# Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming

^{1}

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^{2}Group, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, 10154 Torino, Italy

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^{4}

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}nanocomposite coating with the model created by combining genetic programming and genetic algorithm methods in gene expression programming (GEP).

## 2. Optimization Method and Materials

#### 2.1. Functionally Graded Materials

**Figure 1.**2D-FGM rectangular plate for different boundary conditions [54].

#### 2.2. Optimization Method

#### 2.2.1. Genetic Programming

_{actual}is y values from the data set, y

_{pred}is the estimated y value obtained by entering the values of the solution. The complexity of the model is calculated as in Equation (4) in proportion to the depth of the tree and the number of nodes. C is tree complexity, d is the depth of the solution tree, and n is the number of nodes at depth.

#### 2.2.2. Multi Gene Genetic Programming

#### 2.2.3. Experimental Design

_{2}and the metal material is Ti-6Al-4V. The dimensions of the plate are l = 100 mm and h = 100 mm and the thickness is t = 1 mm. Since the plate thickness is thinner than the other dimensions, all stresses and deformations occurring throughout the thickness are neglected during the analysis, so the problem is considered as 2D. Two different boundary conditions are applied to the 2D-FG plates. In the first boundary condition (model 1), with an initial temperature of 298 K, a heat flux of 50 kW/m

^{2}from the BD edge is applied to the plate until the temperature at the AC edge is 600 K. In the second boundary condition (model 2), with an initial temperature of 298 K, 50 kW/m

^{2}heat fluxes from the BD and CD edges are applied to the plate until the temperature at the AC edge is 600 K. The heat transfer analysis problem is solved with the time-dependent Fourier heat conduction equations and the temperature distribution matrix for the last time step is transferred to the Navier’s equations of elasticity and the thermo-mechanical behavior is determined. FDM is used in numerical analysis and datasets are obtained for MGGP. The input values of the MGGP are analyzed for 200 different values, with compositional gradient exponents in the range of n and m = 0.0001–1.5. In the analyzes made with FDM, it has been recorded that the CPU time is approximately 2 h. Optimization in thermal stress analysis is made by taking ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ (the greatest of the greatest value of equivalent stress levels), ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{2}$ (the smallest value of the greatest value of the equivalent stress levels), ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{3}$(the greatest value of the smallest value of the equivalent stress levels), ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{4}$ (the smallest value of the smallest value of the equivalent stress levels) value into consideration [59]. These are added as output values into the dataset in MGGP. Determined input and output values, i.e., datasets, are split with randomly selected 70% of the datasets for training and the other 30% for the test set.

## 3. Application in GP Programming

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Simply tree structure of GP [55].

**Figure 4.**(

**a**) candidate solution (

**b**) neighborhood solution (

**c**) received subtree from the neighborhood solution (

**d**) generated candidate solution structure.

**Figure 6.**Comparison of actual values and predicted results of equivalent stress values in the training for Model 1.

**Figure 7.**Comparison of actual values and predicted results of equivalent stress values in the test for Model 1.

**Figure 8.**Comparison of actual values and predicted results of equivalent stress values in the training for Model 2.

**Figure 9.**Comparison of actual values and predicted results of equivalent stress values in the test for Model 2.

**Table 1.**Model 1 ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ example for the illustration of the overall model in MGGP [58].

Gene 1 | Gene 2 | Gene 3 | Gene 4 | |
---|---|---|---|---|

Genotype | ||||

Weight | −64.2 | 5.16 | −3.74 | 24.2 |

Phenotype (simplified) | $\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\right)$ | $\mathrm{tanh}{\left({m}_{1}\right)}^{2}\left({n}_{1}+{m}_{1}{}^{\frac{1}{2}}\right)$ | ${n}_{1}+1.998{m}_{1}{}^{\frac{1}{2}}$ | $\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{4}}\right)$ |

Weight Bias | 427.0 | |||

Formulation | $-64.2\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\right)+5.16\text{}\mathrm{tanh}{\left({m}_{1}\right)}^{2}\left({n}_{1}+{m}_{1}{}^{\frac{1}{2}}\right)-3.74\left({n}_{1}-1.998{m}_{1}{}^{\frac{1}{2}}\right)+24.2\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{4}}\right)+427$ |

Parameters | GP |
---|---|

Population Size | 100 |

Runs | 100 |

Number of generations | 100 |

Maximum Tree Depth | 4 |

Maximum Gene | 4 |

Crossover Rate | 0.14 |

Mutation Rate | 0.84 |

Direct Reproduction Rate | 0.02 |

Initialization | Ramped Half and Half |

Functions | +, −, *, square, tan, exp, log, sqrt, cube, negexp, neg, abs |

Fitness function | RMSE |

Model 1 | Model 2 | ||||
---|---|---|---|---|---|

Equivalent Stress | Criteria | Training | Test | Training | Test |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ | Fitness | 0.143 | 0.157 | 15.486 | 20.877 |

Max | 0.287 | 0.321 | 26.223 | 62.125 | |

Min | 0.063 | 0.073 | 6.254 | 6.816 | |

S.D | 0.047 | 0.054 | 3.864 | 8.614 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{2}$ | Fitness | 0.925 | 0.916 | 1.860 | 1.979 |

Max | 2.000 | 2.087 | 3.889 | 4.173 | |

Min | 0.237 | 0.260 | 0.934 | 0.898 | |

S.D | 0.391 | 0.377 | 0.647 | 0.767 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{3}$ | Fitness | 1.082 | 1.156 | 0.668 | 0.780 |

Max | 1.343 | 1.380 | 0.800 | 0.929 | |

Min | 0.763 | 0.815 | 0.328 | 0.400 | |

S.D | 0.113 | 0.106 | 0.093 | 0.104 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{4}$ | Fitness | 0.218 | 0.143 | 0.617 | 0.658 |

Max | 0.692 | 0.287 | 1.284 | 1.393 | |

Min | 0.162 | 0.165 | 0.299 | 0.285 | |

S.D | 0.066 | 0.047 | 0.191 | 0.206 |

Equivalent Stress | Criteria | Model 1 | Model 2 |
---|---|---|---|

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ | Node | 22 | 42 |

Complexity | 56 | 129 | |

Tree Depth | 4 | 4 | |

Gen | 4 | 4 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{2}$ | Node | 29 | 21 |

Complexity | 82 | 53 | |

Tree Depth | 4 | 4 | |

Gen | 4 | 4 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{3}$ | Node | 18 | 19 |

Complexity | 44 | 50 | |

Tree Depth | 4 | 4 | |

Gen | 4 | 4 | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{4}$ | Node | 24 | 17 |

Complexity | 64 | 43 | |

Tree Depth | 4 | 4 | |

Gen | 4 | 4 |

Dataset | Equations | |
---|---|---|

Model 1 | ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ | $24.2\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{4}}\right)-64.2\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\right)-3.74{n}_{1}+5.16\text{}\mathrm{tanh}{\left({m}_{1}\right)}^{2}\left({n}_{1}+{m}_{1}{}^{\frac{1}{2}}\right)-7.47{m}_{1}{}^{\frac{1}{2}}+427$ |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{2}$ | $6.56{n}_{1}+8.97{m}_{1}+54.3\mathrm{exp}\left(-2{m}_{1}\right)-162\mathrm{exp}\left(-1.26{m}_{1}\right)\mathrm{exp}\left(-{n}_{1}{m}_{1}\right)\left|{n}_{1}\right|\left|{m}_{1}\right|+116$ | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{3}$ | $760\mathrm{tanh}\left(\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\right)\right)+707\mathrm{exp}\left(-{m}_{1}{}^{\frac{1}{2}}\right)+36.9{m}_{1}\mathrm{tanh}\left(\mathrm{exp}\left({n}_{1}\right)\right)+31.9{n}_{1}\mathrm{tanh}\left({n}_{1}{m}_{1}\right)-613$ | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{4}$ | $10.1\mathrm{tanh}\left({n}_{1}\right)+28.6\mathrm{tanh}\left({m}_{1}\right)\mathrm{tanh}\left({n}_{1}\right)-5.76{m}_{1}{}^{3/2}\mathrm{tan}\mathrm{h}{\left({m}_{1}\right)}^{3}+1.28\mathrm{exp}\left({m}_{1}\right)\mathrm{tanh}{\left({n}_{1}\right)}^{2}+0.655$ | |

Model 2 | ${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{1}$ | $100\mathrm{exp}\left(-\frac{\left(2{m}_{1}\right)}{{n}_{1}+0.0557}\right)+58.4{n}_{1}{}^{3}{m}_{1}\mathrm{exp}\left(-{m}_{1}\right)-5.78{m}_{1}\left({m}_{1}+0.116\right)\left({n}_{1}{}^{2}-9.29\right)+\frac{66.4{m}_{1}}{\left({n}_{1}+0.116\right)\left({n}_{1}+{m}_{1}+0.117\right)}+326$ |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{2}$ | $399-252\mathrm{exp}\left(-{n}_{1}-{m}_{1}{}^{\frac{1}{2}}\right)-29.3\mathrm{exp}\left(-2{m}_{1}\right)-153\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\right)\mathrm{exp}\left(-\mathrm{tanh}\left({n}_{1}\right)\right)-29.1\mathrm{exp}(\mathrm{tan}\mathrm{h}{\left({n}_{1}\right)}^{\frac{1}{2}})$ | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{3}$ | $602{m}_{1}{}^{3/4}-59.4\mathrm{tanh}\left({m}_{1}{}^{\frac{1}{2}}\mathrm{exp}\left(-{m}_{1}\right)\right)-\mathrm{tanh}\left(\mathrm{tanh}\left({n}_{1}\right)\right)\left({m}_{1}-{m}_{1}{}^{\frac{1}{2}}\right)-380{m}_{1}+93.1$ | |

${\left({\mathsf{\sigma}}_{\mathrm{eqv}}\right)}_{4}$ | $107\mathrm{tanh}\left({m}_{1}\right)-4.29\mathrm{exp}\left(-{n}_{1}\right)+54\mathrm{exp}\left(-2{m}_{1}\mathrm{exp}\left(-{n}_{1}\right)\right)-14.6{n}_{1}\mathrm{tanh}\left({n}_{1}{m}_{1}\right)$ |

${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{1}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{2}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{3}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{4}}$ | |
---|---|---|---|---|

RMSE | 0.067 | 0.246 | 3.6078 | 0.1631 |

Min Error | 0.00103 | 0.007 | 0.0014 | 0.0005 |

Max. Error | 0.2129 | 0.9059 | 12.0658 | 0.5662 |

Error Rate (%) | 0.0125 | 0.4884 | 1.395 | 2.752 |

${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{1}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{2}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{3}}$ | ${\left({\mathsf{\sigma}}_{\mathbf{eqv}}\right)}_{\mathbf{4}}$ | |
---|---|---|---|---|

RMSE | 6.428 | 0.960 | 0.350 | 0.295 |

Min Error | 0.065 | 0.020 | 0.001 | 0.002 |

Max Error | 30.90 | 2.478 | 1.384 | 0.96 |

Error Rate (%) | 1.02 | 0.42 | 0.15 | 2.12 |

n-m | Start | End | Time (s) | Average Time (s) | |
---|---|---|---|---|---|

Model 1 | 1.5–1.11 | 13:31:25 | 14:19:40 | 2895.00 | 2049.67 |

0.77–0.66 | 15:12:31 | 15:40:08 | 1657.00 | ||

0.001–0.005 | 16:31:12 | 16:57:49 | 1597.00 | ||

Model 2 | 1.5–1.11 | 09:45:58 | 10:08:33 | 1356.00 | 842.00 |

0.77–0.66 | 10:14:57 | 10:36:36 | 699.00 | ||

0.001–0.005 | 10:46:27 | 11:04:18 | 471.00 |

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**MDPI and ACS Style**

Demirbas, M.D.; Çakır, D.; Ozturk, C.; Arslan, S.
Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming. *Appl. Sci.* **2022**, *12*, 8198.
https://doi.org/10.3390/app12168198

**AMA Style**

Demirbas MD, Çakır D, Ozturk C, Arslan S.
Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming. *Applied Sciences*. 2022; 12(16):8198.
https://doi.org/10.3390/app12168198

**Chicago/Turabian Style**

Demirbas, Munise Didem, Didem Çakır, Celal Ozturk, and Sibel Arslan.
2022. "Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming" *Applied Sciences* 12, no. 16: 8198.
https://doi.org/10.3390/app12168198