A Physic-Informed Data-Driven Relational Model of Plastic Strain vs. Process Parameters during Integrated Heating and Mechanical Rolling Forming of Hull Plates

Abstract

Integrated heat and roll forming (IHMRF) is a process that uses thermal and mechanical loads to produce localized plastic strains in plates to form complex curvature hull plates. The magnitude of the resulting plastic strain depends mainly on the following forming parameters: the machining parameters (power of the heat source, speed of the heat source, and the forming depth of the rollers), the thickness of the plate, and the thermo-physical and mechanical properties of the plate. Finding the correspondence between the plastic strain and forming parameters is the key to selecting the appropriate machining parameters for forming. A data-driven approach is ideal for this purpose. However, due to the characteristics of the IHMRF process, the forming process involves a large number of variables, and different materials have different temperature-dependent yield strengths. These high-dimensional input characteristics create a conflict between the required number of samples and the model training requirements. This paper presents a physically informed data-driven (PIDD) approach for modeling the relationship between forming parameters and plastic strains in IHMRF. Based on dimensional analysis and domain knowledge, the proposed method derives the basic thermal and mechanical relationships between the forming parameters, obtaining a much smaller number of physical parameters. These physical parameters are expressions of the physical knowledge of forming in low-dimensional space. Using the physical parameters yields higher accuracy on fewer sample data points than directly using the forming parameters as input features. Furthermore, the models trained on a variety of commonly used materials and plate thicknesses achieved comparable accuracy to the numerical simulation with unseen materials and plate thicknesses. Experimental and numerical simulations further verify the effectiveness of the proposed method by machining plates of various materials to the same shape.

1. Introduction

Hull plates with complex curvature are both load-carrying members of ships and bear the important responsibility of ensuring the hydrodynamic performance of the ship. Therefore, the forming process is a key link in ship construction [1]. In this process, the plastic strain is a combination of in-plane and out-of-plane strains, and the maximum values of the two types of strains are often comparable. The conventional forming process, commonly known as line heating, applies a combination of the plastic strains required for forming by moving a heat source over the plate’s surface [2]. However, due to the characteristics of the line-heating forming method, such as the limited combination of in-plane and out-of-plane strains generated by the line heating, the shapes obtained by line-heating forming are still limited to a certain extent for thicker plates and areas of greater curvature [3]. Integrated heating and mechanical roll forming (IHMRF) is a new method for forming complex curved plates for ships [4]. The heat source and rollers are combined on one working surface in this method. The machining parameters of the heat source and rollers are changed to account for both in-plane and out-of-plane strain applications, which improves the accuracy and richness of the different plastic strain combinations that can be applied to the plate.

The basic principle of IHMRF is to apply localized plastic strain to the plate and to change the overall shape of the plate by controlling the extension of the localized plastic strain along the planned machining path. In order to realize complex surface forming, the location of the machining path and the machining parameters (power of the heat source, speed of movement of the heat source, and amount of downward pressure of the rollers) need to be derived from the target shape in advance. However, since IHMRF is a die-less, locally loaded forming method, the relationship between the target shape, the machining path, and the machining parameters is relatively complex, and utilizing the intermediate variable is unavoidable. Since plastic strain is both geometrically related to the target shape and physically related to the machining parameters, it is usually regarded as the most suitable intermediate variable [5]. The process of deriving machining paths and machining parameters based on plastic strains consists of two main steps: The first step is to obtain the strain required for forming by optimizing the calculation for flattening the target surface to a planar shape [6]. The second step is to plan the location of the machining path and machining parameters based on the strain distribution [7]. In this regard, the correspondence between the machining parameters, plate thickness, plate thermophysical and mechanical property parameters (hereinafter collectively referred to as the forming parameters), and plastic strain is the key to selecting the appropriate machining parameters for forming. Therefore, finding the correspondence between forming parameters and plastic strain is an inevitable requirement for manufacturing complex curvature hull plates based on the IHMRF process.

IHMRF is a very complex thermoelastic–plastic deformation problem. The relationship between the forming parameters and the plastic strain is highly nonlinear. Therefore, it is difficult to accurately find out the relationship between forming parameters and plastic strain using analytical methods and experimental measurements. With the development of computer technology and finite element methods, numerical simulation techniques have become an important tool for studying forming processes. One of the most commonly used is the thermoelastic–plastic finite element method [8]. In this method, the complete physical process is simplified as a sequential coupling of heat transfer and deformation processes, taking into account nonlinear factors such as thermoelastic–plastic constitutive relationships, geometric nonlinearities, moving heat sources, and nonlinear contact behaviors between the rollers and the plates [9]. TEP-FEM is usually very time-consuming due to the large number of nonlinearities involved in its calculations [7,8]. Using the equivalent load model as a form of load input to the finite element program has improved the computational efficiency of TEP-FEM [10]. However, the response speed of TEP-FEM is far from meeting the demand for the real-time control of intelligent equipment.

Data-driven modeling methods based on neural networks have received increasing attention in smart manufacturing research [11]. Traditional neural network model training methods start directly from the data and extract data features by passing data between multiple hidden layers to automatically learn and infer relationships between variables. However, due to the characteristics of the IHMRF process, the number of variables involved in the forming parameters is large, especially since different materials have different temperature-dependent material properties. If the forming parameters are directly used as input features, traditional training methods may not be able to extract enough information from the complex multidimensional data, resulting in inaccurate predictions when the samples are small, and it is difficult to ensure the generalization ability of the model. However, for IHMRF, it is not feasible to collect sufficiently large samples using experiments or TEP-FEM; this is because it would require significant human, material, and time costs.

In this study, a physic-informed data-driven (PIDD) approach is proposed, which can be used to obtain a relational model of the forming parameters and plastic strains of IHMRF. The method integrates two artificial neural network (ANN) models to simulate the heat transfer process and deformation process of IHMRF, respectively. Based on dimensional analysis and domain knowledge, the proposed method derives the basic thermal and mechanical relationships between the forming parameters, obtaining a much smaller number of physical parameters. These physical parameters are expressions of the inherent thermal and mechanical knowledge of forming in low-dimensional space. This allows the training process of neural network models to be naturally guided by physical laws, which effectively reduces the dependence of model training on the number of samples and improves the generalization ability of the model. Taking several materials commonly used in shipbuilding as an example, plastic strains were obtained using the experimentally validated TEP-FEM for a large range of plate thicknesses and machining parameters. The PIDD model was trained with this dataset. The performance of the PIDD model on unseen materials and plate thicknesses during training was examined. Experiments and TEP-FE simulations were carried out to form machine steel plates with different material properties into the same shape. The deformations of these plates with different material properties were essentially the same.

2. Overview of IHMRF Process

This section reviews the basic components of the IHMRF process. Figure 1 shows a schematic diagram of the IHMRF process’s basic components, with induction heating selected as the heat source in this study. IHMRF is a line loading process in which the loading tool does not move while the plate moves. During this process, the rollers both drive the plate in a specified direction through friction and can apply out-of-plane strains. At the same time, the heat source is located at a fixed distance from the rollers to heat the plate, mainly applying in-plane strains.

The IHMRF method enhances the richness and flexibility of the applied strains by simultaneously loading the heat source and rollers. However, it also leads to a more complicated relationship between machining parameters and plastic strains. On the one hand, the plastic strain produced by the combined action of heating and rolling is not a simple linear superposition of the independent effects of the two, due to the different mechanisms of plastic strain generation by heating and rolling. Temperature changes not only drive the plate deformation, but also cause dramatic changes in the material properties, especially the yield stress. This change, in turn, induces a corresponding change in the out-of-plane strain applied to the plate by the rollers. As for the in-plane strain applied by heating, in addition to the restraining influence of the surrounding colder material, the mechanical stresses in the plate induced by rolling also have an influence. On the other hand, the correlation between certain physical quantities and plastic strains may change with the machining parameters. For instance, a strong correlation exists between the coefficient of the thermal expansion and plastic strain under high temperature loading conditions. However, as the loading temperature decreases, the coefficient of the thermal expansion may gradually lose its status as a key influence factor and become redundant information. In the face of such complex variable relationships, without effective preprocessing of the sample data, coupled with the limitations of the processing capability of the modeling algorithms and the size of the sample dataset, the task of constructing an accurate and reliable model of the relationship between the machining parameters and the plastic strain can become extremely difficult.

If the direction along the machining path is defined as the longitudinal direction and the direction perpendicular to the machining path is defined as the transverse direction, the plastic strain of the plate can be divided into four quantities, i.e., the longitudinal in-plane strain, longitudinal out-of-plane strain, transverse in-plane strain, and transverse out-of-plane strain. Among these, the longitudinal out-of-plane strain is much smaller than the remaining three components and can be neglected. Therefore, this study focuses on the longitudinal in-plane strain, transverse in-plane strain, and transverse out-of-plane strain.

3. Development of a Physics-Informed Data-Driven (PIDD) Model

The modeling strategy here is to integrate two artificial neural networks (ANN) to simulate the heat transfer and deformation processes, respectively, as shown in Figure 2. The two neural networks are named thermal ANN and mechanical ANN, respectively. The thermal ANN simulates the relationship between machining parameters, material thermophysical property parameters, plate thickness, and the maximum temperature of the heated surface. The mechanical ANN simulates the relationship between the machining parameters, plate thickness, material thermophysical property parameters, mechanical property parameters, the maximum temperature of the heated surface (thermal ANN prediction result), and plastic strain.

In order to alleviate the dependence of model training on the size of the sample dataset, dimensional analysis and domain knowledge were utilized here to reduce the dimensionality of the sample input space. The basic idea of this process is as follows: assuming that the m physical variables involved have k dimensions, the physical variables are converted into m − k dimensionless π factors based on the π theorem; then, the π factors are further merged based on the knowledge of thermal variables and mechanics to constitute the sample input features of lower dimensionality, and the dimensionality-reduced dataset is utilized to train the ANN. The physical parameters (β_i and γ_i) in Figure 2 represent the input features after dimensionality reduction.

Figure 3 shows the framework for training the thermal and mechanical ANNs; it includes:

(1) Sample data obtained using TEP-FEM, that is, the maximum temperature and plastic strain of the heated surface corresponding to the forming parameters for different materials and plate thicknesses obtained;

(2) The forming parameters, converted into physical parameters β_i and γ_i based on dimensional analysis and domain knowledge;

(3) Training, validation, and evaluation of the artificial neural networks.

3.1. TEP-FEM for Sample Data Generation

The TEP-FE analysis of the IHMRF process was performed based on ABAQUS software 6.14. Due to the negligible influence of mechanical processes on thermal effects in IHMRF, a decoupled thermal–mechanical analysis scheme was adopted. However, the contribution of the transient temperature field to the stresses produced through thermal expansion was considered. The solution procedure consisted of two steps: Firstly, the temperature history of the induction heating was obtained by heat transfer analysis. Thermal radiation and convective heat transfer boundary conditions were considered in the heat transfer analysis. The surface heat source model was used to simulate the induced heat source. Then, the temperature history and mechanical action of the rollers were applied to analyze the mechanical response of the plate. In the mechanical analysis, the mechanical action of the rollers was modeled through equivalent displacement loads. To prevent rigid displacements, a spring constraint was used.

The same mesh, with a size of 10 × 10 mm, was used in the heat transfer and mechanical analyses. The ABAQUS element S4R was used for the mechanical analyses and the element DS4 was used for the heat transfer analyses. The Mises yield criterion was used to characterize the yielding behavior of the material. The temperature-dependent thermo-physical and mechanical properties were considered in the decoupled analysis. Since the main focus of the study is on the PIDD model, only a brief description of the TEP-FEM is provided here, but more information is available in reference [10].

3.2. Extraction of Physical Parameters

Table 1. Dimension of the forming parameters.

	Plastic Strain	Maximum Surface Temperature	Thermal Loading Parameters		Mechanical Loading Parameters	Geometric Parameter
Variable	[ε]	[Tmax]	[P]	[v]	[d]	[h]
Dimension	none	L0F0t0θ1	L1F1t−1θ0	L1F0t−1θ0	L1F0t0θ0	L1F0t0θ0
	Thermal physical properties		Mechanical properties
Variable	[Cp]	[k]	[E]	[α]	[σs]	[σT]
Dimension	L−2F1t0θ−1	L0F1t−1θ−1	L−2F1t0θ0	L0F0t0θ−1	L−2F1t0θ0	L−2F1t0θ0

displays the fundamental dimensions of the forming parameters involved in the IHMRF process. Using σ_T in Equation (1) as the variable representing the variation of yield stress with temperature,

$${\sigma }_T={\displaystyle \frac{{\displaystyle \underset{T_0}{\overset{T_{\max }}{\int }}{\sigma }_s\left(T\right)\mathrm{d}T}}{{\displaystyle \underset{T_0}{\overset{T_{\max }}{\int }}\mathrm{d}T}}}$$

(1)

where σ_T represents the average value of the material yield stress from room temperature to the processing temperature history, σ_s(T) represents the temperature-dependent yield stress, T₀ represents the room temperature, and T_max represents the maximum temperature of the heating surface.

The functional relationship describing the thermal process can be expressed as:

$$F\left(T_{\max },C_p,k,Q,v,h\right)=0$$

(2)

The functional relationship describing the deformation process can be expressed as:

$$G\left(\epsilon ,C_p,k,\alpha ,{\sigma }_s,E,{\sigma }_T,d,P,v,T_{\max },h\right)=0$$

(3)

As can be seen from Table 1, the thermal process under study involves 4 dimensions (length L, force F, time t, and temperature θ) and 6 physical variables. According to the π theorem, four basic variables need to be determined, and the remaining two variables can be combined with each of these four basic variables to form a dimensionless π factor. The basic variables are selected as follows:

• The power P and velocity v represent the magnitude of the heat source energy and are therefore naturally included as basic variables.
• The plate thickness h is an important geometrical parameter of the heated plate.
• The specific heat C_p is related to the ability of a material to absorb heat.

After determining the basic variables, the dimension matrix can be listed:

$$A=\left[\begin{array}{cccccc}1& 1& -2& 1& 0& 0\\ 1& 0& 1& 0& 1& 0\\ -1& -1& 0& 0& -1& 0\\ 0& 0& -1& 0& -1& 1\end{array}\right]$$

(4)

where rows 1 to 4 of matrix A are the four basic dimensions of length L, force F, time t, and temperature θ, in that order; columns 1 to 4 are the power-square values of the basic variables corresponding to the basic dimensions; column 5 is the power-square value of the thermal conductivity corresponding to the basic dimension; and column 6 is the power-square value of the maximum temperature corresponding to the basic dimension. After the row primitive transformation of the matrix, the π-factor of the heat transfer process can be obtained as:

$$\begin{array}{cc}{\pi }_{T1}={\displaystyle \frac{k}{C_{p}hv}}& {\pi }_{T2}={\displaystyle \frac{P}{C_{p}v{h}^2{T}_{\max }}}\end{array}$$

(5)

Similarly, for the mechanical process, the yield stress σ_s of the material, the thickness of the plate h, the maximum temperature of the heated surface T_max, and the velocity v are chosen as the basic variables. Then, the π factor of the mechanical process can be derived as:

$$\begin{array}{ccc}{\pi }_{m1}={\displaystyle \frac{{\sigma }_s}{E}}& {\pi }_{m2}=\alpha T_{\max }& {\pi }_{m3}={\displaystyle \frac{{\sigma }_K}{{\sigma }_s}}\\ {\pi }_{m4}={\displaystyle \frac{\lambda T_{\max }}{{\sigma }_{s}hv}}& {\pi }_{m5}={\displaystyle \frac{C_p{T}_{\max }}{{\sigma }_s}}& {\pi }_{m6}={\displaystyle \frac{d}{h}}\\ {\pi }_{m7}={\displaystyle \frac{P}{{\sigma }_s{h}^{2}v}}& & \end{array}$$

(6)

The heat transfer equation reveals the physical significance of the π_T1 factor in Equation (5) as a parameter associated with the heat source’s moving speed. Therefore, the π_T1 factor is used as the physical parameter β₁ for training the thermal ANN. In the π_T2 factor, which contains the output of the thermal ANN, the construction of the physical parameter β₂ related to the maximum temperature of the heated surface is considered:

$${\beta }_2={\displaystyle \frac{{\pi }_{T2}}{T_{\max }}}={\displaystyle \frac{P}{C_{p}v{h}^2}}$$

(7)

where the value of P/(vh²) is related to the maximum temperature of the heating surface. Moreover, according to the literature [12], the relationship between the π_T1 factor and the π_T2 factor is completely different in thin and thick plates. Therefore, the plate thickness h is used here as the physical parameter β₃ for training the thermal ANN.

Table 2. Input characteristics of the thermal ANN.

Physical Parameters	Equation	Physical Meaning
β1	k/(Cpvh)	Dimensionless velocity
β2	P/(Cpvh2)	Values related to the maximum temperature of the heating surface
β3	h	Plate thickness

displays the input features related to the thermal ANN.

In IHMRF, the thermal expansion and mechanical bending action are the two types of driving forces that cause the deformation of plates. Therefore, the inputs used to train the mechanical ANN should contain terms for the temperature, coefficient of thermal expansion, and amount of downward pressure. According to the literature [13], the temperature at which the material yields during the heating process is approximately equal to σ_s/(αE). The physical parameter γ₁ is constructed based on the π_m1 factor and π_m2 factor:

$${\gamma }_1={\displaystyle \frac{{\pi }_{m2}}{{\pi }_{m1}}}={\displaystyle \frac{T_{\max }\alpha E}{{\sigma }_s}}$$

(8)

The physical significance of the constructed physical parameter γ₁ is to characterize the magnitude of the heating-induced thermal expansion force relative to the material yield stress. The physical parameter γ₂ is constructed based on the π_m6 factor, the π_m8 factor, and the plate thickness h, considering the distribution of the temperature field in the plate thickness direction:

$${\gamma }_2={\displaystyle \frac{{\pi }_{m7}h}{{\pi }_{m5}}}={\displaystyle \frac{P}{C_{p}vh{T}_{\max }}}$$

(9)

The two parameters γ₁ and γ₂ together reflect the intrinsic effect of the temperature field. For the mechanical bending effect of the rollers, with reference to the slat beam theory, the forming depth can be characterized in the following dimensionless form:

$${\gamma }_3=2{\pi }_{m1}{\pi }_{m6}={\displaystyle \frac{2d{\sigma }_s}{Eh}}$$

(10)

On the other hand, the mechanical properties of the material, particularly the yield stress, are intrinsic factors that determine the magnitude of plastic deformation of the material. Therefore, we also use π_m1 and π_m3 in Equation (6) as input features for training the mechanical ANN. The input features related to the mechanical ANN are shown in

Table 3. Input characteristics of the mechanical ANN.

Physical Parameters	Equation	Physical Meaning
γ1	TmaxαE/σs	Magnitude of the thermal expansion force relative to the yield stress of the material
γ2	P/(CpvhTmax)	Values related to the temperature gradient in the direction of the plate thickness
γ3	2dσs/(Eh)	Dimensionless forming depth
γ4	σs/E	Dimensionless yield stress
γ5	σT/σs	Dimensionless thermal softening

.

3.3. Artificial Neural Network (ANN)

A feedforward neural network with backpropagation was used in this study. This is a supervised learning algorithm that maps a given set of inputs to outputs using the function f: Ω_j → Ω_o, where j is the number of dimensions for input and o is the number of dimensions for output. The purpose of training is to learn the function f using the sample dataset. The function f is found by minimizing the cost function L, which is a measure of how well the neural network performs with respect to the network predicted output f(x) and the target values y.

Usually, a neural network model can be divided into an input layer, the hidden layers, and an output layer. The input layer is composed of a set of neurons representing the input features {x|x₁, x₂, x₃, …, x_j }. The output layer provides the predicted output, f(x). Depending on the complexity of the problem, one or more hidden layers are designed between the input and output layers to transform and process the data. The data transformation from the input layer to the 1st hidden layer can be expressed as:

$$g_1=\phi \left(W_1^{T}x+b_1\right)$$

(11)

where g₁ is the output of the 1st hidden layer, W₁ is a linearly transformed weight matrix, b₁ is a bias vector, x is an input feature vector, and φ is a nonlinear activation function. Similar data transformations are performed between two subsequent neighboring layers (e.g., the rth hidden layer and the r + 1th hidden layer), but the weight matrix W_r, the bias vector b_r, and the activation function φ may change.

At the beginning of the first forward pass of the artificial neural network, the weights and biases are initialized, and the input vectors enter the network for initial prediction. The backpropagation algorithm computes the loss function and then iteratively adjusts the weights and biases to minimize it [14]. The backpropagation algorithm determines the impact of each weight and bias in the model on the predicted output values by computing the partial derivatives of the loss function relative to the weights and biases, as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial L}{\partial W_r}}={\displaystyle \frac{\partial L}{\partial S_r}}{\displaystyle \frac{\partial S_r}{\partial W_r}}\\ {\displaystyle \frac{\partial L}{\partial b_r}}={\displaystyle \frac{\partial L}{\partial S_r}}{\displaystyle \frac{\partial S_r}{\partial b_r}}\end{array}$$

(12)

where S_r is the output of the rth layer. The new weights and biases are computed using the gradient descent method:

$$\begin{array}{l}{W}_i^{*}=W_i-\eta {\displaystyle \frac{\partial L}{\partial W_i}}\\ b_i^{*}=b_i-\eta {\displaystyle \frac{\partial L}{\partial b_i}}\end{array}$$

(13)

The ANN models in this study are based on the Pytorch implementation. As shown in Figure 4, both the trained thermal ANN and mechanical ANN contain two hidden layers. The activation function between the input and hidden layers is a widely used Tanh function. Since the study is a regression problem, a linear activation function is used between the last hidden layer and the output layer. The loss function L is defined as the average variance of the predicted output value f(x) and the actual output value y. The loss function L is defined as the average variance of the predicted output value f(x) and the actual output value y:

$$L={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-f\left(x_i\right)\right)}^2}$$

(14)

where n is the number of samples fed into the network, y_i is the actual value, and f(x_i) is the predicted value. To avoid the suppression of small values by high values in the input features, all data are normalized in the range [−1, 1]. The normalized value x* for each input feature x is calculated as:

$$x^{*}={\displaystyle \frac{2\left(x-x_{\min }\right)}{x_{\max }-x_{\min }}}-1$$

(15)

where x_min is the minimum value of the input feature and x_max is the maximum value of the input feature. Before training, the sample dataset is divided into the training set (80%) and the testing set (20%). About 20% of the training set is further divided into a validation set. The training set is used for the training and learning of the ANN, and the validation set is used to examine the performance of the ANN during training. The test set is not involved in network training and is used to evaluate the predictive ability of the final trained ANN. The weights and biases of the neural network are initialized using the Kaiming uniform distribution, and the weights and biases are optimized using the Adams algorithm during the training process. The network divides the training data into multiple batches of 20 inputs during one training epoch.

The genetic algorithm optimally tunes the hyperparameters of the ANN model (number of neurons in the hidden layer and initial learning rate). The genetic algorithm is an intelligent stochastic global search algorithm inspired by natural evolution, which is less likely to fall into local minima than traditional search methods. This is because the genetic algorithm uses a population of individuals to explore all regions of the solution space. In the genetic algorithm optimization process, the loss function L of the model on the validation set is minimized as the optimization objective. Due to the strong randomness of single random sampling, a 5-fold cross-validation method is used to assess the performance of ANN on the validation set. That is, the loss function L is the evaluated value of the 5-fold cross-validation. The individual in this study has a set of parameters, including the size of the first hidden layer, the size of the second hidden layer, and the initial learning rate. The range of values for each parameter in the initial population is as follows:

• Size of the first hidden layer: $1\le n_1\le 20$
• Size of the second hidden layer: $1\le n_2\le 20$
• Initial learning rate: $0.001\le l_r\le 0.1$

4. Results and Discussion

4.1. Training the Thermal ANN and Mechanical ANN

A thermal ANN and a mechanical ANN make up the PIDD model, as explained in Section 3. Here, the thermal ANN and the mechanical ANN are trained first. The materials used for training include Q235, Q345, and EH40. These three materials are widely used in shipbuilding. The thermophysical and mechanical property parameters of the materials are referred to [10,15,16], as shown in Figure 5. In this study, for each material, TEP-FE calculations were performed for three plate thicknesses and a suitable range of machining parameters to obtain the corresponding maximum temperature and plastic strain at the heated surface. Details of the TEP-FEM used are given in reference [10].

Table 4. Design of the training samples.

Input Parameters	Levels
Thickness h (mm)	15, 25, and 35
Forming depth d (mm)	2, 3, 4, and 5
Power P (kW)	0–110
Velocity v (mm/s)	1, 2, 4, 6, 8, 13, 18, and 23
Material	Q235, Q345, and EH40

summarizes the plate thickness and machining parameters for each material, with a total of 650 samples. Out of the total set of samples, the training set is made up of about 520 and the test set about 130. The genetic algorithm optimizes the hyperparameters of the thermal and mechanical ANNs. Figure 6 displays the population average fitness curves and the best individual curves during evolution.

Table 5. Hyperparameters of the ANN.

Task	n1	n2	lr
Thermal ANN	8	10	0.043
$\mathrm{Mechanical} \mathrm{ANN} \mathrm{for} {\epsilon }_L^{in}$	14	11	0.081
$\mathrm{Mechanical} \mathrm{ANN} \mathrm{for} {\epsilon }_T^{in}$	15	13	0.046
$\mathrm{Mechanical} \mathrm{ANN} \mathrm{for} {\epsilon }_T^{out}$	15	13	0.075

displays the optimal hyperparameters of the ANN after about 11 generations of evolution. In Table 5, n₁ is the number of neurons in the first hidden layer, n₂ is the number of neurons in the second hidden layer, and lr is the initial learning rate. Optimized training takes about 40 min.

Figure 7 compares the prediction results of the PI-ANN model to those calculated by TEP-FEM. It can be seen that the prediction results of the training and test datasets are basically within ±10%.

Table 6. Performance metrics of the trained ANN on the sample data set.

Metrics	Tmax	${\mathit{\epsilon }}_{\mathit{L}}^{\mathit{i}\mathit{n}}$	${\mathit{\epsilon }}_{\mathit{T}}^{\mathit{i}\mathit{n}}$	${\mathit{\epsilon }}_{\mathit{T}}^{\mathit{o}\mathit{u}\mathit{t}}$
R2	99.9%	98.2%	98.8%	99.5%
RMSE	9.6	1.1 × 10−4	2.0 × 10−4	2.9 × 10−4

summarizes the correlation coefficients and root-mean-square errors of PI-ANN. According to Table 5, the thermal ANN and the mechanical ANN have sufficiently high accuracy. After training the thermal and mechanical ANNs, they can be used to form a PIDD relationship model for forming parameters and plastic strains.

4.2. Dependency on the Size of Training Datasets

In order to validate the performance of the proposed method for training ANN models with small sample sizes, the accuracy of training the ANN models directly with forming parameters as input features (the traditional method) and training ANN models with physical parameters as input features (the proposed method) is compared here. The widely used performance metric R² (correlation coefficient) in machine learning was employed to assess the accuracy of the model. This was done by training the ANN with data randomly selected from the sample with a ratio of N_train to the total number of samples, and then using the ANN to predict the samples that are not involved in the training and calculating their R².

Figure 8 shows the variation of the correlation coefficient R² with the number of samples for the mechanistic ANN trained by the two methods. In Figure 8, the black dots represent the ANN model trained by the traditional method (CO-ANN), and the blue triangular dots represent the ANN model trained by the proposed method (PI-ANN). It can be seen that the R² of CO-ANN was around 20% when trained using 10% of the total number of samples, while the R² of PI-ANN reached more than 80%. Furthermore, although CO-ANN’s R² shows an increasing trend as the number of samples increased, its growth rate gradually slowed down and never reached the accuracy level of PI-ANN. These results show that the proposed method can effectively alleviate the dependence of model training on the number of samples.

4.3. Performance of the PIDD Model on Unseen Plate Thickness

In this section, the PIDD model was constructed using the trained thermal and mechanical ANNs from Section 4.1 and the performance of the PIDD model was examined for plate thicknesses that did not occur during training (unseen thicknesses). The plastic strains corresponding to the machining parameters were calculated using TEP-FEM for plate thicknesses of 20 mm and 40 mm. Then, the plastic strains corresponding to the same machining parameters were predicted using the PIDD model and TEP-FEM, respectively. Figure 9 presents the plastic strains obtained by both methods. The consistency between the PIDD model and the TEP-FEM results is evident.

4.4. Performance of the PIDD Model on Unseen Materials

The performance of the PIDD model on materials not used for training (unseen materials) was investigated using the materials EH36 and HY80 as examples. The thermophysical and mechanical performance parameters of the materials are referenced in the literature [17,18], as shown in Figure 10. Among these, the material performance parameters of EH36 are within the range of the material performance parameters in training, while the yield stress of HY80 is outside the range of the training material parameters. Figure 11 compares the results of the TEP-FEM and the PIDD model.

As shown in Figure 11, for material EH36, the PIDD model obtained results consistent with TEP-FEM. The accuracy of the PIDD model decreased for material HY80. These results indicate that the PIDD model can learn the correspondence between forming parameters and the plastic strain from a small set of materials and can transfer the learned knowledge to unseen materials. When some of the performance parameters of the unseen materials are beyond the range of the performance parameters of the training materials, the accuracy of the PIDD model decreases. Therefore, in the practical application of the model, if the material performance parameters of the processed plate are out of the range of the known material performance parameters in the training process, it is necessary to use TEP-FEM to supplement the samples and retrain the model. However, due to the migration learning capability of the model, only a small number of additional samples need to be added to achieve a significant increase in forecasting accuracy. Figure 12 shows the prediction results of the retrained PIDD model after adding a small amount of data (approximately 20) for material HY80 to the sample. The output of the PIDD model significantly improves the agreement with the TEP-FEM.

4.5. Application: Machining Plates of Different Materials to the Same Shape

As mentioned earlier, the PIDD model has sufficiently high accuracy and generalization capabilities. The goal of establishing the model is to ascertain the necessary machining parameters for the forming process. This section further verifies the effectiveness of the proposed method through an engineering case involving the forming of a complex hull plate. Similar forming experiments and numerical simulations were performed for the location of the machining paths for hull plate forming given in reference [19]. This was done by keeping the location of the machining path and the required strain on the machining path constant, changing the material of the plate, and then utilizing the PIDD model to determine the machining parameters required to apply the same strain on a plate of a different material. The matching steps for processing parameters are as follows:

(1) The plastic strains of the target material and plate thickness for a range of appropriate processing parameters are obtained using the PIDD model to form a strain database;

(2) The Euclidean distance between the strain needed on each machining path and the strain in the database is calculated;

(3) The processing parameter corresponding to the smallest Euclidean distance is used as the output.

Figure 13 depicts the hull plate’s shape and machining path arrangement. Ships typically use this type of plate in the transition area between the bow and the parallel midbody. The geometrical dimensions of the plates are 2000 mm × 1000 mm × 20 mm, and the materials include Q235, Q345, and EH36. Virtual forming was carried out using TEP-FEM from the literature [10]. In addition, forming experiments for the material EH36 were also carried out using the experimental setup shown in Figure 14.

Table 7. Machining parameters selected based on the PIDD model.

Material	Power (kW)	Velocity (mm/s)	Forming Depth (mm)
Q235	11.5	1.0	2
Q345	21.0	1.8	0
EH36	24.4	1.8	2

summarizes the selected machining parameters based on the PIDD model for plates of different materials.

Due to the high temperature and mechanical loading conditions present in the IHMRF process, it is difficult to measure plastic strains directly. Therefore, it was verified here by comparing the deflection deformation. Figure 15 shows the deflection curves of plates of different materials after machining. Despite a slight deflection difference, the plates of different materials essentially converged to the same shape. These results indicate that the machining parameters selected based on the PIDD model can produce essentially the same plastic strain in plates of different materials. Complex factors like the crossed effect [20] and the edge effect [21] affect strain during loading, which explains the difference in deflection. Here, the matching process for processing parameters fails to account for these complex factors. Therefore, future work will explore how to incorporate the complex factors affecting strain into the processing parameter matching process. Overall, the PIDD model enables the matching of appropriate processing parameters to the strain needed for forming, laying the groundwork for the development of intelligent equipment.

5. Conclusions

This study proposes a physically informed data-driven (PIDD) modeling approach to model the relationship between forming parameters and plastic strains. The model was trained based on the sample data obtained from TEP-FEM. The performance of the model on unseen plate thicknesses and materials was examined. Experimental and numerical simulations of steel plates made of different material properties machined to the same shape further verify the feasibility of the proposed method. The main conclusions are as follows:

(1) The proposed method is able to obtain an accuracy comparable to that of TEP-FEM under small sample conditions. Several physical parameters were obtained by dimensional analysis and domain knowledge. These physical parameters reveal the inherent thermal and mechanical relationships between machining parameters, plate thickness, material thermophysical property parameters, and mechanical property parameters.

(2) The data-driven model trained based on the proposed method has sufficiently high accuracy and generalization capability. Based on the knowledge learned from the materials and plate thicknesses in training, the model also predicts the plastic strains of the unseen materials and plate thicknesses with high accuracy.

(3) The model trained using the proposed method has the ability to transfer knowledge. Even if the properties of the unseen material are out of range during training, it is only necessary to add a small amount (approximately 20) of additional sample data points before retraining the model to obtain the same accuracy as TEP-FEM. Expanding a small number of samples continuously improves the model, enhancing its reliability in practical applications.