Least-squares stabilized collocation method for the parameter identification in transient inverse heat conduction problems

Abstract

The inverse heat conduction problem (IHCP) has significant applications across multiple disciplines. Traditional methods for IHCPs often require tedious and low-accuracy iteration, which frequently fails to meet engineering demands. Therefore, developing highly efficient and accurate methods for IHCP solutions is required. A novel meshfree least-squares stabilized collocation method (LSCM) for solving transient IHCPs is proposed in this paper. LSCM uses subdomain integration to incorporate physical information on Gauss points into the collocation equations, which enhances accuracy and stability of solution. The least-squares scheme is implemented to avoid the iteration process for overdetermined problems arising from multiple measurement conditions. To address the non-linear characteristics of transient IHCPs, an integral transformation that linearizes the governing equations is introduced in LSCM, which avoids the iteration for nonlinear problems and improves efficiency. Convergence analysis of the proposed LSCM indicates that optimal accuracy can be achieved with appropriate weights on the boundaries and additional conditions. Stability analysis demonstrates that the LSCM provides robust stability for the time integration process of IHCPs. Dispersion analysis shows that the LSCM possesses small dispersion errors and good stability. Finally, numerical examples demonstrate that the proposed LSCM yields reliable parameter solutions even when input data includes up to 10 % and 20 % noise, with convergence rates reaching 1.5 and 0.7, respectively (compared to 1.8 without noise). Additionally, the identification of a source control parameter in a 3D cooling tower model further showcases the method's strong capability in handling complex engineering scenarios.