Certified reduced basis methods for parametrized parabolic partial differential equations with non-affine source terms

Klindworth, Dirk; Grepl, Martin; Vossen, Georg

doi:10.1016/j.cma.2011.10.010

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…Based on [29], we consider the problem of controlling the heat flow in a welding process. The non-dimensionalized temperature distribution within the workpiece is governed by the non-dimensionalized unsteady convection-diffusion equation…”

Section: Control Of a Welding Processmentioning

confidence: 99%

Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations

Saskia¹,

Grepl²

2021

Preprint

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Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems -arising, e.g., through discretization of partial differential equations -requires the solution of high-dimensional optimal control problems and thus poses immense computational effort.We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

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Section: Control Of a Welding Processmentioning

confidence: 99%

Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations

Saskia¹,

Grepl²

2021

Preprint

Self Cite

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“…They also compared the geometry simulated to experimental data obtained with a high speed camera. One of the main drawbacks of FE-based methods is the large computational cost, although improvement are investigated to speed up such models as demonstrated by Klindworth et al [24].…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Temperature Prediction in Direct Metal Deposition of Ti6Al4V

Batut

Fergani²,

Brøtan

et al. 2017

JMMP

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Direct Metal Deposition (DMD) is an additive manufacturing (AM) process capable of producing large components using a layer by layer deposition of molten powder. DMD is increasingly investigated due to its higher deposition rate and the possibility to produce large structural components specifically for the aerospace industry. During fabrication, a complex thermal history is experienced in different regions of the workpiece, depending on the process parameters and part geometry. The thermal history induces residual stress accumulation in the buildup, which is the main cause of distortions. In order to control the process and enhance the product quality, the understanding of the workpiece temperature is substantial. In this study, two methods to predict temperature evolution during the DMD process are introduced based on analytical and finite element methods. The objective is to compare these methods to experimental results and to provide more insights about their capabilities to predict accurately the temperature gradient, the cooling rate, and the melt pool geometry. A comparison of the computational time is also provided. Based on the results of the investigation, It appears that the analytical method provides an effective and accurate method to understand the influence of the process on the workpiece temperature.

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“…We precise also, that the later error bound is obtained by applying classical energetic methods to the discrete equation in the difference between the full numerical solution and the reduced one, so one can bound the ROM-error by the already precised a posteriori residual norm. This technique was applied in the context of offline-online procedures, in the following cases: parameterized linear elliptic PDE which is coercive and not symmetric [16]; nonaffine linear elliptic and parabolic equations and nonlinear elliptic and parabolic equations [17]; parameterized linear parabolic equation with a nonaffine source term [18]; the model Burgers equation [19,20].…”

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confidence: 99%

On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation

Akkari

Hamdouni

Liberge

et al. 2014

Adv. Model. and Simul. in Eng. Sci.

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Background:In what follows, we consider the Proper Orthogonal Decomposition (POD) technique of model order reduction, for a parameterized quasi-nonlinear parabolic equation. Methods: A POD basis associated with a set of reference values of the characteristic parameters is considered. From this basis, a parametric reduced order model (ROM) projecting the initial equation is constructed. Results: A mathematical a priori estimate of the parametric squared L 2 -error induced by this projection is developed. This later estimate is based on both, the parametric behavior of the squared L 2 -ROM-error thanks to the resolution of a Ricatti differential inequality in the parametric ROM-error, and the convergence rate of the parametric ROM to the full problem, via the augmentation of the basis dimension. Indeed, under restrictive conditions on the solutions regularity of such equations, we are able to precise the slope of the logarithm of the squared L 2 -norm of the ROM error, as a function of the logarithm of the basis modes number. Numerical experiments of our theoretical estimate, are presented for the 2D Navier-Stokes equations in the case of an unsteady and incompressible fluid flow in a channel around a circular cylinder. Conclusion: A mathematical a priori estimate of the parametric squared L 2 -error induced by the model reduction by POD is developped for a parameterized quasi-nonlinear parabolic equation. This estimate is obtained thanks to the resolution of a Ricatti differential inequality.

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Certified reduced basis methods for parametrized parabolic partial differential equations with non-affine source terms

Cited by 6 publications

References 31 publications

Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations

Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations

Analytical and Numerical Temperature Prediction in Direct Metal Deposition of Ti6Al4V

On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation

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