# Hierarchical Wavelet-Aided Neural Intelligent Identification of Structural Damage in Noisy Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Theories

#### 2.1. Wavelet Packet Transform (WPT)

#### 2.2. Nonlinear Principal Component Analysis

## 3. Damage Identification Paradigm

#### 3.1. Formation of Damage Features Using AANNs

#### 3.1.1. Wavelet Packet Node Energies (WPNEs)

#### 3.1.2. Damage Feature Extraction

#### 3.2. Damage Pattern Recognition

#### 3.3. Hierarchical Neural Network Model

- (i)
- WPNEs are much more sensitive to damage than WPT coefficients, natural frequencies, and mode shapes.
- (ii)
- AANNs acting as a smart NLPCA tool can extract damage features from WPNEs. Such extracted damage features have lower dimensionality than WPNEs while preserving enough damage information.
- (iii)
- LMNNs can capture the underlying relations between damage features and damage states, on which they can recognize structural damage patterns.
- (iv)
- The special structure of the hierarchical neural network model requires a small set of training samples of damaged cases to produce accurate prediction results of damage identification with great noise robustness.
- (v)
- The hierarchical neural network model is easily implemented in a computational language, e.g., Matlab, to create an automatic program of intelligent damage assessment.

## 4. Numerical Verification

#### 4.1. Damage Cases

#### 4.2. Damage Feature Extraction

#### 4.3. Damage Pattern Recognition

_{Ele 1}, S

_{Ele 2}, S

_{Ele 3}, S

_{Ele 4}, S

_{Ele 5}], among which S

_{Ele k}denotes the SRR of the $k\mathrm{th}$ element of the bridge. S

_{Ele k}takes the value of the interval [0, 0.3], suggesting that the maximum severity of damage considered is SRR = 30%. When the SRR varies from 1% to 30% for each element, 150 damaged cases are elaborated, as listed in Table 3. These damaged cases and the intact case are divided into a training set and a testing set of samples. The training set consists of 16 cases including 15 damaged cases that the SRR = 10%, 20%, 30% for each element and the intact case; the testing set comprises 135 damaged cases with SRRs differing from those in the training set. The training sample set is used to train the LMNNs with the cost function of MSE to control the training.

_{Ele 1}, S

_{Ele 2}, S

_{Ele 3}, S

_{Ele 4}, S

_{Ele 5}] for every sample. The difference between the estimated vectors and the target output vectors indicates the error of the damage identification. For instance, for the result corresponding to Ele 1, Figure 8a presents the output of Ele 1 estimated by the LMNNs for all the test samples; Figure 8b presents the associated target outputs; Figure 8c depicts the difference between the estimated values and the target outputs, i.e., the damage identification error. Clearly, the Ele 1 output of the LMNN approximates the target very well. For all the test samples, the damage identification results are given in Figure 9a–c for the estimated values, targets outputs, as well as the identification error. Firstly, the damage location can be detected with great accuracy; there are few incorrect judgments of the damage location. For the aspect of damage severity, some errors obstruct from obtaining a very precise prediction of the severity of damage. However, those errors are within a small range. As seen in Figure 9c, the errors are all below an upper limit of about seven, indicating that the detection may only fail for some minor damage. In a word, the results show that the proposed hierarchical neural network model can effectively detect both the location and severity of damage.

#### 4.4. Robustness Against Noise

## 5. Comparison with Traditional Methods

## 6. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of wavelet packet transform (WPT) of a time-domain signal. A: approximate component signal; D: detail component signal.

**Figure 2.**Auto-associative neural networks (AANNs) for damage feature extraction from wavelet packet node energies (WPNEs). (

**A**) mapping layer; (

**B**) bottleneck layer; (

**C**) remapping layer.

**Figure 3.**Levenberg-Marquardt neural networks (LMNNs) for damage pattern recognition. (

**A**) input layer; (

**B**) hidden layer; (

**C**) output layer.

**Figure 6.**Extracted nonlinear principle components (NPCs) from wavelet packet node energies (WPNEs) using AANNs. (

**a**) WPNEs; (

**b**) Normalized WPNEs; (

**c**) Normalized NPCs.

**Figure 7.**Visualization of the capability of NPCs to characterize damage by a progressively zoomed-in cube.

**Figure 8.**Identification of damage in Ele 1 of the bridge. (

**a**) Damage estimate; (

**b**) Target output; (

**c**) Damage identification error.

**Figure 9.**Identification of damage in the bridge in noise-free condition. (

**a**) Damage estimate; (

**b**) Target output; (

**c**) Damage identification error.

**Figure 10.**Identification of damage in the bridge in severely noisy conditions. (

**a**) signal-to-noise ratio (SNR) = 10 dB; (

**b**) SNR = 20 dB; (

**c**) SNR = 30 dB.

**Figure 11.**Identification of damage using a typical method in the noise-free condition. (

**a**) Damage estimate; (

**b**) Identification error.

Properties | Values |
---|---|

Mass density | $7800$ ${\mathrm{kg}/\mathrm{m}}^{2}$ |

Young’s modulus | $200$ ${\mathrm{GN}/\mathrm{m}}^{2}$ |

Cross-sectional area | $5\times {10}^{-2}$ ${\mathrm{m}}^{2}$ |

Moment of inertial | $4.17\times {10}^{-6}$ ${\mathrm{m}}^{2}$ |

Damping ratio | $0.01$ |

Case Number | Damaged Element | SRR (%) |
---|---|---|

1–30 | 1 | 1, 2, …, 30 |

31–60 | 2 | 1, 2, …, 30 |

61–90 | 3 | 1, 2, …, 30 |

91–120 | 4 | 1, 2, …, 30 |

121–150 | 5 | 1, 2, …, 30 |

Element | Training Set | Testing Set |
---|---|---|

SRR (%) | SRR (%) | |

1 | 10 | 1, 2, …, 9 |

20 | 11, 12, …, 19 | |

30 | 21, 22, …, 29 | |

2 | 10 | 1, 2, …, 9 |

20 | 11, 12, …, 19 | |

30 | 21, 22, …, 29 | |

3 | 10 | 1, 2, …, 9 |

20 | 11, 12, …, 19 | |

30 | 21, 22, …, 29 | |

4 | 10 | 1, 2, …, 9 |

20 | 11, 12,…, 19 | |

30 | 21, 22, …, 29 | |

5 | 10 | 1, 2, …, 9 |

20 | 11, 12, …, 19 | |

30 | 21, 22, …, 29 |

Noise | SNR (dB) | |||

10 | 20 | 30 | $\mathrm{\infty}$ | |

MSE | 4.892 | 4.94 | 7.138 | 3.309 |

Maximum error | 13.1 | 7.048 | 10.462 | 7.044 |

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**MDPI and ACS Style**

Cao, M.-S.; Ding, Y.-J.; Ren, W.-X.; Wang, Q.; Ragulskis, M.; Ding, Z.-C. Hierarchical Wavelet-Aided Neural Intelligent Identification of Structural Damage in Noisy Conditions. *Appl. Sci.* **2017**, *7*, 391.
https://doi.org/10.3390/app7040391

**AMA Style**

Cao M-S, Ding Y-J, Ren W-X, Wang Q, Ragulskis M, Ding Z-C. Hierarchical Wavelet-Aided Neural Intelligent Identification of Structural Damage in Noisy Conditions. *Applied Sciences*. 2017; 7(4):391.
https://doi.org/10.3390/app7040391

**Chicago/Turabian Style**

Cao, Mao-Sen, Yu-Juan Ding, Wei-Xin Ren, Quan Wang, Minvydas Ragulskis, and Zhi-Chun Ding. 2017. "Hierarchical Wavelet-Aided Neural Intelligent Identification of Structural Damage in Noisy Conditions" *Applied Sciences* 7, no. 4: 391.
https://doi.org/10.3390/app7040391