×

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems. (English) Zbl 1423.35431

Summary: This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.

MSC:

35R30 Inverse problems for PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
65L20 Stability and convergence of numerical methods for ordinary differential equations

Software:

BOBYQA; COMSOL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hào DN, Thanh PX, Lesnic D . Determination of the heat transfer coefficients in transient heat conduction. Inverse Probl. 2013;29(9):095020. · Zbl 1296.65124
[2] Hào DN, Thanh PX, Lesnic D . Determination of the ambient temperature in transient heat conduction. IMA J Appl Math. 2015;80:24-46. · Zbl 1309.35171
[3] Martin T, Dulikravich G . Inverse determination of steady heat convection coefficient distributions. J Heat Transfer. 1998;120(2):328-334.
[4] Hào DN, Huong BV, Thanh PX, et al . Identification of nonlinear heat transfer laws from boundary observations. Appl Anal. 2015;94:1784-1799. · Zbl 1331.35386
[5] Wikström P, Blasiak W, Berntsson F . Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method. Appl Therm Eng. 2007;27(14):2463-2472.
[6] Beck JV, Blackwell B, Clair Jr. CRS . Inverse heat conduction: ill-posed problems. New York (NY): John Wiley and Sons; 1985. · Zbl 0633.73120
[7] Kim SK, Lee WI . A maximum entropy solution for a two-dimensional inverse heat conduction problem. J Heat Transfer. 2003;125(6):1197-1205.
[8] Alifanov OM . Inverse heat transfer problems. Berlin: Springer-Verlag; 1994. · Zbl 0979.80003
[9] Cattani L, Maillet D, Bozzoli F, et al . Estimation of the local convective heat transfer coefficient in pipe flow using a 2D thermal quadrupole model and truncated singular value decomposition. Int J Heat Mass Transfer. 2015;91:1034-1045.
[10] Kanzow C, Fukushima M, Yamashita N . Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties. Technique Report. Department of Applied Mathematics and Physics, Kyoto University; April, 2002. · Zbl 1064.65037
[11] Yamashita N, Fukushima M . On the rate of convergence of the Levenberg-Marquardt method. In: Alefeld G, Chen X , editors. Topics in numerical analysis. Vol. 15, Computing Supplementa. Springer-Verlag; 2001. p. 239-249. · Zbl 1001.65047
[12] Jin B, Zou J . Numerical estimation of the Robin coefficient in a stationary diffusion equation. IMA J Numer Anal. 2010;30(3):677-701. · Zbl 1203.65232
[13] Cahlon B, Schochetman I, Shillor M . Convective cooling and optimal placement of electronic components with variable ambient temperature I. The linear model. J Comput Appl Math. 1993;47(3):351-367. · Zbl 0786.65103
[14] Onyango T, Ingham DB, Lesnic D, et al . Determination of a time-dependent heat transfer coefficient from non-standard boundary measurements. Math Comput Simul. 2009;79(5):1577-1584. · Zbl 1169.65091
[15] Slodicka M, Lesnic D, Onyango T . Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem. Inverse Probl Sci Eng. 2010;18(1):65-81. · Zbl 1186.65132
[16] Ditchfield C, Tadini CC, Singh R, et al . Velocity and temperature profiles, heat transfer coefficients and residence time distribution of a temperature dependent Herschel-Bulkley fluid in a tubular heat exchanger. J Food Eng. 2006;76(4):632-638.
[17] Alimoradi A, Veysi F . Prediction of heat transfer coefficients of shell and coiled tube heat exchangers using numerical method and experimental validation. Int J Therm Sci. 2016;107:196-208.
[18] Muraka P, Barrow G, Hinduja S . Influence of the process variables on the temperature distribution in orthogonal machining using the finite element method. Int J Mech Sci. 1979;21(8):445-456. · Zbl 0408.73072
[19] Pacheco CC, Orlande HR, Colaco MJ, et al . Real-time identification of a high-magnitude boundary heat flux on a plate. Inverse Probl Sci Eng. 2016;24(9):1661-1679. · Zbl 1348.74216
[20] Manual C . Comsol multiphysics user’s guide. Version 3; 2005. p. 1-622.
[21] Cardiff M, Kitanidis PK . Efficient solution of nonlinear, underdetermined inverse problems with a generalized PDE model. Comput Geosci. 2008;34(11):1480-1491.
[22] Brito R, Carvalho S, Silva SLE . Experimental investigation of thermal aspects in a cutting tool using comsol and inverse problem. Appl Therm Eng. 2015;86:60-68.
[23] Sousa J, Villafane L, Lavagnoli S, Paniagua G . Inverse heat flux evaluation using conjugate gradient methods from infrared imaging. In: 11th International Conference on Quantitative Infrared Thermography. Naples, Italy; 2012.
[24] Li R, Fang L, Deng Y, et al . Multi-parameter inverse analysis research based on Comsol Multiphysics and Matlab. In: 2010 International Conference on Computer Design and Applications (ICCDA). Vol. 2. IEEE, V2-164; 2010.
[25] Onyango TT, Ingham DB, Lesnic D . Restoring boundary conditions in heat conduction. J Eng Math. 2008;62(1):85-101. · Zbl 1153.80005
[26] Onyango T, Ingham DB, Lesnic D . Reconstruction of heat transfer coefficients using the boundary element method. Comput Math Appl. 2008;56(1):114-126. · Zbl 1145.65327
[27] Jin B, Lu X . Numerical identification of a Robin coefficient in parabolic problems. Math Comput. 2012;81(279):1369-1398. · Zbl 1255.65170
[28] Cui M, Yang K, Xu X-L, et al . A modified Levenberg-Marquardt algorithm for simultaneous estimation of multi-parameters of boundary heat flux by solving transient nonlinear inverse heat conduction problems. Int J Heat Mass Transfer. 2016;97:908-916.
[29] Jiang D, Talaat TA . Simultaneous identification of Robin coefficient and heat flux in an elliptic system. Int J Comput Math. 2017;94(1):185-196. · Zbl 1375.35165
[30] Abdelhamid T . Simultaneous identification of the spatio-temporal dependent heat transfer coefficient and spatially dependent heat flux using an MCGM in a parabolic system. J Comput Appl Math. 2018;27(328):164-176. · Zbl 1375.65125
[31] Abdelhamid T, Deng X, Chen R . A new method for simultaneously reconstructing the space-time dependent Robin coefficient and heat flux in a parabolic system. Int J Numer Anal Model. 2017;14(6):893-915. · Zbl 1422.65234
[32] Chrysafinos K, Gunzburger MD, Hou LS . Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE. J Math Anal Appl. 2006;323(2):891-912. · Zbl 1259.49045
[33] Reuther JJ, Jameson A, Alonso JJ, et al . Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1. J Aircraft. 1999;36(1):51-60.
[34] Plessix R-E . A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int. 2006;167(2):495-503.
[35] Alifanov OM . Solution of an inverse problem of heat conduction by iteration methods. J Eng Phys Thermophys. 1974;26(4):471-476.
[36] Hansen P, O’Leary D . The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput. 1993;14(6):1487-1503. · Zbl 0789.65030
[37] Stanimirovic IP, Zlatanovic ML, Petkovic MD . On the linear weighted sum method for multi-objective optimization. Facta Acta Univ. 2011;26(4):49-63. · Zbl 1313.90213
[38] Chowdhury S, Dulikravich GS . Improvements to single-objective constrained predator-prey evolutionary optimization algorithm. Struct Multidiscipl Optim. 2010;41(4):541-554. · Zbl 1274.90504
[39] Powell MJD . The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge: University of Cambridge; 2009. p. 1-39. (Cambridge NA Report NA2009/06).
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.