# A Semi-Supervised Based K-Means Algorithm for Optimal Guided Waves Structural Health Monitoring: A Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Guided Waves Based SHM

#### 2.2. Damage Detection Approach

## 3. K-Means Based Method

#### 3.1. Classical K-Means Clustering

- Randomly choose k points (centers) from the input data set;
- Extract feature vectors from data;
- Assign each feature vector to the closet center;
- Compute the new centers of the formed clusters.

#### 3.2. Proposed Online Damage Detection Method

- ${\mathrm{TH}}_{\mathrm{d}}$: threshold of limit value of the distance that can be considered in the same cluster;
- i: counter of signal distances that exceed ${\mathrm{TH}}_{\mathrm{d}}$;
- k: number of clusters;
- N: persistence number to reach a new cluster;
- d: the Euclidian distance between the new signal feature vector and the centroid of its cluster.

## 4. Experiments and Database Building

## 5. Results and Discussion

#### 5.1. Classical K-Means

_{OPT}= 4. In the second case as shown in Figure 13b, which concerns the feature vector (RMS, mean), only the gap criterion indicates that the optimal number of clusters is five, which is the real number of data groups. This confirms that the feature vector (RMS, mean) is the most relevant for damage detection and will be retained in the following development.

#### 5.2. Novel Proposed Method

^{−3}to ${\mathrm{TH}}_{\mathrm{d}}$ = 8 × 10

^{−3}with a step of 1 × 10

^{−3}. This range was graphically determined from the previous results shown in Figure 12. The persistence number was fixed at N = 3. The result in Figure 15d shows that a high value of ${\mathrm{TH}}_{\mathrm{d}}=8\times {10}^{-3}$ kept all the signals in the same cluster and hence the number of clusters remained at k = 1. Therefore, this value of ${\mathrm{TH}}_{\mathrm{d}}$ did not allow the detection of the damaged state clusters. When ${\mathrm{TH}}_{\mathrm{d}}=6\times {10}^{-3}$, as shown in Figure 15c, the resulting clustering process was close to the real cluster of signals.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Experimental setup of the guided waves monitoring technique of a tube, and photography showing the adopted method to create a structural defect.

**Figure 6.**Schematic diagram of the proposed experimental setup shown in Figure 5.

**Figure 7.**Circumferential displacement distribution: for decreased cross-section change (CSC) (

**a**,

**c**), and decreased CSC (

**b**,

**d**), and reflection coefficient versus CSC in both cases (material loss and added material (

**e**,

**f**)). (r, z) are the radial and axial coordinates of the axisymmetric tube.

**Figure 8.**Example of a measured signal from a damaged pipe (top) and its time-frequency representation (bottom).

**Figure 10.**(left) Original signal (

**a**), noise (

**b**), and corrupted signal (

**c**), and (right), the collected data after removing the excitation signal and the end of pipe echo and after adding noise.

**Figure 12.**Clustering of collected data using two feature vectors: original clusters for (RMS, variance) (

**a**), k-means clusters for (RMS, variance) (

**b**), original clusters for (RMS, mean) (

**c**), and k-means clusters for (RMS, mean) (

**d**).

**Figure 13.**Variation of the criterion of optimal cluster number: (

**a**) (RMS, variance) and (

**b**) (RMS, mean).

**Figure 15.**Clustering results with different threshold distance ${\mathrm{TH}}_{\mathrm{d}}$ and a fixed persistence number N = 3. (

**a**) ${\mathrm{TH}}_{\mathrm{d}}=4\times {10}^{-3}$; (

**b**) ${\mathrm{TH}}_{\mathrm{d}}=5\times {10}^{-3}$; (

**c**) ${\mathrm{TH}}_{\mathrm{d}}=6\times {10}^{-3}$; (

**d**) ${\mathrm{TH}}_{\mathrm{d}}=8\times {10}^{-3}.$

**Figure 16.**Clustering results with different threshold distance ${\mathrm{TH}}_{\mathrm{d}}$ and a fixed persistence number N = 10. (

**a**) ${\mathrm{TH}}_{\mathrm{d}}=4\times {10}^{-3}$; (

**b**) ${\mathrm{TH}}_{\mathrm{d}}=6\times {10}^{-3}$; (

**c**) ${\mathrm{TH}}_{\mathrm{d}}=6.33\times {10}^{-3}$; (

**d**) ${\mathrm{TH}}_{\mathrm{d}}=10\times {10}^{-3}.$

**Table 1.**Matching matrix of the clustering results plotted in Figure 12b.

Predicted Cluster (%) | Healthy | Defect 1 | Defect 2 | Defect 3 | Defect 4 | |
---|---|---|---|---|---|---|

Real Cluster (%) | ||||||

Healthy | 52.5 | 47.5 | 0 | 0 | 0 | |

Defect 1 | 47.5 | 52.5 | 0 | 0 | 0 | |

Defect 2 | 0 | 0 | 99 | 1 | 0 | |

Defect 3 | 0 | 0 | 0.5 | 99.5 | 0 | |

Defect 4 | 0 | 0 | 0 | 0 | 100 |

**Table 2.**Matching matrix of the clustering results plotted in Figure 12d.

Predicted Cluster (%) | Healthy | Defect 1 | Defect 2 | Defect 3 | Defect 4 | |
---|---|---|---|---|---|---|

Real Cluster (%) | ||||||

Healthy | 71 | 29 | 0 | 0 | 0 | |

Defect 1 | 35 | 65 | 0 | 0 | 0 | |

Defect 2 | 0 | 0 | 98 | 2 | 0 | |

Defect 3 | 0 | 0 | 1.5 | 98.5 | 0 | |

Defect 4 | 0 | 0 | 0 | 0 | 100 |

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**MDPI and ACS Style**

Bouzenad, A.E.; El Mountassir, M.; Yaacoubi, S.; Dahmene, F.; Koabaz, M.; Buchheit, L.; Ke, W.
A Semi-Supervised Based K-Means Algorithm for Optimal Guided Waves Structural Health Monitoring: A Case Study. *Inventions* **2019**, *4*, 17.
https://doi.org/10.3390/inventions4010017

**AMA Style**

Bouzenad AE, El Mountassir M, Yaacoubi S, Dahmene F, Koabaz M, Buchheit L, Ke W.
A Semi-Supervised Based K-Means Algorithm for Optimal Guided Waves Structural Health Monitoring: A Case Study. *Inventions*. 2019; 4(1):17.
https://doi.org/10.3390/inventions4010017

**Chicago/Turabian Style**

Bouzenad, Abd Ennour, Mahjoub El Mountassir, Slah Yaacoubi, Fethi Dahmene, Mahmoud Koabaz, Lilian Buchheit, and Weina Ke.
2019. "A Semi-Supervised Based K-Means Algorithm for Optimal Guided Waves Structural Health Monitoring: A Case Study" *Inventions* 4, no. 1: 17.
https://doi.org/10.3390/inventions4010017