Active Infrared Thermography in Non-Destructive Evaluation of Surface Corrosion 2: Heat Exchange Between Specimen and Environment

Comparing Laplace and Helmholtz approaches to identify the rear-surface profile of a plate by means of periodic thermal data

“…The Domain Derivative is a mathematical tool introduced in [10] and frequently used to approach inverse problems (see for example [2], [11], [12]). If we define W ¼ u 0 , we can show (see [13]) that W solves Equation (2) in 0 with the following boundary conditions (x 2 [0, 1], y 2 [0, a] and t 4 t 0 ):…”

“…When goes to zero, the Robin boundary conditions, Equations (13), (14), degenerate to the Neumann ones and the Laplace equation becomes clearly incompatible with the positive flux in Equation (14) (for a mathematical proof, see for example [6], Chapter 3). After consideration of the Biot number (see for example [2]) this situation is very frequent for metals in low conducting materials like air. This is a first rough reason for disregarding the Laplace equation and concentrating our attention on the Helmholtz equation, which accounts for the frequency of the forcing term .…”

“…This flexibility comes from the fact that the Helhmoltz model takes directly into account the period of the heating lamp which, as it was pointed out in [2], is a key factor in active infrared thermography (the attribute 'active' here stresses the fact that the flux is produced ad hoc by somebody); Figure 3 shows results for aluminium with different choices of the period. From Equation (20) we obtain the value 0 ¼ 4.71 s for the period of the heating lamp (in case of an aluminium plate of thickness a ¼ 0.01) and we reconstruct function using 2 0 (Figure 3(a)), 0 (Figure 3(b)), 0.5 0 (Figure 3(c)) and 0.25 0 (Figure 3(d )); it is evident that the quality of reconstruction decreases with the period.…”

“…This task requires the numerical solution of an inverse problem for the heat equation (see for example [1][2][3][4] and [5], Chapter 8). In this paper we deal with the formulation of a specific mathematical model, which accounts for the available a priori information about the specimen and the defect.…”

Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

“…In many applications I (t) represents: (i) the boundary of a cavity or a corroded part of Ω (t), [21]- [28], [47], [48], [72], [76], [115], [138], [143] or (ii) a privileged isothermal surface, such as a solidification front of Ω (t), that is not accessible to direct inspection, [13], [19], [20], [39], [55], [71], [117], [129], [131].…”