Machine learning prediction in distortion behavior of unsymmetric laminates under hygrothermal environment

Abstract

This paper deals with the effects of curing process and hygrothermal environment on the distortion behaviors of continuous carbon fiber reinforced polyamide 6 (CF/PA6) unsymmetric laminates. To accurately characterize the shape of CF/PA6 unsymmetric laminates in the water absorption process, the 3D model reconstruction with different water content is carried by combining 3D scanner and secondary development in Abaqus. A full-field displacement comparison method is proposed to calculate the equivalent thermal/moisture expansion coefficient, and the effectiveness of numerical simulation is verified. The dataset with 2816 instances is further constructed through finite element method. Through grid search and five-fold cross validation, the ANN model is trained and validated according to R ² and MSE criterion. The well-trained ANN model builds the mapping relationship between lay-up design parameters, hygrothermal environment and the distortion parameters of unsymmetric laminates.

Introduction

Continuous carbon fiber reinforced polyamide 6 (CF/PA6) have a wide application prospect over thermoset composites in aviation, aerospace and automobile fields,^1,2 due to their high performances of high toughness, rapid prototyping, recycling and easy repair. ³ For unsymmetrical laminates after hot-pressing process, their cured shape types can be divided into cylindrical shape, twisted shape, and saddle shape, ⁴ in which the cylindrical shape can be applied to fuselage segment, ⁵ and the twisted shape are widely used for variable wing structures, ⁶ thanks to the good bi-stability. Despite these benefits, CF/PA6 unsymmetrical laminates in service environment face serious hygrothermal aging problems due to the coupling effect of temperature and humidity, resulting in distortion of composite structures, which greatly reduces their service life. Hence, the distortion behaviors of CF/PA6 unsymmetrical laminates under hygrothermal environment should be researched to realize their full potential.

Extensive investigations have been conducted on the cure-induced deformation of unsymmetrical laminates. Ochinero and Hyer ⁷ experimentally investigated several factors which can influence the dimensional stability of CFRP laminates, including ply thickness, fiber waviness, uneven resin distribution and ply shifting. Various techniques were developed to measure the shape of laminates, such as deflection/chord measurement, ⁸ co-ordinates measurement using a dial gauge, ⁹ and an optical method (fringe projection method). ¹⁰ Hyer^11–13 extended classical laminate theory by incorporating non-linear strain-displacement, followed by minimizing the total potential energy of laminates by means of the Rayleigh-Ritz method, which can predict the cured shape of square cross-ply laminates. Based on the theory developed by Hyer, Telford ¹⁴ validated the accuracy of the analytical predicted results, with benchmark the curvatures of square cross-ply laminates calculated by numerical models.

Numerical models offer flexibility in the analysis of the cure process and corresponding cured shapes, and these models can be divided into physical-based models, ¹⁵ phenomenological-numerical models ¹⁶ and semi-empirical models.^17,18 Physical-based models aim to describe the entry curing process in detail, and they focus on the influence of process parameters (temperature, degree of cure and viscoelastic properties of resin) and lay-up sequence.^15,19 Based on a phenomenological-numerical model, Kappel ²⁰ proposed a novel construct in form of a ”spring-in reference curve” to provide input for the determination of zone-specific simulation parameters, which can fast and accurately predict the process-induced distortions. Obviously distinct from the above-mentioned methods, the analysis of curing process was simplified based on the empirical method, and it was denoted as the semi-empirical model. Various theoretical models are proposed to investigate the process-induced distortion (PID), including elastic model, ²¹ cure-hardening instantaneously linear elastic model, ²² and viscoelastic model. ²³ Parambil^24–26 systematically investigated the processing induced thermal residual stress through finite element model, in which cooling-rate effects on crystallinity, temperature-dependent elastic modulus, and temperature-dependent coefficient of thermal expansion were taken into account. Compared with computationally and time-consuming simulation, machine learning methods can provide accurate and fast solutions to the regression ²⁷ or classification tasks, ⁴ and these methods have widely employed to map the relationship between numerous factors and the mechanical properties of composites. ²⁸ Luo^4,29 developed an ANN model to predict the maximum PID of thermosetting-matrix composites, as well as the cured shape types under different ply-stacking sequences.

PA6 resin is highly sensitive to the hygrothermal environment, due to the presence of amide groups -CO-NH- in the amorphous phase, and absorbed water content can reach about 10 wt%, ³⁰ which has significant effects on the dimensional stability and mechanical properties (Figure 1). Extensive literatures^31,32 have been reported to investigate the mechanical degradation and reveal the hygrothermal aging mechanism of CF/PA6 composites. There are relatively few studies on dimensional stability. Obeid ³³ found that the swelling coefficient of PA6 obey a nonlinear behavior, and built the relationship function with water content. In our previous work, ³⁴ the dimensional change of CF/PA6 composites in the longitudinal and transverse directions was monitored, and the moisture swelling strain is about two orders of magnitude higher than the thermal swelling strain, which indicated that deformation of CF/PA6 laminates induced by water absorption cannot be ignored.

Figure 1.

Schematic diagram of hygroscopic effect.

The existing literature have extensively studied the curing deformation of composite laminates. In contrast, few works have been reported on the effect of hygrothermal environment on the distortion behaviors of laminates, especially for CF/PA6 unsymmetric laminates, which is vital to extent their application range. Additionally, complex processing and environmental factors influencing residual stress and distortion behaviors. Therefore, based on the machine learning techniques, it is necessary to build the mapping relationship between numerous factors and the distortion parameters of unsymmetric laminates.

This study develops a data-driven computational methodology based on FEM-ANN approach to investigate the distortion behaviors of CF/PA6 unsymmetric laminates before and after hygrothermal aging. To obtain the data for the machine learning, the curing deformation and hygroscopic deformation of unsymmetric laminates are studied by combining hygroscopic deformation experiment and numerical simulation. The thermal/moisture expansion coefficient is modified based on the full-field displacement comparison method, and the effectiveness of numerical simulation is verified. Then, the dataset with 2816 instances is constructed through finite element method. The optimal neural network structure and its optimal hyperparameters are automatically found by combining grid search and five-fold cross validation, according to R ² and MSE criterion. Lastly, the well-trained ANN model builds the mapping relationship between lay-up design parameters, hygrothermal environment and the distortion parameters of unsymmetric laminates.

Methods

Experiment

CF/PA6 prepregs with a unit weight of $400\mathrm{g}/{\mathrm{m}}^2$ are prepared by a melt impregnation technique and their thickness is about 0.3 mm. Then, three laminate samples in size of $320 \text{mm}\times 120 \text{mm}$ are fabricated with unsymmetric cross-ply angles [0/90/90/90], [0/0/0/90] and [-45/45/-45/-45] through the high temperature compressing molding process, ³⁴ as shown in Figure 2. Practically, the mismatch in thermal expansion coefficients along the longitudinal and transverse directions of a unidirectional ply can lead to warping, and the cured shape can be classified into four types, namely undeformed, saddle, cylindrical and twist types, as shown in Figure 3.

Figure 2.

Schematic diagram of experimental process.

Figure 3.

Types of cured shape of CF/PA6 laminates during the curing process: (a) unformed type, (b) saddle type, (c) cylindrical type, (d) twist type.

The CF/PA6 laminates are highly sensitive to the moist environment because the absorbed water can obviously change the stress and moisture fields in the laminates, which bring obvious effect on the cured shape of the laminates. To this end, a moisture absorption experiment is conducted for the prepared CF/PA6 laminates. In the experiment, the laminates are immersed into the distilled water in a water bath with specific temperature (i.e. 50°C) until saturation and their masses are measured at a given time interval using an analytical balance with an accuracy of 1 mg, according to the standard ASTM D5229. Then, the moisture absorption curve can be plotted by recording the mass change of the laminate over time. Simultaneously, to analyze the deformation evolution during the moisture absorption, the instant warping configurations of the laminates are recorded by a 3D scanner Reeyee from the wiiboox company with an accuracy of 50 μm and then converted into the corresponding 3D original geometry models in STL format. Subsequently, the 3D scanning models are exported into CATIA software to build the real models and the geometric parameters such as the deflection $\omega$ and the radius of curvature $R$ are obtained by the inbuilt measuring tools. Figure 2(c) displays the variation of the two geometric parameters with respect to the absorbed water content. Finally, with the written Python codes by us, the real geometric models of the unsymmetric laminates are transformed into deformation images in ABAQUS. In this work, the method of full-field displacement comparison is proposed, and it is realized by comparing the displacement comparison of the corresponding nodes between the actual model and the FEM model, which is the research foundation for verification of FEM results and the parameter correction in the following sections.

Finite element simulation

To analyze the change in shape during the cured process and subsequent moisture absorption process, a finite element model is developed using the commercial software Abaqus and the simulation is completed based on the coupled stress-temperature/moisture analysis, as shown in Figure 4.

Figure 4.

Schematic diagram of finite element simulation for cured deformation and moisture-absorption deformation.

Step 1

cured deformation.

The main reason of warping deformation in the cured process is the non-uniform cooling contraction during the cooling down stage, and the details in numerical simulation of cured deformation are as follows: (1) the geometric model of laminates is a 3D deformable shell in size of $320 \text{mm}\times 120 \text{mm}$ ; (2) the material parameters of simulations are obtained by mechanical testing and characterization according to relevant test standards; (3) a thermal load of 160°C is applied through a predefined field and the temperature reduces to 25°C; (4) the boundary conditions are the full constraints of the three nodes in the center of the plate to eliminate the rigid body translation and rotation around the fixed axis; (5) the model is meshed with S4R elements in size of 5 mm.

There are so many factors influencing the accuracy of curing deformation, including cooling rate, shrinkage deformation of materials, and component/mold interaction. The cooling rate affects the crystallization behavior of resin and thus the resulting microstructure, which in turn has a significant effect on the mechanical properties of composites (e.g., elastic modulus, coefficient of thermal expansion, etc.). ²⁴ It is worth noting that the effect of crystallinity and temperature on the layer properties are ignored in this paper, and all factors are included in the coefficient of thermal expansion ( ${\alpha }_{11}^{eff}$ , ${\alpha }_{22}^{eff}$ ) through a full-field displacement comparison method. The accuracy of curing deformation in the FEM model is ensured by modifying ${\alpha }_{11}$ and ${\alpha }_{22}$ , which will be discussed in detail in the following section.

Step 2

moisture-absorption deformation.

Similarly, the non-uniform hygroscopic expansion of different layers can further induce the warping deformation of cured laminates, and this is the reverse process of cured deformation analysis. After the cured deformation analysis, the cured shapes of unsymmetric laminates can be obtained. Subsequently, a constant concentration is applied on their edges, and the moisture content increase gradually until to the saturated state. The moisture diffusion parameters are correspondingly input in the stress-temperature coupled module to simulate the hygroscopic expansion process, due to the similarity between the moisture diffusion equation and the heat conduction equation. It is worth noting that the differences are the moisture filed boundary mentioned above and the mechanical properties depended on the absorbed moisture content. Moisture environment also decreases the matrix-dominated properties of CF/PA6 composites, and the quantitative relationship between the mechanical properties and the absorbed moisture content has been determined in the previous work. ³⁵ Thus, a degradation factor dependent on moisture content is introduced into the constitutive model. After hygrothermal aging, the effect on the longitudinal modulus of unidirectional laminates can be ignored, while the transverse modulus decrease significantly and the retention rate of modulus can be described by equation (1). ³⁵ In the moisture absorption process, the modulus of laminates with different laying angles can be determined by the transformation equation ( $E_{xy}^{*}=E_{xy}\cdot k\cdot \cos \hspace{0.17em}\theta$ ).

$k=\frac{E_{22}^{*}}{E_{22}}=\left(30.27+\frac{100-30.27}{1+\exp \left[183\times \left(M_t-1.78\right)\right]}\right)\times \frac{1}{100}$

(1)

where $E_{22}$ and $E_{22}^{*}$ are the retention rate of transverse modulus before and after hygrothermal aging, $M_t$ is the absorbed moisture content.

All the material parameters of CF/PA6 laminates are summarized in Table 1. The crystallinity of PA6 is 33.1%. The unidirectional tensile and in-plane shear properties are measured following ASTM D3039 and ASTM D3518 standards, respectively. In addition, the coefficients of thermal expansion in longitudinal and transverse directions are measured by a DIL-L75 thermal expansion instrument according to ASTM D696 standard. In order to measure the moisture expansion coefficient in the water absorption test, dimensional changes of the samples with the size of $5 \text{mm}\times 5 \text{mm}\times 25 \text{mm}$ are measured using a digital caliper of spiral micrometer with an accuracy of 1 μm, and the moisture expansion coefficient in different directions can be determined.

Table 1.

Effective elastic parameters of CF/PA6 composite.

Machine learning framework

As one of popular machine learning algorithms, the back propagation neural network (BPNN) algorithm is a multi-layer network with error back propagation, ³⁶ which is used to regulate the weight value and bias value to minimize the total error of the output computed by the network.

As shown in Figure 5, a particular BPNN model usually consists of three or more layers containing multiple neurons, including one input layer, one output layer, and one or more hidden layers. A neuron is regarded as an individual calculation unit, and it contains a fully connected adder and an activation function. In each neuron, a weighted sum of the input signals $x_i^{\left(k-1\right)}$ is firstly computed; and then an activation function $f\left(z\right)$ acts on the sum $z_i^{\left(k\right)}$ and transfer the result to the connected neuron j, which can be represented as equation (2).

$y_j^{\left(k\right)}=f\left(z_i^{\left(k\right)}\right)=f\left({\displaystyle \sum }_{i=1}^N{w}_{ij}^{\left(k-1\right)}{x}_i^{\left(k-1\right)}+b_i^{\left(k-1\right)}\right)$

(2)

where $x_i^{\left(k-1\right)}$ denotes the input of neuron i, $y_j^{\left(k\right)}$ denotes the output of the neuron j in the kth hidden layer, $f\left(z\right)$ is the activation function, $w_{ij}^{\left(k-1\right)}$ indicates the weight, and $b_i^{\left(k-1\right)}$ is the bias.

Figure 5.

Schematic diagram of the ANN architecture. (a) BPNN model, (b) individual neurons in the hidden layer.

To train the BPNN algorithm, the discrepancy of the output and the real target is minimized by error back propagation and the values of weight and bias in equation (2) are regulated correspondingly. The BPNN performance is evaluated in terms of the following two parameters: mean squared error (MSE) and coefficient of determination ( $R^2$ ). MSE is a ratio that indicates the overall deviation degree of the predicted values. $R^2$ ranging from 0 to 1 normally provides a measurement of how well observed outcomes are replicated by the predicted model. Moreover, the value of MSE closer to zero and $R^2$ closer to one, signifies the better accuracy of the BPNN model.

$\text{MSE}\left(y,y^{\prime }\right)=\frac{1}{N}{{\displaystyle \sum }_{i=1}^N\left(y-y_i^{\prime }\right)}^2$

(3)

$R^2\left(y,y^{\prime }\right)=1-\frac{{{\displaystyle \sum }_{i=1}^N\left(y_i-y_i^{\prime }\right)}^2}{{{\displaystyle \sum }_{i=1}^N\left(y_i-\overline{y}\right)}^2}$

(4)

where $y_j^{\prime }$ is the actual output value estimated by the ANN model, $y_i$ is the corresponding target output value, $\overline{y}$ is the mean of the target output values, and N is the number of neuron nodes.

Results and discussion

Experimental results

The maximum water content of CF/PA6 composites under hygrothermal environment is about 5%, and it is mainly depended on the matrix volume fraction due to hydrophilic nature of the PA6 matrix. As the water content increases, the absorbed water can lead to plasticization of the PA6 matrix, which further results in matrix swelling and dimensional stabilities of CF/PA6 composites. Especially for unsymmetric laminates, the differences in moisture absorption expansion between the longitudinal and transverse direction of a unidirectional ply can significantly affect their shapes, which are quantitively described by ${\omega }_{\max }$ and R these two parameters. As shown in Figure 6(a), as the absorbed water content increases, the [0/90/90/90] laminate tends to be flat in shape. The value of ${\omega }_{\max }$ decreases linearly from 115.27 mm at unaged state to 17.46 mm at statured state, while R increases exponentially from 51.27 mm to 712.69 mm. There is a similar phenomenon for the [90/0/0/0] laminate in Figure 6(b); ${\omega }_{\max }$ decreases linearly from 28.16 mm to 2.64 mm, while R increases exponentially from 58.5 mm to 679.86 mm. For the [–45/45/–45/–45] laminate in Figure 6(c), the effect of water content on ${\omega }_{\max }$ is only analyzed in the water diffusion process, because its shape is twist type and the edge of the laminates is a space curve not an arc.

Figure 6.

Effect of water content on radius of curvature and maximum offset for [0/90/90/90] laminates with different lay-up, (a) [0/90/90/90], (b) [90/0/0/0], (c) [-45/45/-45/-45].

In addition, through the comparison of the curve fitting coefficients between the [0/90/90/90] and [90/0/0/0] laminate, the coefficient of ${\omega }_{\max }$ (–23.41) and R (25.9) in the former is obviously higher than the corresponding latter ${\omega }_{\max }$ (–5.09) and R (1.71) respectively. This indicates that the absorbed water content has a greater influence on the deformation of the [0/90/90/90] laminate, because they have more 90° plies dominating the moisture expansion deformation of laminates.

Numerical results and validation

The traditional index to evaluate the accuracy of curing deformation is to calculate the relative error between simulation results and experimental data by comparing $\omega$ values. As shown in Figure 7, the experimental $\omega$ is 115.27 mm measured by 3D EinScan-SP scanner, while the simulation result is 103.37 mm calculated by the a coupled stress–temperature analysis with thermal expansion coefficients of CF/PA6 samples ( ${\alpha }_{11}=4.8\mathrm{E}-6$ , ${\alpha }_{22}=1.05\mathrm{E}-4$ ) directly measured by a thermal expansion instrument. Only by comparing $\omega$ values, numerical simulation results are close to the experimental measured values, and the relative error is 10.23%, which indicates FEM model has acceptable calculation accuracy. However, numerically predicted shapes significantly deviate from the corresponding experimental shapes, in which the span of the former is 6.656 mm while the latter is 75.1480 mm. Here, the relative error is up to 91.14%. Therefore, only using $\omega$ as the evaluation index of FEM calculation accuracy will lead to the distortion and poor predicted accuracy of laminate shape.

Figure 7.

The differences between the FEM predicted shape and the experimental shape.

It can be seen that the FEM model produces excessive deformation, which indicates its thermal expansion coefficients are higher than the actual. The reason can be explained as follows: the curing deformation of laminates is influenced by many factors, including cooling rate, shrinkage deformation of materials, and component/mold interaction, which are not considered in the linear elastic model.

In this work, a full-field displacement comparison method is proposed to find the difference between the experimental model and the FEM model in the displacement component ( $d_{ij}$ ) of the corresponding node, by subsequently calculating the relative distance ( $\sum d_{ij}$ ) of all nodes and the average value ( ${\overline{d}}_{global}=\sum d_{ij}/N_{global}$ ). ${\overline{d}}_{global}$ is taken as the evaluation index, and the higher calculation accuracy of the FEM model with the smaller value of ${\overline{d}}_{global}$ . The influence of various factors mentioned above is unified into the equivalent coefficient of thermal expansion ( ${\alpha }^{eff}$ ), the suitable ${\alpha }^{eff}$ is calculated by an optimization method based on the full-field displacement comparison method.

The specific calculation process is shown in Figure 8, and the values of ${\alpha }_{11}$ are $4.8\mathrm{E}-6$ , $3.6\mathrm{E}-6$ , $2.4\mathrm{E}-6$ , $1.2\mathrm{E}-6$ and 0, while the values of ${\alpha }_{22}$ are $1.05\mathrm{E}-4$ , $9.5\mathrm{E}-5$ , $8.5\mathrm{E}-5$ , $7.5\mathrm{E}-5$ and $6.5\mathrm{E}-5$ . ${\overline{d}}_{global}$ under 25 different cases are calculated, and the discrete points are plotted in Figure 8(b). Subsequently, the 3D surface is obtained by fitting the discrete points, and the bottom end of the surface corresponds to the horizontal and vertical coordinates are the equivalent coefficient of thermal expansion ( ${\alpha }_{11}^{eff}$ and ${\alpha }_{22}^{eff}$ ), which can be calculated by the extremum method. Specially, the bottom edges of the surface are projected to x-z and y-z plane respectively, as shown in Figure 8(c) and (d). After transforming into plane curves, the curves are fitted by the third order Fourier formula, and the equivalent coefficient of thermal expansion can be obtained by find the extremum. Lastly, values of ${\alpha }_{11}^{eff}$ and ${\alpha }_{22}^{eff}$ are $3.773\mathrm{E}-6$ and $8.495\mathrm{E}-5$ respectively, and corresponding ${\overline{d}}_{global}$ is 0.312 mm, which is obviously lower than the unmodified result (21.44 mm).

Figure 8.

Calculation flow of optimal solution for the thermal expansion coefficient.

The coefficient of moisture expansion in the longitudinal ( ${\beta }_{11}$ ) and transverse ( ${\beta }_{22}$ ) direction is $1.253\mathrm{E}-4$ and $1.829\mathrm{E}-3$ respectively, which are determined by a spiral micrometer monitoring its dimensional change in the water absorption test. However, the corresponding warping deformation shape obviously deviates from the experimental data, if ${\beta }_{11}$ and ${\beta }_{22}$ are directly selected as material parameters in the FEM model, which is attributed to the poor measurement accuracy of sample dimension.

The former section has obtained the deformation cloud images of [0/90/90/90], [90/0/0/0] and [-45/-45/45/-45] laminates with different water content. The parameter calibrated process is repeated for the hygroscopic laminate shapes, to obtain equivalent moisture expansion coefficients ( ${\beta }_{11}^{eff}$ and ${\beta }_{22}^{eff}$ ) at different moisture absorption stage, and finally these discrete points are plotted in Figure 9. It is found that the curves of $M_t-{\beta }_{11}^{eff}/{\beta }_{22}^{eff}$ exhibit a linear increase in the initial stage and an apparent plateau at the saturate state, which is similar to the trend of moisture absorption curves. Through the curve fitting of discrete points, the following formula and fitting parameters are obtained, which can be used to accurately calculate the moisture expansion coefficients with different water content.

$\beta \left(M_{\mathrm{t}}\right)=\frac{{\beta }_{\max }}{1+\exp \left[-\lambda \times \left(M_{\mathrm{t}}-M_{\mathrm{c}}\right)\right]}$

(5)

where ${\beta }_{\max }$ denotes the maximum swelling coefficient, $M_{\mathrm{t}}$ is the absorbed water content, $\lambda$ and $M_{\mathrm{c}}$ are fitting parameters controlling the shape of fitting curves.

Figure 9.

Hygrothermal expansion coefficient in the function of the water content.

Machine learning analysis

Preparation of dataset

In this study, the BPNN analysis is performed using Matlab. The machine learning process is shown in Figure 10, and its steps are mainly divided into three parts: preparing the dataset, building & training the network, and forward prediction.

Figure 10.

Schematic representation of BPNN training process.

Firstly, the dataset is constructed through the validated finite element simulation. The several features of input and output as listed in Table 2. The output feature y is the maximum offset of the hygroscopic warping deformation for the different laying laminates. Input features x1-x4 and x8 are user-defined parameters, while the remaining input features x5-x7 are determined according to the laminate theory. More precisely, input features x1, x2, x3, and x4 denote the plying angles for layer one, layer two, layer three, and layer four, respectively. Moreover, there are four types of plying angles in this paper, including −45°, 0°, 45° and 90°. Input feature x5 represents the layup types of laminates, and it can be described by values 1∼6. Here, value 1 and value 2 represent symmetric and antisymmetric laying laminates, respectively. Value 3 and value 4 correspond to cross-ply and angle-ply laminates, respectively. Values 5 denotes repeat-ply laminates, while value 6 represents the rest laminates. Input feature x6 represents the number of non-repeated ply angles, ranging from 1 to 4. Input feature x7 represents the deformation shape of the laminate, and it is described by values 1∼4, where values 1, 2, 3 and 4 represent the plate shape, saddle shape, cylindrical shape and twist shape respectively. Input feature x8 denotes the absorbed water content of the laminates in the hygrothermal environment, and the values ranges from 0 to 5 with an interval of 0.5. A total of 2816 instances are obtained through the finite element analysis.

Table 2.

Several typical input and output examples.

Number	x1	x2	x3	x4	x5	x6	x7	x8	y (mm)
Job16	−45	−45	0	0	5	2	4	0	63.81
Job18	45	0	45	0	2	2	4	0.5	23.50
Job43	90	90	90	0	3	2	3	1.5	72.19
Job218	45	90	45	−45	6	3	4	2.5	53.14
Job248	−45	45	−45	−45	4	2	4	2	61.70

Secondly, based on the feature selection method, the high-dimensional datasets are preprocessed for dimensionality reduction. The data were randomly divided into training set and testing set with a ratio of 8:2, which is achieved by the bulit-in randperm function of matlab. The training set is used to search for the optimal ANN model and its performance is evaluated by MSE and $R^2$ . The testing set is used to verify the accuracy of the structural parameters and hyperparameters determined by training set. Moreover, the input parameters are standardized by the normalization method to ensure that all the input features are on the same scale.

Training and testing of ANN model

In this section, different structural parameters and hyperparameters of the ANN model are systematically analyzed to search for the best ANN model considering 11 architectures, 3 input/output activation functions, and 5 training algorithms.

The number of hidden layers and the number of neurons in hidden layers are important structural parameters for ANN models. In this paper, the hidden layer is defined as one layer to find the best number of neurons to avoid underfitting and overfitting in data prediction. The number of neurons in the hidden layer ( $N_{\mathrm{h}}$ ) can be determined by an empirical formula, ³⁷ and the value of $N_{\mathrm{h}}$ is 32 calculated by equation (6). In order to further find the optimal value of $N_{\mathrm{h}}$ , the variation range is defined as 25 ∼ 35.

$N_{\mathrm{h}}=\frac{N_{\mathrm{s}}}{\alpha \times \left(N_{\mathrm{i}}+N_{\mathrm{o}}\right)}$

(6)

where $N_s$ denotes total number of training set samples, $N_i$ and $N_o$ denote the number of neurons in input layer and output layer respectively, and $\alpha$ is constant of 10.

In the training process, the structural parameters and hyperparameters of the ANN model are optimized to obtain the best prediction accuracy. The number of neurons, different training algorithms (Table 3) and input/output activation functions (Table 4) are adopted to evaluate their influences on the performances of final output. The optimal neural network structure and its optimal hyperparameters are automatically found by combining grid search and five-fold cross validation on the training set. In this process, 495 combinations are used. The L2 regularization cost function and the early stop operation are used to prevent the model from overfitting and improve the generalization performance of the ANN model. Finally, the performance of the trained ANN model is evaluated by MSE and $R^2$ , with formulars as shown in equations (3) and (4). It worth noting that the neural network training follows the following basic principles. On the premise of meeting the prediction accuracy, the number of neurons in the hidden layer should be reduced as much as possible. Simultaneously, under the premise that $R^2$ of the training network is good, the MSE should be smaller as far as possible.

Table 3.

The common training algorithms.

Train Algorithm	Abbreviation
Gradient descent backpropagation	Traingd
Gradient descent with momentum backpropagation	Traingdm
Levenberg-marquardt backpropagation	Trainlm
Gradient descent with momentum and adaptive learning rate backpropagation	Traingdx
BFGS quasi-Newton backpropagation	Trainbfg

Table 4.

The common activation functions.

Activation function	Abbreviation	Equation	Derivative
Linear	Purelin	$f\left(x\right)=x$	$f^{\prime }\left(x\right)=1$
Log-sigmoid	Logsig	$f\left(x\right)=\frac{1}{1+e^{-x}}$	$f^{\prime }\left(x\right)=f\left(x\right)\left(1-f\left(x\right)\right)$
Hyperbolic tangent sigmoid	Tansig	$f\left(x\right)=\frac{2}{1+e^{-2x}}-1$	$f^{\prime }\left(x\right)=1-f{\left(x\right)}^2$

MSE and $R^2$ of predicted results with different hyperparameters and the number of neurons are shown in Figure 11. The ANN model with the input activation function of purelin shows the worst performance, with $R^2$ less than 0.7 and MSE up to about 600, which is attributed to the inherent nonlinear behavior of the moisture absorption deformation problem. There are slight differences in the other combinations of input/output activation functions, the tansig/logsig activation function is the best among them ( $R^2$ = 0.993, MSE = 22.2). The prediction accuracy of traingd and traingdm training algorithms is poor, with $R^2$ less than 0.6 and MSE up to about 860, but among which trainlm training algorithm is the best among them. As the number of neurons in the hidden layer increases, the prediction performance of the ANN model also improves steadily. When the number exceeds 32, the prediction accuracy tends to be stable, and 33 is selected as the best number of neurons in the hidden layer. Therefore, the optimal training algorithm, input/output transfer function and the number of neurons in the hidden layer are trainlm, tansig/logsig and 33 respectively.

Figure 11.

Comparison of hyperparameter calculation results for neural network model: (a) training algorithm, (b) input/output activation functions, (c) number of hidden layer neurons.

The maximum hygroscopic deformation ( ${\omega }_{\max }$ ) is predicted by using the well-trained BPNN model. As shown in Figure 12(a), for the training set containing 2252 randomly selected samples, the maximum relative error ( ${\delta }_{\max }$ ) and average relative error ( $\overline{\delta }$ ) the training set are 12.35% and 4.94%, respectively, in which the number of samples with ${\delta }_{\max }$ greater than 5% accounts for 15.37%. The comparison of the testing set containing 564 randomly selected samples about ${\omega }_{\max }$ is shown in Figure 12(b), the corresponding values of ${\delta }_{\max }$ and $\overline{\delta }$ are 7.34% and 3.5% respectively, and the proportion of samples with ${\delta }_{\max }$ greater than 5% is 9.8%. Figure 12(c) shows the relationship between the real value and the predicted value of the testing set, and well-trained BPNN model exhibits good performance with high correlation coefficient ( $R^2=0.9929$ ). Moreover, to reduce the probability of overfitting, the ANN model completes network training after the 87th iteration in Figure 12(d).

Figure 12.

Prediction results of maximum deformation after water absorption: (a) comparison of training set, (b) comparison of testing set, (c) regression prediction, (d) learning curve.

In order to quantify the contribution rate of input features to output feature, the weight contribution rate analysis is further conducted after training process. The weight matrix can be obtained after ANN model training, and the contribution rate of each input feature variable to the output prediction result is calculated, which can be represented as equations (7) and (8).

$W_i={\displaystyle \sum }_{j=1}^{33}\left|w{1}_{ij}\right|\left|w{2}_j\right|\left(\frac{\left|w{1}_{ij}\right|}{{\sum }_{i=1}^5\left|w{1}_{ij}\right|}\right)$

(7)

$P_i=W_i/{\displaystyle \sum }_{i=1}^5{W}_i$

(8)

Figure 13 shows the weight contribution rate of different input features to the output feature ( ${\omega }_{\max }$ ). The absorbed moisture content of laminates has the largest weight contribution rate (19.15%), indicating that it has the greatest influence on ${\omega }_{\max }$ . The second is the laminate lay-up angle, the weight contribution rate fluctuates around 15%, while the number of non-repeated ply angles has the least influence (4.37%).

Figure 13.

Weight contribution rate of input features.

Forward prediction

The well trained BPNN model is conserved and utilized to predict the ${\omega }_{\max }$ of composites with new ply orientation angle, as listed in Table 5. The FEM model has the reliable accuracy after the calibration of thermal and moisture expansion coefficients through the full-field displacement comparation, and it can be used to verify the accuracy of the value predicted by BPNN. Some new ply angles (30°, 60°, 120°) are taken into accounted in the validation process, and the MSE and $R^2$ of predicted values are 3.28 and 0.9947 respectively. Therefore, the well trained BPNN model has high prediction accuracy and high generalization capacity, which builds the mapping relationship between the lay-up design parameters, hygrothermal environment and ${\omega }_{\max }$ .

Table 5.

Comparation between the value predicted by BPNN and simulation result.

Input								Output
x1	x2	x3	x4	x5	x6	x7	x8	ANN	FEM
30	60	90	120	6	4	4	3	56.12	50.71
150	60	30	90	6	4	4	0.5	79.51	73.82
0	30	90	30	6	3	4	0	13.52	15.03
30	60	60	30	1	2	1	1	1.2341	0
30	60	30	60	2	2	4	2	17.65	19.96
90	90	60	60	5	2	4	5	13.21	11.41
MSE								3.28
$R^2$								0.9947

Conclusion

This study develops a data-driven computational methodology based on FEM-ANN approach to investigate the distortion behaviors of CF/PA6 unsymmetric laminates before and after hygrothermal aging. Through this study, the following conclusions can be drawn.

(1) To accurately characterize the shape of CF/PA6 unsymmetric laminates in the water absorption process, the 3D model reconstruction with different water content is carried by combining 3D scanner and secondary development in Abaqus. The absorbed water content has significantly effects on the geometric parameters including deflection ( $\omega$ ) and radius of curvature (R), in which $\omega$ decreases linearly as water content increases, but R shows an exponential downward trend.

(2) A full-field displacement comparison method is proposed to calculate the equivalent thermal/moisture expansion coefficient, considering inevitable experimental errors. Compared with unmodified result ( ${\overline{d}}_{global}$ = 21.44 mm), the corresponding ${\overline{d}}_{global}$ is only 0.312 mm, which validate the accuracy of the numerical simulation, and the FEM model can be used to prepare the 2816 data samples in the following machine learning.

(3) 495 combinations are conducted to search for the optimal neural network structure and optimal hyperparameters by combining grid search and five-fold cross validation on the training set, and the optimal training algorithm, input/output transfer function and the number of neurons in the hidden layer are trainlm, tansig/logsig and 33 respectively. The well-trained ANN model exhibits good performance ( $R^2$ = 0.993, MSE = 22.2) in predicting of the maximum hygroscopic deformation ( ${\omega }_{\max }$ ).

Overall, this study provides a new way to the forward performance prediction of unsymmetric laminates under hygrothermal environment. In the future work, the present data-driven design method is highly expected to be extended for further multi-objective performance prediction and inverse design between the lay-up design parameters, hygrothermal environment and mechanical properties.