1. Introduction
The automotive industry witnessed a significant shift with the emergence of giga-casting, a revolutionary approach pioneered by Tesla [
1]. This technology utilises colossal, high-pressure die-casting (HPDC) machines, aptly named Giga Presses, to manufacture large, single-piece chassis components from aluminium [
2,
3]. Prior to giga-casting, car bodies were traditionally constructed by welding together numerous smaller parts. By eliminating the need for extensive welding processes, giga-casting offers the possibility of significantly reduced production times and costs [
4]. Additionally, using single-piece components can enhance structural integrity and improve vehicle performance metrics like weight and, potentially, fuel efficiency [
5]. As the electric vehicle market continues its rapid expansion, giga-casting’s ability to optimise production for lighter, more efficient car bodies positions it as a potentially disruptive force [
6].
Despite the numerous advantages of giga-casting, the process has its challenges. One significant hurdle lies in the inherent tendency for these large, thin-walled components to warp and distort during solidification or subsequent heat treatment processes [
7]. Several factors contribute to this phenomenon. The immense size of the castings creates uneven cooling rates across the part, leading to thermal stresses that can cause warping [
8]. Additionally, the high pressures employed during casting and the potential for non-uniform heating during heat treatment can induce residual stresses within the material [
9]. These combined effects can result in components that deviate from their intended geometry.
Straightening these distortions is crucial for several reasons. Firstly, deviations from the designed shape can compromise the structural integrity of the component [
10]. Uneven load distribution and stress concentrations can arise due to warping, potentially leading to safety hazards. Secondly, distorted components can create challenges during downstream assembly processes. Misaligned components can require additional processing and can even lead to production line disruptions [
11]. Finally, deviations from the intended form can negatively impact the aesthetic qualities of the final product.
Traditional straightening methods, which involve mechanical pressing, often rely on custom-designed die blocks that match specific sections of the component. These die blocks are expensive to manufacture and can only be used for a particular component geometry [
12]. This dependence on custom tooling renders traditional methods incapable of addressing the general complex geometries encountered in giga-castings. The sheer variety of shapes and sizes within these large components would necessitate an extensive and costly collection of die blocks, making traditional straightening impractical for large-scale production. In addition, most of the straightening process has been traditionally carried out by workers’ manual handwork [
13], and a systematic approach for the straightening process has yet to be proposed.
Given the limitations of traditional techniques and the growing need for efficient straightening solutions for versatile geometries, this paper proposes a novel approach utilising a straightening machine with multiple straightening pins. This innovative method offers several advantages over conventional methods. Unlike die blocks, which require a custom design for each specific geometry, a straightening machine with multiple straightening pins can readily adapt to a wide range of complex shapes encountered in giga-castings. By strategically pressing the component with each straightening pin, this method allows the distortion to be removed after straightening. It enables the machine to address distortions in specific areas of the giga-casting without introducing new ones elsewhere, making it a highly versatile solution.
Machine learning has emerged as a powerful tool for tackling complex challenges in various mechanical engineering domains [
14]. This study utilised two prominent algorithms, deep neural network (DNN) and XGBoost, to address the intricate problem of giga-casting straightening. With their ability to learn complex relationships from vast datasets [
15], deep neural networks are well suited for capturing the non-linear interactions of geometric variations in giga-castings. XGBoost, a robust gradient boosting algorithm, excels in handling high-dimensional data [
16] and modelling intricate feature interactions [
17], making it suitable for predicting stiffness variations within the component. Compared to traditional methods that rely on simpler models or that lack the ability to adapt to complex geometries, incorporating DNN and XGBoost can enhance the straightening performance, leading to more accurate and efficient correction of distortions in these large, complex components.
Numerous researchers have successfully adopted machine learning algorithms for mechanical engineering problems. Salb et al. [
18] proposed a method for enhancing IoT network security by combining CNNs for feature extraction with XGBoost for intrusion detection. They further introduced a modified Reptile Search algorithm for hyperparameter optimisation, leading to a more robust defence against emerging threats in IoT security. Park et al. [
19] proposed a method for designing patterns in tubular robots that utilises deep neural networks to extract key features and metaheuristics optimisation to achieve desired mechanical properties, surpassing previous designs in performance and efficiency. Ref. [
20] developed a system for the intelligent fault diagnosis of rotary machinery using a convolutional neural network (CNN) with automatic hyperparameter optimisation via Bayesian optimisation, achieving accurate fault detection without manual network configuration. Lin et al. [
21] combined finite element simulation to generate data on self-piercing riveted joints and utilised the XGBoost algorithm to analyse it, achieving highly accurate predictions of their cross-tension strength with an impressive error rate of only 7.6%, which offered a significant advancement in predicting joint performance, potentially replacing traditional testing methods. Hashemi et al. [
22] utilised machine learning to create surrogate finite element models. The surrogate finite element models could efficiently predict the dynamic response of mechanical systems, significantly reducing the time and resources needed compared to traditional full-scale finite element analysis.
This paper introduces the concept of a “straightening stroke decision algorithm” to achieve precise straightening and overcome the challenges associated with complex geometries. This innovative algorithm plays a crucial role in determining the optimal straightening stroke for each individual pin within a straightening machine. The algorithm utilises the following two key components: The first component is a polynomial model representing the global stiffness of the giga-casting component. This model provides a baseline understanding of the component’s overall stiffness to the deformation. The second component is a machine learning model that captures the stiffness changes arising from the current geometry of the specific component being straightened. This model accounts for the deviations of the stiffness of the current geometry from the global stiffness. By combining these two models, the straightening stroke decision algorithm calculates the optimal stroke length required for each straightening pin. The following sections will delve deeper into the details of the straightening stroke decision algorithm, explaining the underlying models and their role in achieving efficient and accurate straightening of giga-cast components.
This paper presents the results of comprehensive numerical experiments to evaluate the effectiveness of the proposed straightening approach. Finite element analyses were employed to simulate the straightening process for various target component geometries. These target components included a simple box, a centre spine, and a side member, each representing different levels of geometric complexity encountered in giga-castings.
The straightening performance was meticulously assessed by comparing the following four different approaches: The first and second approaches are the straightening algorithms with the deep neural network (DNN) and the XGBoost [
23] models. These models utilise machine learning algorithms to capture the intricate stiffness variations due to the geometry deviations. The third approach is the straightening algorithm with only the polynomial model for the global stiffness, which means that the model does not consider the effect of the geometry deviations. The last approach is the naive L-BFGS-B method. It serves as a benchmark representing a traditional optimisation technique commonly used for solving engineering problems. By comparing the straightening performance of these approaches across various component geometries, this paper aims to demonstrate the effectiveness of the proposed straightening stroke decision algorithm.
4. Results and Discussion
The numerical experiments were conducted for the three target components in
Figure 6 as mentioned in the previous section. For each target component, 1500 straightening strokes were simulated, and
and the machine learning models for
were trained as shown in the left side of
Figure 7. Ten randomly distorted geometries, which are not included in the training database, were generated for each component to evaluate the performance of the proposed straightening stroke decision algorithm.
The performances of the proposed algorithm with the machine learning model for and without it (which means only was considered) were compared. The DNN model was selected for the performance comparison with the algorithm with only.
The maximum distortion measurements of the simple box, after the straightening stroke decided using only
and the surrogate model with the DNN model, are depicted in
Figure 10. For all ten randomly distorted shapes, the surrogate model with the DNN model showed better performance than the results that considered
only as the maximum distortion measurements of the straightened geometries were low with the surrogate model for all shapes. The average of the maximum distortion measurements after the straightening stroke was 0.0935 mm for the model with
only and 0.0751 mm for the surrogate model with the DNN model. The distortion measurement distributions of the simple box Shape no. 10 after the straightening strokes are presented on the right side of
Figure 10. The distortion measurement distributions were considerably different since the model that considered
only could not take account of the change in the stiffness due to the geometry. In addition, this made the straightening stroke determined with only
less effective than the one determined by the surrogate model with the DNN model.
Figure 11 shows the maximum distortion measurements of the centre spine after the straightening stroke determined with only
and the surrogate model with the DNN model. The average of the maximum distortion measurements after the straightening stroke was 0.1156 mm for the model with
only and 0.0909 mm for the surrogate model with the DNN model. Similar to the simple box cases, the straightening strokes determined with the DNN model were more effective for all ten shapes. The magnitude of the distortion measurements for the straightened geometries was higher than the simple box since the centre spine has a complex geometry, and the number of straightening pins to determine is greater than for the simple box. On the right side of
Figure 11, the distortion measurement distributions of the centre spine Shape no. 8 after the straightening are presented. Similar to the simple box, the overall distributions were different from each other, and the DNN model showed better performance than the model with only
.
Similar results were derived for the side member as shown in
Figure 12. The average of the maximum distortion measurements after the straightening stroke was 0.1550 mm for the model with
only and 0.1192 mm for the surrogate model with the DNN model. The side member showed the highest magnitude of the distortion measurements for the straightened geometries since the geometry’s size and complexity were the most significant among the target geometries. The distortion measurement distributions of the side member Shape no. 1 are presented on the right side of
Figure 12. As shown in the simple box and the centre spine cases, the distortion measurement distributions were significantly different from each other since there is a difference in whether the current geometry was considered or not.
The straightening results derived from the model with only and from the surrogate model with the DNN model showed that the surrogate model was superior to the model with only . Considering the current geometry’s effect on the component’s stiffness is crucial since the surrogate model provided more satisfactory straightening results from all target geometries. The surrogate model showed that the average of the maximum distortion measurements after the straightening stroke improved by 21.4% for all target geometries compared with the model with only . However, it should be noted that establishing the global stiffness matrix is also an essential part of the surrogate model since the model is also based on global stiffness. An improper global stiffness matrix will make the algorithm decide the straightening stroke inappropriately, whether the effect of the current geometry is considered or not.
The straightening stroke determined from the naive L-BFGS-B mentioned in
Section 3.2 could be treated as a local optimum solution since the naive L-BFGS-B does not utilise any surrogate model and attempts to find the optimal point from the finite element analysis results directly. In this study, the straightening results obtained from the naive L-BFGS-B were considered global optimum solutions because the search space was too broad to conduct a global optimisation process without a surrogate model. The performances of the surrogate models with the DNN model and with the XGBoost model were evaluated by comparing their results with those of the naive L-BFGS-B.
Figure 13 shows the maximum distortion measurements of the simple box after the straightening stroke determined using the naive L-BFGS-B and the surrogate models with the DNN model and with the XGBoost model. The average of the maximum distortion measurements after the straightening stroke was 0.0609 mm for the naive L-BFGS-B and 0.0763 mm for the surrogate model with the XGBoost model. The DNN model showed slightly better performance than the XGBoost model based on the straightening results of the simple box. The distortion measurement distributions of the simple box Shape no. 8 are presented on the right side of
Figure 13. The surrogate models showed similar distortion measurement distributions, whilst the distribution of the naive L-BFGS-B was dissimilar from that of the surrogate models.
The maximum distortion measurements of the centre spine, after the straightening stroke decided using the naive L-BFGS-B and the surrogate models with the DNN model and with the XGBoost model, are shown in
Figure 14. The average of the maximum distortion measurements after the straightening stroke was 0.0720 mm for the naive L-BFGS-B and 0.0911 mm for the surrogate model with the XGBoost model. The DNN model performance was slightly better than that of the XGBoost model. However, the difference in the performance between the surrogate models was negligible. The distortion measurement distributions of the centre spine Shape no. 1 are shown on the right side of
Figure 14. Similar to the simple box results, the distortion measurement distributions of the DNN and the XGBoost model were similar to each other. In contrast, the naive L-BFGS-B showed a different pattern of distribution.
The side member’s maximum distortion measurements after the straightening stroke decided using the naive L-BFGS-B and the surrogate models with the DNN model and with the XGBoost model are shown in
Figure 15. The average of the maximum distortion measurements after the straightening stroke was 0.0886 mm for the naive L-BFGS-B and 0.1205 mm for the surrogate model with the XGBoost model. The magnitude of the distortion measurements was the largest among the target geometries for the same reason discussed above. Like the centre spine, the DNN model showed slightly better performance than the XGBoost model. The distortion measurement distributions of the side member Shape no. 2 are depicted on the right side of
Figure 15. The straightening behaviour was similar between the DNN and XGBoost models, whereas the naive L-BFGS-B showed different behaviour.
The maximum distortion measurements after the straightening stroke for each method and model are organised in
Appendix A. Student’s
t-tests were conducted to evaluate the efficiency of each proposed model for the straightening process. The naive L-BFGS-B was excluded from the tests since it was used to make reference results for each target geometry because the naive L-BFGS-B can be considered to determine the optimum straightening stroke. In the statistical analysis, the significance level was set to be
and the standard deviations were assumed to be different.
Table 2 shows the
p-values of the two-tailed tests with the null hypothesis of the same mean for the surrogate models with the DNN and XGBoost. In
Table 2,
is the null hypothesis, and
and
are the means of the maximum distortion measurements after the straightening stroke for the surrogate models with the DNN and XGBoost, respectively. For all geometries, the
p-values are greater than the significance level
, which means that the null hypothesis should not be rejected and the means of the maximum distortion measurements were statistically identical.
Table 3 shows the
p-values of the left-tailed tests with the null hypothesis of
:
, where
and
are the means of the maximum distortion measurements after the strengthening stroke for the surrogate model with the DNN and the model with the global stiffness only, respectively. The
p-values are less than the significance level for all geometries, which means the null hypothesis should be rejected. It shows that the straightening performances were statistically better with the DNN model since the mean is smaller for the surrogate model with the DNN model.
The surrogate model with global stiffness could only be used for straightening. However, the straightening performance can be improved when the DNN and XGBoost models are utilised in the surrogate model with the global stiffness matrix, and the statistical analysis results support this. In addition, the maximum distortion measurements after the straightening stroke were reduced by 26.6% on average when the DNN or XGBoost model was adopted in the surrogate model.
Compared to the straightening results from naive L-BFGS-B, the maximum distortion measurements after the straightening stroke from DNN and XGBoost models differed by 28.0% and 29.3%, respectively, on average. The surrogate models with the DNN model and with the XGBoost model showed similar performances for the target geometries, while the DNN model showed slightly better performance on average. On the other hand, it cannot be said that the DNN model is always preferable to the XGBoost model since the DNN model did not show better straightening performance for all cases. In addition, the statistical analysis results indicate that the performances of the DNN and XGBoost models were statistically identical. The straightening results showed that the DNN and XGBoost models are adequate to model , the change in the component’s stiffness. In addition, the surrogate models could decide the straightening strokes so that the maximum distortion after the straightening becomes 0.02% of the largest dimension of each target geometry.
The distortion measurement distributions from the surrogate models were similar for all target components, while the distribution from the naive L-BFGS-B was different. This shows that the DNN and XGBoost models estimated similarly, whilst the naive L-BFGS-B estimated it directly from the finite element analyses. The estimated stiffness matrix can differ from the real stiffness matrix of the current geometry. However, the naive L-BFGS-B is computationally expensive, which means that it cannot be utilised to operate the straightening process. On the contrary, the surrogate model can provide admissible solutions for the straightening process with an adequately fitted and a trained machine learning model.