Recovering of the heat transfer coefficient in transmission problems with imperfect contact conditions

S. G. Pyatkov ; V. A. Belonogov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 3, pp. 331-350.

Résumé

We consider systems of parabolic equations and well-posedness questions in Sobolev spaces of inverse problems of recovering the heat transfer coefficients at the interface which are included in the transmission condition of the imperfect contact type. Under certain conditions on the data, it is demonstrated that there exists a unique solution to the problem. The proof employs a priori estimates and the fixed-point theorem.

Détail

Keywords: inverse problem, transmission problem, heat transfer coefficient, parabolic system, heat and mass transfer.

Bibliographie

[1] Baehr H. D., Stephan K., Heat and Mass Transfer, Springer-Verlag, Berlin, Heidelberg, 2006

[2] Ladyzhenskaya O.A. Uraltseva N.N. Solonnikov V.A., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968 | MR

[3] Alifanov O.M., Artyukhin E.A., Nenarokomov A.V., Inverse problems in the study of complex heat transfer, Yanus-K Publ., Moscow, 2009 (In Russ.)

[4] Tkachenko V.N., Mathematical modeling, identification and control of technological processes of heat treatment of materials, Naukova Dumka Publ., Kiev, 2008 (In Russ.) | MR

[5] Artyukhin E. A., Nenarokomov A. V., “Deriving the thermal contact resistance from the solution of the incerce heat-conduction problem”, Journal of Engineering Physics, 46:4 (1984), 495–499 | DOI

[6] Drenchev L. B., Sobczak J., “Inverse heat conduction problems and application to estimate of heat paramters in 2-D experiments”, Proceedings of International Conference on High Temperature Capillarity (Cracow, Poland, 29 June — 2 July 1997, Foundry Research Institute, Krakow (Poland)), 1998, 355–361

[7] Huang C., Ju T., “An inverse problem of simultaneously estimating contact conductance and heat transfer coefficient of exhaust gases between engine's exhaust valve and seat”, International Journal for Numerical Methods in Engineering, 38 (1995), 735–754 | DOI

[8] Loulou T., Scott E., “An inverse heat conduction problem with heat flux measurements”, International Journal for Numerical Methods in Engineering, 67:11 (2006), 1587–1616 | DOI | MR | Zbl

[9] Abreu A., Orlande H. R. B., Naveira-Cotta C. P., Quaresma J. N. N., Cotta R. M., “Identification of contact failures in multi-layered composites”, Proceedings of the International Design Engineering Technical Conferences Computers and Information in Engineering Conference (August 28-31, 2011. DETC2011-47511), 2011, 479–487

[10] Abreu L. A. S., Colaco M. J., Alves C. J. S., Orlande H. R. B., Kolehmainen V., Kaipio J., “A comparison of two inverse problem techniques for the identification of contact failures in multi-layered composites”, 22nd International Congress of Mechanical Engineering (COBEM 2013) (November 3-7), 2013, 5422–5432

[11] Zhuo L., Lesnic D., “Reconstruction of the heat transfer coefficient at the interface of a bi-material”, Inverse Problems in Science and Engineering, 28:3 (2020), 374–401 | DOI | MR | Zbl

[12] Matsevitiy Yu.M., Kostikov A.O., Safonov N.A., Ganchin V.V., “To solving of non-stationary nonlinear boundary-value inverse problems of heat conduction”, Problems of mechanical engineering, 20:4 (2017), 15–23 (In Russ.)

[13] Nenarokomov A. V., Salosina M. O., Alifanov O. M., “Optimal design of multi-layer thermal protection of variable thickness”, International Journal of Numerical Methods for Heat Fluid Flow, 27:5 (2017), 1040–1055 | DOI

[14] Kostin A.B., Prilepko A.I., “On some problems of restoration of a boundary condition for a parabolic equation. I”, Differential. equations, 32:1 (1996), 113–122 | MR | Zbl

[15] Kostin A.B., Prilepko A.I., “Some problems of restoring the boundary condition for a parabolic equation. II”, Differential. equations, 32:11 (1996), 1515–1525 | MR | Zbl

[16] Pyatkov S.G., Baranchuk V.A., “Determination of the heat transfer coefficient in mathematical models of heat and mass transfer”, Mathematical Notes, 113:1 (2023), 93–108 (In Russ.) | DOI | MR | Zbl

[17] Belonogov V.A., Pyatkov S.G., “On some classes of inverse problems of recovering the heat transfer coefficient in stratified media”, Siberian Mathematical Journal, 63 (2022), 206–223 | DOI | MR | Zbl

[18] Triebel H., Interpolation theory. Functional spaces. Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 | MR

[19] Denk R., Hieber M., Prüss J., “Optimal $L_{p}-L_{q}$-estimates for parabolic boundary value problems with inhomogeneous data”, Mathematische Zeitschrift, 257:1 (2007), 193–224 | DOI | MR | Zbl

[20] Belonogov V.A., Pyatkov S.G., “On solvability of conjugation problems with non-ideal contact conditions”, Russian Mathematics, 64:7 (2020), 13–26 | DOI | MR | Zbl

[21] Mikhaylov V.P., Partial differential equations, Nauka Publ., Moscow, 1976 (In Russ.)

[22] Nikol'skiy S.M., Approximation of functions of several variables and embedding theorems, Nauka Publ., Moscow, 1977 (In Russ.) | MR

[23] Prüss J., Simonett G., Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhäuser Publ., Basel, 2016 | MR | Zbl

[24] Amann H., “Compact embeddings of vector-valued Sobolev and Besov spaces”, Glasnik Matematicki, 35:1 (2000), 161–177 | MR | Zbl

[25] Grisvard P., “Equations differentielles abstraites”, Annales scientifiques de l'École normale supérieure, $4^{e}$ series, no. 2, 1969, 311–395 | MR | Zbl