Introduction

Molding technology has been widely used to form composite structures in the aerospace and automotive industries because of its ability to reduce both the number of parts and the production cost1,2,3,4,5,6. However, composite components manufactured using molding usually introduce curing-induced deformation and residual stresses7. To overcome this problem, experimental and empirical analyses are generally used to evaluate the curing process to ensure that the deformation meets the manufacturing requirements. The traditional method is to make repeated adjustments and compensatory designs to the curing process curves and molds according to experience and processing tests8,9. However, this approach may not work for composite structures with relatively complex shapes, those consisting of new materials, or if the experiments are inefficient and time-consuming. To maintain the quality of manufacturing, reliable curing analyses should be complemented to replace numerous repeated experiments while reducing the cost.

The curing of composite structures molded with fibers, resin, and molding materials involves complicated coupled multi-physics processes that can be easily affected by temperature, and curing deformations and residual stresses can ultimately influence the molding quality10,11. Numerous studies have been conducted on analytical models for the curing of composites. Zhang et al.12 reviewed the generation mechanism and methods of controlling the residual stress and curing deformation. Jiang et al.13 observed the resin’s viscosity and measured the heat flow during the curing process to estimate the relative curing degree, and the resin’s critical curing degree transferring from liquid to solid was determined. Sung et al.14 put forward a curing model for a composite laminate to investigate its curability by varying the material parameters with the DoC and temperature. Fisher et al.15 investigated the influence of a range of parameters on the material, geometry and processing conditions in a thermochemical model of the curing of composite laminates. Bogetti and Giliespie16 developed a two-dimensional heat transfer model for composites and investigated the development of curability using a coupled model of curing kinetics. Sekmen et al.17 studied the viscoelasticity of a material in relation to temperature, curing degree and frequency. Kang et al.18 studied the curing degree and proposed a method for continuous 3D printing. Nevertheless, the curing of composites is sophisticated, and the processing parameters significantly affect the development of curability. Analytical analysis and experiments seem less efficient and time-consuming, extremely for thick composite structures.

The continuous development of computing technologies has enabled numerical simulations to replace experimental testing19,20. The curing of composites involving coupled multi-physics fields is efficient for determining the degree of curing and coupled stresses and strains via FE simulations. Pantelelis et al.21 established a one-dimensional nonlinear FE model to obtain the temperature change law for composite laminates. Loos et al.22 developed a one-dimensional FE model to analyze the effects of temperature and resin flow on the curing temperature history and curing curve of carbon fiber-reinforced epoxy resin prepreg AS4/3501-6. By predicting the curing-induced deformation of components through the virtual design of the curing process and then optimizing the parameters of processing and structural design, the consumption of time, and raw materials, the production cost can be significantly reduced23. The simulation of the curing process has been well developed. However, it should be noted that the simulation itself is not the cure model itself; rather, it involves understanding initial conditions such as fiber volume fraction, initial degree of cure (DoC), and convective boundary conditions. Furthermore, the impact of curing can be crucial for the residual stress development24,25,26. Therefore, studying the degree of cure is essential for accurately assessing the curing-introduced deformation and residual stress.

In recent decades, both academia and industry have utilized numerical simulations as alternatives to empirical methods. Artificial intelligence (AI) methods have been applied to materials to reduce design time and cost27,28,29. For instance, Hou et al.30 used three different machine learning methods to establish prediction of temperature difference and curing degree. Okafor et al.31 reviewed advances in machine learning based reinforced composite design. Liu et al.32 used machine learning methods to predict the macroscopic thermal conductivity of polymer graphene-reinforced composites. Huang et al.33 adopted FE and machine learning methods to predict the transverse modulus of unidirectional composites. Shabley et al.34 predicted the failure strength of composite materials using machine learning methods. Models based on machine learning characterized by nonlinearity, robustness, and high learning capability have also been used to predict damage and fatigue assessment in composites35,36,37,38,39. The proposed framework integrating FE methods and machine learning also proves that AI methods are more efficient in comparison to manual experiments40,41. There is a large amount of data on composites due to structural and material nonlinearity as well as anisotropy. Therefore, the data-driven approach is suitable candidate for the mechanical analysis of composites. Herein, a simulation-data-driven method is developed to predict the DoC-time curve subjected to a given process curve (temperature over time), which transforms complex problems of heat transfer coupled with cure kinetics into intuitive data and will benefit the subsequent curing simulation to predict the residual stress and curing-induced deformations.

The curing simulation is rather a well established problem and here this paper focuses on the data-driven approach. A theoretical background focusing on heat conduction theory, curing mechanisms and kinetics for the curing analysis of composites is given after the introduction. In the third section, the FE curing model with subroutines for a fiber-reinforced composite laminate constructed with AS4/3501-6 is introduced, and the data generation according to the FE-based curing simulation of the AS4/3501-6 laminate is described, followed by a description of machine learning methods and prediction model development for the DOC-time curve. Finally, conclusions and suggestions for future work are given.

Theoretical background

It has been known that the composites are composed of resin matrix and fiber, and the curing process of composites is characterized by resin’s internal solidification and exothermic heat generation, along with the absorption of external heat transfer including both endothermic and exothermic reactions. During the curing of composite structures, the resin typically undergoes a sequence of heating, insulation, and cooling. Initially, the resin is in a liquid state, but it transitions to a solid state due to cross-linking reactions during the heating phase. For the anisotropic composites, uneven heat conduction in all directions occurs during the heat release, leading to varying degrees of curing due to different temperature gradients within the composite. The phase transformation of the resin introduces variations of the mechanical properties that cause volume shrinkage in different directions of the composite. Simultaneously, thermal stress and chemical shrinkage stress also arise. Following the cooling process, residual stress remains within the composites due to uneven heat dissipation, ultimately leading to curing deformation.

The simulation of cure has been extensively studied. Basically, it involves three parts: thermo-chemical model, curing kinetic equation and curing constitutive model. In the curing simulation, the thermo-chemical model is mainly used to describe the temperature change and chemical reaction of the composite material; the curing kinetic equation describes the deformation and stress change, and the curing constitutive equation defines the constitutive relationship and stress increment equation. Before setting up an FE-based curing simulation, heat transfer analysis according to Fourier thermal conductivity equations and curing kinetics of the resin are essential to theoretically determine the curing degree and temperature distribution of the resin during the curing reaction of the composite laminate. In this section, the thermo-chemical and curing kinetic models are briefly introduced42.

Thermo-chemical analysis

According to the Fourier heat conduction equation and the curing kinetics of the resin, combined with heat transfer analysis, the curing degree and temperature history of the resin can be determined. The transient conduction equation in the curing process of the composite materials can be determined as follows40:

$$\rho C_{p} \frac{\partial T}{{\partial t}} = \frac{\partial }{\partial x}\left( {k_{x} \frac{\partial T}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {k_{y} \frac{\partial T}{{\partial y}}} \right) + \frac{\partial }{\partial z}\left( {k_{z} \frac{\partial T}{{\partial z}}} \right) + \frac{\partial Q}{{\partial t}}$$
(1)

where ρ is the composites density, Cp presents the specific heat, T denotes the curing temperature, t is the curing time, and k illustrates the thermal conductivity of the composite material, and x, y and z define the axial directions. The density ρ can be determined by the density of the resin, the density of the fiber, and the volume fraction of the fiber. Similarly, the Cp is also determined by the specific heat capacities of the resin and the fiber, the densities of the resin and the fiber, and the fiber volume fraction.

Q in Eq. (1) denotes the heat released during the resin solidification and can be calculated using the equation illustrated as follows:

$$\frac{\partial Q}{{\partial t}} = \rho_{m} \left( {1 - V_{f} } \right)H_{R} \frac{d\alpha }{{dt}}$$
(2)

where Vf represents the fiber volume fraction, HR is the total heat released at the completion of curing that could be measured via a differential scanning calorimetry (DSC) under dynamic and isothermal conditions; α is the curing degree of resin; and dα/dt is the rate of resin curing, which can be determined via the curing kinetic equations.

Curing kinetic model

The curing behavior of resin can be described using the curing kinetic model, and consequently the corresponding curing rate and curing degree can be determined. The phenomenological model focuses primarily on the overall reaction rather than the kinematic mechanism in the curing reaction process. It utilizes a simplified equation to characterize the state of the system before and after the curing reaction occurs. The general expression of the model can be expressed as:

$$\frac{{{\text{d}}\alpha }}{{{\text{dt}}}} = {\text{k}}\left( T \right)f\left( \alpha \right)$$
(3)

where \({\text{k}}\left( T \right)\) is a function of the curing rate, which follows the Arrhenius equation and can be expressed as:

$$k_{i} \left( T \right) = A_{i} \exp \left( { - \frac{{\Delta E_{i} }}{RT}} \right),\left( {i = 1,2,3} \right)$$
(4)

where \(A_{i}\) is the frequency factor, \(\Delta E_{i}\) represents the activation energy, and R is the constant of gas.

The curing degree function, f(α), can be formulated as two types of phenomenological models: n-step reaction kinetics model and autocatalytic model. The n-level reaction kinetic model adopts a specific formulation expressed as follows:

$$\frac{{{\text{d}}\alpha }}{{{\text{dt}}}} = {\text{k}}\left( T \right)\left( {1 - \alpha } \right)^{n}$$
(5)

where n represents the number of reactions.

Moreover, the specific form of the autocatalytic model is:

$$\frac{{{\text{d}}\alpha }}{{{\text{dt}}}} = {\text{k}}_{1} \left( T \right)\alpha^{{\text{m}}} \left( {1 - \alpha } \right)^{{\text{n}}}$$
(6)

or

$$\frac{{{\text{d}}\alpha }}{{{\text{dt}}}} = \left[ {k_{2} \left( T \right) + k_{3} \left( T \right)\alpha^{{\text{m}}} } \right]\left( {1 - \alpha } \right)^{{\text{n}}}$$
(7)

where m and n are reaction numbers.

Therefore, simulation of cure is not the cure model itself, but rather knowing the initial conditions such as fiber volume fraction, initial degree of cure, convective boundary conditions etc.

FE simulation of cure

Solving the heat transfer coupled with the cure kinetics may present additional requirement for time, making AI tools promising for this problem. This paper focuses on developing simulation-data-driven approach to quickly predict the DoC curve. The curing process is one of many factors contributing to the molding quality of composite structures and is usually an endothermic and exothermic process involving heating, insulation, and cooling. In this process, the temperature gradient formed by the external temperature and the internal curing reaction temperature of the composite materials will produce complex residual stress and ultimately deformation, which seriously affects the molding accuracy and quality of the composite components. Employing the FE method to predict the curing degree and analyze the impact of the processing parameters of composite materials on the curing process can effectively lower the production cost and enhance the molding quality. In this section, an FE model of a composite plate constructed with AS4/3501-6 is created to simulate the curing process, and the influences of the processing parameters, mainly the heating rate, holding time and cooling rate, on the curing degree are analyzed.

FE simulation and validation

The simulation of cure for a three-dimensional laminated plate is executed in ABAQUS, and analysis step of heat transfer invoking the subroutines is conducted to derive the temperature field and DoC curve. For easy validation the simulation results, we used an identical model as presented in reference42, which was constructed with a unidirectional AS4/3501-6 laminate and the dimensions of the composite plate were 15.24 cm × 15.24 cm × 2.54 cm. To reduce the computing cost, a model of one-eighth plate (7.62 cm × 7.62 cm × 1.27 cm) was created considering the symmetry of the plate. The kinetic parameters for AS4/3501-6 listed in Table 1 are utilized for the curing simulation. A total of 2304 hexahedral mesh elements of DC3D8 are chosen for the FE model. Subroutines of HETVAL and USDFLD defining the thermal-physical properties and curing kinetic parameters are programmed in a visual studio configured with a Fortran environment for the thermo-chemical analysis. Simultaneously, the initial curing degree is defined in the USDFLD subroutine. Before submitting the job, the subroutines are invoked. Consequently, the degree of cure and curing temperature of the composite plate can be obtained.

Table 1 Curing kinetic properties of AS4/3501-643.
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The curing kinetic equation for the AS4/3501-6 composite plate can be expressed as43:

$$\left\{ \begin{gathered} \frac{d\alpha }{{dt}} = \left( {K_{1} + K_{2} \alpha } \right)\left( {1 - \alpha } \right)\left( {0.47 - \alpha } \right),\left( {\alpha < 0.3} \right) \hfill \\ \frac{d\alpha }{{dt}} = K_{3} \left( {1 - \alpha } \right),\left( {\alpha \ge 0.3} \right) \hfill \\ \end{gathered} \right.$$
(8)

where \(K_{i} = A_{i} \exp \left( { - E_{i} /RT} \right),\left( {i = 1,2,3} \right)\), K represents a function of the reaction rate for the 3501-6 resin, A defines the frequency factor, E denotes the activation energy and i is the i-th element.

The curing process is shown in Fig. 1a: (1) heating from 20 ℃ (room temperature) to 120 ℃ at a heating rate of 2.5 ℃ /min, and holding this temperature at 120 ℃ for 1 h; (2) heating from 120 to 177 ℃ at a heating rate of 3.8 ℃/min and keeping the temperature at177 ℃ for 2 h; and (3) cooling to room temperature at a cooling rate of 2.5 ℃/min. Correspondingly, the DoC curve for center-point node is obtained by running the FE simulation, as illustrated in Fig. 1b. Additionally, temperature and degree of cure gradients develop throughout the thickness of the laminate during the curing cycle. These gradients can subsequently induce residual stresses. Furthermore, variations in the degree of cure result in chemical hardening effects, causing mechanical properties to evolve nonuniformly across the laminate’s thickness. To examine the variation of DoC at different locations within composite plates throughout their thickness under a specified temperature profile (refer to Fig. 1a), we plot the curing degrees as a function of laminate thickness at 18,000 s in Fig. 1c,d. Here, the curing temperature profile refers to Fig. 1a, and the laminate has a thickness of 0.8 cm and 1.27 cm, respectively. Figure 1c,d demonstrate that both composite plates achieve complete curing by 18,000 s with only minor DoC variations occurring from the innermost surface (closest to the heating element) to the outermost surface of the composite plates throughout the laminate thickness. Therefore, the DoC at the center-point node is considered to reduce the variation of DoC introduced by the throughout thickness. Moreover, to validate the accuracy of the subroutines formulated, we present the DoC curve for center-point node as a function of the curing time in Fig. 1b, corroborating these results with literature43. The simulated curing degrees over time exhibit great consistency with findings published in reference43, which align with experimental data presented in44. As shown in Fig. 1b, a maximum discrepancy of approximately 3.24% is illustrated, indicating that the developed subroutines are effective for the curing simulation.

Fig. 1
figure 1

Curing processing curve and validation of the simulation results. (a) Temperature profile. (b) Degree of cure profile for center-point node. (c) DoC vs laminate thickness. Results are for a 0.8 cm thick composite plate from the innermost surface to the outermost surface. (d) DoC vs laminate thickness. Results are for a 1.27 cm thick composite plate from the innermost surface to the outermost surface.

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Effect of curing temperature on the curing degree

According to the analytical model, the curing temperature affects the DoC curve of composite structures. The influence of curing temperature on the curing process of the carbon/epoxy composite plate described in “FE simulation and validation” is explored by changing the insulation temperature. As shown in Fig. 2a, the time history of temperature in the curing process is given, and the insulation temperatures are set to 120 ℃, 130 ℃, 140 ℃ and 150 ℃. Consequently, the corresponding curing degree curves at these curing temperatures can be obtained and are given in Fig. 2b. The temperature and curing degree of the plate gradually increase during the whole curing process. Before the initial curing reaction, the insulation temperature has not yet played a role in the resin interior, so no obvious difference can be observed. As the reaction proceeds, the insulation temperature is transferred from the surface of the plate to the interior of the component, and the heat released by the chemical shrinkage of the resin due to different insulation temperatures causes a difference in the curing degree. Therefore, the difference in curing degree gradually increases with the increment of the temperature.

Fig. 2
figure 2

(a) Temperature–time curves with various curing temperatures, (b) curing degree curves for the composite plate subjected to various curing temperatures.

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Effect of holding time on the curing degree

According to “Effect of curing temperature on the curing degree”, the effects of the holding temperature on the curing degree are investigated. To investigate the effect of the temperature–time curve on the curing of the composite plate constructed with AS4/3501-6, the curing degree curves subjected to various holding times were obtained by running the FE models with different temperature–time curves. As shown in Fig. 3a, the composite materials were heated from 20 to 120 ℃ at a rate of 2.5 ℃ /min and held at 120 ℃ for 3600 s, 5400 s, 7200 s and 9000 s. Then, the temperature was increased from 120 to 200 ℃, and the temperature was held constant for 3600 s, 5400 s, 7200 s and 9000 s. The corresponding DoC-time curves are given in Fig. 3b, where the DoC curves remain identical at the first heating stage, and then increase to 0.99 at various reaction rates. The curve in black with less holding time completes the curing at the highest rate.

Fig. 3
figure 3

(a) Temperature–time curves with various holding times, (b) curing degree curves for the composite plate subjected to various holding times.

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Data preparation

The processing temperature and time described by the process curve (temperature–time curve) affect the curing degree. However, how they influence the DoC curve is still unclear. Thus, to further investigate the mapping relationship between the temperature–time and DoC curves, a simulation-data-driven approach is developed. The distributions of the temperature and curing degree over time obtained from the FE curing simulations can be taken as datasets for machine learning. An overall flow chart of the simulation-data-driven approach utilized to predict the DoC curve is shown in Fig. 4, where the FE simulations are run to obtain the DoC and curing temperature of the composite laminate as input data for machine learning.

Fig. 4
figure 4

Flow chart of the data preparation for machine learning.

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Processing of the DoC-time curve

The curves of curing degree over time obtained from FE simulations constitute high-dimensional datasets encompassing numerous data points. Employing unprocessed curability-time curves as input data for machine learning algorithms can significantly elevate the complexity of neural networks. This can potentially affect their performance and efficiency, leading to excessive consumption of computational resources and, consequently, a reduction in the learning efficiency of neural networks. From this point of view, this paper simplifies the DoC-time curve in two steps (see Fig. 5): (1) use 27 points with fixed horizontal coordinates of the curve to characterize a curability-time curve; (2) between different curability-time curves, the horizontal coordinates of the 27 points are repetitive information, so they can be temporarily omitted, and only the information of the vertical coordinates is retained. Consequently, one DoC-time curve can be expressed as a vector in 20 dimensions.

Fig. 5
figure 5

Data preprocessing of the DoC–Time curve.

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Datasets generation for machine learning

Curing parameters such as the processing curve which describes the history of temperature, have been shown to significantly affect the curability of composites. A machine learning method suitable for complex problems is utilized to construct a model to express the DoC-time curve as a function of the processing curve. Herein, we investigated the effects of room temperature, heating temperature and holding time on the processing curve. Assuming that the room temperature can vary from 20 to 25 ℃, the heating temperature can be varied from 100 to 220 ℃, and the holding time can vary from 0.5 to 2.5 h. The carbon/epoxy AS4/3501-6 composite plate was heated and insulated to achieve complete curing. The result files obtained from Python and ABAQUS are generated in batches, and the DoC-time curves are executed as datasets for machine learning. Two thousand datasets are divided into training and prediction sets at a ratio of 9:1.

Model construction based on machine learning

This section aims to construct prediction models using machine learning techniques according to the curing simulation data to express the DoC-time curve as a function of the processing curve. Initially, the SVR algorithm and BP neural network were utilized to construct models to estimate the DoC-time curve for a carbon/epoxy laminate subjected to a given processing curve. Nevertheless, for the BP neural network, the initial values are generally randomly generated, and it might be prone to local minima during the training process45. Thus, the genetic algorithm is used to optimize the weights and thresholds of the BP neural network, which improves its performance and accuracy. Three prediction models are constructed and evaluated in this section.

Model construction using SVR

SVR, as a regression model suitable for small sample sets is considered46. The 2000 sets of data obtained from the FE simulations are separated into two groups: 1800 for training and 200 for verification. For the established datasets, the following equation can be used to fit the nonlinear regression model:

$$f\left( x \right){ = }{{\varvec{\upomega}}}\phi \left( x \right) + b$$
(9)

where \(\phi \left( x \right)\) represents the nonlinear transformation of the original data, \({{\varvec{\upomega}}}\) defines the normal vector and \(b \in R\) denotes the value of the threshold. In practical applications, there are normally some anomalies, such as noisy data or outliers, which increase the difficulty in achieving perfect prediction results. Thus, we introduce two relaxation variables \(\xi_{i} , \, \xi_{i}^{*}\) to convert them into the original objective function as given in Eq. (10) to fit the complex data.

$$\begin{gathered} \min \, \frac{1}{2}\left\| w \right\|^{2} + C\sum\limits_{i}^{l} {\left( {\xi_{i} + \xi^{*}_{i} } \right)} \hfill \\ s.t. \, \left\{ \begin{gathered} y_{i} - w\phi (x_{i} ) - b \le \varepsilon + \xi_{i} {, }i = 1,2, \ldots l \hfill \\ - y_{i} + w\phi (x_{i} ) + b \le \varepsilon + \xi_{i}^{*} \hfill \\ \xi_{i} \ge 0,\xi_{i}^{*} \ge 0 \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$$
(10)

where C represents the punishment function, \(\left\| w \right\|\) defines the hyperplane’s parametrization, and \(\varepsilon\) illustrates the area around the hyperplane that is insensitive. Lagrange transformations include Lagrange multipliers several times to address the duality issue, along with the kernel function to address the regression function indicated as Eq. (11).

$$f(x) = \sum\limits_{i = 1}^{l} {(a_{i} - a^{*}_{i} )} K(x_{i} ,x) + b^{*}$$
(11)

where l represents the sample number, \(K(x_{i} ,x)\) illustrates the kernel function and the radial basis function is used as the kernel function here.\({\varvec{a}}_{{\varvec{i}}} ,\user2{ a}_{{\varvec{i}}}^{\user2{*}}\) denote the Lagrange multiplier. xi defines the supporting vectors when \(({\varvec{a}}_{{\varvec{i}}} - {\varvec{a}}_{{\varvec{i}}}^{\user2{*}} )\) is nonzero. \(b^{*}\) stands for the bias value.

The primary usage of kernel functions is the mapping of input features to higher-dimensional spaces. The three primary types are radial basis kernel (RBF) functions, sigmoid kernel functions, and linear kernel functions, which are used to find nonlinear correlations between features and better fit data47. One kernel function that can handle nonlinear situations is the radial basis function. To attain nonlinear approximation, the Gaussian distribution function is utilized to map the nonlinear relation into a high-dimensional feature space. The RBF function in this paper is expressed as:

$$k(xi,xj) = \exp \left( { - \frac{{\left\| {xi - xj} \right\|^{2} }}{{2\sigma^{2} }}} \right)$$
(12)

where \(\sigma\) is the variance of the radial basis function. The optimal combination of parameters C and σ is searched through multiple rotations to train and test the model performance, and the selection of these two parameters has a great influence on the model accuracy. Herein, the values of C and σ are set to 0.5 and 22.62, respectively. An analysis of the actual and predicted values for the validation datasets revealed 1573 support vectors with a bias value of -0.2096.

Model construction using BP neural network

A BP neural network is a complex neural network with self-learning and inductive capabilities that is capable of replacing linear relationships between simple conditional inputs and outputs via nonlinear correlation mapping48. In general, a BP neural network is constructed with three or more layers, including input, hidden and output layers49. For three-layer BP neural network, the matrix form is expressed as:

$$f(x) = {\mathbf{W}}^{(o)} \tan sig({\mathbf{W}}^{(h)} {\varvec{x}} + {\varvec{b}}^{(h)} ) + b^{(o)}$$
(13)

where W is the weight matrix, x and b are vectors, b is the threshold value, and superscripts o and h represent the output layer and the input layer, respectively.

The input variable in this work is the coordinate value of the temperature–time curve with a total of 8 classes as the eigenvalues of the input layer, and the output variable is the coordinate value of the curing degree-time curve with a total of 27 classes as the eigenvalues of the output layer. Therefore, the input and output layers are vectors of 8 × 1 and 27 × 1, respectively. The number of hidden layers is normally set to 1, which has the advantage of simplicity over multi-hidden layer networks. Although increasing the number of hidden layers can reduce the error of the neural network and improve the accuracy, it will also enhance the network complexity, leading to increment of the training time and a tendency towards “overfitting”. Therefore, a three-layer structure is selected, and the structure of the BP neural network is shown in Fig. 6.

Fig. 6
figure 6

Architecture of a three-layer BP neural network.

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The prediction model of the BP neural network is established in Matlab. The Mapminmax function was applied to normalize the data, the activation function of the output layer was set to purelin (see Eq. 14), and the activation function of the hidden layer neurons was defined as the tansig function (see Eq. 15). The conjugate gradient algorithm was selected to train the model with a training error of 0.00001, a learning rate of 0.1, and an iteration number of 1000. Once the BP neural network model is established, the data can be imported for training.

$$f(x) = kx$$
(14)
$$\tan sig(x) = \frac{2}{{1 + e^{ - 2x} }} - 1$$
(15)

Model construction using GA-BP neural network

As already known that BP neural networks have several drawbacks, such as long training time and being prone to local minima50, a GA can be utilized to optimize the initial weights and thresholds of BP neural networks. As an optimization algorithm based on the principle of biological evolution, the GA which has the advantages of simplicity, adaptability, and weak constraints51, is commonly used to solve complex search and optimization problems. The flowchart to optimize the BP neural network employing the GA is shown in Fig. 7. Initially, the overall architecture of the BP neural network is determined, and its weights are initialized. Subsequently, it assesses the fitness of individual solutions according to the training error of the BP neural network. Finally, the GA is used to determine the optimal weights and thresholds, which are then employed to update the initial weights and thresholds of the BP neural network. The parameters of the weights of the GA optimization are given in Table 2.

Fig. 7
figure 7

Flowchart of GA-BP neural network optimization.

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Table 2 Main parameters of the GA utilized to optimize the BP neural network.
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Results and model evaluation

Considering the challenge of quantitatively assessing the curve prediction on a global scale, this section selects a case study on the DoC-time curve. The evaluation of the model is subsequently conducted using the following indices52: (1) the mean square error (MSE) is used to measure the deviation between the observed value and the true value, and a larger value indicates a worse model; (2) the root mean square error (RMSE) reflects the actual situation of the prediction error, and a larger value illustrates a worse model; (3) the mean absolute error (MAE) is used to measure the degree of dispersion of a set of numbers themselves, and the larger the MAE value, the worse the model; and (4) the coefficient of determination (R2) reflects the degree of fit of the model to the sample, and a larger value indicates that the network model regression effect is better. The evaluation results of the four indices are listed in Table 3. The MSE, RMSE and MAE of the GA-BP neural network are quite small, and R2 is close to 1, showing that the GA-BP neural network yields the highest degree of accuracy.

Table 3 Evaluation indices for three different machine learning methods.
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To evaluate the GA-BP neural network model, the true value and predicted value for the DoC curve at t = 21,000 s are selected as the benchmark. The true and predicted values remain great consistency, indicating the accuracy of the model. In Fig. 8a, the input process curve is given: (1) the composites are heated from 25 ℃ (the room temperature) to 130 ℃ at a heating rate of 2.1 ℃/min, and maintained the temperature of 130℃ for 0.5 h; (2) then, the temperature is increased to 210 ℃ at a heating rate of 1.6 ℃/min and kept at 210 ℃ for 0.5 h; and (3) the samples are cooled from 210 ℃ to room temperature at a cooling rate of 2.5 ℃/min. In Fig. 8b, the predicted curves of the DoC-time using the SVR, BP and GA-BP neural networks are given and compared with the true curve. In Fig. 8c, the fitness of the GA-BP neural network in the 65th generation is 0.0021236 when the MSE is the lowest. It is noted that the GA effectively converges the fitness value to the global optimum.

Fig. 8
figure 8

Curing processing curve and the corresponding DoC curves predicted using the SVR, BP and GA-BP neural networks. (a) Curing processing curve. (b) Degree of curing curves. (c) MSE as a function of the iteration number.

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Conclusion

The simulation of the curing process has been extensively developed. However, it is noted that the focus of cure simulation lies not in the cure model itself, but rather in understanding the initial conditions such as fiber volume fraction, initial DoC, and convective boundary conditions. Additionally, in the simulation of cure, the heat transfer coupled with cure kinetics were addressed, which may necessitate additional time, making the utilization of data-driven approach for these problems a promising opportunity. Therefore, in this study, a simulation-data-driven prediction method is proposed to predict the curing degree over time subjected to a specific processing curve, so that it can accelerate the simulation for analyzing the curing-induced deformation and residual stress. The conclusions are as follows:

  1. (1)

    An FE model of curing simulation was established to simulate the curing process of a carbon/epoxy composite plate. The subroutines of HETVAL and USDFLD, which define the thermal-physical properties and curing kinetic parameters, are implemented within a Visual Studio environment configured for Fortran to facilitate thermo-chemical analysis. Concurrently, the initial degree of curing is specified in the USDFLD subroutine. The obtained result of the DoC curve was validated by existing publications with a maximum error of 3.24%, indicating that the developed codes are efficient for curing simulations.

  2. (2)

    The temperature–time curves are used as the input data to compute the corresponding DoC curves in ABAQUS evoked by the subroutines. Subsequently, the DoC curves obtained from the FE simulations of cure are preprocessed as datasets for training the prediction models using SVR, BP and GA-BP neural networks. The evaluation index values of R2 for the SVR, BP and GA-BP networks are 0.7459, 0.979 and 0.988, respectively. Evaluation and validation analysis demonstrate that the DoC curve predicted by the GA-BP neural network has the highest prediction accuracy and is least prone to local optima.

Furthermore, the DoC curves present the solidification of the resin as a function of time, which consequently affect the residual stress development as well as the deformation introduced by cure of the composite structures. Therefore, in the future, the developed simulation-data-driven approach will be implemented to evaluate the deformations and residual stresses caused by cure.