Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware
Abstract
:1. Introduction
2. Literature Review
2.1. Laser Heating/Cutting Simulation
2.2. FFTBased Laser Heating Simulation
2.3. Conclusions of the Literature Review
3. Methodology
3.1. Heat Transfer Equation for Laser Heating on Thin Plates
3.2. Analytic Solution
3.3. Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
3.4. Scheme 1—Discrete Sine Transform (DST)
Algorithm 1 Retrieve temperature using a 2D DST 
Require:$\mathsf{\Theta}\in {\mathbb{R}}^{(M2)\times (N2)},\phantom{\rule{1.em}{0ex}}{u}_{\infty}\in \mathbb{R}$ Ensure:$U\in {\mathbb{R}}^{M\times N}$

3.5. Scheme 2—FFT Padded with Zeros
Algorithm 2 Retrieve temperature using a Fast Fourier Transform (FFT) with zero padding 
Require:$\mathsf{\Theta}\in {\mathbb{R}}^{(M2)\times (N2)},\phantom{\rule{1.em}{0ex}}{u}_{\infty}\in \mathbb{R}$ Ensure:$U\in {\mathbb{R}}^{M\times N}$

3.6. Scheme 3—OddSymmetry 1D FFT
Algorithm 3 Retrieve temperature using 1D FFTs by applying odd symmetry to the original coefficients 
Require:$\mathsf{\Theta}\in {\mathbb{R}}^{(M2)\times (N2)},\phantom{\rule{1.em}{0ex}}{u}_{\infty}\in \mathbb{R}$ Ensure:$U\in {\mathbb{R}}^{M\times N}$

3.7. Scheme 4—OddSymmetry 2D FFT
Algorithm 4 Retrieve temperature using 2D FFTs by applying odd sine symmetry and even cosine symmetry to the original coefficients 
Require:$\mathsf{\Theta}\in {\mathbb{R}}^{(M2)\times (N2)},\phantom{\rule{1.em}{0ex}}{u}_{\infty}\in \mathbb{R}$ Ensure:$U\in {\mathbb{R}}^{M\times N}$

3.8. Complexity Analysis
4. Results
4.1. Numerical Validation
4.2. Computational Performance
4.2.1. CPU Performance Measurements
4.2.2. GPU Performance Measurements
4.2.3. Comparison of CPU and GPU Performance
4.2.4. Comparison against State of the Art
4.3. Interactive Simulator Prototype
5. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
Abbreviations
PDE  Partial Differential Equation 
DST  Discrete Sine Transform 
DFT  Discrete Fourier Transform 
FFT  Fast Fourier Transform 
$a,b,\Delta z$  Width, height and thickness of the thin plate (m^{3}). 
${T}_{f}$  Total simulation time (s). 
$\overrightarrow{x},t$  Spatial $\overrightarrow{x}=(x,y)\in [0,a]\times [0,b]$ and temporal $0\le t\le {T}_{f}$ coordinates. 
$u=u(\overrightarrow{x},t)$  Temperature field $u:[0,a]\times [0,b]\times [0,{T}_{f}]\to \mathbb{R}$ on the metal plate (K). 
$\rho $  Plate density (kg/m^{3}). 
${c}_{p}$  Plate specific heat (J/kg K). 
$\kappa $  Plate thermal conductivity (W/m K). 
R  Plate reflectivity ($0\le R<1$). 
$q=q\left(u\right)$  Temperaturedependent heat convection field $q:\mathbb{R}\to \mathbb{R}$ (W/m^{2}). 
h  Natural convection coefficient at the plate surface (W/(m^{2} K)) 
${u}_{\infty}$  Ambient temperature (K). 
${\overrightarrow{x}}_{0}={\overrightarrow{x}}_{0}\left(t\right)$  Laser spot location at a given time ${\overrightarrow{x}}_{0}\left(t\right)=({x}_{0}\left(t\right),{y}_{0}\left(t\right))$. 
$f=f(\overrightarrow{x},t)$  Power Density Field $f:[0,a]\times [0,b]\times [0,{T}_{f}]\to \mathbb{R}$ for the laser beam (W/m^{2}). 
P  Laser power (W). 
r  Laser spot radius (m). 
$M\times N$  2D plate discretization size ($M,N\in \mathbb{N}$). 
${\theta}_{mn}\left(t\right)$  ${m}^{th},{n}^{th}$ Fourier coefficient ($m,n=0,1,\dots $) for the temperature solution u at time t. 
${\alpha}_{m},{\beta}_{n}$  Coefficients ${\alpha}_{m}=(m+1)\pi /a$ and ${\beta}_{n}=(n+1)\pi /b$ for the Fourier basis in the X and Yaxis, respectively. 
${\gamma}_{m},{\delta}_{n}$  ${\gamma}_{m}=m\pi /M$ and ${\delta}_{n}=n\pi /N$ are the discrete equivalent of ${\alpha}_{m}$ ($m=0,1,\dots ,M1$) and ${\beta}_{n}$ ($n=0,1,\dots ,N1$), respectively. 
${\omega}_{mn}$  ${m}^{th},{n}^{th}$ eigenvalue of the heat (Laplace) operator defined on the rectangular plate. 
${\overrightarrow{C}}_{i}\left(t\right)$  Piecewise linear discretization of the laser trajectory ${\overrightarrow{x}}_{0}\left(t\right)$. 
Appendix A. Scheme 1—Discrete Sine Transform (DST)
Appendix B. Scheme 2—FFT Padded with Zeros
Appendix C. Scheme 3—OddSymmetry 1D FFT
Appendix D. Scheme 4—OddSymmetry 2D FFT
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Line  Description  Number of Operations  Dominant Term  Complexity Order 

1, 2  Memory initialization  $\mathcal{O}\left(MN\right)$  
4  FFT of a column  $2M+2Mlog\left(2M\right)$  $2Mlog\left(2M\right)$  $\mathcal{O}(MlogM)$ 
5  Extract complex part  $2M$  $\mathcal{O}\left(M\right)$  
3–6  Loop through columns  $(N2)(4M+2Mlog(2M\left)\right)$  $2MNlog\left(2M\right)$  $\mathcal{O}(MNlogM)$ 
8  FFT of a row  $2N+2Nlog\left(2N\right)$  $2Nlog\left(2N\right)$  $\mathcal{O}(NlogN)$ 
9  Extract complex part  $2N$  $\mathcal{O}\left(N\right)$  
7–10  Loop through rows  $(M2)(4N+2Nlog(2N\left)\right)$  $2MNlog\left(2N\right)$  $\mathcal{O}(MNlogM)$ 
11, 12  Memory initialization  $\mathcal{O}\left(MN\right)$  
13  Sum of matrices  $MN$  $\mathcal{O}\left(MN\right)$  
TOTAL  $\mathcal{O}(MNlogM+MNlogN)$$=\mathcal{O}(MNlog(MN\left)\right)$ 
Parameter  Description  Value  Units 

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 
Library  Python Package  Hardware 

FFTPACK  scipy.fft  CPU 
MKL  numpy.fft  CPU 
FFTW  pyfftw  CPU 
cuFFT  pyCUDA, scikitcuda  GPU 
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MejiaParra, D.; Arbelaiz, A.; RuizSalguero, O.; LalindePulido, J.; Moreno, A.; Posada, J. Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware. Appl. Sci. 2020, 10, 3281. https://doi.org/10.3390/app10093281
MejiaParra D, Arbelaiz A, RuizSalguero O, LalindePulido J, Moreno A, Posada J. Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware. Applied Sciences. 2020; 10(9):3281. https://doi.org/10.3390/app10093281
Chicago/Turabian StyleMejiaParra, Daniel, Ander Arbelaiz, Oscar RuizSalguero, Juan LalindePulido, Aitor Moreno, and Jorge Posada. 2020. "Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware" Applied Sciences 10, no. 9: 3281. https://doi.org/10.3390/app10093281