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# The Inverse Problem of Determining Heat Transfer Coefficients by the Meshless Local Petrov-Galerkin Method

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia, e-mail: sladek@savba.sk
Department of Civil Engineering, Queen Mary University of London, Mile End, London E14NS, U.K.
Department of Mathematics, City University of Hong Kong, Hong Kong

Computer Modeling in Engineering & Sciences 2009, 48(2), 191-218. https://doi.org/10.3970/cmes.2009.048.191

## Abstract

The meshless local Petrov-Galerkin (MLPG) method is used to solve the inverse heat conduction problem of predicting the distribution of the heat transfer coefficient on the boundary of 2-D and axisymmetric bodies. Using this method, nodes are randomly distributed over the numerical solution domain, and surrounding each of these nodes, a circular sub-domain is introduced. By choosing a unit step function as the test function, the local integral equations (LIE) on the boundaries of these sub-domains are derived. To eliminate the time variation in the governing equation, the Laplace transform technique is applied. The local integral equations are nonsingular and take a very simple form. The spatial variations of the Laplace transforms of temperature and heat flux are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The truncated singular value decomposition (TSVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after the MLS approximation. The Stehfest algorithm is then applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions. The effectiveness of the MLPG method for this inverse problem is demonstrated by some numerical examples.

## Keywords

Sladek, J., Sladek, V., Wen, P., Hon, Y. (2009). The Inverse Problem of Determining Heat Transfer Coefficients by the Meshless Local Petrov-Galerkin Method. CMES-Computer Modeling in Engineering & Sciences, 48(2), 191–218. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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