Inverse solutions of temperature, heat flux and heat source by the Green element method

In two spatial dimensions, inverse heat conduction problems of temperature, heat flux and heat source recovery are solved in homogeneous and heterogeneous media for steady and transient cases by the Green element method (GEM). The formulation of GEM employed is presented in Taigbenu (2012) [27] and it uses a second-order difference expression to approximate the internal normal fluxes and, therefore, gives accuracy comparable to the flux-based formulation. The Tikhonov regularization with the singular value decomposition (SVD) are used to solve in a least square sense the over-determined, ill-conditioned discrete equations arising from the element-by-element implementation of the singular integral equations. With seven numerical examples, the numerical characteristics of the GEM are evaluated for inverse problems where it is required to recover the temperature, heat flux and heat source from available data. In some of the examples, the performance of the formulation is evaluated when random errors are introduced into the measured data. Excellent results are obtained from the simulated numerical examples, and more especially that these results are obtained with coarse grids.