×

zbMATH — the first resource for mathematics

Fast three-dimensional temperature reconstruction. (English) Zbl 1225.80008
Summary: The work presented here deals with the reconstruction of the thermal field inside a three-dimensional structure when only some pointwise temperature measurements along a time interval \([0, T]\) are available. The model-based reconstruction procedure builds upon optimal control theory applied to the determination of the unknown boundary conditions. The proposed dual approach enables one to reduce the on-line computational cost so that the resulting algorithm can be part of a real-time process. The complexity of the resulting algorithm does not depend on geometry. The paper details a novel methodology that enables to implement the reconstruction procedure using standard finite element tools, despite the difficulty to define the pointwise values of a three-dimensional field in the usual functional spaces where finite element methods converge.

MSC:
80A23 Inverse problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)
Software:
Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C.R. Farrar, F. Hemez, D. Shunk, D. Stinemates, B. Nadler, A Review of Shm Literature: 1996-2001, Los Alamos National Laboratory Internal Reports.
[2] Farrar, C.R.; Worden, K., An introduction to structural health monitoring, Philos. trans. R. soc. A, 365, 303-315, (2007)
[3] M. Basseville, F. Bourquin, L. Mevel, H. Nasser, F. Treyssède, A statistical nuisance rejection approach to handling the temperature effect for monitoring civil structures, in: Proceedings of the Fourth IASC World Conference on Structural Control and Monitoring, San Diego, CA, 2006.
[4] Balmes, E.; Basseville, M.; Bourquin, F.; Mevel, L.; Nasser, H.; Treyssède, F., Merging sensor data from multiple temperature scenarios for vibration monitoring of civil structures, Struct. health monit., 7, 2, 129-142, (2008)
[5] Basseville, M.; Bourquin, F.; Mevel, L.; Nasser, H.; Treyssède, F., Handling the temperature effect in vibration monitoring: two subspace-based analytical approaches, J. engrg. mech., 136, 3, 367-378, (2009)
[6] Lions, J.-L., Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, (1968), Dunod, Translation by S.K. Mitter: Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971 · Zbl 0179.41801
[7] Glowinski, R.; Lions, J.-L.; He, J., Exact and approximate controllability for distributed parameter systems, (2008), Cambridge University Press
[8] Engl, H.W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1994), Kluwer Academic Publishers · Zbl 0711.34018
[9] Alifanov, O.M., Inverse heat transfer problems, (1994), Springer-Verlag New York · Zbl 0979.80003
[10] Alifanov, O.M.; Artyukhin, E.A.; Rumyantsev, S.V., Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems, (1995), Begell House, Inc. · Zbl 1006.35001
[11] Beck, J.V.; Blackwell, B.; Clair, C.S., Inverse heat conduction, ill-posed problems, (1985), Wiley Interscience New York · Zbl 0633.73120
[12] Jarny, Y.; Özisik, M.N.; Bardon, J.P., A general optimization method using adjoint equation for solving multidimensional inverse heat conduction, Int. J. heat mass transfer, 34, 11, 2911-2919, (1991) · Zbl 0729.73930
[13] Alifanov, O.M.; Egerov, Y.V., Algorithm and results of solving inverse heat-conduction boundary problems in a two-dimensional formulation, J. engrg. phys., 48, 489-496, (1985)
[14] Huang, C.-H.; Özisik, M.N., Inverse problem of determining unknown wall heat flux in laminar flow through a parallel plate duct, Numer. heat transfer, part A, 21, 55-70, (1992)
[15] Silva Neto, A.J.; Özisik, M.N., Two-dimensional heat conduction problem of estimating the time-varying strength of a line heat source, J. appl. phys., 71, 11, 5357-5362, (1992)
[16] Huang, C.-H.; Wang, S.-P., A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method, Int. J. heat mass transfer, 42, 18, 3387-3403, (1999) · Zbl 0977.74594
[17] F. Bourquin, A. Nassiopoulos, Temperature assimilation with accurate final state, Int. J. Heat Mass Transfer (submitted for publication). · Zbl 1219.80107
[18] A. Nassiopoulos, Identification rapide de la température dans les structures du génie civil, Ph.D. Thesis, Ecole Nationale des Ponts et Chaussées, 2008.
[19] Saitta, S.; Kripakaran, P.; Raphael, B.; Smith, I.F.C., Improving system identification using clustering, J. comput. civ. engrg., 22, 8, 292-302, (2008)
[20] Catbas, F.N.; Ciloglu, S.K.; Hasancebi, O.; Grimmelsman, K.; Aktan, A.E., Limitations in structural identification of large constructed structures, J. struct. engrg., 133, 8, 1051-1066, (2007)
[21] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationnels, (1974), Dunod Gauthier-Villars · Zbl 0281.49001
[22] Lions, J.-L.; Magenes, E., Problfmes aux limites non homogfnes et applications, 1, 2, 3, (1968), Dunod · Zbl 0165.10801
[23] Bourquin, F.; Branchet, B.; Collet, M., Computational methods for the fast boundary stabilization of flexible structures. part 1: the case of beams, Comput. methods appl. mech. engrg., 196, 4-6, 988-1005, (2007) · Zbl 1120.74637
[24] Brézis, H., Analyse fonctionnelle: théorie et applications, (1999), Dunod
[25] J.M. Urquiza, Contrôle d’équations des ondes linéaires et quasilinéaires, Ph.D. Thesis, Université Paris VI, 2000.
[26] Ciarlet, P., The finite element method for elliptic problems, (1978), North Holland Amsterdam
[27] Thomée, V., Galerkin finite element methods for parabolic problems, (1984), Springer-Verlag Berlin, New York · Zbl 0528.65052
[28] F. Bourquin, Approximate normal stresses for the boundary control of flexible structures, in: European Workshop on Smart Structures in Engineering and Technology, Giens, France, 2002.
[29] Flament, B.; Bourquin, F.; Neveu, A., Synthèse modale: une méthode de sous-structuration dynamique pour la modélisation des systèmes thermiques linéaires, Int. J. heat mass transfer, 36, 6, 1649-1662, (1993) · Zbl 0775.73062
[30] T. Mathworks™, Matlab®, 2007. Available from: <http://www.mathworks.com>.
[31] SDTools, INRIA, Openfem. Available from: <http://www-rocq.inria.fr/openfem/>.
[32] Vogel, C.R., Computational methods for inverse problems, SIAM front. appl. math., (2002) · Zbl 1008.65103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.