reduced Basis Method for quadratically nonlinear transport equations

Jung, Nicolai H; Haasdonk, Bernard; Kröner, Dietmar

doi:10.1504/ijcsm.2009.030912

Cited by 12 publications

(9 citation statements)

References 11 publications

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“…While the former is directly applicable in the multi-dimensional parameter domain, the latter is most often applied only in the one-dimensional Reduced basis methods can be effectively applied also to nonlinear problems [51, 17,74], although this typically introduces both numerical and theoretical complications, and many questions remain open. For classical problems with a quadratic nonlinearity, there has been substantial progress, e.g., Navier-Stokes/Boussinesq and Burgers' equations in fluid mechanics [123,156,155,31,128,35,148] and nonlinear elasticity in solid mechanics.…”

Section: Historical Background and Perspectivesmentioning

confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

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The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-22470-1International audienceThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples

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Section: Historical Background and Perspectivesmentioning

confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

View full text Add to dashboard Cite

show abstract

“…They can be effectively applied also to nonlinear problems [37,58,59], even if this in turn introduces both numerical and theoretical complications, and many open research issues are still to be faced. Classical problems arising in applied sciences are, for example, Navier-Stokes/Boussinesq and Burgers' equations in fluid mechanics [16-18, 47, 48, 60, 61] and nonlinear elasticity in solid mechanics.…”

Section: Extension To Complex Problemsmentioning

confidence: 99%

Certified reduced basis approximation for parametrized partial differential equations and applications

2011

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Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis methods (built upon a high-fidelity 'truth' finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for realtime simulation and many-query contexts (for example, optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some re-A Quarteroni · G Rozza ( ) · A Manzoni

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“…An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25…”

Section: Discussionmentioning

confidence: 99%

Time‐space PGD for the rapid solution of 3D nonlinear parametrized problems in the many‐query context

Néron

Boucard

Relun³

2015

Numerical Meth Engineering

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SUMMARYThis work deals with the question of the resolution of nonlinear problems for many different configurations in order to build a 'virtual chart' of solutions. The targeted problems are three-dimensional structures driven by Chaboche-type elastic-viscoplastic constitutive laws. In this context, parametric analysis can lead to highly expensive computations when using a direct treatment. As an alternative, we present a technique based on the use of the time-space proper generalized decomposition in the framework of the LATIN method. To speed up the calculations in the parametrized context, we use the fact that at each iteration of the LATIN method, an approximation over the entire time-space domain is available. Then, a global reduced-order basis is generated, reused and eventually enriched, by treating, one-by-one, all the various parameter sets. The novelty of the current paper is to develop a strategy that uses the reduced-order basis for any new set of parameters as an initialization for the iterative procedure. The reduced-order basis, which has been built for a set of parameters, is reused to build a first approximation of the solution for another set of parameters. An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25 in the industrial examples that are presented.

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Reduced Basis Method for quadratically nonlinear transport equations

Cited by 12 publications

References 11 publications

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified reduced basis approximation for parametrized partial differential equations and applications

Time‐space PGD for the rapid solution of 3D nonlinear parametrized problems in the many‐query context

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