Multi-Mode Guided Waves Based Reference-Free Damage Diagnostic Imaging in Plates

Probability-based diagnostic imaging (PDI) is one of the most well-known damage identification methods using guided waves. It is usually applied to diagnose damage in plates. The previous studies were dependent on the certain damage index (DI) which is always calculated from the guided wave signals. In conventional methods, DI is simply defined by comparing the real-time data with the baseline data as reference. However, the baseline signal is easily affected by varying environmental conditions of structures. In this paper, a reference-free diagnostic imaging method is developed to avoid the influence of environmental factors, such as temperature and load conditions. The DI is defined based on the mode conversion of multi-mode guided waves with realtime signals without baseline signals. To improve the accuracy of diagnosis, two terms are included in the reference-free DI. One is called energy DI, which is defined based on the feature of signal energy. The other is called correlation DI and is defined based on the correlation coefficient. Then the PDI algorithm can be carried out instantaneously according to the reference-free DI. The real-time signals which are used to calculate DI are collected by the piezoelectric lead zirconate titanate (PZT) transducers placed on both sides of a plate. The numerical simulations by the finite element (FE) method on aluminum plates with PZT arrays are performed to validate the effectiveness of the reference-free damage diagnostic imaging. The approach is validated by two different arrays: a circle network and a square network. The results of diagnostic imaging are demonstrated and discussed in this paper. Furthermore, the advantage of reference-free DI is investigated by comparing the accuracy of defined reference-free DI and energy DI.

using a new DI based on mode conversion [Kim and Sohn (2007)]. Mode conversion can occur during the process of guided wave scattering in the case of non-symmetric thickness variation [Cho (2002)]. These studies could identify the presence of damage and the approximate location in the direct path between transducers. Nevertheless, the mode conversion has not been used for damage diagnostic imaging yet. In this paper, a novel reference-free PDI method on the basis of the mode conversion of multi-mode guided waves is developed to identify damage in plates, which can avoid the environmental influence on diagnostic results. Firstly, the extraction method of modes in a T-shaped stringer is proposed. The pristine modes and converted modes can be extracted from real-time signals. Then, a new reference-free DI with two terms is defined in terms of the extractive modes, which is used in the PDI algorithm for the imaging. Finally, the numerical results show that the method is capable of damage diagnostic imaging in plates.

Damage diagnostic imaging approach 2.1 The extraction of converted modes
Mode conversion phenomena of guided waves in plates were found in the past years [Cho and Rose (1996)]. Mode conversion occurs due to the non-symmetric thickness variation and damage in plates almost belongs to non-symmetric thickness variation [Cho (2002)]. Therefore, if guided waves propagating along a plate encounter a sudden thickness discontinuity such as cracks or corrosion, some modes would convert to other modes. Some mode conversion phenomena in damaged plates are shown in Fig. 1. Even same mode can be divided into two different types according to the deformed shapes of the surface of plates. The mode, whether S0 or A0, can only convert to the mode with the same deformed shapes. The additional modes are named M1 and M2 form due to the crack and signals M1 and M2 are both between signals S0 and A0. Note that M1 and M2 could be either the converted S0 or A0 according to the relative position of the crack and the actuating and sensing transducers [Kim and Sohn (2007)]. In the previous study, it has been demonstrated that M1 and M2 can be extracted from the signals AC, AD, BC and BC which are shown in Fig. 2 [Lee, Kim and Sohn (2008)].

Figure 2: The phases of modes in signals
It is observed that the signals AC, AD, BC and BC are all drawn as the superpositions of signals S0, A0, M1 and M2. Reversely, the signals S0, A0, M1 and M2 can be isolated by additions and subtractions of signals AC, AD, BC and BC, which are expressed in matrixes 0 1 2 0 1 1 1 1 1 1 1 1

Reference-free DI calculation
In this study, the DI consists of two terms. The first term of DI is energy DI. It is defined based on the the feature of signal energy which is calculated as The second term of DI is correlation DI. It is defined based on the correlation coefficient, which has been broadly applied in previous research Zhao et al. [Zhao, Gao, Zhang et al. (2007)]. A correlation coefficient  of two signals can be define in a simple mathematical formula. The second term is given by

PDI algorithm
In PDI approaches, the monitoring area of a structure is meshed into grids and projected to an image, which concerns the probability of presence of damage. The probability of damage occurring at each grid is yielded with appropriate features extracted from guided wave signals and the probability at certain grid ( ) , xy can be calculated as W R x y   are respectively the DI and the weight distribution function of the ith sensing path. The DI can be calculated by the approach in the last section, and the weight distribution function is the non-negative linear decreasing function which can be written as This weight value increases with a decrease in the relative distance between grid ( ) , xy and the direct sensing path in which damage can cause the most significant signal change [Zhao, Gao, Zhang et al. (2007)].     3 Numerical simulations

FE modeling
The commercial finite element program ABAQUS/IMPLICIT is used to simulate the reference-free PDI algorithm. An aluminum plate with PZT transducers as the simulation object is modeled in this work. The size of the aluminum plate is 450 mm×450 mm×3 mm and the dimensions of PZT transducers are 8 mm in diameter and 0.2 mm in thickness. The material properties of the aluminum plate are illustrated as follows: mass density of aluminum is 2700 kg/m 3 , Young's modulus is 71 GPa and Poisson' ratio is 0.33. The material properties of PZT are illustrated in Tab. 3. To reduce the reflection from boundary, the absorbing region is set up by absorbing layers with incremental damping [Moreau and Castaings (2008)] as shown Fig. 6. The plate is discretized by C3D6 elements, the absorbing region is discretized by C3D8R elements and the PZT transducers are discretized by C3D8E elements. Some works were investigated to discuss the effect of the element number per wavelength on FE calculation result [Zhang, Zou and Madenga (2006)]. The element size of the plate depends on the minimum wavelength and usually 20 nodes are taken in a minimum wavelength [Li, Jing and Jin (2017)]. The size of the elements used to the plate is 1.5 mm and the size of the elements used to the PZT transducers was 1 mm in order to ensure that a sufficient number of nodes in one transducer were available for collecting signals.  A 5 cycle Hanning windowed sinusoidal signal of which the central frequency is 200 kHz is used as the input signal in this paper. The propagation velocity used in this simulation is determined by the dispersion curves of the aluminum plate with the thickness of 3 mm in Fig. 7. It is observed that the mode S0 and mode A0 can be excited at 200 kHz without other modes. Two types of damage are set on the aluminum plates individually as illustrated in Fig. 8. They are both nonpenetrating, which are belong to non-symmetric thickness variation. The size of the square damage is 20 mm×20 mm×2 mm, and the size of the circle damage is φ20 mm×2 mm. The reference-free damage diagnostic imaging of the two plates is carried out individually using the process above.  Fig. 9 shows the static displacement nephograms of different models. It is observed that guided waves convert to new modes after encountering damage because the nonpenetrating damage belongs to the non-symmetric thickness variation along the direction of guided wave propagation. The signals with individual mode can be isolated from the response signals by Eq. (1), and the results are shown in Fig. 10. The reference-free DI of all damage types calculated from the isolated signals are illustrated in Fig. 11 and Fig. 12 in which the energy DI and combinational DI are both shown for the comparison. And the comparison of the energy DI and combinational DI indicates that the combinational DI can reduce the impact of non-damaged sensing paths. Therefore the combinational DI can improve the precision of diagnosis. The results of the PDI calculated on the basis of the DIs are shown in Fig. 13 and Fig. 14 for circle and square networks respectively. Fig. 13(a) and Fig. 13(c) are the diagnostic images only using energy DI, and Fig. 13(b) and Fig. 13(d) using combinational DI. Correspondingly Fig. 14 is same as Fig. 13.   SDHM, vol.13, no.1, pp.41-59, 2019

Conclusion
This paper has presented a novel reference-free damage diagnostic imaging method based on the probability-based diagnostic imaging (PDI) without baseline data vulnerable to the labile environment. The reference-free PDI algorithm in this paper is on the basis of the damage index (DI) like the previous PDI algorithms. However, the DI is defined by the converted modes extracted from real-time multi-mode guided wave signals. Thus the dependence on the baseline data is eliminated on the strength of mode conversion. Futhermore, the reference-free DI consists of two terms in order to improve the accuracy of diagnosis. They are the energy DI defined based on the feature of signal energy and the correlation DI defined based on the correlation coefficient. By applying the referencefree DI to the PDI algorithm, diagnostic images of two different configurations, circle network and square network, are clear and accurate. The effectiveness of this method is assessed over the numerical simulations by the FE method. Noted that with the combinational DI adopted instead of the energy DI, higher localization accuracy is obtained. More tests at low-temperature environment need to be carried out experimentally to further prove the effectiveness of the proposed method without baseline data.