1. Introduction
Fiber-reinforced composites are considered as important materials in various engineering applications as they are superior materials possessing high specific strength, high specific modulus, light weight, and good resistance to aging and corrosion. With the increasing demand for high-performance composite materials, advanced fibers, matrices, and their composites have been developed with remarkable mechanical or non-mechanical properties [
1,
2,
3,
4]. After impregnating the fibers with a matrix to form a single ply, the ply is stacked in specific directions to meet the design requirements. Such laminated structures are widely used in the civil and aerospace engineering fields. The anisotropic characteristics of composite laminates provide wide flexibility in their load-carrying capacity, which has promoted the development of a proper design methodology for laminates [
5,
6]. Many researchers have recently started to investigate the layup design of composite laminates. For example, Kharghani [
7] used numerical stacking optimization to reduce the free edge effect around the composite plate hole, and Maung [
8] completed the Wageningen B of a series of marine propellers. To optimize the ply design, Abdallah [
9] evaluated the ply angle and stacking sequence using LS-DYNA finite element software and discovered that the optimized blade design with a curved fiber stack resulted in a 20% reduction in Tsai–Hill failure index under the same pitch change. Tensile and radial compressive loads affect the mechanical properties of glass/phenolic composite tubes. Nebe [
10] investigated the effect of stacking sequence and circumferential layer drop position on the mechanical response of an internal-pressured type IV composite pressure vessel and the results indicated that the mechanical properties of the laminate were significantly influenced by the laminate design. At the same time, the difficulty in studying laminate design is that there is no universal optimal solution for the best laminate design under different working conditions. A method for optimizing the layup design applicable to various operating conditions awaits to be explored.
In recent years, with the rapid advancement of machine learning (ML) algorithms and accessible open-source libraries, data-driven methods have frequently been utilized for efficient and effective structural design and characterization including composite laminate structures [
11]. For example, Wanigasekara [
12] established an automated fiber placement unidirectional composite laminate prediction model using artificial neural networks (ANN) to estimate the quality and integrity of the manufactured laminate. Bharata [
13] combined the finite element method (FEM) and ML to analyze the buckling of an inclined laminated composite plate. Qiu [
14] proposed a novel characterization method for composite fracture toughness using ML to extract information from the indirect measurement data. Erban [
15] adopted ML technology to accelerate the design process of composites by replacing the time-consuming FEM analysis. Veivers [
16] used particle swarm optimization (PSO) to optimize the layup design of generic tubular geometries under simultaneous thermal and mechanical loading conditions, and Cai [
17] studied the application of ML methods to analyze the dynamic strength of 3D-printed polypropylene (PP) composites. A number of results in the literature show the successful application of machine learning in composite materials, which provides a good idea for the integration of machine learning and composites.
Even though ML performs well in terms of predicting the accuracy for complex nonlinear problems and is efficient in real-time evaluation, it requires the input of large amounts of data, which is a vexing problem. The easiest way to obtain training data is through numerical simulations, but a reliable and low-cost simulation technique should be developed. An alternative method of acquiring data can be from the literature or experiments, which are exhausting work. To alleviate the problems related to data, a variety of methods have been adopted by many scholars. Data augmentation [
18], which artificially generates training data, helps expand the available data samples. Specialized learning algorithms [
19] such as transfer learning, which transfer knowledge from the domain where training data are abundant to the target domain where data are scarce. Model architecture design [
20], which improves the prediction accuracy with limited training data by constraining the parameter space with prior knowledge. For the application of ML to engineering problems, researchers have established physical-informed neural network (PINN) [
21] or theory-guided machine learning (TGML) [
22], which focus on how to import the corresponding domain knowledge to help the ML model better extract the physical laws hidden behind less training data. Theoretical guidance is implemented by employing pre-known governing equations as constraints or logically designing the model architecture according to the related theories [
21,
23,
24]. Therefore, a combination of the theory of composite materials with machine learning is considered to address the issue that machine learning requires a large dataset.
Designing composite laminates is not an easy task. On one hand, the multi-scale feature of the composite shows complex behavior, indicating difficulty in the forward evaluating structural response [
24]. On the other hand, multiple design variables such as constitutive materials, microstructure, and layups need to be determined, therefore requiring time-consuming inverse optimization [
25]. The use of an appropriate neural network model and optimization algorithm can significantly reduce the fitting and optimization time. However, according to the current literature, most of the time, the usage of the ML-based design method lies in the training data generation. Data generation accounts for a large portion of the overall optimization design time, which is a common problem in optimizing the composite performance using ML [
26]. Our research aims to improve the efficiency of data utilization throughout the design process using the TGML model. Therefore, in this paper, we attempted to improve the performance of the model with the help of theoretical guidance using ML-based models with a few training data samples. For this purpose, we built a multi-layer interconnected neural network system following the logical sequences of composite theories regarding the stiffness and strength. With the forward prediction by the NN system and the inverse optimization by genetic algorithm (GA), the TGML model aims to provide optimum composite layup sequences under multiple design constraints.
In this paper, the TGML model was utilized for the efficient layup design of composite tubes with minimum deformation and maximum strength values under the loading conditions of torsion or bending.
Section 2 describes the target problem and the model development in detail. In
Section 3, the model performance and the effect of theory guidance are discussed. Our conclusions are presented in
Section 4.
3. Results and Discussion
To better observe the accelerating effect and the accuracy provided by the TGML models, we conducted research from two aspects. First, we analyzed the training performance of the TGML model, and the design scheme was generated accordingly. Second, the effect of the theory-guided model was investigated by comparing the performance of the TGML models trained on a small training set with direct NN systems trained on a large training set. The TGML models could achieve an accurate regression performance with a very small training set by designing efficient machine learning models that greatly reduced the overall machine learning time.
3.1. TGML Model Performances
It should be noted that all NNs in
Figure 2 were the same, and the ANN configuration of the models without the theory-guided model were also the same. The model without theoretical guidance referred to the direct fitting and the regression of the training dataset derived from FEM. The layup sequence was the input to the without theory-guided models, and the output was the strength or stiffness result, without calculating the intermediate variables.
To realize the intelligent optimization design, “Network 1” took the layup as the input and the in-ply stresses as the output, which was a NN that fits the HFT. By connecting “Network 1” and “Network 3” serially, the entire strength TGML model could randomly generate a combination of layups that met the strength requirements and find the optimal global solution through the GA module. Similarly, in the stiffness TGML model, the inputs of “Network 2” were the layup, and the outputs were the laminate modulus, a NN that fits the CLT. The role of “Network 2” in the stiffness TGML model was similar to that of “Network 1” in the strength TGML models.
Each of these two problems generated 3000 random data points, which implies 3000 different layups and the corresponding properties. A total of 10% of these 3000 sets of data were utilized as the test dataset, which was used to test the regression performance of the models. The models were trained with 2700 data points. After training the models, the model output the predicted values with the test dataset as inputs, and a comparison was made between the predicted values and the output values of the test set. The training data for “Network 3” were obtained from the mathematical formulations of Equations (1) and (2) containing 2700 random inputs and the corresponding outputs. The construction process of the stiffness models was the same as that of the strength models. However, the training data of “Network 4” were obtained from the mathematical formulations of Equations (3) and (4).
After training, the predicted performance of the models is shown in
Figure 3.
Figure 3a,b shows the regression performance of “Network 1” and “Network 3”, respectively, which indicates the regression performance of the strength model. Similarly, the regression performance of the stiffness models is shown in
Figure 3c,d. The coefficient of regression
R2 > 0.95 demonstrates that the network models can accurately predict the properties, provided the material and layup are known. Therefore, the ML model was meticulously designed to generate data samples that would allow the network model to produce satisfactory results.
3.2. Effect of the Theory-Guide
In this section, we compared the models with and without the theory guide at different amounts of training data. The models without the theory guide were directly trained using the layup sequences as inputs and the stiffness or strength as the outputs. According to the fitting performance, only the strength models are shown here due to the small difference between the stiffness and strength models. The comparison of the iteration convergence times of the mean squared error (MSE) values in
Figure 4a showed that the difference between the models with under 2700 and 270 training data were marginal. The MSE iteration of the models with or without theory-guided both converged. However, for the training data, the TGML model fit better for the sample case with 300 data.
Figure 4b reveals that under the 270 training dataset, the
R2 value of the prediction performance of the strength TGML models was greater than that of the strength ML models. The regression performance of the TGML models was significantly better than the models without the theory-guided model.
As shown in
Figure 4c, it can be seen that the TGML models exhibited different performances at the270, 900, and 2700 training sets, respectively. The TGML model presents a high regression performance and low loss values for both large (2700 data points) and small (270 data points) data volumes. The
R2 values of the TGML models in 270 data points were similar to those of the
R2 values with 2700 data points. Under the requirement of the regression performance of the models, the TGML models with 270 data points can be selected instead of those with 2700 data points for the optimal design of the layup sequences. However, the regression performance of the models without the theory-guided model under the different amounts of training data was significantly different. The MSE without the theory-guided model with 270 data points was more than ten times that of the without the theory-guided model with 2700 data points, indicating the poor prediction performance of the model without theoretical guidance in small data training. It is worth mentioning that the models with and without theoretical guidance exhibited higher
R2 values and lower MSEs under training with 3000 datasets, thanks to the careful design of the NN configuration.
Combined with the optimization process in
Figure 4d, it can be interpreted that the accuracy of the model is crucial in that the data points close to the optimal result are very dense. Otherwise, the optimal solution cannot be obtained. The GA module searches for the optimal layup solution, which is the solution closest to the origin on the diagonal of the coordinate axis, as shown in
Figure 4d. The total design space is defined as all of the possible layup combinations with the angle increment of 1 degree. The optimal layup sequence was [±30/±30/±30/±25]s for the composite tube design problem. That is, the failure indicator reached a minimum value when the layup sequence was [±30/±30/±30/±25]s under a bending stress of 1300 N and a torsional stress of 300 N·m.
3.3. Calculation Efficiency
As can be seen in the figure above, the prediction performance of the TGML models with 270 and 2700 training sets was similar, which is another advantage of the TGML models. The TGML models still maintained the original regression performance despite greatly reducing the training datasets. The total computer run time spent training the TGML models on training sets with numbers of 270 and 2700 was compared in this study. The calculations were performed on a PC with an Intel i7-6700 CPU (Intel Co., Santa Clara, CA, USA). The total running time of the entire design process is listed in
Table 1. The FEM generation data took up most of the time during the entire run. It took 135 min to calculate 2700 sets of data by finite element and 13.5 min to calculate 270 sets of data. The training of NNs did not take much time and completed the training of 2700 sets of data points in only 2 min. It took nearly 1.5 min to complete the training of 270 sets of data points. Obviously, the training time of the neural network and the amount of data have a nonlinear relationship. We predicted that the training time was not much different due to the small volume of training data. At the same time, the computing power of the computer will also affect the training time of the NNs. The ML models spent roughly 3 min in the optimization design stage with the GA module. Therefore, a significant reduction in data is a significant reduction in time cost. TGML is a path proposed to solve the shortcomings of traditional ML requiring a large amount of data.
3.4. Solutions Provided by the TGML Models
Some layup designs were generated through the TGML models, as shown in
Table 2 and
Table 3. The TGML models can be used to design layups with the highest strength or stiffness under different load conditions. The three cases in
Table 2 represent the corresponding layups when the failure indices reached the minimum values under different load conditions. The three cases in
Table 3 represent the layups with the lowest fiber modulus required to meet the different stiffness requirements.
Table 2 and
Table 3 show the optimal solutions under the corresponding conditions.
It is worth mentioning that for the layup design that satisfied the target stiffness in
Table 3, the
E1M is the single-ply modulus in the fiber direction in the output of the TGML models. This is because of the combination of the CLT, but the strength calculation results did not yield the
E1M. FEM further verified these results. The results were slightly different from those listed in the table, but the error was within an acceptable range. The experiments in the next section will further validate the results.
4. Experimental Validation
In this section, the composite tube specimens were prepared to use the optimized layup sequence produced by the TGML stiffness model, and two experiments of bending and torsion were conducted, respectively. Verification of the accuracy of the TGML model involves comparing the stiffness of the composite pipe under the specified laying sequence to the stiffness produced by the model. The layup sequences used were the results of the stiffness models shown in
Table 3. Three specimens were tested for each ply combination.
The output of the stiffness TGML model from
Table 3 revealed that the layup sequences and fiber orientation modulus were included. However, a fixed fiber direction modulus
E1T = 115 GPa was used in the experiments. Therefore, the dimensionless stiffness to the modulus ratio was utilized for evaluation.
Three layup sequences were used for the experiments, which were the combination of [±0/±35/±25/±15]s and
E1T; [±25/±10/±40/±5]s and
E1T; and [±30/±25/±20/±20]s and
E1T.
Figure 5 displays the torsion–torsion angle curves for the torsion test as well as the force–displacement curves for the three composite tubes under the cantilever bending test. The stiffness can be obtained by simply calculating the curvature of the curve in the elastic phase. The torsion angle needs to be converted into radians before the torsional stiffness can be calculated.
Table 4 lists all of the calculation results. The stiffness results of Case 1 and Case 2 were nearly identical on account of the fact that the gap in the target bending stiffness design of the model was not large enough. No significant differences in bending stiffness were reflected in the results. It can be observed in
Table 4 that the [±30/±25/±20/±20]s layup sequence had a more significant effect in improving the torsional stiffness, probably due to the fact that the torsional stiffness of Case 3 was greatest when the design target torsional stiffness was applied.
The predicted solution calculated by the model was close to the actual solution. The model, as far as we can tell, is generally feasible for material design. The ratio of the stiffness to fiber modulus in
Table 4 can also indicate that the theoretical value is close to the actual test result. Case 2 was best for improving the flexural stiffness performance of the pipe at the same fiber modulus, while the laminate design used Case 3,which was best suited to improving the torsional stiffness properties of the pipe under the same fiber modulus. Therefore, we can see that lamination has a significant effect on the structural properties of the laminate.
According to the experimental results, the TGML model overpredicted the bending and torsional stiffness. This overprediction is due to the fact that the design parameters such as the angle and modulus of the model are theoretical values, but the actual specimen contains flaws. Simultaneously, the design of the loading experiment scheme affected the experimental results, however, the errors in the model were within the acceptable limits. Because of its accuracy and practicability, the TGML model can be used as a reference in engineering applications.