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Technical Brief

On Elliptic Inverse Heat Conduction Problems

[+] Author and Article Information
M. Tadi

Department of Mechanical Engineering,
University of Colorado at Denver,
Campus Box 112,
P.O. Box 173364,
Denver, CO 80217-3364
e-mail: mohsen.tadi@ucdenver.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 13, 2016; final manuscript received February 7, 2017; published online April 4, 2017. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 139(7), 074504 (Apr 04, 2017) (4 pages) Paper No: HT-16-1660; doi: 10.1115/1.4036006 History: Received October 13, 2016; Revised February 07, 2017

This note is concerned with a new method for the solution of an elliptic inverse heat conduction problem (IHCP). It considers an elliptic system where no information is given at part of the boundary. The method is iterative in nature. Starting with an initial guess for the missing boundary condition, the algorithm obtains corrections to the assumed value at every iteration. The updating part of the algorithm is the new feature of the present algorithm. The algorithm shows good robustness to noise and can be used to obtain a good estimate of the unknown boundary condition. A number of numerical examples are used to show the applicability of the method.

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References

Hensel, A. , and Hills, R. , 1987, “ Steady-State Two-Dimensional Inverse Heat Conduction,” Inverse Probl. Sci. Eng., 15(2), pp. 227–240.
Taler, J. , and Duda, P. , 2006, Solving Direct and Inverse Heat Conduction Problems, Springer-Verlag, Berlin.
Hao, D. H. , and Dinh, N. H. , 1998, Methods for Inverse Heat Conduction, Peter Lang GmbH, Frankfurt, Germany.
Lattes, R. , and Lions, J. , 1960, Method of Quazi-Reversibility: Application to Partial Differential Equations, Elsevier, Amsterdam, The Netherlands.
Wu, X.-H. , and Tao, W.-Q. , 2008, “ Meshless Method Based on the Local Weak-Forms for Steady-State Heat Conduction Problems,” Int. J. Heat Mass Transfer, 51(11–12), pp. 3103–3112. [CrossRef]
Gu, Y. , Chen, W. , and He, X.-Q. , 2012, “ Singular Boundary Method for Steady-State Heat Conduction in Three Dimensional General Anisotropic Media,” Int. J. Heat Mass Transfer, 55(17–18), pp. 4837–4848. [CrossRef]
Kanjanakijkasem, W. , 2015, “ A Finite Element Method for Prediction of Unknown Boundary Conditions in Two-Dimensional Steady-State Heat Conduction Problems,” Int. J. Heat Mass Transfer, 88, pp. 891–901. [CrossRef]
Wroblewska, A. , Frackowiak, A. , and Cialkowski, M. , 2016, “ Regularization of the Inverse Heat Conduction Problem by the Discrete Fourier Transform,” Inverse Probl. Sci. Eng., 24(2), pp. 195–212. [CrossRef]
Mohebbi, F. , and Sellier, M. , 2016, “ Parameter Estimation in Heat Conduction Using a Two-Dimensional Inverse Analysis,” Int. J. Comput. Methods Eng. Sci. Mech., 17(4), pp. 274–287. [CrossRef]
Olson, L. , and Throne, R. , 2000, “ A Comparison of Generalized Eigensystem, Truncated Singular Value Decomposition, and Tikhonov Regularization for the Steady Inverse Heat Conduction Problem,” Inverse Probl. Eng., 8(3), pp. 193–227. [CrossRef]
Hao, D. N. , Thanh, P. H. , Lesnic, D. , and Johansson, B. T. , 2012, “ A Boundary Element Method for a Multi-Dimensional Inverse Heat Conduction Problem,” Int. J. Comput. Math., 89(11), pp. 1540–1554. [CrossRef]
Tadi, M. , and Sritharan, S. S. , 2012, “ Identification of Far-Field Electric Field Based on Near Field Distributed Measurements,” Int. J. Comput. Appl. Math., 7(3), pp. 235–249.
Hamad, A. , and Tadi, M. , 2016, “ A Numerical Method for Inverse Source Problems for Possion and Helmholtz Equations,” Phys. Lett. A, 380(44), pp. 3707–3716. [CrossRef]
Tadi, M. , Nandakumaran, A. K. , and Sritharan, S. S. , 2011, “ An Inverse Problem for Helmholtz Equation,” Inverse Probl. Sci. Eng., 19(6), pp. 839–854. [CrossRef]
Wang, Y. , Tadi, M. , and Radenkovic, M. , “ A Numerical Method for an Inverse Problem for Helmholtz Equation With Separable Wave Number,” Int. J. Comput. Sci. Math. (in press).

Figures

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Fig. 1

A 2D domain for the error field with the over-specified boundary conditions, ey|y=0=∇nu−∇nû=η1(x)

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Fig. 2

The reduction in error for the first example as a function of the number of iterations. Only one set of data is used. The error is the difference between the given data and the computed normal derivative at the boundaries, i.e., ∫[∇nu(x)−∇nû(x)]2dx, for x ∈ Γ2.

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Fig. 3

The recovered function for Example 1 at different iterations. The figure also compares them to the actual function.

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Fig. 4

A cylindrical geometry

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Fig. 5

The reduction in error for Example 2 as a function of the number of iterations

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Fig. 6

The recovered function for Example 2 at different iterations. The figure also compares them to the actual function.

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Fig. 7

The reduction in error for Example 3 as a function of the number of iterations

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Fig. 8

The recovered function for Example 3 at different iterations. The figure also compares them to the actual function.

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