Flow Discharge Prediction Study Using a CFDBased Numerical Model and Gene Expression Programming
Abstract
:1. Introduction
2. Materials and Methods Methods
2.1. Flow3D Numerical Modeling
Estimation of Uncertainty in a CFD Application
2.2. Gene Expression Programming Approaches
Performance Criteria
3. Discussion
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
GEP  Gene expression Programming 
RMSE  Root Mean Square Error 
${R}^{2}$  Coefficient of Determination 
MAE  Mean Absolute Error 
Appendix A
Case Number  Equation 

Case 1 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=3.87\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+(d\left(3\right)/atan((((d{\left(1\right)}^{2})\ast d(2))+(3.4)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+gep3Rt(((({(d\left(3\right)+d\left(3\right))}^{2})+((8.61\ast 8.61)+d\left(3\right)))/2.0))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 2 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=min\left(\right(\left(min\right((3.95+9.42),min\left(d\right(3),3.95))+(4.6d\left(2\right)\left)\right)/2.0),(2.4\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+max\left(atan\right(exp\left(atan\right(d\left(1\right)\left)\right)),(\left(\right(d\left(3\right)+d\left(2\right))+(d\left(2\right)/d\left(1\right)\left)\right)/2.0\left)\right)\hfill \\ & Y=Y+tanh\left(\right(\left(\right(min(4.18,d(2\left)\right)+\left(d\right(1)\ast d(1\left)\right))/2.0)\left(\right(d\left(3\right)3.3)+2.52)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 3 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=(1.0/>((({(((d\left(1\right)d\left(2\right))\ast (d{\left(1\right)}^{2})+reallog(0.51))/2.0)}^{2})))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(\left(\right((7.2+d(3\left)\right)/2.0)+(d\left(2\right)d\left(2\right)\left)\right)/2.0)((7.57+2.09)\left(\right(d\left(2\right)+7.57)/2.0)\left)\right)\hfill \\ & Y=Y+reallog\left(\right(\left(\right(reallog\left(d\right(2\left)\right)+(4.157+d(2\left)\right))+(\left(\right(3.74+d\left(3\right))/2.0))/2.0))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 4 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=(1.0/(atan\left(gep3Rt\right(\left(\right(reallog\left(d\right(2)+(6.03\left)\right)+\left(\right(d\left(2\right)+(6.03))/2.0)\left)\right)\left)\right))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(1.0max\left(reallog\right(d\left(2\right)),(5.78\left)\right))\ast (\left(d\right(2)d(3\left)\right)+(1.0/(2.28\left)\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+d\left(3\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 5 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=\left(d\right(3)(\left(atan\right(2.39)(d\left(1\right)d\left(1\right)\left)\right)(1.0/(d\left(1\right)\left)\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+tanh\left(\right(\left(max\right(2.86,d\left(1\right))+(7.93+d\left(1\right)\left)\right)+\left(\right(d\left(1\right)+d\left(1\right))d(2\left)\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+tanh(((((6.7(1.05)+(5.52\ast (7.54)))/2.0)((d{\left(1\right)}^{2})d\left(3\right))))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 6 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=\left(\right(1.0/\left(d\right(2\left)\right))1.58)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+d\left(3\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+(1.0/(\left(\right(\left(\right(d\left(1\right)d\left(3\right))\ast (8.1+(1.55))+(1.0\left(\right(3.74+1.30)/2.0)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 7 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=gep3Rt\left(\right(\left(min\right(d\left(3\right),d\left(1\right))/tanh(d\left(1\right)\left)\right)+\left(min\right(2.68,2.34)d(1\left)\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+tanh\left(tanh\right(\left(\right((3.98d(3\left)\right)\left(d\right(3)(7.59\left)\right)+\left(d\right(1)\ast d(1\left)\right)\left)\right))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(\left(\right(\left(d\right(2)/d(1\left)\right)+\left(\right(7.26+d\left(1\right))/2.0))/2.0)+\left(\right(d\left(3\right)/0.97)\ast atan(d\left(3\right)\left)\right))/2.0)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 8 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=\left(\right(gep3Rt\left(\right(1.0tanh\left(d\right(2\left)\right)\left)\right)+\left(\right(d\left(2\right)+reallog\left(d\right(3\left)\right))/2.0))/2.0)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(min\left(gep3Rt\right(\left(d\right(1)6.6)),(\left(d\right(3)+(8.98)/2.0))+d(3\left)\right)/2.0)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(gep3Rt\left(d\right(3\left)\right)/\left(d\right(1\left)\right))+atan(tanh\left(3.3\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 9 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=\left(\right(\left(\right(d\left(3\right)7.21)(4.07d\left(3\right)\left)\right)+\left(\right(1.0/\left(d\right(1\left)\right))+(1.0/\left(d\right(2\left)\right)\left)\right))/2.0)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+atan\left(\right(\left(\right(tanh(1.7)\ast d\left(2\right))+d(1\left)\right)*d\left(1\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+atan\left(\right(\left(9tanh\right(1.706)(d\left(1\right)d\left(3\right)\left)\right)+atan\left(\right(d\left(3\right)d\left(1\right)\left)\right)\left)\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

Case 10 
$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & Y=gep3Rt(6.92)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+(1.0((({((d\left(2\right)\ast (0.327)+reallog\left(d\left(2\right)\right)/2.0)\ast (d\left(3\right)d\left(2\right)))}^{2}))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & Y=Y+\left(\right(d\left(2\right)+d\left(2\right))/2.0)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & result=Y\hfill \end{array}$$

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Values  Values  Values  

N${}_{1}$, N${}_{2}$, N${}_{3}$  1780000, 305212, 222500  1780000, 305212, 222500,  1780000, 305212, 222500 
r${}_{21}$  1.72  1.72  1.72 
r${}_{32}$  1.56  1.56  1.56 
Q${}_{1}$  5.43  10.29  11.805 
Q${}_{2}$  5.51  10.2  11.69 
Q${}_{3}$  5.37  10.08  11.58 
P  2.48  1.79  1.2 
Q${}_{ext}^{21}$  5.38  10.34  11.93 
Q${}_{a}^{21}$  1.47%  0.87%  0.97% 
Q${}_{ext}^{21}$  0.5%  0.48%  1.04% 
Q${}_{fine}^{21}$  0.66%  0.67%  1.31% 
Setting Parameters  Value 

Functions set  +, −, /, x${}^{2}$, exp, ln, cube root, Atan, Tanh 
Chromosomes  30 
Head size  8 
Number of genes  3 
Linking function  Addition 
Fitness function error type  RMSE 
Mutation rate  0.044 
Inversion rate  0.1 
Onepoint recombination rate  0.3 
Twopoint recombination rate  0.3 
Gene recombination rate  0.1 
Gene transposition rate  0.1 
Type  Operator  Train Phasing  Test Phasing  

${\mathit{R}}^{2}$  RMSE  MAE  ${\mathit{R}}^{2}$  RMSE  MAE  
Option 1  +, −, *, /  0.9  1.57  1.07  0.86  1.73  1.13 
Option 2  +, −, *, /, x${}^{2}$  0.91  1.53  1.05  0.86  1.73  1.12 
Option 3  +, −, *, /, x${}^{2}$, exp  0.9  1.65  1.03  0.86  1.59  1.01 
Option 4  +, −, *, /, x${}^{2}$, exp, ln, cube root  0.9  1.59  1.06  0.87  1.61  1.06 
Option 5  +, −, *, /, x${}^{2}$, exp, ln, cube root, Atan, Tanh, min, max  0.972  0.85  0.64  0.912  1.42  1.12 
Case Number  Train Phasing  Test Phasing  

${\mathit{R}}^{2}$  RMSE  MAE  ${\mathit{R}}^{2}$  RMSE  MAE  
Case 1  0.9  1.611  0.95  0.86  1.49  0.97 
Case 2  0.92  1.43  0.85  0.89  1.71  1.28 
Case 3  0.96  0.94  0.74  0.86  1.19  0.87 
Case 4  0.94  1.21  0.95  0.71  2.11  1.3 
Case 5  0.91  1.48  0.83  0.87  1.89  1.01 
Case 6  0.92  1.47  0.94  0.85  1.49  0.93 
Case 7  0.96  0.97  0.7  0.87  1.47  1.07 
Case 8  0.93  1.32  0.86  0.82  1.73  1.016 
Case 9  0.91  1.48  0.95  0.89  1.51  1.02 
Case 10  0.91  1.54  1.02  0.86  1.69  1.06 
Parameters  Value  Coefficients  Value 

${d}_{0}$  $\frac{{L}_{e}}{{H}_{o}}$  G1C1  2.02 
${d}_{1}$  $\frac{{H}_{e}}{{H}_{o}}$  G3C6  0.06 
${d}_{2}$  $\frac{{H}_{e}}{p}$     
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Mozaffari, S.; Amini, E.; Mehdipour, H.; Neshat, M. Flow Discharge Prediction Study Using a CFDBased Numerical Model and Gene Expression Programming. Water 2022, 14, 650. https://doi.org/10.3390/w14040650
Mozaffari S, Amini E, Mehdipour H, Neshat M. Flow Discharge Prediction Study Using a CFDBased Numerical Model and Gene Expression Programming. Water. 2022; 14(4):650. https://doi.org/10.3390/w14040650
Chicago/Turabian StyleMozaffari, Sevda, Erfan Amini, Hossein Mehdipour, and Mehdi Neshat. 2022. "Flow Discharge Prediction Study Using a CFDBased Numerical Model and Gene Expression Programming" Water 14, no. 4: 650. https://doi.org/10.3390/w14040650