Comptes Rendus
Nodal predictive error model and Bayesian approach for thermal diffusivity and heat source mapping
[Modèle nodal basé sur l'erreur de prédiction et approche bayésienne pour la cartographie de diffusivité thermique et de terme source]
Comptes Rendus. Mécanique, Inverse problems, Volume 338 (2010) no. 7-8, pp. 434-449.

This article aims at solving a two-dimensional inverse heat conduction problem in order to retrieve both the thermal diffusivity and heat source field in a thin plate. A spatial random heat pulse is applied to the plate and the thermal response is analysed. The inverse approach is based on the minimisation of a nodal predictive error model, which yields a linear estimation problem. As a result of this approach, the sensitivity matrix is directly filled with experimental data, and thus is partially noisy. Bayesian estimators, such as the Maximum A Posteriori and a Markov Chain Monte Carlo approach (Metropolis–Hastings), are implemented and compared with the Ordinary Least Squares solution. Simulated temperature measurements are used in the inverse analysis. The nodal strategy relies on the availability of temperature measurements with fine spatial resolution and high frequency, typical of nowadays infrared cameras. The effects of both the measurement errors and of the model errors on the inverse problem solution are also analysed.

Un problème de conduction 2D est traité pour estimer un champ de diffusivité thermique et terme source sur une plaque mince. Un flash photothermique spatiallement aléatoire est appliqué à la plaque et on analyse le champ de réponse thermique. L'inversion est abordée en minimisant un modèle basé sur l'erreur de prédiction, ce qui permet de rester dans un cadre d'estimation linéaire. Une conséquence directe est que la matrice de sensibilité est construite avec les données expérimentales, et donc contient des erreurs de mesures. On développe des estimateurs bayésiens tels que l'estimateur du Maximum à Postériori ou dans le cadre des algorithmes de Monte Carlo et Chaînes de Markov (Metropolis–Hastings), pour les comparer à la solution des Moindres Carrés Ordinaires. Les résultats sont obtenus à partir de données simulées. L'approche nodale est liée à l'obtention de mesures par thermographie infrarouge. On analyse à la fois l'effet des erreurs de mesures et des erreurs de modèles sur les résultats de l'estimation.

Publié le :
DOI : 10.1016/j.crme.2010.07.015
Keywords: Heat transfer, Thermal diffusivity mapping, Heat source mapping, Infrared image processing, Metropolis–Hastings, Inverse problem, Bayesian estimation
Mots-clés : Transferts thermiques, Champ de diffusivité thermique, Cartographie de terme source, Traitement d'images infrarouges, Metropolis–Hastings, Problèmes inverses, Estimation bayésienne

H. Massard 1, 2, 3 ; Olivier Fudym 2, 3, 4 ; H.R.B. Orlande 1 ; J.C. Batsale 4

1 PEM/COPPE, Federal University of Rio de Janeiro, Brazil
2 Université de Toulouse, mines Albi, CNRS, campus Jarlard, 81013 Albi cedex 09, France
3 École des mines Albi, centre RAPSODEE, campus Jarlard, 81013 Albi, France
4 TREFLE-ENSAM, UMR 8508 CNRS, 33405 Talence cedex, France
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H. Massard; Olivier Fudym; H.R.B. Orlande; J.C. Batsale. Nodal predictive error model and Bayesian approach for thermal diffusivity and heat source mapping. Comptes Rendus. Mécanique, Inverse problems, Volume 338 (2010) no. 7-8, pp. 434-449. doi : 10.1016/j.crme.2010.07.015. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.015/

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