# Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware
## 4 Results
### Numerical Validation  
Section 3 validates the mathematical correctness of the presented schemes. However, a numerical validation is presented in this section with numerical and graphical results for a \(0.01\times 0.01\times 0.001\) rectangular plate. Laser and material parameters are presented in Table 2 while the laser trajectory used for the tests is the same presented in Figure 2a. As a groundtruth, we choose the method presented in References [23; 24; 25]. This method already solves the problem presented in Equation (1) using a brute-force approach, which requires \(\mathcal{O}(M^{2}N^{2})\) operations (as already discussed in Section 3.3. Figure 7 plots the temperature distribution results obtained with this brute-force method.  
Figure 8a plots the temperature distribution at the end of the laser trajectory, computed with the DST algorithm for a \(1024\times 1024\) plate discretization. Figure 8b plots the same result computed with the zero padding FFT algorithm. The absolute error for the DST and the zero padding FFT result (w.r.t. the brute-force approach) is presented in Figure 8c,d, respectively. The measured absolute error is below \(10^{-10}\) (K) in both cases. It is worth pointing out that this error is evenly distributed through  
\begin{table}
\begin{tabular}{c l l l l} \hline \hline
**Name** & **Description** & **Number of Operations** & **Dominant Term** & **Complexity Order** \\ \hline
1, 2 & Memory initialization & & & \(\mathcal{O}(MN)\) \\
4 & FFT of a column & \(2M+2M\log(2M)\) & \(2M\log(2M)\) & \(\mathcal{O}(M\log M)\) \\
5 & Extract complex part & \(2M\) & & \(\mathcal{O}(M)\) \\
3-6 & Loop through columns & \((N-2)(4M+2M\log(2M))\) & \(2 MN\log(2M)\) & \(\mathcal{O}(MN\log M)\) \\
8 & FFT of a row & \(2N+2N\log(2N)\) & \(2N\log(2N)\) & \(\mathcal{O}(N\log N)\) \\
9 & Extract complex part & \(2N\) & & \(\mathcal{O}(N)\) \\
7–10 & Loop through rows & \((M-2)(4N+2N\log(2N))\) & \(2 MN\log(2N)\) & \(\mathcal{O}(MN\log M)\) \\
11, 12 & Memory initialization & & & \(\mathcal{O}(MN)\) \\
13 & Sum of matrices & \(MN\) & & \(\mathcal{O}(MN)\) \\ \hline \hline
**TOTAL** & & & & \(\mathcal{O}(MN\log M+MN\log N)\) \\  & & & & \(=\mathcal{O}(MN\log(MN))\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational complexity analysis for Algorithm 2 (FFT padded with zeros). Complexity simplification rules applied are: (a) \(\mathcal{O}(kf(n))=\mathcal{O}(f(n))\), (b) \(\mathcal{O}(f(n)+g(n))=\mathcal{O}(\max(f(n),g(n)))\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Value** & **Units** \\ \hline \(a\) & Plate width & \(0.01\) & m \\ \(b\) & Plate height & \(0.01\) & m \\ \(\Delta z\) & Plate thickness & \(0.001\) & m \\ \(\rho\) & Plate density & \(8030\) & kg/m\({}^{3}\) \\ \(c_{p}\) & Specific heat & \(574\) & J/(kg K) \\ \(\kappa\) & Thermal conductivity & \(20\) & W/(m K) \\ \(R\) & Plate reflectivity & \(0\) & \(1\) \\ \(h\) & Convection coefficient & \(20\) & W/(m\({}^{2}\) K) \\ \(u_{\infty}\) & Ambient temperature & \(300\) & K \\ \(p\) & Laser power & \(500\) & W \\ \(r\) & Laser spot radius & \(0.0003\) & m \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the physical simulation.  
the 2D plate, which means that such error is not sensitive to the laser path or any other geometric features (such as the domain boundaries).  
Similarly, Figure 9a,b plot the temperature distributions at the end of the laser trajectory for the 1D symmetric FFT and the 2D symmetric FFT algorithms, respectively. Figure 9c,d plot the absolute error for the 1D symmetric and 2D symmetric FFTs, respectively. Again, the error is below \(10^{-10}\), evenly distributed through the 2D plate.  
Figure 8: Temperature and absolute error distributions (w.r.t. the brute-force approach [24]) on the thin plates for the DST and the zero padding FFT simulations.  
Figure 7: Temperature solution for the laser trajectory presented in Figure 2a obtained by the brute-force method [24]. No FFT or Discrete Sine Transform (DST) is used.