# A Method of Defect Depth Estimation for Simulated Infrared Thermography Data with Deep Learning
## 2 Thermal Consideration and FEM Stimulation
### Finite Element Modeling with Transient Heat Transfer  
Finite Element Modelling (FEM: COMSOL, etc.) [12] has become an important and economical platform to evaluate the thermal response of pulsed thermography, which builds up models for the platform of components that allow us to flexibly examine all specific physical aspects of thermal data such as geometries and properties of materials. Simulated thermograms matched well with experimental data. In this work, COMSOL will be utilized as a 3D based simulation to build up models for the synthetic data to provide for depth estimation of artificial defects. A Carbon Fiber Reinforced Polymer (CFRP) structure-based specimen containing artificial defects of different shapes (flat-bottom holes) is modeled as shown in Figure 2. A high thermal pulse is projected on the surface of specimens. Due to the existence of the temperature gradients in the sample, a thermal front propagates from the high temperature region on the surface to the region underneath. A delamination or discontinuities create a lower thermal diffusion rate to the heat flow and then reflect the unnormal thermal patterns on the surface. The COMSOL software is utilized as the heat transfer simulation model for obtaining the temperature evaluations of the surface of each sample, as indicated below [13]:  
\[\rho C_{P}\left(\frac{\partial T}{\partial t}\right)-\ \
abla\cdot(k\
abla T)=0 \tag{2}\]  
\[\frac{\partial}{\partial x}\left(w_{x}\frac{\partial T}{\partial x}\right)+ \frac{\partial}{\partial y}\left(w_{y}\frac{\partial T}{\partial y}\right)+ \frac{\partial}{\partial z}\left(w_{z}\frac{\partial T}{\partial z}\right)= \rho C_{P}\frac{\partial T}{\partial t} \tag{3}\]  
\[n\cdot(w\cdot\
abla T)=h_{conv}(T_{amb}-T)+\sigma_{Tc}\varepsilon\left(T_{amb }^{4}-T^{4}\right) \tag{4}\]  
where in Equations (2)-(4), the density is \(\rho\left(\frac{kg}{m^{3}}\right)\), constant specific heat is \(c_{\rho}\left(\frac{J}{kg\cdot m}\right)\), and the absolute temperature is T(K). Time variable is set as t(s). A rectangular coordinate system (x, y, z) as in Equation (3) in anisotropic media can lead to numerous possible solutions. \(w\), \(w_{x}\), \(w_{y}\), \(w_{z}\left(\frac{w}{m}\cdot K\right)\) are the main conductivity rates and conductivity rates which respectively for three coordinates (x, y, z) in 3D thermal modeling. The convection heat transfer is given by \(h_{conv}\left(\frac{w}{m^{2}k}\right)\). In Equation (4), a boundary condition that regarding to the thermal heat transfer from radiation and convection between the all the specimen surfaces and the ambient temperature has been indicated, where \(T_{amb}\) is an initial environment temperature which consider all the external environment as the same temperature; \(\sigma_{Tc}\left(\frac{w}{m^{2}k^{2}}\right)\) is the Stefan-Boltzmann physical constant which links the temperature with energy; \(\varepsilon\) represents surface emissivity. Figure 2: Heat transfer during the thermal excitation and cooling process during FEM stimulation. COMSOL's heat transfer module are adapted in this work which uses the 3D transient heat equation and provides the specimen temperature distribution, to simulate the specimen's thermal response after a heat pulse is applied to the specimen surface. A Gaussian power density distribution was employed to model the heat source, which closely approximates a real heat pulse from two flashes, i.e., a slightly larger heat accumulation at the center of the specimen. The heat pulse is applied as a radiative source that is partly absorbed by the specimen surface and transferred inside the specimen by heat conduction. Table 1 briefly illustrates the physical properties of CFRP specimens and pulsed thermography parameters involved in the COMSOL simulation in this experiment. All properties and parameters in Table 1 are cited from the literature [14]. In this table, \(T_{i}\) is the initial temperature of an experimental specimen that was texted by PT.