[Paper Review] A procedure for detecting hidden surface defects in a plate from real thermal data by means of active thermography

Gabriele Inglese, Roberto Olmi+1 more|arXiv (Cornell University)|Mar 9, 2017

Numerical methods in inverse problems10 references3 citations

TL;DR

This paper presents a formal analytical procedure for detecting hidden surface defects in metallic plates using active thermography. By combining domain derivative theory and thin plate approximation, it derives an explicit inversion formula to reconstruct subsurface damage from real thermal data, validated through synthetic simulations and a real laboratory experiment with high accuracy in defect localization and depth estimation.

ABSTRACT

Let $Ω_ε$ be a metallic plate whose top inaccessible surface has been damaged by some chemical or mechanical agent. We heat the opposite side and collect a sequence of temperature maps $u^ε$. Here, we construct a formal explicit approximation of the damage $εθ$ by solving a nonlinear inverse problem for the heat equation in three steps: (i) smoothing of temperature maps, (ii) domain derivative of the temperature, (iii) thin plate approximation of the model and perturbation theory. Our inversion formula is tested with realistic synthetic data and used in a real laboratory experiment.

Motivation & Objective

• To develop a non-invasive, non-destructive method for detecting hidden surface defects in metallic plates using thermal imaging.
• To address the challenge of reconstructing inaccessible surface damage from measured thermal contrast on the accessible side of a plate.
• To provide a formal, explicit approximation of the damage profile εθ using perturbation theory and domain derivative techniques.
• To validate the method with realistic synthetic data and real experimental thermal measurements.

Proposed method

• The method begins with smoothing of measured temperature maps to reduce noise and improve signal quality.
• It applies domain derivative theory to linearize the heat equation's response to small perturbations in the boundary condition caused by damage.
• A thin plate approximation is constructed by expanding the temperature field in powers of plate thickness, enabling asymptotic analysis of the thermal response.
• The inverse problem is solved order-by-order using perturbation theory, yielding explicit formulas for the damage function εθ at first and higher orders.
• The reconstruction formula relates the thermal contrast on the heated side to the damage depth and shape via the domain derivative and boundary flux terms.
• The method is tested using both synthetic data with known defects and real thermal data from a laboratory experiment with a 1000W spotlamp heating source.

Experimental results

Research questions

• RQ1 Can a formal analytical approximation be derived for the inverse problem of reconstructing hidden surface defects from thermal data in a thin metallic plate?
• RQ2 How accurately can the domain derivative and thin plate approximation methods reconstruct the depth and shape of subsurface damage from real thermal measurements?
• RQ3 What is the impact of non-uniform heat flux on the reconstruction accuracy, and can a constant equivalent flux be used as a valid approximation?
• RQ4 How does the proposed method perform in distinguishing between different types of surface damage, such as uniform corrosion or pitting, using thermal contrast?

Key findings

• The proposed method successfully reconstructs the damage profile εθ with high spatial resolution using only thermal contrast data from the accessible side of the plate.
• The first-order thin plate approximation provides an explicit analytical formula for εθ₀(x,t) in terms of the measured temperature gradient and boundary flux, enabling direct inversion.
• Higher-order terms (εθ₁, εθ₂) improve reconstruction accuracy by capturing second-order thermal effects, particularly in regions with strong thermal gradients.
• The method demonstrates robustness under non-uniform heat flux conditions, with the reconstructed damage profile closely matching the true defect shape when using a constant equivalent flux approximation.
• In the real laboratory experiment, the method correctly identified the presence, location, and approximate depth of a hidden surface defect, confirming its practical applicability.
• The reconstruction error was minimized through smoothing of temperature maps and careful handling of boundary conditions, especially at the top surface where damage occurs.