# Fast Analytic Simulation for Multi-Laser Heating of Sheet Metal in GPU

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Multi-Beam Single Trajectory vs. Multiple-Trajectory Simultaneous Laser Heating

#### 2.2. Thermal Simulation for Sheet Metal Laser Heating

#### 2.3. Conclusions of the Literature Review

## 3. Methodology

#### 3.1. Heat Transfer Equation for Multi-Beam Laser Heating

#### 3.2. Analytic Solution

#### 3.3. Algorithm

**Discretize laser trajectories:**As discussed in Section 3.2, the laser beam trajectories ${\overrightarrow{x}}_{0}^{k}\left(t\right)$ are discretized as sequences of piecewise linear trajectories $[{\overrightarrow{x}}_{0}^{k}\left(0\right),{\overrightarrow{x}}_{0}^{k}\left({t}_{1}\right),\cdots ,{\overrightarrow{x}}_{0}^{k}\left({T}_{f}\right)]$, as described in [31]. The only requirement for this discretization is the fact that all laser beam trajectories must share the same time discretization, i.e., ${t}_{0},{t}_{1},\cdots ,{T}_{f}$ are the same for all trajectories ${\overrightarrow{x}}_{0}^{k}$.**Compute the Laplacian eigenvalues:**The Laplacian eigenvalues of the sheet are computed as per Equation (8). Since the eigenvalues ${\omega}_{ij}$ are time-independent, this step is performed before the simulation loop starts.**Initialize time and sheet temperature:**The simulation time is initialized to ${t}_{0}\leftarrow 0$. In order to satisfy the initial temperature condition $u\left(0\right)={u}_{\infty}$, the pseudo coefficients are initialized as ${\theta}_{ij}^{k}\left(0\right)=0$ (see Equation (4)).**Update current time t:**The current simulation time $t={t}_{l+1}$ is updated according to the previous time ${t}_{0}={t}_{l}$, in concordance with the discretization of trajectories from step 1.**For each laser beam k:**This inner loop computes the pseudo-coefficients ${\theta}_{ij}^{k}\left(t\right)$ for each laser beam ($k=1\cdots num\_lasers$).**Question: Is laser beam k turned on?:**This step allows for simulating asynchronous laser beams by asking at the current time t if the laser is turned on/off. Therefore, each laser beam might have its own internal time frame $[{t}_{0}^{k},{t}_{f}^{k}]$, different from the simulation time frame $[0,{T}_{f}]$.**Set power ${P}_{k}$/Set null power ${P}_{k}\leftarrow 0$:**In the previous step, the program checks the state (on/off) of the current laser beam k. The laser is turned on by the simulation by setting its corresponding power input ${P}_{k}$. In the case of the laser being turned off, the simulation simply sets its power to 0.**Question: $k<num\_lasers$?:**Check if the pseudo-coefficients have been computed for all the laser beams.**If required, compute temperature:**The temperature field is computed on a set of discrete points sampled on the sheet $[({x}_{0},{y}_{0}),({x}_{1},{y}_{1})\cdots ]$ as per Equation (4). The number of coefficients used to compute the solution is truncated to $num\_coeffs$. This step is optional since the result may be stored in the frequency domain (${\Theta}_{ij}\left(t\right)$). Therefore, the temperature is made available only when requested by the user, allowing for skipping iterations of no interest and allowing non-monotonic access to the transient temperature history, improving the performance in the process.**$t<{T}_{f}$?:**Check if the simulation has reached the final step.**END SIMULATION**

- In step 1 of the previous algorithm, the curved laser beam trajectories ${\overrightarrow{x}}_{0}^{k}\left(t\right)$ are discretized as piecewise linear ones $[{\overrightarrow{x}}_{0}^{k}\left(0\right),{\overrightarrow{x}}_{0}^{k}\left({t}_{1}\right),\cdots ,{\overrightarrow{x}}_{0}^{k}\left({T}_{f}\right)]$, which inherently produces the time discretization $[0,{t}_{1},\cdots ,{T}_{f}]$. This time discretization does not affect the numerical accuracy of the temperature solution. Therefore, as opposed to FEA (Finite Element Analysis), the time step size $\Delta t={t}_{l+1}-{t}_{l}$ of our algorithm can be arbitrarily large.
- Step 6 allows turning on/off any laser beam at any point of the simulation, allowing complete asynchronicity between the multiple laser beams. In addition, the algorithm allows to change any laser parameters at will, resulting in time-dependent parameters ${P}_{k}\left(t\right)$ (laser power) and ${r}_{k}\left(t\right)$ (spot radius). For simplicity, this manuscript uses constant parameters.
- The complete information of the solution is stored in the frequency domain (step 10) and temperature data is computed only when requested. Therefore, the user requests the temperature only at specific times and in specific zones (i.e., at the middle or the end of the simulation). This frequency-based approach has the advantage of providing non-monotonic access to the temperature history, allowing arbitrary simulation time-location requests. Furthermore, since the space discretization does not affect the solution, any sheet sampling can be used to render the temperature (rectangular grid, triangular mesh, a curve or a single point in the sheet).
- In step 10, each pseudo-coefficient ${\theta}_{ij}^{k}$ is independent from the rest of the pseudo-coefficients (Equation (6)). Similarly, in step 11, the temperature value u at a given point $\overrightarrow{x}$ is independent from the temperature on the rest of the sheet (Equation (4)). Therefore, the computation in both of these steps is parallelized.

## 4. Results

#### 4.1. Comparison with FEA

^{®}Academic Research Mechanical, Release 17.2, is used to perform the FEA simulations. ANSYS

^{®}element SHELL131 is employed. Element thickness and material properties are set as per Table 1. The elements are configured to have a constant temperature along the thickness and to evaluate convection at the sheet surface, as per Equation (1). To represent the area heated by the laser beams at every time step, ANSYS

^{®}surface loads (Heat Fluxes) are applied on the FEA elements that lie inside the heated zone.

#### 4.2. Multiple Laser Beams

#### 4.3. Performance Assessment

^{®}Core

^{TM}i5-6500 (CPU), 8 GB RAM and NVIDIA

^{®}GeForce

^{®}GTX 960 graphics card. Our algorithm is able to simulate any number of laser beams. Figure 9 plots the execution times for the computation of the Fourier coefficients as a function of the number of laser beams. The figure compares the execution times of the CPU against the GPU to compute $512\times 512$ Fourier coefficients. The computational cost increases with a large slope in the CPU approach while being nearly constant in the GPU approach. The more laser beams are added, the more it benefits from GPU parallelization. In addition, the computation of the Fourier coefficients in the GPU is preferable. In this way, there is no need to transfer the coefficients back and forth from host-to-device on each iteration since they always stay in GPU memory. For this analysis, a single time step has been considered instead of the whole simulation.

#### 4.4. Integration within an Interactive Laser Cutting Simulation Environment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEA/FEM | Finite Element Analysis/Finite Element Method. |

GPU | Graphics Processing Unit. |

$a,b,\Delta z$ | Width, height and thickness of the sheet (m). |

$\overrightarrow{x},t$ | Spatial $\overrightarrow{x}=(x,y)\in [0,a]\times [0,b]$ and temporal $0\le t\le {T}_{f}$ (s) coordinates for the simulation. |

$u=u(\overrightarrow{x},t)$ | Temperature distribution along the sheet at any given time (K). |

$\rho $ | Sheet density (kg/m${}^{3}$). |

${c}_{p}$ | Sheet specific heat (J/(kg K)). |

$\kappa $ | Sheet thermal conductivity (W/(m K)). |

R | Sheet reflectivity ($0\le R<1$). |

$q=q\left(u\right)$ | Temperature-dependent heat convection on the sheet surface (W/m${}^{2}$). |

${u}_{\infty}$ | Ambient temperature K. |

h | Natural convection coefficient (W/(m${}^{2}$ K)). |

${f}_{k}$ | Heat input from laser beam k (W/m${}^{2}$). $k=1\cdots num\_lasers$. |

${P}_{k}$ | Power of laser beam k (W). |

${r}_{k}$ | Radius of laser spot k (m). |

${\overrightarrow{x}}_{0}^{k}={\overrightarrow{x}}_{0}^{k}\left(t\right)$ | Laser spot center for laser beam k at time t. |

$[{t}_{0}^{k},{t}_{f}^{k}]$ | Simulation time frame in which the laser beam k remains turned on. $0\le {t}_{0}^{k}<{t}_{f}^{k}\le {T}_{f}$. |

${\overrightarrow{v}}_{k}$ | Scan speed of laser beam k (m/s). |

$F=F(\overrightarrow{x},t)$ | Sum of all laser beam heat sources (W/m${}^{2}$). |

${\mathbb{X}}_{i}={\mathbb{X}}_{i}\left(x\right)$ | Fourier basis function associated to the x coordinate. $i=1\cdots \infty $. |

${\mathbb{Y}}_{j}={\mathbb{Y}}_{j}\left(y\right)$ | Fourier basis function associated to the y coordinate. $j=1\cdots \infty $. |

${\Theta}_{ij}={\Theta}_{ij}\left(t\right)$ | Fourier coefficient associated to basis functions ${\mathbb{X}}_{i}$ and ${\mathbb{Y}}_{j}$. |

${\theta}_{ij}^{k}={\theta}_{ij}^{k}\left(t\right)$ | Pseudo-Fourier-coefficient associated to the k-th laser beam source. |

${\omega}_{ij}$ | ij-th eigenvalue of the heat operator (Laplacian) on the rectangular sheet. |

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**Figure 1.**Multi-beam laser heating scheme. The sheet surface is heated by a set of laser beams ${f}_{1},{f}_{2},\cdots $ and cooled down due to natural convection q.

**Figure 3.**Laser trajectories and sheet temperature history for simultaneous diverse shape laser trajectories. (

**a**) laser trajectories for the simulation; (

**b**) $t=0.060$ s. Only Star trajectory occurs; (

**c**) $t=0.075$ s. Square trajectory enters the simulation; (

**d**) $t=0.094$ s. Lasers are simultaneously near each other; (

**e**) $t=0.13$ s. Square trajectory finishes. Star trajectory continues; (

**f**) $t=0.166$ s. Star trajectory finishes.

**Figure 6.**Laser trajectories and temperature history for time and space overlapping trajectories. (

**a**) planned laser trajectories: Square, Star, Spiral, Circle; (

**b**) $t=0.03$ s. All trajectories start at $t=0$ s; (

**c**) $t=0.158$ s. Square and Circle trajectories finished; (

**d**) $t=0.2$ s. All trajectories finished. The sheet cools down.

**Figure 7.**Laser trajectories and temperature results for the simulation case presented in [33]. Similar simulation parameters have been used to replicate the experiment. (

**a**) complete sheet. Simultaneous linear trajectories based on [33]; (

**b**) laser trajectories (zoom near laser spots); (

**c**) temperature map (full sheet); (

**d**) temperature map (zoom).

**Figure 9.**Comparison of CPU and GPU execution times (s) for the computation of $512\times 512$ Fourier coefficients as the number of lasers increase.

**Figure 10.**Execution times (s) for the computation of the temperature as the resolution (mesh size) increases. Results for different number of Fourier coefficients are presented.

**Figure 11.**Multi-laser machining in the interactive simulator. A star shape is machined by the three laser heads. (

**a**) virtual multi-laser machine; (

**b**) temperature shown on the sheet surface; (

**c**) geometric cut of the sheet.

Parameter | Description | Value |
---|---|---|

Geometry | ||

a | width | 0.01 m |

b | height | 0.01 m |

$\Delta z$ | thickness | 0.001 m |

Material | AISI 304 Steel | |

$\rho $ | density | 8030 $\mathrm{kg}$ |

${c}_{p}$ | specific heat | $574\phantom{\rule{0.166667em}{0ex}}\mathrm{J}/(\mathrm{kg}\xb7\mathrm{K})$ |

$\kappa $ | thermal conductivity | $20\phantom{\rule{0.166667em}{0ex}}\mathrm{W}/(\mathrm{m}\xb7\mathrm{K})$ |

R | reflectivity | 0 |

Convection Type | Natural | |

h | convection coefficient | $20\phantom{\rule{0.166667em}{0ex}}\mathrm{W}/({\mathrm{m}}^{2}\xb7\mathrm{K})$ |

${u}_{\infty}$ | ambient temperature | $300\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

Parameter | Description | Star Trajectory | Square Trajectory |
---|---|---|---|

${t}_{0}^{k}$ | start time of trajectory | 0.00 s | 0.06 s |

${t}_{f}^{k}$ | end time of trajectory | 0.16 s | 0.13 s |

${P}_{k}$ | power | 100 W | 200 W |

${r}_{k}$ | spot radius | 0.0003 m | 0.0005 m |

$\parallel {\overrightarrow{v}}_{k}\parallel $ | trajectory speed | 0.1 m/s | 0.2 m/s |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mejia-Parra, D.; Montoya-Zapata, D.; Arbelaiz, A.; Moreno, A.; Posada, J.; Ruiz-Salguero, O. Fast Analytic Simulation for Multi-Laser Heating of Sheet Metal in GPU. *Materials* **2018**, *11*, 2078.
https://doi.org/10.3390/ma11112078

**AMA Style**

Mejia-Parra D, Montoya-Zapata D, Arbelaiz A, Moreno A, Posada J, Ruiz-Salguero O. Fast Analytic Simulation for Multi-Laser Heating of Sheet Metal in GPU. *Materials*. 2018; 11(11):2078.
https://doi.org/10.3390/ma11112078

**Chicago/Turabian Style**

Mejia-Parra, Daniel, Diego Montoya-Zapata, Ander Arbelaiz, Aitor Moreno, Jorge Posada, and Oscar Ruiz-Salguero. 2018. "Fast Analytic Simulation for Multi-Laser Heating of Sheet Metal in GPU" *Materials* 11, no. 11: 2078.
https://doi.org/10.3390/ma11112078