The generalized finite difference method for an inverse boundary value problem in three-dimensional thermo-elasticity

In this study, a new framework for the efficient and accurate solutions for an inverse problem associated with three-dimensional (3D) coupled thermo-elasticity equation is presented. The ill-conditioned problem is solved here with the generalized finite difference method (GFDM) together with the first-order Tikhonov regularization technique and the L-curve criterion. The GFDM uses the Taylor series expansions and the moving least squares approximation to derive explicit formulae for the required partial derivatives of unknown variables. The method is truly meshless that can be applied for solving problems merely defined over irregular clouds of points. In addition, for problems involving complex geometries, a new distance criterion for adaptive selection of nodes in the GFDM simulations is proposed. Preliminary numerical experiments show that the regularized GFDM proposed in this study are very promising for accurate and efficient inverse thermo-elasticity simulations, even with a comparatively large level of noise added into the input data.