Numerical Algorithms for the Reconstruction of Space-Dependent Sources in Thermoelasticity

TL;DR

This paper explores numerical algorithms, including Landweber and Sobolev gradient methods, for reconstructing space-dependent sources in thermoelastic systems, demonstrating their effectiveness through computational examples.

Contribution

It introduces and compares various numerical methods for inverse source problems in thermoelasticity, including a novel application of the Sobolev gradient approach.

Findings

Sobolev gradient method improves source reconstruction when boundary values are unknown.

Numerical methods successfully reconstruct sources using FEniCSx platform simulations.

Different approaches have varying convergence properties and applicability.

Abstract

This paper investigates the inverse problems of determining a space-dependent source for thermoelastic systems of type III under adequate time-averaged or final-in-time measurements and conditions on the time-dependent part of the sought source. Several numerical methods are proposed and examined, including a Landweber scheme and minimisation methods for the corresponding cost functionals, which are based on the gradient and conjugate gradient method. A shortcoming of these methods is that the values of the sought source are fixed ab initio and remain fixed during the iterations. The Sobolev gradient method is applied to overcome the possible inaccessibility of the source values at the boundary. Numerical examples are presented to discuss the different approaches and support our findings based on the implementation on the FEniCSx platform.