A stable and well-posed iFEM framework for inverse heat transfer in energy applications

Accurate inverse heat transfer analysis is critical for optimizing energy systems, including renewable technologies and industrial thermal management. This study presents the extension of the Inverse Finite Element Method (iFEM) originally developed for displacement field reconstruction in structural mechanics to inverse heat transfer analysis. The iFEM formulation employs a weighted least-squares variational functional based on Fourier’s Law, which minimizes the difference between numerically computed and experimentally measured temperature gradients. The resulting discrete system matrix is symmetric positive definite once a minimal temperature datum is imposed, ensuring existence, uniqueness, and stability of the reconstruction. The framework eliminates the need for prior knowledge of heat fluxes, thermal properties, or heat transfer coefficients, addressing key limitations in practical thermal diagnostics. A rigorous numerical validation is performed across multiple energy-relevant scenarios, considering multi-mode boundary conditions encompassing conduction, convection, and radiation. Additionally, a Thermal Index (TI) is introduced as a novel diagnostic metric for detecting, localizing, and quantifying thermal anomalies from boundary sensor data. The framework enables real-time full-field thermal monitoring and diagnostics in energy systems.