Linear Heat Diffusion Inverse Problem Solution with Spatio-Temporal Constraints for 3D Finite Element Models

High-voltage ceramic insulators are routinely exposed to short-duration overvoltages such as lightning impulses, switching surges, and partial discharges. These events occur on microsecond to millisecond timescales and can produce highly localized thermal spikes that are difficult to measure directly but may compromise long-term material integrity. This paper addresses the estimation of the internal temperature distribution immediately after a lightning impulse by solving a three-dimensional inverse heat conduction problem (IHCP). The forward problem is modeled by the transient heat diffusion equation with constant thermal diffusivity, discretized using the finite element method (FEM). Surface temperature measurements are assumed available from a 12 kV ceramic post insulator and are used to reconstruct the unknown initial condition. To address the ill-posedness of the IHCP, a spatio-temporal regularization framework is introduced and compared against spatial-only regularization. Numerical experiments investigate the effect of measurement time ($T=60$ s, 600 s, and 1800 s), mesh resolution (element sizes of 20 mm, 15 mm, and 10 mm), and measurement noise ($\sigma =1$ K and 5 K). The results show that spatio-temporal regularization significantly improves reconstruction accuracy and robustness to noise, particularly when early-time measurements are available. Moreover, it is observed that mesh refinement enhances accuracy but yields diminishing returns when measurements are delayed. These findings demonstrate the potential of spatio-temporal IHCP methods as a diagnostic tool for the condition monitoring of ceramic insulators subjected to transient electrical stresses.