Thermal diffusion recast as energy-density transport: divergence-saliency thermography robust depth indication for nondestructive evaluation

Thermal diffusion governs active thermography nondestructive evaluation (NDE), but many existing indicators remain heuristic and brittle under nonuniform heating, curvature, and low signal-to-noise ratio (SNR). This paper establishes a physics-consistent methodology by recasting the thermal diffusion equation into a conservative transport form for the energy density e = ρcT . By defining an equivalent energy-density velocity field uₑ = q/e = −α ∇ lnT , with q = −k ∇ T and α = k/(ρc) , the diffusion process becomes ∂ₜe + ∇ ·(e uₑ) = S . We show that ∇· uₑ exhibits a robust source–sink signature around subsurface anomalies, and sign-reversal locus forms a narrow band that appears as a distinct contour in divergence maps, referred to as the divergence-saliency interface ring (DSIR). Building on this representation, divergence-saliency thermography is developed through log-time TSR regularization, cross-time saliency aggregation, DSIR-guided boundary reconstruction, and calibrated diffusion-similarity depth reporting. The square-root depth-time form d=γαt_peak is retained, but the prefactor is explicitly treated as a calibrated effective scale. Experiments on flat, curved, L-corner, and honeycomb composite coupons demonstrate improved boundary closure and more stable depth indication than threshold/peak-contrast baselines in well-conditioned cases, while residual, curl, and DSIR-closure diagnostics identify low-confidence regimes near strong geometric transitions and lateral heat leakage.