# Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware
## 1 Introduction  
Based on virtual modelling and simulation of physical phenomena, Industry 4.0 solutions aim to integrate interactive virtual worlds with their equivalent physical part (e.g., using digital twins). These solutions enable the development of decision making tools that can be of great use in the optimization of manufacturing processes. In this context, engineering solutions use extensively Finite Element Analysis (FEA) for simulation of such physical phenomena (e.g., acoustics, heat transfer, structural analysis, fluid flow, etc.). However, FEA approaches require a great amount of computation resources. In contrast, spectral analysis and spectral methods are competitive alternatives to numerical simulations. These methods provide frequency-domain solutions (infinite sum of trigonometric functions) to the Partial Difference Equations (PDEs) that model such physical phenomena. In the particular sub-domain of laser heating/cutting simulation, frequency-based algorithms have been developed for heat transfer analysis on rectangular plates. These algorithms are faster than traditionalnumerical methods (such as Finite Element Methods) at the cost of some model simplifications. In addition, these methods provide some advantages over FEA, such as allowing to zoom into asynchronous time intervals without computing or storing the complete history of the solution. This property makes frequency-based algorithms more adequate for decision making tools that require rapid response times, allowing to be more flexible towards changes in the heating/cutting manufacturing process. Fast simulation of the laser heating/cutting problem is very important for different engineering problems, such as tool path planning, laser parameter optimization, waste and resources optimization, and so forth. Moreover, interactive simulation and visualization of laser machining processes contributes to many different challenges and opportunities currently present in the Industry 4.0 framework [1]. The aforementioned methods for laser heating/cutting simulation allow simulation of complex laser trajectories on rectangular plates, including parametric trajectories and the introduction of multiple laser beams simultaneously. However, there are no Fast Fourier Transform (FFT)-based solutions to the laser heating/cutting problem in the current state of the art. The FFT is a widely used algorithm not only in the context of PDEs simulation, but also in other areas such as signal analysis and image processing. Thus, its development has been refined and studied extensively in the literature. Several FFT algorithms exist in the literature that further optimize the computation in function of the input signal properties (e.g., symmetry, real/imaginary, size, etc.). In general, the FFT is a key algorithm that retrieves the original spatial-based solution by performing a factorization of the Discrete Fourier Transform (DFT) and avoids redundant computations, reducing the computational complexity of the original DFT problem [2]. This article presents four different schemes that cast the laser heating/cutting problem into DST (Discrete Sine Transform) and DFT (Discrete Fourier Transform) problems. Such casting enables the use of FFT libraries to implement these schemes. The test results show a significant improvement over existing methods in the computational time, both in CPU and GPU, due to the computational complexity reduction. This manuscript is an extension of the work presented in Reference [3], where only two schemes were briefly introduced for the FFT computation of the laser heating problem. The current research discusses in more detail each of the four schemes, including mathematical and algorithmic descriptions but also the intuition behind the schemes followed by illustrations. Furthermore, a different simulation case is designed and tested. Finally, the presented schemes are in the process of being applied in an Industry 4.0 application prototype.