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Research Papers: Conduction

Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems

[+] Author and Article Information
Krzysztof Grysa

Professor
Chair of Mathematics
Faculty of Management
and Computer Modelling,
Kielce University of Technology,
Kielce 25–314, Poland
e-mail: grysa@tu.kielce.pl

Artur Maciag

Professor
Chair of Mathematics
Faculty of Management
and Computer Modelling,
Kielce University of Technology,
Kielce 25–314, Poland
e-mail: maciag@tu.kielce.pl

Justyna Adamczyk-Krasa

Kielce University of Technology,
Kielce 25–314, Poland
e-mail: justynadamczyk1@interia.pl

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 11, 2013; final manuscript received May 13, 2014; published online June 24, 2014. Assoc. Editor: Wilson K. S. Chiu.

J. Heat Transfer 136(9), 091302 (Jun 24, 2014) (9 pages) Paper No: HT-13-1471; doi: 10.1115/1.4027770 History: Received September 11, 2013; Revised May 13, 2014

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

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Figures

Grahic Jump Location
Fig. 1

Solution obtained in Ref. [49] for (a) t = 0.4, t = 0.7, t = 1, (b) t = 1.4, t = 1.7, t = 2

Grahic Jump Location
Fig. 2

Exact (solid line) and approximate solution (dashed line) obtained by means of Trefftz functions for (a) t = 0.4, (b) t = 0.7, (c) t = 1, (d) t = 1.4, (e) t = 1.7, and (f) t = 2

Grahic Jump Location
Fig. 3

Relative error of approximation of the boundary condition T(1, t) for (a) ε = 0.1, (b) ε = 0.2, and (c) ε = 0.3

Grahic Jump Location
Fig. 4

Exact solution and the approximate one obtained by means of Trefftz functions for M = N = 10 and (a) t = 0.3, (b) t = 0.6, and (c) t = 1

Grahic Jump Location
Fig. 5

Relative error of approximation of the boundary condition T(1, t) (disturbed data) for (a) ε = 0.1, (b) ε = 0.2, and (c) ε = 0.3

Grahic Jump Location
Fig. 6

Relative error of approximation of the boundary condition T(1, y, t) for (a) ε = 0 (direct problem), (b) ε = 0.2, and (c) ε = 0.3

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