## 4 Numerical Experiments  
In this section, the implementation of (25) is presented and numerical experiments are discussed. For the minimization process  
\[\min_{\mathbf{L}\in\mathbb{R}^{n}_{+}}||F_{n}(\mathbf{L})-\mathbf{\varphi}_{\mathbf{\hat{T}}, \text{meas}}||_{2}^{2},\]  
the function _lsqnonlin_ from the MATLAB Optimization Toolbox is used, which provides a solver for nonlinear least-square problems by utilizing a reflective trust-region algorithm. The function requires an initial value \(\mathbf{L}_{0}\in\mathbb{R}^{n}_{+}\). Optionally, also a lower and upper bound (\(\mathbf{L}_{\text{lb}},\mathbf{L}_{\text{ub}}\in\mathbb{R}^{n}_{+}\)) can be specified, where the order relation is to be understood component-wise. The reader is referred to the _lsqnonlin_ documentation website for further details.  
The following numerical simulations investigate the one-layered, the two-layered, and the three-layered system, i.e., \(n\in\{1,2,3\}\) coating layers of different paints on top of a thermally thick aluminum substrate material modeling the painting line in the automotive industry. The thermophysical properties of the included materials are known and given by Table 1:  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Material**} & **Thermal Diffusivity** & **Thermal Effusivity** \\  & \(\left[\times\mathbf{10}^{-6}\text{ m}^{2}\text{s}^{-1}\right]\) & \(\left[\times\mathbf{10}^{3}\text{ Ws}^{1/2}\text{m}^{-2}\text{K}^{-1}\right]\) \\ \hline Paint layer 1 & 0.210 & 3.163 \\ Paint layer 2 & 0.222 & 1.570 \\ Paint layer 3 & 0.354 & 1.243 \\ \hline Substrate layer (aluminum) & 93.3 & 24.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermal properties used in the experiments.  
These values are needed in order to be able to determine the thermal diffusion lengths of every material and the effective reflection coefficients on the interfaces.  
In each experiment, the substrate layer is the same, and the coating layers are chosen in ascending order. Therefore, the case \(n=1\) represents the paint layer 1 on the aluminum substrate, whereas for \(n=2\), the paint layer 2 is between the paint layer 1 and the substrate. For \(n=3\), also paint layer 3 is between paint layer 2 and the substrate. For reasons of simplicity, each coating layer possesses the same thickness of \(L_{\text{leexact}}=5\times 10^{-5}\) m (\(i=1,\ldots,n\)), i.e.,  
\[\mathbf{L}_{\text{exact}}=(L_{\text{le exact}},\ldots,L_{\text{ ne exact}})^{T}. \tag{29}\]  
In order to vary the penetration depth of the thermal waves, the excitation frequency \(f\) (and therefore the angular frequency \(\omega=2\pi f\)) can be adjusted. In this context, it is worth noting that it is recommended to use sufficiently small frequencies leading to deeper penetration of all materials in every setting. In order to test whether it is possible to reconstruct the exact layer thicknesses from the phase angle data, the data need to be generated by forward calculation (23), i.e., for given frequencies \(\boldsymbol{\omega}=(\omega_{1},\ldots,\omega_{m})\) one computes  
\[F_{n}(\mathbf{L}_{\text{exact}})=:\boldsymbol{\varphi}_{\mathbf{T},\text{meas }}\in\mathbb{R}^{m}. \tag{30}\]  
For every experiment, 20 initial values are chosen in the interval of \([1,\ 9]\times 10^{-5}\) with equidistant spacing. The results in the following tables show the reconstructed \(L_{\text{lecalculated}},\ i=1,\ldots n\) as well as the average relative error of the phase angles  
\[\overline{\text{error}}(\omega_{1},\ldots,\omega_{m}):=\frac{1}{m}\sum_{j=1}^ {m}\text{error}(\omega_{j}), \tag{31}\]  
where \(\text{error}(\omega_{j})\) is the relative error of the phase angle associated with the \(j\)-th frequency. The relative error is calculated by the absolute value of the difference of the phase angle of the exact thickness and the calculated thickness, divided by the phase angle of the exact thickness.