×

zbMATH — the first resource for mathematics

Numerical reconstruction of a non-smooth heat flux in the inverse radial heat conduction problem. (English) Zbl 1448.65141
Summary: When gun barrels are multiple fired, monitoring the non-smooth heat flux of the gun with the additional input data makes it possible to identify the corrosion of the gun. Based on this background, we study the identification of the non-smooth heat flux in a 1-dimensional heat transfer system. Aiming to overcome the non-smoothness of the unknown heat flux and to simplify the choice of regularization parameter, we transform the inverse problem to a constraint optimization problem and prove the existence and convergence properties of the regularization solution. Then an algorithm based on minimizing the data-match term and penalty term alternatively is proposed. Finally, a numerical experiment supports our regularization scheme quantitatively, especially for input data with high noise level.
MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
80M50 Optimization problems in thermodynamics and heat transfer
65K10 Numerical optimization and variational techniques
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
35K05 Heat equation
49J20 Existence theories for optimal control problems involving partial differential equations
49N45 Inverse problems in optimal control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zhang, L. H.; Chen, Z. Z.; Wen, D. H., Estimation of the time-varying high-intensity heat flux for a two-layer hollow cylinder, Energies, 11, 12, 3332 (2018)
[2] Cheng, W., Stability estimate and regularization for a radially symmetric inverse heat conduction problem, Bound. Value Probl., 2017, 53 (2017) · Zbl 1360.65228
[3] Seidman, T. I.; Eldén, L., An ’optimal filtering’ method for the sideways heat equation, Inverse Problems, 6, 4, 681-696 (1990) · Zbl 0726.35053
[4] D. Lesnic, L. Elliott, D.B. Ingham, Boundary element method for solving an inverse problem in radial heat conduction, in: Proceedings of the ASME/JSME Thermal Engineering Joint Conference, Hawaii, USA, 19-24 March (1995) pp. 71-78.
[5] Wen, S.; Qi, H.; Li, Y., An on-line extended kalman filtering technique for reconstructing the transient heat flux and temperature field in two-dimensional participating media, Int. J. Therm. Sci., 148, Article 106069 pp. (2020)
[6] Tautenhahn, U., Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19, 377-398 (1998) · Zbl 0907.65049
[7] Garshasbi, M.; Dastour, H., Estimation of unknown boundary functions in an inverse heat conduction problem using a mollified marching scheme, Numer. Algorithms, 68, 4, 769-790 (2014) · Zbl 1312.65151
[8] Hon, Y. C.; Wei, T., A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28, 5, 489-495 (2004) · Zbl 1073.80002
[9] Eldén, L.; Berntsson, F.; Regińska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21, 6, 2187-2205 (2000) · Zbl 0959.65107
[10] Regińska, T.; Eldén, L., Solving the sideways heat equation by a wavelet-Galerkin method, Inverse Problems, 13, 4, 1093-1106 (1997) · Zbl 0883.35123
[11] Wróblewska, A.; Frackowiak, A.; Cialkowski, M., Regularization of the inverse heat conduction problem by the discrete Fourier transform, Inverse Probl. Sci. Eng., 24, 2, 195-212 (2016)
[12] Qian, Z.; Zhang, Q., Differential-difference regularization for a 2d inverse heat conduction problem, Inverse Problems, 26, 9, Article 095015 pp. (2010) · Zbl 1200.35332
[13] Elkins, B. S.; Keyhani, M.; Franke, J. I., Global time method for inverse heat conduction problem, Inverse Probl. Sci. En., 20, 5, 651-664 (2012) · Zbl 1253.80003
[14] Xiong, X. T.; Hon, Y. C., Regularization error analysis on a one-dimensional inverse heat conduction problem in multilayer domain, Inverse Probl. Sci. Eng., 21, 5, 865-887 (2013) · Zbl 1308.65161
[15] Chapko, R.; Johansson, B. T.; Vavrychuk, V., A projected iterative method based on integral equations for inverse heat conduction in domains with a cut, Inverse Problems, 29, 6, Article 065003 pp. (2013) · Zbl 1273.65130
[16] Xie, J. L.; Zou, J., Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43, 4, 1504-1535 (2005) · Zbl 1101.65097
[17] Acar, R.; Vogel, C. R., Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10, 1217-1229 (1994) · Zbl 0809.35151
[18] Jin, B. T.; Lu, X. L., Numerical identification of a robin coefficient in parabolic problems, Math. Comp., 81, 279, 1369-1398 (2012) · Zbl 1255.65170
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.