The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

Abstract

Abstract. An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.

Missing boundary data reconstruction for the heat equation

“…This can be observed by the numerous theoretical and numerical methods introduced to solve this problem. Indeed, several approaches have been investigated for many types of equations, we mention the methods based on the fundamental solutions, we refer to Marin et al, 3 the domain decomposition techniques, we can cite previous works 4,5 and Ben Belgacem and El Fekih, 6 quasi reversibility methods, see Bourgeois and Chesnel, 7 iterative regularizing approaches, see Cemetière et al, 8 another process based on the minimization of the energy‐like functional, we cite the recent paper of Caubet and Dardé 9 and Andrieux et al 10 …”

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

“…A typical situation in applications is when some physical condition is known to hold throughout the boundary of a body but additional data needed can only be measured on a portion due to, for example, a hostile environment. Examples in thermoelasticity where such incomplete data render ill‐posed problems are given in [].…”