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ARTICLE

Inverse Green Element Solutions of Heat Conduction Using the Time-Dependent and Logarithmic Fundamental Solutions

Akpofure E. Taigbenu¹

1 School of Civil and Environmental Engineering, University of the Witwatersrand. P. Bag 3, Johannesburg, WITS 2050. South Africa. E-mail: akpofure.taigbenu@wits.ac.za

Computer Modeling in Engineering & Sciences 2014, 102(4), 271-289. https://doi.org/10.3970/cmes.2014.102.271

Abstract

The solutions to inverse heat conduction problems (IHCPs) are provided in this paper by the Green element method (GEM), incorporating the logarithmic fundamental solution of the Laplace operator (Formulation 1) and the timedependent fundamental solution of the diffusion differential operator (Formulation 2). The IHCPs addressed relate to transient problems of the recovery of the temperature, heat flux and heat source in 2-D homogeneous domains. For each formulation, the global coefficient matrix is over-determined and ill-conditioned, requiring a solution strategy that involves the least square method with matrix decomposition by the singular value decomposition (SVD) method, and regularization by the Tikhonov regularization method. Comparisons of the two formulations are made using five numerical examples of transient IHCPs. Using the same spatial and temporal discretizations, the GEM with the logarithmic fundamental solution is generally more superior in accuracy and computational speed than the formulation with the time-dependent fundamental solution.

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