Galerkin method for solving combined radiative and conductive heat transfer

Gabsi, Mohamed; Roche, Jean Rodolphe; Asllanaj, Fatmir; Boutayeb, Mohamed

doi:10.1016/j.ijthermalsci.2015.10.011

Cited by 21 publications

(22 citation statements)

References 30 publications

(45 reference statements)

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“…The DG method with the upwind numerical fluxes is used for the spatial discretization of the RTE for all direction, [15,11,7]. The finite element method is used to solve the non-linear heat equation with the homogeneous Dirichlet boundary condition.…”

Section: Observer Synthesis Methodsmentioning

confidence: 99%

On reduced order observer for radiative conductive heat transfer systems

Gabsi¹,

Boutayeb²,

Roche³

2015

2015 Proceedings of the Conference on Control and Its Applications

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This contribution deals with state observer design for a class of PDE non linear systems described by Radiative transfer equation (RTE) coupled with nonlinear heat equation (NHE) in two dimensional domain. Observations are made though sensors placed at the upper boundary of the two dimensional domain. We explored the Galerkin method for a semi-discretization to obtain a large scale in finite dimensional. Thanks to the special structure of the obtained state system, we show through the Differential Mean Value Theorem (DMVT) that there always exists an observer gain matrix that assures asymptotic convergence. Both full order and reduced order state estimators are provided.

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Section: Observer Synthesis Methodsmentioning

confidence: 99%

On reduced order observer for radiative conductive heat transfer systems

Gabsi¹,

Boutayeb²,

Roche³

2015

2015 Proceedings of the Conference on Control and Its Applications

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“…For a fuller treatment of the dimensionless form of a combined nonlinear radiation-conduction heat transfer system in a gray, absorbing and non-scattering medium, we refer the readers to [2] and their references. Recently, the existence and uniqueness results in three-dimensional case with angular varying on the unit sphere i.e., β ∈ S 2 have been proposed by Ghattassi et al in [1].…”

Section: Model Problemmentioning

confidence: 99%

“…We previously proved the existence and uniqueness of the solution of the considered PDE system, see [1]. Moreover, a large number of numerical results has been presented in Ghattassi et al [2]. In fact, the choice of DG methods is based on the attractive properties for the numerical approximation of hyperbolic problems, compared to both classical FEM and FVM.…”

Section: Introductionmentioning

confidence: 95%

Analysis of a full discretization scheme for 2D radiative–conductive heat transfer systems

Gabsi

Roche

Schmitt

2019

Journal of Computational and Applied Mathematics

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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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“…The DG method with the upwind numerical fluxes is used for the spatial discretization of the RTE for all direction, [16,11,9]. The finite element method is used to solve the non-linear heat equation with the homogeneous Dirichlet boundary condition.…”

Section: Problem Statementmentioning

confidence: 99%

Observer based control for a class of coupled parabolic hyperbolic systems

Gabsi¹,

Boutayeb²

2015

2015 Proceedings of the Conference on Control and Its Applications

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This paper investigates the control problem for a class of highly nonlinear-coupled partial differential equations that describe radiative-conductive heat transfer systems. Thanks to the special structure of the obtained state system, using the Galerkin method for the semi-discretization of PDE and to the differential mean value theorem (DMVT), a new LMI condition is provided for the observer-based controller design. The observer and controller gain are computed simultaneously by solving linear matrix inequality (LMI), i.e a convex problem. Also we provide a reduced order observer based controller that assures global asymptotic stability. IntroductionRadiative-conduction heat transfer phenomena is usually described by a set of nonlinear partial differential equations (PDEs) with mixed or homogeneous boundary conditions [1]. Due to infinite-dimensional nature of these PDE systems, it is very difficult to apply directly the design methods of lamped parameter systems for their controller design that can be readily implemented in real time with reasonable computing power. Until now, modeling and control of nonlinear PDE systems is still being one of the most challenging areas in control theory. Motivated by the fact that the eigenspectrum of the parabolic spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement, most of the existing results on the control design for parabolic PDE systems involve initially the application of the spatial discretization techniques (predominantly Galerkin's method) to the PDE system to derive ordinary differential equation (ODE) systems that describe the dominant dynamics of the PDE system. These ODE systems are subsequently used as the basis for the synthesis of the finite-dimensional controllers. For example, [2,17,3] studied the finite-dimensional control problems of linear parabolic PDE systems. In [18], Ray also discussed the linearization method for the control design of a class of nonlinear parabolic PDE systems. However,

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Galerkin method for solving combined radiative and conductive heat transfer

Cited by 21 publications

References 30 publications

On reduced order observer for radiative conductive heat transfer systems

On reduced order observer for radiative conductive heat transfer systems

Analysis of a full discretization scheme for 2D radiative–conductive heat transfer systems

Observer based control for a class of coupled parabolic hyperbolic systems

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