# Design Optimization of Chute Structure Based on E-SVR Surrogate Model

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## Abstract

**:**

## 1. Introduction

## 2. Simulation of the Material-Conveying Process

## 3. Optimization Model and Flowchart

#### 3.1. Optimization Model

#### 3.2. Flowchart of Chute Structure Optimization

## 4. E-SVR Model

#### 4.1. SVR Model

- (1)
- Multiple correlation coefficient ${R}^{2}$$${R}^{2}=1-\frac{{{\displaystyle \sum _{i=1}^{m}({y}_{i}-{\widehat{y}}_{i})}}^{2}}{{{\displaystyle \sum _{i=1}^{m}({y}_{i}-\overline{y})}}^{2}}=1-\frac{MSE}{Var}$$

- (2)
- Relative average absolute error ($RAAE$)$$RAAE=\frac{{\displaystyle \sum _{i=1}^{m}\left|{y}_{i}-{\widehat{y}}_{i}\right|}}{m\times STD}$$

- (3)
- Relative maximum absolute error ($RMAE$)$$RMAE=\frac{\underset{i=1,\dots ,m}{\mathrm{max}}\left|{y}_{i}-{\widehat{y}}_{i}\right|}{STD}$$

#### 4.2. E-SVR Surrogate Model

#### 4.3. Numerical Examples

- (1)
- Brain-Hoo function [33]:$$f(x,y)={(y-5.1{x}^{2}/4{\pi}^{2}+5x/\pi -6)}^{2}+10(1-1/8\pi )\mathrm{cos}(x)+10,x\in \left[-5,10\right],y\in \left[0,15\right]$$

- (2)
- Hartman3 function [32]$$\begin{array}{l}f(x)=-{\displaystyle \sum _{i=1}^{4}{c}_{i}}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-{\displaystyle \sum _{j=1}^{n}{\mathit{a}}_{\mathit{i}\mathit{j}}{({\mathit{x}}_{\mathit{j}}-{p}_{ij})}^{2}}\right\}\\ \mathit{x}=({\mathit{x}}_{1},{\mathit{x}}_{2},\cdots ,{\mathit{x}}_{\mathit{n}}),{\mathit{x}}_{\mathit{i}}\in \left[0,1\right]\end{array}$$

## 5. Design Optimization of Chute Structure

#### 5.1. Design of Experiment

#### 5.2. Solution and Verification

#### 5.3. Comparison of Single/Ensemble-Kernel-Based SVR

## 6. Conclusions

- (1)
- The optimization model of the chute structure parameter was established and solved in this paper. The maximum impact force of coke on the chute and the material crushing rate were effectively reduced within the allowed speed range, which improved the utilization rate of coke and prolonged the service life of the chute.
- (2)
- The E-SVR model was developed to integrate the advantages of both the Poly-SVR model and the RBF-SVR model. The effectiveness of the E-SVR model was verified through numerical examples.
- (3)
- By using the E-SVR surrogate model in the chute structural parameter optimization models to replace the implicit relationship between the maximum impact force, maximum conveying speed, and design variables, the computational cost of design optimization was reduced.
- (4)
- Through comparison with the results by using the Poly-SVR model and results by using the RBF-SVR model, the optimal results obtained by the proposed E-SVR model are accurate and effective, which is of great significance for the design of chute structure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters | Value | |
---|---|---|

Spherical material property | Diameter (mm) | 80 |

Poisson ratio | 0.28 | |

Shear modulus (GPa) | 1.98 | |

Density (Kg/m^{3}) | 500 | |

Chute properties | Poisson ratio | 0.3 |

Shear modulus (GPa) | 80 | |

Density (Kg/m^{3}) | 7850 | |

Collision properties between materials | Coefficient of restitution | 0.40 |

Static friction coefficient | 0.40 | |

Coefficient of rolling friction | 0.05 | |

Collision properties between material and chute | Coefficient of restitution | 0.50 |

Static friction coefficient | 0.30 | |

Coefficient of rolling friction | 0.10 | |

Simulation parameter setting | Number of particles between materials/(s) | 5000 |

Fixed time step(s) | 6.8 × 10^{4} |

Kernel Function | Formulas |
---|---|

Linear kernel function | $k(x,{x}^{\prime})={x}^{T}{x}^{\prime}$ |

Sigmoid kernel function | $k(x,{x}^{\prime})=\mathrm{tanh}(ax\ast {x}^{{}^{\prime}}+c)$ |

Radial basis kernel function | $k(x,{x}^{\prime})=\mathrm{exp}\left[-{\left|x-{x}^{\prime}\right|}^{2}/(2{s}^{2})\right]$ |

Polynomial kernel function | $k(x,{x}^{\prime})={({x}^{T}{x}^{\prime}+h)}^{d}$ |

Evaluation Index | Poly-SVR | RBF-SVR | E-SVR |
---|---|---|---|

R^{2} | 1 | 0.9575 | 1 |

RAAE | 0.0026 | 0.1078 | 0.0038 |

RMAE | 0.0276 | 1.3177 | 0.031 |

i | a_{ij} | c_{i} | p_{ij} | ||||
---|---|---|---|---|---|---|---|

1 | 3.0 | 10 | 30 | 1.0 | 0.3689 | 0.1170 | 0.2673 |

2 | 0.1 | 10 | 35 | 1.2 | 0.4699 | 0.4387 | 0.7470 |

3 | 3.0 | 10 | 30 | 3.0 | 0.1091 | 0.8732 | 0.5547 |

4 | 0.1 | 10 | 35 | 3,2 | 0.03815 | 0.5743 | 0.8828 |

Evaluation Index | Poly-SVR | RBF-SVR | E-SVR |
---|---|---|---|

R^{2} | 0.2628 | 0.7461 | 0.7581 |

RAAE | 0.6744 | 0.3854 | 0.3689 |

RMAE | 3.3907 | 1.8152 | 2.1837 |

R_{1}, R_{2}, θ | R_{1}, R_{2}, θ | R_{1}, R_{2}, θ | R_{1}, R_{2}, θ | R_{1}, R_{2}, θ |
---|---|---|---|---|

2168/2034/27.9 | 2350/2125/29.1 | 2406/2153/29.4 | 2042/1971/27.1 | 2296/2098/28.7 |

2375/2137/29.2 | 1877/1888/26.1 | 1738/1819/25.2 | 1961/1931/26.6 | 2338/2119/29.0 |

1907/1903/26.3 | 1934/1917/26.5 | 1979/1939/26.7 | 2144/2022/27.8 | 1830/1865/25.8 |

2052/1976/27.2 | 1782/1841/25.5 | 2428/2164/29.6 | 1763/1832/25.4 | 2088/1994/27.4 |

1852/1876/25.9 | 2205/2052/28.2 | 2220/2060/28.3 | 2495/2198/30.0 | 2010/1955/26.9 |

1705/1802/25.0 | 2281/2090/28.6 | 2460/2180/29.8 | 2248/2074/28.4 | 2116/2008/27.6 |

Structure Parameter | Maximum Impact Force | Maximum Conveying Speed | |
---|---|---|---|

Before optimization | 2100, 2000, 27.5 | 2460 N | 7.59 m/s |

After optimization | 2259.6, 1813.4, 29.9 | 2040 N | 7.09 m/s |

Structure Parameter | Maximum Impact Force | Maximum Conveying Speed | |
---|---|---|---|

Poly-SVR | 2296.7, 1742.4, 29.3 | 2420 N | 7.14 m/s |

RBF-SVR | 2173.5, 1909.5, 28.2 | 2370 N | 7.18 m/s |

E-SVR | 2259.6, 1813.4, 29.9 | 2040 N | 7.09 m/s |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Jiang, Q.; Long, Y.; Chen, Z.; Zhao, W.; Ming, W.; Cao, Y.; Ma, J.
Design Optimization of Chute Structure Based on E-SVR Surrogate Model. *Metals* **2023**, *13*, 635.
https://doi.org/10.3390/met13030635

**AMA Style**

Li X, Jiang Q, Long Y, Chen Z, Zhao W, Ming W, Cao Y, Ma J.
Design Optimization of Chute Structure Based on E-SVR Surrogate Model. *Metals*. 2023; 13(3):635.
https://doi.org/10.3390/met13030635

**Chicago/Turabian Style**

Li, Xiaoke, Qianlong Jiang, Yu Long, Zhenzhong Chen, Wenbo Zhao, Wuyi Ming, Yang Cao, and Jun Ma.
2023. "Design Optimization of Chute Structure Based on E-SVR Surrogate Model" *Metals* 13, no. 3: 635.
https://doi.org/10.3390/met13030635