1 Introduction
With the exponential increase in the number of civil infrastructures, including buildings and bridges, it has become imperative to monitor their structural integrity. A number of techniques exist for monitoring structural health. Ultrasonic guided wave testing (UGWT) is one such popular, nondestructive method. Ultrasonic guided waves can scan large areas and are sensitive to damage. Hence they are a popular option for damage localization systems. An UGWT setup consists of a spatial array of sensors that can transmit and / or receive acoustic signals. Based on these acoustic signals, multiple techniques have been developed for damage localization michaels2008detection .
Wave physics based techniques are a common choice for solving the inverse problem of guided wave based localization fan2011vibration . Typically, such techniques for damage localization use a theoretical model of wave propagation. By comparing the signal received at receiver sensors with the output of the physical model at all of the possible damage locations, the structural damage is localized. Yet, there exists uncertainty in guided wave propagation because of variations of external factors, such as temperature, humidity, air pressure and random noise sohn2006effects . This uncertainty is challenging to incorporate in a physical wave propagation model because of the complex and dispersive nature of guided waves friswell2006damage .
To tackle these uncertainties, researchers have explored datadriven approaches for damage localization with guided waves harley2017managing
. Simultaneously, there has been a growing interest in machine learning models for physical parameter estimation problems
raissi2019physics karpatne2017physics . To physically model real world phenomena in a system, tackling sources of uncertainty due to external factors is critical oberkampf2004challenge . Accurate physical characterization of uncertainties in real world scenario for further use in machine learning models remains a challenging and open research problem.In this work, we investigate the potential for tackling uncertainty in guided wave propagation using deep learning models. We simulate guided wave data with uncertainty in velocity of wave. We also simulate random noise by adding Gaussian noise to the data. We then build a deep neural network (DNN) that learns representations which are robust to the uncertainties in the data. We further motivate the use of dropout regularization as a tool to tackle uncertainty in physics inspired machine learning models. We validate our results on test dataset and conclude our discussion with future scope.
2 Modelbased damage localization and challenges
2.1 Damage localization setup
Lamb waves are a specific case of guided waves commonly used in guided wave based damage localization algorithms worden2001rayleigh . The farfield Lamb wave model is given by
(1) 
where X in (1
) is the frequency domain representation of the signal and is modeled as the summation across
wave modes. The function is the frequency and mode dependant wavenumber (known as the dispersion relation) and is the distance travelled by the wave.Consider the following setup for damage localization. An array of sensors is placed on a grid of dimensions . Every unique sensor pair acts as a transmitterreceiver pair. There exists a damage at some point in the grid that needs to be localized. The physical model for wave propagation in (1) is calculated for all possible damage locations on the grid. The location at which maximum correlation is obtained between the physical model and received data is the location estimate.
2.2 Challenges with modelbased damage localization
Localization heatmaps: (a) in noiseless conditions and (b) in noisy conditions (illustrated here with a signaltonoise ratio (SNR) of 5 dB).
The dispersion relation between and is critical to understanding the behaviour of waves in structures. This relation is dependent on the properties of the material in which the wave propagates. The accurate recovery of dispersion relation thus becomes complicated in the presence of external uncertainties such as temperature, humidity, air pressure and random noise which affect material properties. We observe that uncertainty in the dispersion relation leads to uncertainty in (1). Moreover, as group velocity (velocity of a wave packet) is given by
(2) 
uncertainty in dispersion relation also leads to uncertainty in velocity. Similarly, random noise also affects the performance of model based localization. Figure 1 (a) and Figure 1 (b) show the performance in ideal (noiseless) conditions and noisy conditions, respectively, as a heatmap. The preceding discussion motivates our research direction of dealing with uncertainty in wave propagation.
3 Deep neural network based localization framework
3.1 Simulation setup
For damage localization, we assume a plate with unit dimensions and the damage modeled at a random point. The damage acts as a point scatterer of incident waves. We use a sparsearray configuration for the sensors. We use random sensor placement to unbias the results based on sensor configuration. We simulate guided waves using (1) with sensors ( unique sensor pairs) placed at random locations on the plate and equally spaced frequencies from 0 to 1000 kHz.
We refer to the timedomain wave data matrix and the corresponding damage location as one inputlabel pair. We do simulations to build a guided wave dataset with samples. We divide this dataset into train and test set. As shown in Figure 2, we add uncertainty and random noise to the simulated wave data. We introduce a random multiplicative effect on the dispersion curves such that modified dispersion curves are defined by
. This creates velocity uncertainty in the simulations. The random variable
is randomly sampled from a Gaussian distribution truncated between
andand a standard deviation of
. The random noise is modeled as additive white Gaussian noise (AWGN).Next, the timedomain wave data is preprocessed appropriately for the DNN to ensure optimum performance. This includes standardizing it and flattening the matrix into a 1D vector to be used as input to the DNN. The DNN has 3 fully connected hidden layers. First hidden layer has
= 300 nodes, second hidden layer has = 200 nodes, and third hidden layer has = 50 nodes. The output layer has 2 nodes, one each for anddimension localization. We choose loss function as the Euclidean distance between prediction of DNN and the actual damage location as shown in the extreme right of Figure 2. We train the DNN using Keras package
chollet2015kerasfor 50 epochs.
3.2 How does DNN help tackle uncertainty in guided wave propagation ?
The DNN has fully connected layers (every node from previous layer is connected to every node in the next layer). At the input layer, this enables the network to learn crossfrequency relationships for the dispersive wave data. Having multiple hidden layers in the DNN, likely also helps to learn more complex representations for the inverse mapping between the wave data and the damage location.
For a physical science problem, overfitting is analogous to having an overcomplex model that explains the available data but does not explain well on unseen data. We want to ensure that the network does not learn overly complex representations yet it should also account for the uncertainty. Overfitting is tackled in machine learning community using regularization techniques such as dropout srivastava2014dropout . When using dropout, nodes in the DNN are randomly dropped out of the network while training, which is equivalent to learning multiple models. Researchers have presented theory that casts dropout as a measure of model uncertainty gal2016dropout . This motivates our research direction of using deep learning models and tools to deal with uncertainty in wave propagation.
4 Results
The performance of the localization algorithms is quantified with average localization error (ALE) on the test dataset (wave data with uncertainties and random noise)
(3) 
where T is the number of samples in the dataset and , are the actual and predicted damage locations respectively.
Figures 3 ((a)(b)) are reported for test dataset (wave data with uncertainties and random noise). The xaxis represents the signaltonoise ratio of the samples in the test dataset and yaxis represents the localization error metric (arrowheads representing the standard deviation). Figure 3 (a) compares the localization performance of DNN trained with uncertainty and random noise with that of DNN trained on ideal data. At 25 dB, DNN’s trained with and without uncertainty have localization errors of 0.0321 m and 0.0742 m respectively. Figure 3 (a) thus illustrates that model trained with uncertainty is able to learn representations that are robust to the uncertainty better than a model trained without any explicit uncertainty in the SNR range of 5 to 25 dB.
Figure 3 (b) compares the localization performance of the DNN trained with uncertainty and random noise with that of a physical model based technique friswell2006damage . The physical model based technique uses a known wave propagation model which does not reflect the uncertainty present in a realistic setup. This leads to a highly variable performance trend for model based technique when tested on data with uncertainty. Figure 3 (b) illustrates that the DNN trained with uncertainty and noise has a superior performance compared to the physical model based technique.
5 Conclusions
We discussed the challenges posed by uncertainty due to external factors and random noise in ultrasonic wave based damage localization. We simulated wave data with uncertainty in velocity to reflect wave propagation in a realistic scenario. We further modeled random noise in environment as Gaussian noise. We trained a DNN on this simulated data and validated the simulation results on a test dataset.
Based on the initial results, we can conclude that deep learning models can help deal with physical uncertainty in ultrasonic wave propagation. These results also provide further motivation for research of deep learning models and tools as a way of incorporating uncertainty in physical science problems.
6 Acknowledgments
This research is supported by the Air Force Office of Scientific Research under award number FA95501710126 and the National Science Foundation under award number 1839704.
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