# Prediction of Thermal Barrier Coatings Microstructural Features Based on Support Vector Machine Optimized by Cuckoo Search Algorithm

^{1}

^{2}

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## Abstract

**:**

^{2}) of CS-SVM model showed that the prediction accuracy reached by over 95%, and the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) were less than 0.1, which also verified the reliability of the CS-SVM model. Finally, this study proposed a novel and efficient microstructural feature prediction that could be potentially employed to improve the performance of TBCs in service.

## 1. Introduction

## 2. Experimental Procedures and Modeling Methods

#### 2.1. Coatings Fabrication and Microstructural Features Characterization

_{2}8 wt.% Y

_{2}O

_{3}(8YSZ) powders (Beijing Sunspraying Technology Co., Ltd., Beijing, China) were received with two batches of the nominal particle size distribution of 40–96 µm and 15–55 µm, 24, 30, 33, and 36 kW were chosen as the spray power during TBCs samples preparation, respectively; the spray distance was chosen from 70 to 120 mm. Disk-shaped, grit-blasted, carbon steel substrate plates (Ø 25.4 × 3.1 mm

^{2}) were used as the substrates [3].

_{f}, aspect ratio AR, and circularity C

_{r}, which were considered to reflect the morphology of micropores more comprehensively, were chosen to perform statistical analysis, we obtained the average value as the modeling output according to the frequency weighting of the respective microstructural feature distribution, and 5 different discontiguous SEM images were counted for each processing parameter to ensure the reliability of the statistical results. The definitions of these chosen microstructural features are listed in Table 2, and the schematic illustration of these chosen microstructural features could be found in [35,36].

#### 2.2. Methodological Background and Model Construction

#### 2.2.1. Support Vector Machine

#### 2.2.2. Cuckoo Search Algorithm

_{th}nest of the t

_{th}bird’s nest is ${x}_{i}^{(t)}$, and the machine search path is $L\left(\mathsf{\lambda}\right)$, then the update formula of the cuckoo’s path and position for finding the bird’s nest can be expressed as follows:

_{a}, the nest position ${x}_{i}^{(t+1)}$ is changed, otherwise it keeps constant. Finally, the group of nest positions ${y}_{i}^{(t+1)}$ with the best-retained effect is still recorded as ${x}_{i}^{(t+1)}$.

_{th}nest position; ${d}_{\mathrm{max}}$ is the maximum distance between the optimal position and other nests.

#### 2.2.3. SVM Parameter Optimization Based on CS

- The SVM parameters determine the learning and generalization ability of the SVM model. Two crucial and decisive RBF parameters are C and $\mathsf{\sigma}$, respectively. The former determines the balance between the complexity of the SVM model and empirical error, the latter determines the complexity of the sample data distribution. Hence, in this study, MATLAB software (R2017a, MathWorks. Inc) is used for modeling implementation, the CS algorithm is applied to optimize the SVM model parameters C and $\mathsf{\sigma}$ as follows [34]: Gather the training set samples, and preprocess the training set samples to acquire SVM learning samples. As a matter of experience, set the value range of SVM parameters C and $\mathsf{\sigma}$, the minimum step ${\mathrm{step}}_{\mathrm{min}}$ and the maximum step ${\mathrm{step}}_{\mathrm{max}}$ of the CS algorithm, and the number of iterations N [34].
- Set the probability P
_{a}= 0.25 and the number of nests n = 20 from the beginning, stochastically generate the position ${p}_{i}^{(0)}={\left[{x}_{1}^{(0)},{x}_{2}^{(0)},\cdots ,{x}_{n}^{(0)}\right]}^{\mathrm{T}}$ of the n nests, each nest corresponds to a set of parameters $\left(C,\mathsf{\sigma}\right)$, calculate the fitness evaluation function of each set of nest positions corresponding to the training set, and find the optimum nest position ${x}_{b}^{(0)}$ and fitness evaluation function ${F}_{\mathrm{max}}$ at present. - Keep the position ${x}_{b}^{(0)}$ of the optimal nest of the previous generation, calculate the Levy flight step length according to Equations (5) and (6), use the Levy flight to update the positions of other nests to acquire a new set of nest positions, and calculate their fitness evaluation function F.
- According to the fitness evaluation function F, the position of the new bird’s nest is compared with the position P
_{i}_{−1}of the previous generation bird’s nest, and the poorer bird’s nest position is replaced with a better bird’s nest position to acquire a new set of bird’s nest position ${p}_{t}={\left[{x}_{1}^{(t)},{x}_{2}^{(t)}\cdots ,{x}_{n}^{(t)}\right]}^{\mathrm{T}}$. - Use random number r to compare with ${P}_{\mathrm{a}}$, keep the bird’s nest with the smaller probability of being found in P
_{t}, update the bird’s nest with the higher probability of discovery, calculate the fitness evaluation function of the new nest, and make comparison with the fitness evaluation function of the position P_{t}, and replace the bad position with a better bird’s nest position to get a set of the latest and better bird’s nest position P_{t}. - Find the optimal nest position b in Step (5), determine whether the fitness evaluation function F is up to the standard. If it is up to the standard, the search is stopped, and output the global best fitness evaluation function F and its optimal nest t; if it is not up to the mustard, return back to Step (3) to continue the optimization.
- Set the SVM parameters according to the optimal parameters $\left(C,\mathsf{\sigma}\right)$ corresponding to the optimal bird’s nest position ${x}_{b}^{(t)}$.

#### 2.2.4. Model Performance Indicators

^{2}), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). Their definitions are as follows:

## 3. Results and Discussion

#### 3.1. Microstructure Features

#### 3.2. Analysis of the Training and Prediction Process of CS-SVM Model

^{2}of CS-SVM training model on all these microstructural features reached by over 95%, and all these error performance indices (RMSE, MAE, MAPE) had low values, all these indicators meant that the CS-SVM training model obtained by 12 random samples had high accuracy and reliability in microstructural features prediction.

^{2}of CS-SVM training model on all these microstructural features also reached by over 95%, and all these error performance indices (RMSE, MAE, MAPE) also had low values, all these indicators meant that the CS-SVM training model obtained by 12 random samples had high accuracy and reliability in microstructural features prediction. It indicated that the CS-SVM model was very accurate in predicting all the microstructure features of APS coatings, even though some microstructure features changed slightly when the processing parameters changed significantly, so the CS-SVM model could meet the requirements in microstructure feature prediction.

## 4. Conclusions

_{f}, aspect ratio AR, circularity C

_{r}, owing to the variation of melting indices during the preparation process. An SVM model had been optimized by the CS algorithm, employed to build regression models, and trained to predict the coating microstructural features using the APS processing parameters. Fifteen samples were conducted to set up the data set for the training and prediction of the CS-SVM model. Five CS-SVM models with high R

^{2}values (>95%) and low error values (<0.1), in terms of five kinds of microstructure features prediction, were achieved by 12 random samples. The accuracy and reliability of the CS-SVM models have been further verified by the remaining three random samples test sets, where the R

^{2}values also reached by over 95% and error values were less than 0.1. Undoubtedly, these indicators proved that the proposed novel CS-SVM model in this work is very suitable for small sample regression prediction and its performance could meet the accuracy needed in actual microstructure features prediction. Additionally, this novel hybrid machine-learning method will potentially be extensively employed to establish the relationship among the process parameters, microstructure features, and service performance, and to monitor and ensure the integrity and safety of TBCs.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Implementation of thresholding and opening operation for microstructural features extraction.

**Figure 3.**The testing results and the prediction results of the microstructural features of the CS-SVM training model. (

**a**) Porosity; (

**b**) pore-to-crack ratio; (

**c**) maximum Feret’s diameter; (

**d**) aspect ratio; (

**e**) circularity.

**Figure 4.**The testing results and the prediction results of the microstructural features of the CS-SVM prediction model. (

**a**) Porosity; (

**b**) pore-to-crack ratio; (

**c**) maximum Feret’s diameter; (

**d**) aspect ratio; (

**e**) circularity.

**Table 1.**The atmospheric-plasma-sprayed (APS) process parameters for deposition of the 8 wt.% yttria partially stabilized zirconia (YSZ) ceramic top coatings.

APS Parameters | Optional Indicators |
---|---|

Particle size (μm) | 40–96, 15–55 |

Spray distance L (mm) | 70, 80, 90, 100, 110, and 120 |

Spray power (kW) | 24, 30, 33, and 36 |

Powder feed rate (L/min) | 10 |

gun speed (cm/s) | 15 |

Microstructure Feature | Definition | Symbol |
---|---|---|

Porosity | The volume percent of pore | ω |

Pore-to-crack ratio | The ratio of the globular pore area to crack network area | k |

Maximum Feret’s diameter | Maximum distance between two points on the boundary | D_{f} |

Circularity | 4πA/p^{2} (A is the pore area and p is the pore perimeter) | C_{r} |

Aspect ratio | Ratio of the longest diameter to shortest diameter (best fitting ellipse) | AR |

**Table 3.**Database for the training, validation, and test of the cuckoo search–support vector machine (CS-SVM) model.

No. | APS Process Parameters | Microstructural Features (Average Value) | ||||||
---|---|---|---|---|---|---|---|---|

Particle Size (μm) | Spray Distance (mm) | Spray Power (kW) | Porosity | Pore-to-Crack Ratio | Maximum Feret’s Diameter (μm) | Aspect Ratio | Circularity | |

1 | 15–55 | 70 | 36 | 11.83% ± 1.04% | 0.4062 ± 0.0222 | 1.2949 ± 0.2160 | 1.6312 ± 0.0519 | 0.8591 ± 0.0231 |

2 | 15–55 | 80 | 36 | 11.37% ± 1.63% | 0.5005 ± 0.0479 | 1.6076 ± 0.3802 | 1.6983 ± 0.0517 | 0.8436 ± 0.0130 |

3 | 15–55 | 90 | 24 | 12.98% ± 1.98% | 0.5419 ± 0.0278 | 1.7304 ± 0.1842 | 1.7283 ± 0.0136 | 0.8286 ± 0.0115 |

4 | 15–55 | 90 | 30 | 11.91% ± 1.02% | 0.5337 ± 0.0346 | 1.9900 ± 0.1723 | 1.7126 ± 0.0155 | 0.8039 ± 0.0052 |

5 | 15–55 | 90 | 33 | 11.25% ± 1.65% | 0.4336 ± 0.0517 | 1.6842 ± 0.3143 | 1.7333 ± 0.0431 | 0.8130 ± 0.0048 |

6 | 15–55 | 90 | 36 | 8.99% ± 0.96% | 0.3582 ± 0.0387 | 1.7011 ± 0.0591 | 1.7281 ± 0.0297 | 0.7899 ± 0.0134 |

7 | 15–55 | 100 | 36 | 10.14% ± 1.92% | 0.3998 ± 0.0301 | 1.8026 ± 0.1241 | 1.7658 ± 0.0295 | 0.8028 ± 0.0078 |

8 | 15–55 | 110 | 36 | 13.02% ± 1.44% | 0.4305 ± 0.0147 | 1.8072 ± 0.1407 | 1.7800 ± 0.0131 | 0.8049 ± 0.0049 |

9 | 15–55 | 120 | 36 | 16.83% ± 2.05% | 0.5902 ± 0.0271 | 2.0997 ± 0.2245 | 1.8294 ± 0.0384 | 0.7968 ± 0.0083 |

10 | 40–96 | 70 | 36 | 15.02% ± 1.77% | 0.3690 ± 0.0571 | 2.4856 ± 0.1337 | 1.7172 ± 0.0271 | 0.7899 ± 0.0163 |

11 | 40–96 | 80 | 36 | 15.38% ± 1.56% | 0.3480 ± 0.0194 | 2.2385 ± 0.1580 | 1.7898 ± 0.0167 | 0.7710 ± 0.0254 |

12 | 40–96 | 90 | 36 | 15.93% ± 1.85% | 0.3620 ± 0.0201 | 2.6294 ± 0.2081 | 1.7981 ± 0.0635 | 0.7811 ± 0.0169 |

13 | 40–96 | 100 | 36 | 18.31% ± 2.23% | 0.3720 ± 0.0288 | 2.6478 ± 0.2335 | 1.7821 ± 0.0215 | 0.7638 ± 0.0140 |

14 | 40–96 | 110 | 36 | 20.62% ± 1.79% | 0.3990 ± 0.0302 | 2.9361 ± 0.3542 | 1.8054 ± 0.0457 | 0.7529 ± 0.0234 |

15 | 40–96 | 120 | 36 | 23.01% ± 2.42% | 0.4100 ± 0.0256 | 3.0536 ± 0.2861 | 1.8350 ± 0.0529 | 0.7322 ± 0.0134 |

Range Analysis | Particle Size (Factor A) | Spray Distance (Factor B) | Spray Power (Factor C) | Influence Rank |
---|---|---|---|---|

Porosity | 0.0601 | 0.0771 | 0.0379 | B > A > C |

Pore-to-crack ratio | 0.0894 | 0.1142 | 0.1298 | C > B > A |

Maximum Feret’s diameter | 0.9188 | 0.6865 | 0.5078 | A > B > C |

Aspect ratio | 0.0538 | 0.1580 | 0.0508 | B > A > C |

Circularity | 0.0506 | 0.0600 | 0.7907 | C > B > A |

**Table 5.**The prediction performance of the 12 remaining random samples obtained by the CS-SVM model.

Training Results | R^{2} | RMSE | MAE | MAPE |
---|---|---|---|---|

Porosity | 0.9773 | 0.0050 | 0.0046 | 0.0360 |

Pore-to-crack ratio | 0.9900 | 0.0118 | 0.0117 | 0.0273 |

Maximum Feret’s diameter (μm) | 0.9709 | 0.0794 | 0.0739 | 0.0382 |

Aspect ratio | 0.9569 | 0.0106 | 0.0102 | 0.0058 |

Circularity | 0.9759 | 0.0059 | 0.0057 | 0.0072 |

Prediction Results | R^{2} | RMSE | MAE | MAPE |
---|---|---|---|---|

Porosity | 0.9922 | 0.0061 | 0.0057 | 0.0303 |

Pore-to-crack ratio | 0.9798 | 0.0076 | 0.0063 | 0.0159 |

Maximum Feret’s diameter (μm) | 0.9955 | 0.0954 | 0.0943 | 0.0408 |

Aspect ratio | 0.9767 | 0.0140 | 0.0130 | 0.0074 |

Circularity | 0.9953 | 0.0061 | 0.0047 | 0.0059 |

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## Share and Cite

**MDPI and ACS Style**

Ye, D.; Wang, W.; Xu, Z.; Yin, C.; Zhou, H.; Li, Y.
Prediction of Thermal Barrier Coatings Microstructural Features Based on Support Vector Machine Optimized by Cuckoo Search Algorithm. *Coatings* **2020**, *10*, 704.
https://doi.org/10.3390/coatings10070704

**AMA Style**

Ye D, Wang W, Xu Z, Yin C, Zhou H, Li Y.
Prediction of Thermal Barrier Coatings Microstructural Features Based on Support Vector Machine Optimized by Cuckoo Search Algorithm. *Coatings*. 2020; 10(7):704.
https://doi.org/10.3390/coatings10070704

**Chicago/Turabian Style**

Ye, Dongdong, Weize Wang, Zhou Xu, Changdong Yin, Haiting Zhou, and Yuanjun Li.
2020. "Prediction of Thermal Barrier Coatings Microstructural Features Based on Support Vector Machine Optimized by Cuckoo Search Algorithm" *Coatings* 10, no. 7: 704.
https://doi.org/10.3390/coatings10070704