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Based on the recently developed MFS-based particular solution for two-dimensional problems in isotropic linear thermoelastic materials [1],we investigate the following inverse problems associated with two-dimensional isotropic linear thermoelastic solids:(i)Inverse boundary value problems,i.e.the numerical reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary in the case of two-dimensional linear isotropic thermoelastic materials from over-prescribed noisy measurements taken on the remaining accessible boundary [2,3,4].(ⅱ)Inverse geometric problem,i.e.the numerical reconstruction of smooth star-shaped voids contained in a two-dimensional isotropic linear thermoelastic material from the knowledge of a single non-destructive measurement of the temperature field and both the displacement and traction vectors(over-specified boundary data)on the outer boundary [5].The stabilisation of the former inverse problem is achieved by using several singular value decomposition(SVD)-based regularization methods,such as the Tikhonov regularization method [6],the damped SVD and the truncated SVD [7],whilst the optimal regularization parameter is selected according to Morozovs discrepancy principle [8],the generalized cross-validation criterion [9] and Hansens L-curve method [10].The latter inverse problem reduces to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data,penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void.The interior source points are anchored to and move with the void during the iterative reconstruction procedure.In the method adopted herein the interior source points are free to move during the reconstruction procedure which is a further novel feature compared with earlier approaches.This is a joint work with Andreas Karageorghis(Department of Mathematics and Statistics,University of Cyprus,1678 Nicosia,Cyprus)and Daniel Lesnic(Department of Applied Mathematics,University of Leeds,Leeds LS2 9JT,UK).