A surrogate model for finite element simulation of elastic wave propagation in composite materials

Deep learning (DL) has been increasingly applied to the prediction and analysis of static properties of composites, such as strength, stiffness, and fracture toughness. By training deep neural networks on large datasets of material properties and microstructural data, these models can learn to extract patterns and relationships between various factors that influence the static properties of composites. For example, convolutional neural networks (CNNs) have been used to analyze the microstructural images of composites and predict their mechanical properties with high accuracy.^1–4 Other DL approaches, such as recurrent neural networks and long short-term memory networks, have been used to model the time-dependent deformation and failure behavior of composites under various loading conditions.^5–10 The application of DL in predicting the static properties of composites has the potential to significantly reduce the time and cost associated with traditional experimental testing methods and enable more efficient and accurate material design and optimization.^11–14 

Previous research has mainly focused on the static behavior of composites. In this study, we use DL to predict the dynamic response of composites. First, we collect a large dataset of input–output pairs using finite element modeling (FEM), where the input is a description or simulation of a composite bar and the output is the corresponding waveform generated by the bar. Second, we preprocess the data to ensure that they are in a suitable format for the CNN. Third, we divide the data into a training set, validation set, and testing set. Fourth, we design the architecture of the CNN, including the number of layers, filters, and activation functions. Fifth, we train the CNN on the training set using a suitable optimization algorithm (e.g., stochastic gradient descent) and a suitable loss function (e.g., mean squared error). Sixth, we evaluate the performance of the model on the validation set and adjust the model parameters as necessary to improve performance. Finally, we evaluate the final performance of the model on the testing set and report the results. In addition, we deployed Fourier transform to reduce the dimension of the variables.^15,16 The workflow of this study is demonstrated in Fig. 1.

The simulation setup for data collection is demonstrated in Fig. 2. The composite bar is divided into 30 equal size sections, with each section assigned with either steel or rubber. Each composite design is characterized as a 30-dimensional row vector assigned 0 and 1, which correspond to steel and rubber, respectively A sinusoidal wave with an amplitude of 1 μm and frequency of 600 Hz is applied at the left end of the composite bar, and the output wave signal is collected at the right end of the bar. To avoid interference of reflected waves, perfect matched layers are placed at the left and right ends of the bar. The materials are assumed to be elastic linear, and the interface between the materials is perfectly bonded.

To implement our CNN, we utilized the Keras open-source Python package with a TensorFlow backend.¹⁷ We kept the batch size fixed at 64 and ran 100 epochs for all experiments. We selected the Mean Squared Error (MSE) as the loss function, and the Adam optimizer was utilized. We used a binary label matrix of 5 × 6 size to represent the unit cell instead of a 30-dimensional vector to leverage the efficiency of CNNs in parameterizing two-dimensional inputs. In the context of predicting the properties of the output elastic wave in composite bars using machine learning, Fourier transform serves as a preprocessing step to extract the frequency components of the wave. During the training process, we convert the output elastic waves to the frequency domain to reduce the dimension from 601 to 60. When predicting new data, we reconstruct the wave signals by performing an inverse Fourier transform, which transforms the wave signals from the frequency domain back into the time domain. By applying the inverse Fourier transform to the Fourier coefficients obtained from the forward transform of the original wave signals, we can obtain time-domain wave signals that accurately represent the behavior of the composite bar under various loading conditions.

The size of the dataset can have a significant impact on the performance of a CNN model. Generally, having a larger dataset can improve the performance of a CNN model, while having a smaller dataset can lead to overfitting and poor generalization. A larger dataset can help prevent overfitting as the CNN model has access to more examples to learn from. The performance of the DL model on the prediction of output elastic waves of binary composite bars over different dataset sizes is demonstrated in Fig. 3. The mean square error decreases significantly as more samples are fed into the DL model and stabilizes at ∼16 000 samples.

Three randomly selected sets of predicted elastic wave outputs are compared to the actual output curve, including the associated composite. These examples were generated separately from the training and evaluation set. As illustrated in Fig. 4, the predicted curves closely match the actual ones, accurately predicting the trend and details of the output elastic waves.

This letter presents a study where convolutional neural networks and Fourier transform were employed to forecast the elastic wave response of composite bars. The microstructures of the bars were taken as inputs to the deep learning model, while the elastic wave response was the output. By identifying significant composite features in the input, the convolutional neural network was able to predict the output elastic waves. The model’s performance was assessed by comparing the mean squared error of the predicted and actual output signals. The study’s findings demonstrate that the deep learning model can rapidly and accurately predict composite elastic wave output, thereby acting as a substitute model for time-intensive finite element simulations.

The authors have no conflicts to disclose.

Hong Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Qingfeng Wang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.