Identification and characterization of damaged fiber-reinforced laminates in a Bayesian framework

Abstract

Non-destructive thermographic testing of damaged composite laminates modeled from the homogenization of fiber-reinforced polymers is a challenge, both because of its underlying complexity and because of the difficulties encountered in the quantification of uncertainties related to the identification and characterization of defects. To provide a rigorous framework that accepts data from different modalities and allows data fusion as well, a Bayesian neural network (BNN) [I. Kononenko, Biological Cybernetics61(5) (1989), 361–370] with two input streams is proposed, with a focus on local inter-layer delaminations identification and characterization.

Keywords

Non-destructive testing fiber-reinforced laminates infra-red thermal imaging Bayesian neural network uncertainty quantification

1. Introduction

Composite materials are widely used in numerous fields, ranging from the automotive industry to green engineering, due to their low cost and lightness. Additionally, their superior strength/ and stiffness/weight ratios vs. traditional materials make them appealing. Yet their non-destructive testing (NdT) may be highly challenging, with the need for more modeling and computational imaging, before envisaging sound laboratory-controlled experimentations.

Active thermography as a contactless procedure is increasingly used for the inspection of isotropic workpieces, refer to [2] and references therein, yet more and more contributions appear on fibered laminates, with a strong emphasis on carbon-fibered ones, those exhibiting high anisotropic behavior due to their thermal conductivity function of orientation, e.g., [3], as illustrated in the recent review [4].

In the context of thermographic inspection, addressing the following threefold challenge involves simplifications. First, intact laminates are assumed horizontally homogeneous with uniaxial anisotropic behavior per ply. Second, damages are simplified as thin delaminations between plies, suitable for thermography. Third, efficient computational imaging tools are required for optimized data utilization. Simplifying further, data fusion uses two infrared cameras capturing temperature evolution on top and bottom surfaces. Simulating data on various damages leads to employing double-stream convolutional neural networks (CNN) for effective data fusion, ensuring resilience if semi-analytical algorithms falter. This approach, as demonstrated in [5], proves highly effective.

So, if data are simulated on a host of damages, double-stream convolutional neural networks (CNN) appear as the way forward, also having insurance that brute force codes (FEM) could always provide proper data, if the semi-analytical algorithms were to fail in some cases. This approach was shown highly effective in [5].

The main contribution of the present study lies in the deployment of a Bayesian Neural Network (BNN) with a dual-output architecture, addressing the challenges inherent in delamination detection. By integrating a regression output for a precise estimation of the delamination parameters and a segmentation output for pixel-wise defect classification, this approach offers a comprehensive solution for characterizing delamination in materials.

Fig. 1.

2D sketch of double-layered damaged piece illuminated by a flash lamp.

2. Dataset

2.1. BNN inputs: Reasons and benefits

The use of a dual input of the heat map evolution from the top and bottom layers of a fiber-reinforced laminate in the context of infrared thermography is proposed as represented in Fig. 1. Indeed, by combining the heat map evolution data from both the top and bottom layers, one can achieve a more comprehensive understanding of the inspected laminate. Defects or anomalies inside a given layer may not be visible or distinguishable in isolation. Yet, by fusing the data, one can identify patterns or thermal signatures that manifest across layers. This improves the accuracy and reliability of defect detection, enabling a better assessment of the laminate’s condition.

A dataset made of 12,500 samples has been developed, each containing two sequences of 30 elements of size 160 by 160, showcasing the evolution of temperature on the top and bottom surfaces of the studied workpiece with a constant temporal step size (Δt = 0.5 s). Each sample has only a single rectangular defect. This dataset has been developed using a method called “Faster Simulation Approach Dedicated to Infrared Thermographic Inspection of Anisotropic Delaminated Planar Pieces” described in [6], denoted as the “convolution approach”. A flash lamp is used, illuminating the top surface of the studied piece, with a spatial support of 60 × 60 pixels for a duration of 10 ms. The method of acquisition is based on a propagation/reflection principle. Heat propagation inside the room is defined as a function of the conductivity of the materials in it. The thermal conductivity in the direction of the carbon fibers is much higher than in the rest of the workpiece, e.g., with values in W. m⁻¹.K⁻¹ as orthogonal to the carbon fiber = 0.61, in the carbon fiber direction = 2.71, and cross-ply = 0.53, heat propagation is highly dependent on the direction of the carbon fibers.

The different conductivities in any direction of the part as a function of the orientation 𝜃 of the carbon fibers of the studied layer, as first introduced in [7], read as $\begin{eqnarray}\bar{\bar{{\sigma}}}=\left(\begin{array}{@{}ccc@{}}{\sigma}_{//}\text{cos}^{2}{\theta}+{\sigma}_{\bot }\text{sin}^{2}{\theta} & {\displaystyle \frac{{\sigma}_{//}-{\sigma}_{\bot }}{2}}\text{sin}2{\theta} & 0\\[10.0pt] {\displaystyle \frac{{\sigma}_{//}-{\sigma}_{\bot }}{2}}\text{sin}2{\theta} & {\sigma}_{//}\text{cos}^{2}{\theta}+{\sigma}_{\bot }\text{sin}^{2}{\theta} & 0\\[10.0pt] 0 & 0 & {\sigma}_{CP}\end{array}\right).\end{eqnarray}$ (1)

The heat transfer in a multi-ply CFRP material involves considering influences from adjacent layers (n − 1 and n + 1), the layer itself (n), and defects at layer interfaces (n and n + 1). These influences are categorized based on the originating layer, requiring three convolution kernels for the n layer and two for the edges. If studying the n layer with both n − 1 and n + 1 layers, five influences are considered, necessitating three convolution kernels.

These kernels are developed as follows: each layer has the same conductivity components in the direction of the fiber and orthogonal to it. Yet, each layer has its proper fiber orientation. With this information, it is possible to define convolution kernels corresponding to each of these layers. Indeed, letting 𝜎_sd be the standard deviation and ${\sigma}_{sd}^{2}$ the variance, one can determine the Gaussian kernel for zero orientation with the parallel and transverse conductivities 𝜎_∕ and 𝜎_∕ as $\begin{eqnarray}G_{2D}(x,y;{\sigma}_{sd})=\frac{1}{2{\pi}{\sigma}_{sd}^{2}}e^{-\frac{{\sigma}_{\bot }\times x^{2}+{\sigma}_{//}\times y^{2}}{2{\sigma}_{sd}^{2}}}.\end{eqnarray}$ (2) To obtain the convolution kernels oriented according to the fiber orientation, it is sufficient to define the fiber orientation for each layer and to perform a rotation of the same angle to the convolution kernel.

Depending on heat propagation direction and carbon fiber orientation, one can associate it with a specific layer using a priori knowledge. Delamination, representing a thin air layer between two layers, disrupts the heat flow, creating hotter zones on the top surface and cooler zones below. A human operator, informed about layer orientations, can identify the delamination location from these thermal patterns. To enhance the accuracy of the system modeling, noise with a signal-to-noise ratio (SNR) of 20 dB is introduced. The convolution method, employed as a direct model, generates a synthetic dataset capturing intricate thermal delamination patterns. This dataset proves crucial for addressing the underlying inverse problem via Bayesian CNNs.

2.2. Bayesian CNN’s use for temporal data

While the dataset is not explicitly organized as a time series, the two sequences of 30 elements representing temperature evolution capture temporal dynamics. When presented sequentially during training, the model can learn patterns and dependencies over time, despite not conforming to the traditional time series format. The choice between a Bayesian Neural Network (BNN) and a Recurrent Neural Network (RNN) [8], such as LSTM [9] or GRU [10], for this task is influenced by several factors. In addition to BNNs providing uncertainty estimation, they exhibit regularization effects through Bayesian inference, aiding in preventing overfitting, particularly in scenarios with limited data.

2.3. Examples of the direct model performances

In this subsection, the aim is to compare the proposed convolution method, employed as a direct model for the simulation of IRT in delaminated CFRPs, and the established Comsol approach. Alignment and coherence of results between these two methodologies are exemplified for the configurations shown in Fig. 2, temperature evolutions at the central pixel being displayed in Fig. 3,

Fig. 2.

Sketches of the different configurations: (a) n°1, (b) n°2, (c) n°3.

Fig. 3.

Central pixel temperature evolution for both top and bottom surfaces with configurations (a) n°1, (b) n°3, (c) n°2 with and without central defect.

2.4. BNN outputs

The Bayesian Neural Network (BNN) produces two key outputs with distinct purposes. The first output is a volumetric object (160 × 160 pixels, depth of 5) representing layer interfaces in the fiber-reinforced laminate. This visualizes structural features and defects. The second set of outputs includes coordinates of delamination key points—four corner points, depth, and width. These outputs provide detailed insights into the location, extent, and geometric features of the delamination within the structure.

By providing both the volumetric object representing the layer interfaces and the precise coordinates of the delamination, the BNN outputs offer a comprehensive characterization of the CFRP’s internal structure and the presence of potential defects.

3. BNN architecture

A suitable choice of architecture for the BNN could be a convolutional neural network (CNN) combined with Bayesian layers. Hereafter is the suggested architecture (Fig. 4):

Fig. 4.

Sketch of the proposed dual-input dual-output BNN for delamination detection.

The model yields two distinct outputs: segmentation and regression. The segmentation output serves the purpose of providing a volumetric representation. It creates a binary mask that discerns which pixels within the input matrices correspond to the detected delamination and which do not. Essentially, this output visually delineates the spatial extent of the identified delamination within the material. Complementing the segmentation output, the regression output offers quantitative insights into the detected delamination. The regression output is succinctly represented as a table of parameters, offering numerical estimates and coordinates associated with the detected delamination.

4. Illustrative numerical simulations

4.1. Quantitative assessment – Regression

To quantitatively appraise the accuracy of the regression output, one employs a relative error metric. This evaluation allows measurement of the discrepancy between predicted coordinates of the delamination and actual ones, providing insights into the precision and reliability of the regression model. The relative error Err obtained with one-defect simulated anisotropic laminates reads as $\begin{eqnarray}Err=\frac{1}{N}\mathop{\sum }_{n}\sqrt{{\displaystyle \frac{\displaystyle \mathop{\sum }_{w,h}|\hat{{\zeta}}_{w,h}^{n}-{\zeta}_{w,h}^{n}|^{2}}{\displaystyle \mathop{\sum }_{w,h}|{\zeta}_{w,h}^{n}|^{2}}}}.\end{eqnarray}$ (3)

The Bayesian CNN network exhibits superior accuracy in relative error metrics for coordinate estimation of the delamination compared to a Res-Net as shown in Table 1, underscoring its efficacy in achieving more precise and reliable predictions in the task of delamination characterization.

Table 1

Err of different networks on the test set (2500 samples)

Network	x₁ , y₁	x₂ , y₂	x₃ , y₃	x₄ , y₄	Depth
Res-Net	0.1236	0.1110	0.1352	0.1417	0.0847 ± 0.0092
Bayesian CNN	0.1150	0.1123	0.1201	0.1484	0.0719 ± 0.0083

4.2. Quantitative assessment – Segmentation

The classification objective is to differentiate between damaged (Class 1) and intact (Class 0) parts. Figure 5 displays classification results for delamination at the 2nd interface (between the 2nd and 3rd layers), presenting maps for each internal interface of the part. Figures 5 and 6 depict classification and probability maps for the studied internal interface. Notably, ground truth defect size classification appears qualitatively incomplete. While the estimation of delamination position at depth is generally accurate, points on internal interfaces 1 and 3 are misclassified as defects. The distribution of these erroneous estimates varies in dimensions across internal interfaces.

Fig. 5.

Classification maps for each internal interface (ground truth defect size in red).

Fig. 6.

Probability maps for each internal interface (ground truth defect size in red).

For example, at interface 1, misclassified points are distributed over a much wider area than at interface 3. This can be explained by the fact that the algorithm can estimate that, based on the temperature maps as training input, such changes in surface temperature flow (high and low) might proceed from, e.g., the presence of a wider defect at a higher interface, or a less extended one but at a deeper interface, directly reflected in the visual interpretation of the classification maps.

4.3. Robust uncertainty estimation in delamination detection with Bayesian neural networks

In the delamination detection approach, robust uncertainty estimation is vital. Metrics like prediction intervals and reliability diagrams contribute to the BNN capacity to discern uncertainties in geometric parameters, ensuring effective alignment with actual outcomes. This collective set of metrics establishes a robust framework for uncertainty quantification.

5. Conclusion

The present investigation, still preliminary in many aspects, aims at pointing out that semi-analytical solutions are capable at least in simple enough cases (thin inter-layer delaminations) to yield proper data to mimic thermographic inspections, once accounted for homogenization of the intact structures. Then, such data can be used and/or fused like was done by the authors and co-workers in breast imaging [5], [11] but in a Bayesian realm that enables uncertainty quantification. Yet, the successful application of BNNs in NdT for fiber-reinforced laminates requires careful model design, appropriate selection of prior distributions, and efficient training algorithms. Additionally, the integration of BNNs with existing NdT techniques and protocols should be further explored to facilitate adoption in industrial settings.

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