This paper presents a novel machine learning method to assess the thermal performance of conformal cooling systems. Creating a sole neural network to predict the time-series thermal behavior of conformal cooling systems is challenging. By employing a combination of a non-linear regression model and a neural network, the thermal performance of conformal cooling systems can be predicted and assessed precisely. Consequently, the determination of the optimal parameters for conformal cooling design can be attained by leveraging the thermal performance assessment results. Figure 3 describes the systematic methodology of building the machine learning method for assessing the thermal performance of conformal cooling systems. It contains three main components: data collection, non-linear regression models, and neural networks, along with an introduction to the conformal cooling system parameters. Detailed explanations of each component are elaborated in the subsequent sections.
Conformal cooling system parameters
To comprehensively evaluate the thermal performance of conformal cooling molding systems, it is imperative to consider two essential sets of parameters and variables: conformal cooling design parameters and molding system parameters. The three key conformal cooling design parameters (D, DtP, and TL) play a significant role in impacting cooling performance. Additionally, molding system variables, including product data, coolant data, and mold insert data, also have a direct influence on cooling efficiency. Product data includes three key parameters (product volume, melt temperature, and product material) that define the total heat energy, melt thermal characteristics, and solidification conductivity and coefficients within the molding product. The two key coolant data are coolant temperature and inlet pressure, both exerting significant influence on cooling efficiency. The temperature and material of mold inserts are two factors influencing heat transmission rates within the metal inserts.
Data collection
To develop robust machine learning models, a substantial volume of sample data is needed for training and validation purposes. In this study, SolidWorks’ thermal simulation (FEA) and flow simulation (CFD) were employed to gather extensive cooling performance sample data from various case studies as shown in Fig. 4a. Figure 4b shows the samples of thermal records in multiple locations and conformal cooling system data. These records serve as sample datasets for the training and validation of machine learning models. Additionally, a real conformal cooling mold was integrated with thermal sensors to gather real-time thermal data during molding processes for validating the regression model and neural networks.
To compile a representative sample dataset for training regression models and neural networks, a varied set of design parameters and molding system variables is incorporated, as outlined in Table.1. These include various case studies, an appropriate range of conformal cooling parameters, diverse product materials (including ABS, nylon, PE, rubber, etc.), coolant configuration data, and mold insert materials.
Non-linear regression models
The objective of building a non-linear regression model is to identify the most effective algorithm for accurately approximating the cooling performance behavior, and extract the features, represented as coefficients, from historical cooling performance data. These coefficients reflect the relationship between cooling time and melt temperature of conformal cooling systems. The inputs for non-linear regression models are the recorded cooling performance data, with cooling time (t) and melt temperature (Temp) as the key variables. The outputs are the determined non-linear regression model and the coefficients (features) derived from it. Subsequently, these coefficients along with their associated conformal cooling system parameters serve as the inputs of neural networks in the subsequent phase. In this study, three non-linear regression algorithms were trained and evaluated for their effectiveness: Polynomial Regression, Logarithmic Regression and Exponential Regression.
Algorithm 1: Polynomial regression
Polynomial regression is chosen due to its simplicity in modeling non-linear relationships. The polynomial equation is defined as follows:
$$Temp = \mathop \sum \limits_{i = 0}^{d} c_{i} t^{i}$$
where, d is the degree; c is polynomial coefficient; t is cooling time; and Temp is the temperature.
As depicted in Fig. 5a, the fitting accuracy and errors of polynomial regression are sensitive to the polynomial degree. The fitting error decreases as the model degree rises. However, the fitting error still remains large (mae = 3.08) even the model degree increases to 5. It is also found that the higher-degree polynomial models introduce extra oscillations, leading to poor fitting performance. As a result, the polynomial regression model fails to precisely capture the thermal performance characteristics of conformal cooling systems.
Algorithm 2: Logarithmic regression
Logarithmic regression method is appropriate for scientific and technical scenarios, where the sample data exhibits a logarithmic pattern. The logarithmic equation is expressed as follows:
$$Temp = \mathop \sum \limits_{i = 0}^{d} c_{i} \log \left( t \right)^{i}$$
where, d is the degree; c is logarithmic coefficient; t is cooling time; and Temp is the temperature.
As shown in Fig. 5b, the logarithmic regression model accurately captures the relationship between Temp and t from the thermal performance data. The logarithmic model with a degree of 4 yields satisfactory fitting performance (mae = 0.46), as further increasing the model degree does not notably enhance fitting performance. Therefore, the logarithmic regression model with a degree of 4 precisely meets the goal for fitting the cooling thermal performance.
Algorithm 3: Exponential regression
The exponential regression method is applicable for scientific and technical situations, where the behavior of the sample data is exponential in nature. The exponential equation is specified as follows:
$$Temp = \mathop \sum \limits_{i = 0}^{d} e^{{c_{i} t^{i} }} or \, Temp = \mathop \prod \limits_{i = 0}^{n} e^{{c_{i} t^{i} }} ,$$
where, d is the degree; c is exponential coefficient; t is cooling time; and Temp is the temperature.
As illustrated in Fig. 5c, the fitting accuracy and errors of the exponential regression model are responsive to the degree of exponential model. A higher model degree is expected to minimize the fitting error. In contrast to the logarithmic regression model, the exponential regression model demonstrates relatively poorer fitting performance.
In conclusion, the logarithmic regression model with a degree of 4 is chosen as the preferred non-linear regression model for this study, as it exhibits the most effective fitting performance with the collected sample datasets, and successfully accomplishes the specified objectives. By employing the logarithmic regression model to process all the historical cooling performance data, the results were consolidated into a unified datasheet as shown in Fig. 6, This dataset includes the 5 coefficients (w0, w1, w2, w3, w4) generated from non-linear logarithmic regression along with their associated conformal cooling system parameters.
Neural networks
The goal of developing a neural network is to establish the correlation between the thermal performance graph, represented by non-linear regression coefficients, and the specified conformal cooling molding system. As depicted in Fig. 6, the inputs for Neural Networks comprise the combined data of two groups: the conformal cooling system parameters as the training input features, and the coefficients generated from the logarithmic regression model as the target features. The 10 conformal cooling system parameters include 3 conformal cooling design parameters (D, DtP, TL), 3 product variables, 2 coolant configuration variables, and 2 insert variables. The 5 target features correspond to the coefficients of the logarithmic regression with a degree of 4. The output is a trained neural network to predict the thermal performance of the given 10 conformal cooling system parameters. With the given conformal cooling system parameters, the trained neural network can predict the output coefficients of the non-linear regression model. Subsequently, the cooling performance graph can be generated precisely using the non-linear regression model. Designers can easily pinpoint the optimal conformal cooling design parameters (D, DtP, TL) with a predetermined target melt cooled temperature and cooling cycle time, relying on the thermal performance results obtained through the non-linear regression model and neural network, eliminating the need for laborious CAE/FEA/CFD analysis.
In this study, three neural network architectures were developed and evaluated: Single ANN, Multiple NNs, and Branch NN.
Method 1: Single ANN
A single Artificial Neural Network (ANN) was first developed as shown in Fig. 7a. It comprises 10 input neurons, 5 hidden layers and 5 output neurons. The training results revealed that the Single ANN was effectively trained and converged within 2000 epochs, yielding low loss and error (val_mae: 3.392 ~ 4.036) with the collected sample datasets. In Fig. 7d, a comparison result for a test example was also presented between the actual cooling performance data and the cooling performance curve generated from the logarithmic regression model using the coefficients predicted by the trained neural network. The results showed that the trained Single ANN can accurately predict the thermal performance of conformal cooling systems.
Method 2: Multiple NNs
Considering the disparate value ranges of the five coefficients observed in Fig. 6, where w0 ranges in [−2.4, 2.5], w1 in [−13, 20], w2 in [−40, 30], w3 in [−111, −9.9], and w4 in [83, 380], a multiple NNs architecture (Fig. 7b) was introduced to learn each coefficient with its dedicated neural network. This strategy is expected to enhance the performance for individual coefficient. Five distinct neural networks (NN1, NN2, NN3, NN4, and NN5) were devised to independently train the coefficients (w0, w1, w2, w3, w4) respectively. Figure 7b displayed the training curves for these distinct neural networks concerning their respective coefficients. All the models exhibit successful convergence for the datasets. However, the training errors showed noticeable deviations, such as val_mae of NN5: 34.349. One possible reason could be that different capacities and configurations of neural networks should be designed and dedicated to different coefficients with distinct behaviors and value ranges. Figure 7d presented the comparison result between the actual collected cooling performance data and the cooling performance curve generated from the logarithmic regression model with coefficients predicted by multiple NNs. The results indicated that the performance of multiple NNs falls short when compared to the Single ANN in terms of the fitting accuracy and validation errors while replicating the actual cooling graph.
Method 3: Branch NN
The primary goal of the machine learning models is to optimize the three key conformal cooling design parameters (D, DtP, TL). Consequently, these three variables are considered more critical compared to the other seven molding system parameters. The idea was sparked by introducing a specialized branch NN to distinguish between these two sets of parameters in initial layers. This innovative approach has the potential to empower the neural network by allowing it to assign distinct weights separately to the design parameters and other molding system parameters, thereby enhancing overall effectiveness. As depicted in Fig. 7c, the three design parameters are interconnected within one branch of neurons, while the remaining seven parameters constitute the other branch. Following three distinct layers, these two branches are merged into a dense layer, ultimately producing the five coefficients as a unified output.
Figure 7c showed the training curve of the branch NN, which exhibits an accurate convergence with very low error (val_mae = 3.057 ~ 3.568). A comparative result was presented between the actual cooling performance data and the cooling performance curve generated from the logarithmic regression model with the coefficients derived from the branch NN in Fig. 7d. The results demonstrated that the branch NN can accurately simulate the cooling performance graph for conformal cooling systems.
It was observed that the performance of the branch NN surpasses that of the Single ANN and multiple NNs. The branch NN is therefore chosen as the preferred neural network for this research due to its superior fitting and validation performance in replicating the actual cooling behavior for the collected sample datasets, effectively achieving the specified objectives. Regarding the other two models, the multiple NNs and Single ANN, additional refinements to their architectures, including neurons, activation functions, and layers, may enhance their overall performance.
Evaluation with real molding cycle data
In this study, the presented machine learning models were evaluated using a real molding cycle data. Figure 8 displayed a partial set of the thermal data recorded through thermal sensors from actual molding cycles, in which a single segment of molding cycle data was extracted for model validation. The comparative results were presented between the actual molding cycle data and the predicted outcomes generated from the branch NN and the logarithmic regression model. The result affirmed the effectiveness of the presented methods in this case study despite the fitting accuracy not reaching perfect. It is anticipated that greater accuracy and performance can be attained by expanding the size of the collected sample data and further optimizing the neural networks.