Quality optimization of liquid silicon lenses based on sequential approximation optimization and radial basis function networks

Quality optimization of liquid silicon lenses based on sequential approximation optimization and radial basis function networks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Quality optimization of liquid silicon lenses based on sequential approximation optimization and radial basis function networks Hanjui Chang, Shuzhou Lu, Yue Sun, Yuntao Lan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4898818/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Feb, 2025 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract In order to obtain competitive advantages in terms of product cost and quality, this study proposes a multi-objective optimization method based on sequential approximation optimization and radial basis function networks. In the optimization process, the radial basis function network replaces the finite element reanalysis and allows the construction of an approximate functional relationship between quality and process conditions. In this study, injection molding of objects was simulated and analyzed while varying the filling time, melt temperature, mold temperature, curing pressure, and curing time schemes to better understand the aspects affecting the optimization process. Using the automobile optical liquid silicone lens as an example, the Pareto boundary is used to determine the residual stress and volume shrinkage, as well as the deviation function and radial basis function network. Because numerical simulations are time-consuming, the radial basis function sequential approximation optimization method is applied. The product had the highest quality when the filling time was 1.57s, the melt temperature was 27.18°C, the mold temperature was 150°C, the curing time was 20.02s, and the curing pressure was 28.79 MPa, according to numerical results. Experiments were carried out to test the efficacy of the proposed approach. Nondestructive analysis is used to determine the target values (residual stress and volume shrinkage). Because nondestructive testing does not damage materials, workpieces, or buildings, the inspection rate of items can be quite high following nondestructive testing. Furthermore, numerical and experimental data demonstrate that the technique effectively reduces residual stress and volume shrinkage. Physical sciences/Optics and photonics/Optical physics/Micro optics Physical sciences/Optics and photonics/Optical physics/Nonlinear optics Physical sciences/Physics/Fluid dynamics Physical sciences/Physics/Optical physics Physical sciences/Physics/Statistical physics thermodynamics and nonlinear dynamics Physical sciences/Mathematics and computing/Scientific data liquid optical silicone lenses multi-objective optimization destructively measure sequential approximate optimization Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1 Introduction High brightness LEDs of the most recent generation can reach extremely high temperatures, which can put pressure on traditional polymers like PMMI, PMMA, and PC. Over a temperature range of -40°C to 200°C, silicones provide great temperature resistance, strong resistance to heat aging and high chemical resistance, purity, transparency, and stable mechanical qualities. Silicones significantly lessen the yellowing effect, making them perfect for usage in outdoor or severe environments. The silicone material for silicone lenses is a highly transparent two-component liquid silicone rubber with excellent injection molding process characteristics: providing fast cure, low viscosity, and high flow. These properties significantly increase productivity by reducing injection time and total cycle time. No additional protection is required, such as in the case of light poles or high grids, and no gaskets for IP insulation are required: no yellowing or heat shrinkage cracking in UV or high and low temperature environments, specifically for street, high pole, and high grill lighting applications. As an isotropic material, silicone offers a high degree of flexibility for silicone lenses that can easily fit into any hazardous and harsh environment. Due to its elastomeric nature, silicone offers ideal compensation for mechanical tolerances in construction, thereby contributing to a precise fit in the final application by virtue of its exceptional functionality. Given its high productivity, cost-effectiveness, lightweight properties, and superior ability to accommodate intricate geometries, the manufacturing process that involves injecting plastic into molds has become an essential method for producing plastic items. The six-stage process involving clamping, filling, packing, cooling, opening, and ejection constitutes a critical cycle in the production of high-quality plastic components, with each stage contributing significantly to achieving the desired outcome. The material, component and mold design, as well as the manufacturing process factors, all affect the quality of plastic injection molded parts. [ 1 ] Product flaws like warpage, shrinkage, sink marks, and residual strains are caused by a range of production-related causes. Hence, a lot of study has been undertaken on the best design of plastic injection molding process parameters. 2 Literature and review Given the significant attention this longstanding issue has garnered from researchers, recent years have seen a proliferation of advanced optimization theories and sophisticated techniques in design aimed at addressing the problem through enhanced understanding and resolution [ 2 ]. The conventional continuous modal testing methods have been replaced by CAE simulation approaches [ 3 ]. Integrating it with the idea of optimization design and using it to optimize the conditions of the injection molding process enhances the quality of the plastic parts while also speeding up the molding process [ 4 ]. R. Joseph Bensingh et al. [ 5 ] investigated the effects of injection molding process parameters such as mold surface temperature, melt temperature, injection time, filling volume percentage V/P switching, filling pressure, and filling time on the volume shrinkage deformation of double aspheric lenses. The scientists used a mixture of computer numerical simulation and optimization approaches to determine the ideal molding parameters that resulted in the least variance of volume shrinkage deformation. Kuo-Ming Tsai et al. developed a lens shape accuracy prediction model using the artificial neural network (ANN) and response surface method in another study (RSM). Following experimental validation and accuracy comparison, the results showed that the injection molding process window obtained using the ANN for determining the cooling time and filling time had a high-order irregular shape, whereas the injection molding process window obtained using the RSM had an oblique ellipse shape. Furthermore, the findings demonstrated that the ANN model outperformed the RSM model in terms of lens shape accuracy [ 6 ]. Currently, the industry mainly relies on the technical experience of designers to find a better solution to the quality problems of plastic products during the injection molding process by repeatedly testing and repairing the mold, and repeatedly adjusting the parameters of the injection molding process, but these methods not only increase the design cost and prolong the production cycle, but also make it difficult to obtain the best design results. Therefore, the optimization of injection molding process parameters plays a crucial role in the quality of products. A approach to examine the impact of cavity pressure distribution on part quality using a neural network was put forth by Jinsu Gim et al. in 2021 [ 7 ]. The relationship between the mold internal pressure and the weight of the workpiece was determined by evaluating the mold internal pressure profile, and the fluctuation rule of the mold internal pressure was transformed into the quality index of the mold. The influence degree of each element on the injection process may be clarified by the analysis of numerous factors, and the injection process can be optimized in accordance with the influence degree of each component on the injection process. In 2020, Ming-Shyan Huang and colleagues [ 8 ] proposed a unique method for predicting the geometry of manufactured parts using a multilayer perceptron (MLP) neural network model with quality indicators. Four factors were used as inputs to the model: the first-stage pressure retention index, pressure integral index, residual pressure drop index, and peak pressure index. Meanwhile, the output of the model was the geometry of the portion in question. The MLP model was trained and used to aid in learning and prediction. The MLP model correctly predicted the geometric width of the pieces, with over 92% accuracy in both training and testing outcomes. As a result, this technology has been demonstrated to be very successful and reliable, showing the potential for wider adoption in the manufacturing business. In 2022, Han-Jui Chang et al. [ 9 ] used a non-explicit genetic algorithm with kriging response surface analysis to optimize the multi-objective design of UAV shells, considering various process parameters as model variables. After obtaining the warpage value, die stamping index, and mathematical relationship, a multi-objective genetic algorithm optimization program is employed to replace the experimental analysis. The model was optimized using the approach, and the model was then tested. The four critical points and mold index had a clear optimization effect, and the error was just 8.48%, which was within the acceptable range for production. Han-Jui Chang [ 10 ] made the suggestion in 2021 that highlighting the product qualities is beneficial in the assessment of identifiability. However it is difficult to establish the reference value due to the influence of many causes, suggesting that one variable can be the dependent variable that is most affected. The information provided gives us a starting point for talking about how these phenomena occur. To create an independent injection molding quality control system that can be recognized and assessed, a novel technique for defect knowledge re-study and virtual measurement-based implementation is also suggested. Machines can now gather and store data throughout the production cycle in the era of information and technology, and with the help of big data management, this data can be processed to create efficient troubleshooting techniques. In conclusion, a number of researchers have automated and intelligently predicted the quality of injection molded parts using artificial intelligence techniques. [ 11 ] The link between the process parameters and the molding quality objective is intricate, time-varying, nonlinear, and tightly connected [ 12 ]. Thus, identifying an acceptable set of optimal process parameters is very challenging. According to the conventional approach to optimization, the injection molded continually modifies the process settings in light of experience to find the ideal set of parameters. However, because the conventional approach is unable to produce the global ideal parameters, the test time, cost, and raw material waste are all significantly increased [ 13 , 14 ]. Hence, for high-precision and high-efficiency manufacturing, it is crucial to swiftly identify the ideal process parameters for microinjection molding on a scientific foundation. Advances in software computing have enabled more efficient determination of process parameters in recent years through the use of numerous methodologies such as evolutionary algorithms, fuzzy systems, expert systems, and artificial neural networks. Process parameter optimization can be divided into two categories: static process parameter optimization and dynamic optimization based on knowledge or previous cases. The former category uses an agent model to get globally optimal results, whereas the latter achieves optimization results gradually through interactions. The majority of research in this area has been on static process parameter optimization utilizing current methodologies, which typically entail three steps: collecting background data through experimental design, building an agent model, and using optimization algorithms. Other models for modeling the link between process parameters and quality indicators include response surface methodology (RSM) [ 15 ], kriging, artificial neural network (ANN) [ 16 ], and support vector regression (SVR). Jian Zhao et al. [ 17 ] tackled the multi-objective optimization problem of injection molding plastic part quality by using a two-stage approach that combined an efficient global optimization algorithm and a genetic algorithm. For example, Gang Xu et al. [ 18 ] proposed a comprehensive approach that integrated multiple optimization methods to achieve excellent performance. Both approaches demonstrated effective solutions in terms of quality and efficiency. With the continuous development and popularity of computer-aided engineering, more and more fitting methods are being applied to optimization systems in various fields. These systems can be used to analyze and optimize various processes by building mathematical models to improve the quality and efficiency of products. Among the fitting methods, RBF neural networks are a commonly used method with the advantages of fast convergence and the ability to handle high-dimensional data. In addition, optimization systems based on fitting methods also include methods based on response surface methods and genetic algorithms. These methods can be applied in different scenarios, such as optimization of process parameters, design optimization, product performance prediction, etc. With these optimization systems, we can better understand and control complex processes and provide effective support for improving product quality and efficiency. Researchers examined the Kriging method [ 19 ], radial basis function neural networks [ 20 ], and sequential approximation optimization (SAO) [ 21 , 22 ] to create high precision response surfaces and identify the global optimal solution. Satoshi Kitayama et al., for example, optimized plastic injection molding process parameters and holding pressure distribution using radial basis function neural networks and sequential approximation optimization (SAO) to reduce warpage and cycle time. Numerical studies and experimental findings confirmed the usefulness of this strategy in reducing warpage and cycle time. Gao and Wang constructed the approximation function of warpage versus process parameters using Kriging approach. Then, SAO was used to minimize warpage in the cell phone casing. However, the various parameters involved in the injection molding process are interdependent. An increase in melt temperature can result in a reduction in viscosity and shear stress, but a longer cooling time may also be required. Higher injection pressure and shorter injection time may decrease temperature differentials between different production areas but may also increase the temperature at the flow front. An increase in holding time can lead to a reduction in sink marks, but an increase in flash points. As the number of factors considered increases, so does the number of optimization targets. Therefore, multi-objective optimization is employed to balance these conflicting parameters. This study aims to perform multi-objective optimization of the holding curve and process parameters in order to minimize residual stress and volume shrinkage. This research focuses on a complete optimization technique for determining the best process parameters for minimizing residual stress and shrinkage. It is roughly separated into the following aspects, as indicated in Fig. 1 : (1) carrying out experiments on residual stress and volume shrinkage in the lens molding process; (2) developing an RBF neural network model; (3) the effect of process parameters on molding quality; and (4) multi-objective optimization of minimum residual stress and minimum volume shrinkage and experimental verification. 3 Material and methodology 3.1 Liquid Silicone Rubber Due to its high clarity and refractive index, LSR materials are well suited for optical applications such as LED lighting, camera lenses, and automotive headlight lenses. In addition, the excellent rheological properties and low viscosity of LSR materials make them suitable to produce high-precision, complex-shaped lenses through injection molding. The advantages of LSR materials over traditional glass or plastic lenses are their light weight, high strength and excellent optical properties that provide higher light transmission and better dispersion characteristics, resulting in improved quality and efficiency of optical components. the benefits of LSR materials include high precision, durability, biocompatibility and high productivity, making them suitable for a wide range of applications such as medical devices, prosthetics and other products requiring high precision and durability. In this experiment, the LSR material with the model number of LSR-1 from the manufacturer CAE is used. Generally, the volume of LSR changes significantly with pressure and temperature. To calculate the shrinkage or warpage of the lens after vulcanization, it is necessary to describe the pressure-volume-temperature (PVT) relationship, as shown in Fig. 2 . Viscosity is an indicator of LSR to flow resistance and depends on temperature, shear rate and pressure, as shown in Fig. 3 . In a complete injection molding cycle, it generally goes through the stages of melt-fill-recharge-shrink-pressure-hold-cooling-eject. As we all know, the filling stage of injection molding, VP switching point and pressure-holding process must be accurately controlled to get high quality and high precision products. To avoid premature solidification of the material during the filling process, the filling rate of the plastic must be controlled. Likewise, the pressure applied to the material during the holding phase must be controlled to compensate for the shrinkage that occurs during cooling and to avoid material spillage. To ensure high quality, accuracy and repeatability in weight and dimensions of the molded product, the optimal pressure profile during the holding phase should be the one that varies isotropically during the cooling of the material. During the injection molding process, the polymer material is heated into the molten state and injected into the mold cavity under high pressure, undergoing a process from high temperature and high pressure to rapid cooling and pressure drop, followed by a change from the molten state to the solid state, while the various physical parameters of the polymer material undergo a series of drastic changes, all of which are highly dependent on T, P and V. In particular, the V of the polymer determines the final weight and size of the product. The final molded product's performance and quality are specifically determined by the polymer's V: if the density is too low, it will result in insufficient strength; if the density is not uniform, it will produce internal residual stresses, causing warpage and deformation, etc. 3.2 Sequential Approximation Optimization Sequential Approximation Optimization (SAO) is an optimization method usually used to solve complex multivariate optimization problems. SAO continuously optimizes the objective function to find the optimal solution through repeated iterations and stepwise approximation. This method has a wide range of applications in many fields, including engineering, machine learning, and optimization algorithms. In SAO, the value of the objective function is usually calculated based on the current parameter configuration, and the parameters are adjusted according to the calculation results. This process is repeated until a set termination condition is reached or converges to some local optimal solution. The core idea of SAO is to approximate the global optimal solution through repeated approximate optimization. This step-by-step optimization approach makes SAO robust and reliable in dealing with complex optimization problems. Overall, Sequential Approximation Optimization (SAO) is a commonly used optimization method for solving various complex multivariate optimization problems. The schematic representation of the sequential steps involved in the implementation of the SAO optimization method is visually depicted in Fig. 4 , showcasing the systematic flow and intricate details of this approach. (Step 1) Initialize with number of sequences = 1. (Step 2) Confirm the optimization objective, constraints, and design variable ranges. (Step 3) A Latin hypercube design (LHD) is used to generate some initial sampling points. (Step 4) Perform numerical simulation using Moldex3D software to numerically calculate the objective function for the sampling points. (Step 5) Approximate all the functions as RBF networks. Here, the approximated objective function is expressed as $$\:{f}_{i}\left(x\right)\le\:0\:\:i=\text{1,2},\dots\:,k$$ (Step 6) If convergence is satisfied, terminate the SAO algorithm and output the result. Otherwise, the number of sequences is added by one and the Pareto optimal solution will be added as a new sampling point and return to step 3. 3.3 Radial Basis Function neural network Owing to their outstanding performance, RBF (Radial Basis Function) neural networks have been extensively employed in a diverse range of fields, including but not limited to pattern recognition, function approximation, system identification, and control. One of the major advantages of RBF neural networks is their fast-learning speed, which is achieved through a simple structure that requires fewer iterations to converge than other types of neural networks. Another advantage of RBF neural networks is their excellent approximation performance, which enables them to approximate complex functions with high accuracy. In addition, RBF neural networks have good generalization ability, which allows them to perform well on new, unseen data. A radial basis function is utilized in the hidden layer of RBF neural networks to transfer the input space into a higher dimensional space where linear separation is achievable. This enables the network to record nonlinear interactions between inputs and outputs, which is frequently required for difficult problem solving. The Gaussian function, which has a bell-shaped curve and is defined by a center and a width parameter, is the most often used radial basis function. Overall, the RBF neural network is an useful tool for handling difficult problems with nonlinear input-output relationships. Its straightforward structure, quick learning speed, high approximation performance, and generalization capabilities make it a popular choice for a wide range of applications in industry and academics. The topology of a radial-basis neural network with numerous inputs and multiple outputs is depicted in Fig. 5 below. The complete interpolation method requires the interpolation function to pass through each sample point, i.e., \(\:F\left({X}^{n}\right)={d}^{n}\) . There is a total of k sample points. The RBF method is to choose k basis functions, each corresponding to a training data, each of which is of the form \(\:\phi\:(\parallel\:X-{X}^{k}\parallel\:)\) , which is called radial basis function since the distance is radially homogeneous. \(\:\parallel\:X-{X}^{k}\parallel\:\) denotes the modulus of the difference vector, or called the 2-parametric number. The interpolation function based on the radial basis function is $$\:F\left(x\right)=\sum\:_{k=1}^{n}{\omega\:}_{k}{\phi\:}_{k}\parallel\:X-{X}^{k}\parallel\:={\omega\:}_{1}{\phi\:}_{1}\parallel\:X-{X}^{1}\parallel\:+\:{\omega\:}_{2}{\phi\:}_{2}\parallel\:X-{X}^{2}\parallel\:+\dots\:{\omega\:}_{n}{\phi\:}_{n}\parallel\:X-{X}^{n}\parallel\:$$ The following Gaussian kernel is generally used as the basis function: \(\:hi\left(x\right)=\text{e}\text{x}\text{p}(-\frac{{\left(x-xi\right)}^{T}(x-xi)}{{r}^{2}j})\) where xi is the i-th sampling point and ri is the width of the i-th basis function. The response yi is computed at the sampling point xi. The learning of RBF networks is usually done by solving. In Fig. 6 , the horizontal coordinate is the design variable dimension, and the vertical coordinate is the solution vector dimension. For each design variable, the high precision numerical solution corresponds to a unique solution vector, as shown by the solid line in the figure. At each time when a new design variable *X needs to be calculated, the data of certain sample points, as shown by the solid dots in the figure, have been obtained in the previous period. Based on these sample points, the variation of the solution vector with the design variables can be predicted by approximate modeling, and the next sample point is sampled based on this prediction, as shown by the star-shaped point in the figure. The results in Fig. 6 demonstrate that, when the iteration format is left unchanged and the optimization is carried out, the distance between the predicted value and the converged value is significantly smaller than the distance between the initial value and the converged value, indicating that using the approximate model acceleration significantly decreases the total amount of iterative steps of the high-precision simulation. 4 Case study A company's selection of liquid silicone lenses for a car lamp is shown in Fig. 7 . The object has a maximum flesh thickness of approximately 22.483 mm, a minimum flesh thickness of approximately 0.101 mm, and a volume of approximately 53,972.35 mm 3 . We model it using modeling software, and the model that is created will be used in the test's mold flow analysis. The essential parameters linked to injection molding allows users to set in this inquiry are shown in Table 1 . Based on the literature analysis, five process factors were chosen and experimentally analyzed: melt temperature (°C), mold temperature (°C), filling time (s), maturation time (s), and maturation pressure (MPa), all of which are substantially connected with the density of residual stress values. Table 1 Experimental optimization parameters for injection molding of liquid silicone lenses for automotive lights Parameters Unit Range Melt temperature ℃ 20 ~ 30 Mold temperature ℃ 150 ~ 170 Cooling temperature ℃ 30 Cooling time s 20 Filling time s 1.8 ~ 2.2 Curing time s 15 ~ 20 Curing pressure MPa 15 ~ 40 Among them, the residual stress detection methods for injection molded parts can be basically categorized as lossy testing and nondestructive testing. The principle of lossy testing is to destroy the original structure of the product, break the original stress equilibrium state to make the product appear deformation or appearance changes, so as to calculate the size and distribution of residual stress. The common methods of lossy stress detection include drilling, layer peeling, solvent method, etc. Non-destructive testing is the use of instrumentation to detect changes in the physical properties of the product due to the presence of stress, and thus calculate the stress distribution. The common polarized stress test is one in which, according to the law of stress optics, the refractive index of a transparent plastic material changes when the material is subjected to stress. This is reflected under the stress polarization equipment by the color distribution presented on the product. Through the color distribution state, it is possible to simply observe the residual stress distribution on the product. It is characterized by a simple method, easy operation and inexpensive equipment, and is a widely used nondestructive stress detection method. The disadvantage is that the results are generally measured qualitatively and cannot be calibrated in quantitative terms. Table 2 Input-output parameters (Experimental data) in this study. Group Melt temperature (℃) Filling time (s) Curing time (s) Curing pressure (MPa) Mold temperature (℃) AverageVon-Mises thermal stress (Mpa) Volume Shrinkage(%) 1 1.76 23.17 19.06 20.34 154.52 21.27 4.97 2 2.44 29.77 15.48 33.60 153.70 21.32 3.75 3 1.89 22.17 15.23 26.63 150.43 24.42 4.18 4 1.93 29.06 19.29 36.12 152.52 18.33 3.74 5 2.04 20.73 16.50 29.07 165.83 22.97 4.05 6 2.16 21.23 16.90 25.09 163.56 22.76 4.42 7 2.21 26.14 16.60 19.18 164.19 23.01 4.96 8 2.30 23.97 17.56 30.76 168.79 21.02 4.04 9 1.98 28.45 18.26 24.64 159.56 20.34 4.63 10 2.38 24.13 17.21 17.52 155.85 22.64 5.14 11 1.75 27.09 17.34 15.65 169.87 22.99 5.34 12 2.11 21.76 15.76 22.09 161.62 24.26 4.60 13 2.34 20.33 19.65 37.34 166.18 19.12 3.59 14 1.81 25.75 17.85 23.08 157.83 21.65 4.69 15 1.60 28.83 16.23 28.18 158.62 22.28 4.19 16 1.56 27.84 18.01 16.73 151.72 22.18 5.27 17 2.07 22.71 19.82 39.22 156.88 18.53 3.46 18 2.47 26.93 18.77 32.43 160.10 19.16 4.00 19 1.66 24.89 15.69 33.89 167.20 22.99 3.66 20 1.52 25.26 18.59 37.79 162.22 19.69 3.52 Table 2 demonstrates the injection molding experimental data, i.e., the effect of variations of five factors on residual stress and volume shrinkage. These data contain a large amount of feature and label information that can be used to train and validate the performance of the model. The data need to be pre-processed and cleaned to ensure the accuracy and consistency of the data before proceeding to modeling, and finally they will be used for the building of RBF networks for LSR lens injection molding. The pre-processed and cleaned data will serve as the foundation for constructing RBF networks for LSR lens injection molding, which constitutes a crucial step in our experimental process. A multilayer radial basis neural network with three network topologies (i.e., input, hidden, and output) was used to map the injection molding process of liquid silicone rubber lenses. The mapping process was performed using MATLAB 2020 software and a neural network toolbox. The hidden layer neurons use Gaussian activation functions, while the output layer neurons use pure linear functions. The input layer of the neural network consists of five neurons corresponding to the number of independent variables, while the output layer has two neurons matching the number of dependent variables. Figure 8 shows the multilayer LSR neural network developed in the MATLAB 2020 environment, which was used to predict process parameters and evaluate the quality of the molded LSR lenses. 5 Result and discussion The energy used in mass production cannot be overlooked, despite the fact that the cost of energy consumption during injection molding is very modest. Up to 30 billion kWh of energy are used annually by the global injection molding industry, which accounts for 10% of all energy used globally and generates a significant amount of carbon emissions. Thus, it is essential to maximize energy efficiency to ensure product quality. The main piece of machinery used in the injection molding process is the injection molding machine. Using high-tech tools like servo motors with high precision and energy-efficient injection molding machines can drastically save energy usage. Using low-energy LSR materials can also significantly reduce the amount of energy needed to heat plastic in the injection molding machine, leading to a more environmentally friendly and energy-efficient manufacturing process. Consequently, it is possible to increase energy efficiency and reduce emissions in the injection molding process while maintaining the quality of the final product by upgrading equipment and employing high-quality materials. When a substance undergoes cooling from a state of high temperature and high pressure to a state of low temperature and low pressure, its volume changes. This change is known as volume shrinkage, which is defined as the percentage difference in volume before and after cooling. Volume shrinkage is positive when the substance undergoes a reduction in volume and negative when the substance expands due to excessive holding pressure. The shrinkage rate of the entire plastic part differs from that of its profile, leading to the formation of internal residual stresses that mimic the effects of external forces. The distribution of residual stress is closely related to the optical properties of the product, and the presence of residual stress can significantly impact its optical performance. Areas of the product with higher residual stress tend to exhibit poorer optical performance than areas with lower residual stress. If the volume shrinkage of a product is uneven, it can compromise not only the dimensional accuracy of the product but also indirectly impact its optical performance. This is due to the potential irregularities in the product's shape caused by the uneven shrinkage, which may result in deviations on the surface of optical components and ultimately affect their optical performance. LSR material is characterized by low shrinkage and warpage, which makes it an ideal material for applications where dimensional accuracy and optical performance are critical. The volume change of a product before and after V/P switching is a reliable indicator of the injection molding process's carbon footprint. Failure to adjust the process parameters correctly can result in higher shrinkage and a subsequent increase in the carbon footprint of the product. To gain a better understanding of the interaction between different factors, response surface plots were employed to visualize the effect of each factor on the residual stress in the plastic part. In this study, five influential factors were considered, including melt temperature, maturation pressure, maturation time, filling time, and mold temperature. Figure 9 displays a 3D surface that demonstrates how different factors influence the residual stress in the plastic part. Figure 9 (a) shows the interaction between melt temperature and maturation pressure on the residual stress values. The results reveal that the residual stress value of the lens is at its minimum at a melt temperature of 25–30°C and a maturation pressure of approximately 40 MPa. Similarly, Fig. 9 (b) illustrates the relationship between maturation time and maturation pressure on the residual stress values. The results demonstrate that the residual stress values decrease as the maturation time increases at a constant maturation pressure. Figure 9 (c) shows the effect of filling time and maturation pressure on the residual stresses, where the effect of filling time is comparatively minor compared to the significant impact of maturation pressure on the residual stresses. Furthermore, Fig. 9 (d) displays the response surface plots of filling time and maturation time on the residual stress values. The results show that the residual stress values of the lenses gradually decrease with increasing curing time. In Fig. 9 (e), the interaction between melt temperature and maturation time on the residual stress values is demonstrated. The results indicate that the residual stress values of the lenses are at their minimum when the maturation time is 18–20 seconds, and the melt temperature is higher. Finally, Fig. 9 (f) presents the effect of melt temperature and mold temperature on residual stresses. The results show that the lenses have the smallest residual stress values when the mold temperature is 150°C and the melt temperature is approximately 30°C. Overall, these response surface plots provide valuable insight into the effects of different factors on residual stress values in plastic parts. Table 3 Ⅰ, Ⅱ and Ⅲ point injection-molding parameter results. Item Filling time (s) Melt temperature (℃) Curing pressure (MPa) Curing time (s) Mold temperature (℃) AverageVon-Mises thermal stress (Mpa) Volume Shrinkage(%) Ⅰ 1.94 26.45 33.97 19.99 150.04 18.42 2.78 Ⅱ 2.23 29.20 39.34 19.49 150.12 17.72 3.75 Ⅲ 1.57 27.18 28.79 20.02 150.00 16.72 4.14 To optimize these parameters and improve the efficiency and quality of injection molding, the optimal Latin hypercube sampling method was used to obtain samples in this study. We selected five input parameters as the input layer of the model, namely melt temperature, mold temperature, holding time, holding pressure and filling time, while a radial basis (RBF) neural network model was constructed with minimum volume shrinkage and minimum residual stress as the output layer. The model has strong nonlinear approximation capability and good accuracy and can effectively perform prediction and optimization with a large amount of data. Through numerical calculations, we determined the Pareto boundary between residual stress and volume shrinkage to obtain the best combination of parameters. As shown in Fig. 10 , we compared three different points on the Pareto boundary and found that point I had a larger value of residual stress and a smaller volume shrinkage, while point II had more average values of residual stress and volume shrinkage. The distribution of stress values at point III was improved and was similar to points I and II. By analyzing the results in Table 3 , we conclude that by constructing the RBF neural network model, the best combination of parameters can be obtained, thus reducing the cost of production trial and error and saving production time and material. It is important to note that there must be some trade-offs between the optimization objectives. In practice, optimizing one objective may lead to the sacrifice of another, so the best balance needs to be found. In injection molding, the balance between the two objectives, minimum residual stress and minimum volume shrinkage, requires careful trade-offs and considerations. In actual production, choosing the best injection molding parameters can not only improve production efficiency and quality, but also significantly reduce production costs. 6 Conclusion Based on polymer injection molding theory, this paper aims to investigate the molding process parameters of optical liquid silicone injection molded products, focusing on their residual stress and volume shrinkage, to improve the quality of the injection molded products of headlight lenses. After analysis, the main parameters that affect the product quality include filling time, melting temperature, curing time, curing pressure and mold temperature. Combined with the mold flow analysis software Moldex-3D and intelligent optimization algorithms, the optimal process parameters for optical plastic lenses were studied in depth, aiming to achieve the dual target optimization of minimum residual stress and optimal volume shrinkage. Usually, numerical simulations of injection molding are intensive and can be optimized using radial basis function (RBF) networks and SAO optimization methods. The research results are as follows: A multi-objective optimization of the maturation pressure distribution and process parameters was carried out to reduce residual stresses and density. The ripening pressure distribution, which varies during the ripening stage, was used, and optimized. In addition to the ripening pressure distribution, melt temperature, injection time, holding pressure, and holding time were used as process parameters for optimization. the intensity of the PIM numerical simulation was high, so the sequential approximation optimization method with radial basis functions was used. The Pareto boundary between residual stress and density was determined by numerical calculations. In order to verify the validity of the numerical calculation results, experiments were conducted. The experimental results are in good agreement with the numerical calculation results. The effectiveness of the method to reduce the residual stresses and density was verified by numerical calculations and experimental results. When it comes to optimization of injection molding process parameters, the sequential approximation optimization method has many significant advantages. First, the sequential approximate optimization method can find the optimal solution more quickly and efficiently than the traditional trial-and-error or empirical methods. Second, the method can obtain sufficient data with fewer trials, thus reducing the cost and time of the trials. In addition, the sequential approximate optimization method can optimize multiple parameters simultaneously, thus achieving diversified process parameter optimization. Finally, the method can be updated and adjusted to suit the needs of process parameter optimization as the optimization process is continuously performed. We propose a new optimization approach for RBF neural networks that combines the concepts of sequential approximation optimization with multi-objective optimization. The original multi-objective optimization problem is simplified into a single-objective optimization problem using this method, making the optimization process more efficient and effective. A series of experiments were carried out to examine the feasibility and effectiveness of the suggested strategy. The results show that the suggested optimization method considerably enhances plastic part production by lowering residual stress values and volume shrinkage by 27.4% and 10.3%, respectively. Additionally, the suggested method minimizes the time required to establish the ideal process parameters, resulting in a more efficient production process. The proposed optimization method can be applied to a wide range of industries that utilize plastic injection molding, as it offers a powerful tool for improving the quality and productivity of plastic parts. Moreover, the method can also be extended to other optimization problems in manufacturing, providing a valuable tool for optimizing complex systems and processes. Declarations Author Contributions: Conceptualization, H.C.; Data curation, Y.L.; Methodology, S.L.; Project administration, Y.S.; Writing—original draft, H. C. All authors have read and agreed to the published version of the manuscript. Funding: Not applicable. Data Availability Statement: The authors declare that the data supporting the results of this study are available in the paper. If any raw data files in other formats are required, they can be obtained from the corresponding author upon reasonable request. Acknowledgments: This research was funded by the 2023 Guangdong Province Science and Technology Special Fund Project—the Guangdong Taiwan Normal University Excellent Project and Technical Support by Xuying Biomedicine Co., Ltd., and Software Support by CoreTech System Co., Ltd., which are gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest. References Chen, W.-C.; Nguyen, M.-H.; Chiu, W.-H.; Chen, T.-N.; Tai, P.-H., Optimization of the plastic injection molding process using the Taguchi method, RSM, and hybrid GA-PSO. The International Journal of Advanced Manufacturing Technology 2015, 83, (9-12), 1873-1886. Lin, W. C.; Fan, F. Y.; Huang, C. F.; Shen, Y. K.; Wang, H., Analysis of the Warpage Phenomenon of Micro-Sized Parts with Precision Injection Molding by Experiment, Numerical Simulation, and Grey Theory. Polymers (Basel) 2022, 14, (9). Huang, C.-T.; Xu, R.-T.; Chen, P.-H.; Jong, W.-R.; Chen, S.-C., Investigation on the machine calibration effect on the optimization through design of experiments (DOE) in injection molding parts. Polymer Testing 2020, 90. Huang, H. Y.; Fan, F. Y.; Lin, W. C.; Huang, C. F.; Shen, Y. K.; Lin, Y.; Ruslin, M., Optimal Processing Parameters of Transmission Parts of a Flapping-Wing Micro-Aerial Vehicle Using Precision Injection Molding. Polymers (Basel) 2022, 14, (7). Bensingh, R. J.; Boopathy, S. R.; Jebaraj, C., Minimization of variation in volumetric shrinkage and deflection on injection molding of Bi-aspheric lens using numerical simulation. Journal of Mechanical Science and Technology 2016, 30, (11), 5143-5152. Tsai, K.-M.; Luo, H.-J., Comparison of injection molding process windows for plastic lens established by artificial neural network and response surface methodology. The International Journal of Advanced Manufacturing Technology 2014, 77, (9-12), 1599-1611. Gim, J.; Rhee, B., Novel Analysis Methodology of Cavity Pressure Profiles in Injection-Molding Processes Using Interpretation of Machine Learning Model. Polymers (Basel) 2021, 13, (19). Ke, K. C.; Huang, M. S., Quality Prediction for Injection Molding by Using a Multilayer Perceptron Neural Network. Polymers (Basel) 2020, 12, (8). Chang, H.; Zhang, G.; Sun, Y.; Lu, S., Non-Dominant Genetic Algorithm for Multi-Objective Optimization Design of Unmanned Aerial Vehicle Shell Process. Polymers (Basel) 2022, 14, (14). Chang, H.-J.; Zhang, G.-Y.; Su, Z.-M.; Mao, Z.-F., Process Prediction for Compound Screws by Using Virtual Measurement and Recognizable Performance Evaluation. Applied Sciences 2021, 11, (4). Chang, H.; Su, Z.; Lu, S.; Zhang, G., Intelligent Predicting of Product Quality of Injection Molding Recycled Materials Based on Tie-Bar Elongation. Polymers (Basel) 2022, 14, (4). Alvarado-Iniesta, A.; Cuate, O.; Schütze, O., Multi-objective and many objective design of plastic injection molding process. The International Journal of Advanced Manufacturing Technology 2019, 102, (9-12), 3165-3180. Hriberšek, M.; Kulovec, S., Preliminary study of void influence on polyamide 66 spur gears durability. Journal of Polymer Research 2022, 29, (6). Lee, J.; Yang, D.; Yoon, K.; Kim, J., Effects of Input Parameter Range on the Accuracy of Artificial Neural Network Prediction for the Injection Molding Process. Polymers (Basel) 2022, 14, (9). Miza, A. T. N. A.; Shayfull, Z.; Noriman, N. Z.; Sazli, S. M.; Hidayah, M. H. N.; Norshahira, R., Optimization of warpage on plastic injection molding part using response surface methodology (RSM) and particle swarm optimization (PSO). 2018. Everett, S. E.; Dubay, R., A sub-space artificial neural network for mold cooling in injection molding. Expert Systems with Applications 2017, 79, 358-371. Zhao, J.; Cheng, G.; Ruan, S.; Li, Z., Multi-objective optimization design of injection molding process parameters based on the improved efficient global optimization algorithm and non-dominated sorting-based genetic algorithm. The International Journal of Advanced Manufacturing Technology 2015, 78, (9-12), 1813-1826. Xu, G.; Yang, Z., Multiobjective optimization of process parameters for plastic injection molding via soft computing and grey correlation analysis. The International Journal of Advanced Manufacturing Technology 2014, 78, (1-4), 525-536. Liu, J.; Chen, X.; Lin, Z.; Diao, S., Multiobjective Optimization of Injection Molding Process Parameters for the Precision Manufacturing of Plastic Optical Lens. Mathematical Problems in Engineering 2017, 2017, 1-13. Kitayama, S.; Yokoyama, M.; Takano, M.; Aiba, S., Multi-objective optimization of variable packing pressure profile and process parameters in plastic injection molding for minimizing warpage and cycle time. The International Journal of Advanced Manufacturing Technology 2017, 92, (9-12), 3991-3999. Kitayama, S.; Hashimoto, S.; Takano, M.; Yamazaki, Y.; Kubo, Y.; Aiba, S., Multi-objective optimization for minimizing weldline and cycle time using variable injection velocity and variable pressure profile in plastic injection molding. The International Journal of Advanced Manufacturing Technology 2020, 107, (7-8), 3351-3361. Chang, H.; Zhang, G.; Sun, Y.; Lu, S., Using Sequence-Approximation Optimization and Radial-Basis-Function Network for Brake-Pedal Multi-Target Warping and Cooling. Polymers (Basel) 2022, 14, (13). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Feb, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 07 Oct, 2024 Reviews received at journal 28 Sep, 2024 Reviewers agreed at journal 16 Sep, 2024 Reviews received at journal 16 Sep, 2024 Reviewers agreed at journal 16 Sep, 2024 Reviewers agreed at journal 16 Sep, 2024 Reviewers invited by journal 16 Sep, 2024 Editor assigned by journal 16 Sep, 2024 Editor invited by journal 22 Aug, 2024 Submission checks completed at journal 20 Aug, 2024 First submitted to journal 12 Aug, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4898818","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":354909631,"identity":"ff0cfd12-1665-4189-9a7d-8014f2aa6918","order_by":0,"name":"Hanjui 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1","display":"","copyAsset":false,"role":"figure","size":245729,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eOptimization method for liquid silica optical lens experiments.\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/5c45e7680c845868fca9d336.png"},{"id":64685824,"identity":"fddbcaa1-2dca-44c1-870b-a051d5af8b83","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":47939,"visible":true,"origin":"","legend":"\u003cp\u003ePVT curves of LSR materials for headlight lenses\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/fc40c3d0073e4c820dbdeef4.png"},{"id":64685825,"identity":"e9685d59-ca7c-4de9-ab3f-8b84ffe15950","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":37572,"visible":true,"origin":"","legend":"\u003cp\u003eViscosity curves of LSR materials used in headlight lenses.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/aeb49d91e8939c0a929d844c.png"},{"id":64686245,"identity":"148cae57-2ac4-4bf7-b2fc-775f7d67e332","added_by":"auto","created_at":"2024-09-17 14:56:05","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":52813,"visible":true,"origin":"","legend":"\u003cp\u003eSAO optimization algorithm flowchart\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/89e97271e730df817941be8b.png"},{"id":64685830,"identity":"e191f3ad-2ba2-4266-8776-f885ee9beb64","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":76361,"visible":true,"origin":"","legend":"\u003cp\u003ethe multi-input and multi-output structure of radial basis neural network\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/a9609b1410d41d67e0f8e98e.png"},{"id":64685831,"identity":"ad770822-291d-4a0b-88bf-86e2c1619904","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":58828,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic representation of the approximate model solving process\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/e3284cb9b783484575ba5de2.png"},{"id":64685833,"identity":"5a38c587-b33f-4699-af3e-4401e56e3042","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":136834,"visible":true,"origin":"","legend":"\u003cp\u003ePhysical and model images of an optical silicone lens array\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/6dc2bf3ab9f7856ec08928d8.png"},{"id":64685826,"identity":"9e99b7b2-27fe-4a69-a4ea-e7f0926f7f32","added_by":"auto","created_at":"2024-09-17 14:48:05","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":32289,"visible":true,"origin":"","legend":"\u003cp\u003eRadial-based neural network for LSR lens injection molding process mapping.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/d522d8c3af70e3eaf3537dd2.png"},{"id":64686247,"identity":"7502d8fd-a173-4dbe-928e-1728fe2f8477","added_by":"auto","created_at":"2024-09-17 14:56:05","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":685086,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of the average Von-Mises thermal stress of the lens under different conditions\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/cb217d847a955ca7d1ccf187.png"},{"id":64686244,"identity":"0b70dc63-16b3-4b57-8f3e-e0fe249466a7","added_by":"auto","created_at":"2024-09-17 14:56:05","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":66510,"visible":true,"origin":"","legend":"\u003cp\u003ePareto boundary between target values\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/2b6506c32b7669bd90611a7a.png"},{"id":75930807,"identity":"9f2d4d1c-3d40-4a88-8e00-9fef1795816a","added_by":"auto","created_at":"2025-02-10 16:13:28","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2313319,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4898818/v1/83d49997-79f6-407e-bc70-e8ff017681b6.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Quality optimization of liquid silicon lenses based on sequential approximation optimization and radial basis function networks","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eHigh brightness LEDs of the most recent generation can reach extremely high temperatures, which can put pressure on traditional polymers like PMMI, PMMA, and PC. Over a temperature range of -40\u0026deg;C to 200\u0026deg;C, silicones provide great temperature resistance, strong resistance to heat aging and high chemical resistance, purity, transparency, and stable mechanical qualities. Silicones significantly lessen the yellowing effect, making them perfect for usage in outdoor or severe environments. The silicone material for silicone lenses is a highly transparent two-component liquid silicone rubber with excellent injection molding process characteristics: providing fast cure, low viscosity, and high flow. These properties significantly increase productivity by reducing injection time and total cycle time. No additional protection is required, such as in the case of light poles or high grids, and no gaskets for IP insulation are required: no yellowing or heat shrinkage cracking in UV or high and low temperature environments, specifically for street, high pole, and high grill lighting applications. As an isotropic material, silicone offers a high degree of flexibility for silicone lenses that can easily fit into any hazardous and harsh environment. Due to its elastomeric nature, silicone offers ideal compensation for mechanical tolerances in construction, thereby contributing to a precise fit in the final application by virtue of its exceptional functionality.\u003c/p\u003e \u003cp\u003eGiven its high productivity, cost-effectiveness, lightweight properties, and superior ability to accommodate intricate geometries, the manufacturing process that involves injecting plastic into molds has become an essential method for producing plastic items. The six-stage process involving clamping, filling, packing, cooling, opening, and ejection constitutes a critical cycle in the production of high-quality plastic components, with each stage contributing significantly to achieving the desired outcome. The material, component and mold design, as well as the manufacturing process factors, all affect the quality of plastic injection molded parts. [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] Product flaws like warpage, shrinkage, sink marks, and residual strains are caused by a range of production-related causes. Hence, a lot of study has been undertaken on the best design of plastic injection molding process parameters.\u003c/p\u003e"},{"header":"2 Literature and review","content":"\u003cp\u003eGiven the significant attention this longstanding issue has garnered from researchers, recent years have seen a proliferation of advanced optimization theories and sophisticated techniques in design aimed at addressing the problem through enhanced understanding and resolution [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The conventional continuous modal testing methods have been replaced by CAE simulation approaches [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Integrating it with the idea of optimization design and using it to optimize the conditions of the injection molding process enhances the quality of the plastic parts while also speeding up the molding process [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. R. Joseph Bensingh et al. [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] investigated the effects of injection molding process parameters such as mold surface temperature, melt temperature, injection time, filling volume percentage V/P switching, filling pressure, and filling time on the volume shrinkage deformation of double aspheric lenses. The scientists used a mixture of computer numerical simulation and optimization approaches to determine the ideal molding parameters that resulted in the least variance of volume shrinkage deformation. Kuo-Ming Tsai et al. developed a lens shape accuracy prediction model using the artificial neural network (ANN) and response surface method in another study (RSM). Following experimental validation and accuracy comparison, the results showed that the injection molding process window obtained using the ANN for determining the cooling time and filling time had a high-order irregular shape, whereas the injection molding process window obtained using the RSM had an oblique ellipse shape. Furthermore, the findings demonstrated that the ANN model outperformed the RSM model in terms of lens shape accuracy [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eCurrently, the industry mainly relies on the technical experience of designers to find a better solution to the quality problems of plastic products during the injection molding process by repeatedly testing and repairing the mold, and repeatedly adjusting the parameters of the injection molding process, but these methods not only increase the design cost and prolong the production cycle, but also make it difficult to obtain the best design results. Therefore, the optimization of injection molding process parameters plays a crucial role in the quality of products.\u003c/p\u003e \u003cp\u003eA approach to examine the impact of cavity pressure distribution on part quality using a neural network was put forth by Jinsu Gim et al. in 2021 [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. The relationship between the mold internal pressure and the weight of the workpiece was determined by evaluating the mold internal pressure profile, and the fluctuation rule of the mold internal pressure was transformed into the quality index of the mold. The influence degree of each element on the injection process may be clarified by the analysis of numerous factors, and the injection process can be optimized in accordance with the influence degree of each component on the injection process. In 2020, Ming-Shyan Huang and colleagues [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] proposed a unique method for predicting the geometry of manufactured parts using a multilayer perceptron (MLP) neural network model with quality indicators. Four factors were used as inputs to the model: the first-stage pressure retention index, pressure integral index, residual pressure drop index, and peak pressure index. Meanwhile, the output of the model was the geometry of the portion in question. The MLP model was trained and used to aid in learning and prediction. The MLP model correctly predicted the geometric width of the pieces, with over 92% accuracy in both training and testing outcomes. As a result, this technology has been demonstrated to be very successful and reliable, showing the potential for wider adoption in the manufacturing business. In 2022, Han-Jui Chang et al. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] used a non-explicit genetic algorithm with kriging response surface analysis to optimize the multi-objective design of UAV shells, considering various process parameters as model variables. After obtaining the warpage value, die stamping index, and mathematical relationship, a multi-objective genetic algorithm optimization program is employed to replace the experimental analysis. The model was optimized using the approach, and the model was then tested. The four critical points and mold index had a clear optimization effect, and the error was just 8.48%, which was within the acceptable range for production. Han-Jui Chang [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] made the suggestion in 2021 that highlighting the product qualities is beneficial in the assessment of identifiability. However it is difficult to establish the reference value due to the influence of many causes, suggesting that one variable can be the dependent variable that is most affected. The information provided gives us a starting point for talking about how these phenomena occur. To create an independent injection molding quality control system that can be recognized and assessed, a novel technique for defect knowledge re-study and virtual measurement-based implementation is also suggested. Machines can now gather and store data throughout the production cycle in the era of information and technology, and with the help of big data management, this data can be processed to create efficient troubleshooting techniques.\u003c/p\u003e \u003cp\u003eIn conclusion, a number of researchers have automated and intelligently predicted the quality of injection molded parts using artificial intelligence techniques. [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] The link between the process parameters and the molding quality objective is intricate, time-varying, nonlinear, and tightly connected [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Thus, identifying an acceptable set of optimal process parameters is very challenging. According to the conventional approach to optimization, the injection molded continually modifies the process settings in light of experience to find the ideal set of parameters. However, because the conventional approach is unable to produce the global ideal parameters, the test time, cost, and raw material waste are all significantly increased [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. Hence, for high-precision and high-efficiency manufacturing, it is crucial to swiftly identify the ideal process parameters for microinjection molding on a scientific foundation. Advances in software computing have enabled more efficient determination of process parameters in recent years through the use of numerous methodologies such as evolutionary algorithms, fuzzy systems, expert systems, and artificial neural networks. Process parameter optimization can be divided into two categories: static process parameter optimization and dynamic optimization based on knowledge or previous cases. The former category uses an agent model to get globally optimal results, whereas the latter achieves optimization results gradually through interactions. The majority of research in this area has been on static process parameter optimization utilizing current methodologies, which typically entail three steps: collecting background data through experimental design, building an agent model, and using optimization algorithms. Other models for modeling the link between process parameters and quality indicators include response surface methodology (RSM) [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], kriging, artificial neural network (ANN) [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], and support vector regression (SVR). Jian Zhao et al. [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] tackled the multi-objective optimization problem of injection molding plastic part quality by using a two-stage approach that combined an efficient global optimization algorithm and a genetic algorithm. For example, Gang Xu et al. [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] proposed a comprehensive approach that integrated multiple optimization methods to achieve excellent performance. Both approaches demonstrated effective solutions in terms of quality and efficiency.\u003c/p\u003e \u003cp\u003eWith the continuous development and popularity of computer-aided engineering, more and more fitting methods are being applied to optimization systems in various fields. These systems can be used to analyze and optimize various processes by building mathematical models to improve the quality and efficiency of products. Among the fitting methods, RBF neural networks are a commonly used method with the advantages of fast convergence and the ability to handle high-dimensional data. In addition, optimization systems based on fitting methods also include methods based on response surface methods and genetic algorithms. These methods can be applied in different scenarios, such as optimization of process parameters, design optimization, product performance prediction, etc. With these optimization systems, we can better understand and control complex processes and provide effective support for improving product quality and efficiency. Researchers examined the Kriging method [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], radial basis function neural networks [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e], and sequential approximation optimization (SAO) [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e] to create high precision response surfaces and identify the global optimal solution. Satoshi Kitayama et al., for example, optimized plastic injection molding process parameters and holding pressure distribution using radial basis function neural networks and sequential approximation optimization (SAO) to reduce warpage and cycle time. Numerical studies and experimental findings confirmed the usefulness of this strategy in reducing warpage and cycle time. Gao and Wang constructed the approximation function of warpage versus process parameters using Kriging approach. Then, SAO was used to minimize warpage in the cell phone casing. However, the various parameters involved in the injection molding process are interdependent. An increase in melt temperature can result in a reduction in viscosity and shear stress, but a longer cooling time may also be required. Higher injection pressure and shorter injection time may decrease temperature differentials between different production areas but may also increase the temperature at the flow front. An increase in holding time can lead to a reduction in sink marks, but an increase in flash points. As the number of factors considered increases, so does the number of optimization targets. Therefore, multi-objective optimization is employed to balance these conflicting parameters. This study aims to perform multi-objective optimization of the holding curve and process parameters in order to minimize residual stress and volume shrinkage. This research focuses on a complete optimization technique for determining the best process parameters for minimizing residual stress and shrinkage. It is roughly separated into the following aspects, as indicated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e: (1) carrying out experiments on residual stress and volume shrinkage in the lens molding process; (2) developing an RBF neural network model; (3) the effect of process parameters on molding quality; and (4) multi-objective optimization of minimum residual stress and minimum volume shrinkage and experimental verification.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"3 Material and methodology","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Liquid Silicone Rubber\u003c/h2\u003e \u003cp\u003eDue to its high clarity and refractive index, LSR materials are well suited for optical applications such as LED lighting, camera lenses, and automotive headlight lenses. In addition, the excellent rheological properties and low viscosity of LSR materials make them suitable to produce high-precision, complex-shaped lenses through injection molding. The advantages of LSR materials over traditional glass or plastic lenses are their light weight, high strength and excellent optical properties that provide higher light transmission and better dispersion characteristics, resulting in improved quality and efficiency of optical components. the benefits of LSR materials include high precision, durability, biocompatibility and high productivity, making them suitable for a wide range of applications such as medical devices, prosthetics and other products requiring high precision and durability.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn this experiment, the LSR material with the model number of LSR-1 from the manufacturer CAE is used. Generally, the volume of LSR changes significantly with pressure and temperature. To calculate the shrinkage or warpage of the lens after vulcanization, it is necessary to describe the pressure-volume-temperature (PVT) relationship, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Viscosity is an indicator of LSR to flow resistance and depends on temperature, shear rate and pressure, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn a complete injection molding cycle, it generally goes through the stages of melt-fill-recharge-shrink-pressure-hold-cooling-eject. As we all know, the filling stage of injection molding, VP switching point and pressure-holding process must be accurately controlled to get high quality and high precision products.\u003c/p\u003e \u003cp\u003eTo avoid premature solidification of the material during the filling process, the filling rate of the plastic must be controlled. Likewise, the pressure applied to the material during the holding phase must be controlled to compensate for the shrinkage that occurs during cooling and to avoid material spillage. To ensure high quality, accuracy and repeatability in weight and dimensions of the molded product, the optimal pressure profile during the holding phase should be the one that varies isotropically during the cooling of the material.\u003c/p\u003e \u003cp\u003eDuring the injection molding process, the polymer material is heated into the molten state and injected into the mold cavity under high pressure, undergoing a process from high temperature and high pressure to rapid cooling and pressure drop, followed by a change from the molten state to the solid state, while the various physical parameters of the polymer material undergo a series of drastic changes, all of which are highly dependent on T, P and V. In particular, the V of the polymer determines the final weight and size of the product. The final molded product's performance and quality are specifically determined by the polymer's V: if the density is too low, it will result in insufficient strength; if the density is not uniform, it will produce internal residual stresses, causing warpage and deformation, etc.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Sequential Approximation Optimization\u003c/h2\u003e \u003cp\u003eSequential Approximation Optimization (SAO) is an optimization method usually used to solve complex multivariate optimization problems. SAO continuously optimizes the objective function to find the optimal solution through repeated iterations and stepwise approximation. This method has a wide range of applications in many fields, including engineering, machine learning, and optimization algorithms. In SAO, the value of the objective function is usually calculated based on the current parameter configuration, and the parameters are adjusted according to the calculation results. This process is repeated until a set termination condition is reached or converges to some local optimal solution. The core idea of SAO is to approximate the global optimal solution through repeated approximate optimization. This step-by-step optimization approach makes SAO robust and reliable in dealing with complex optimization problems. Overall, Sequential Approximation Optimization (SAO) is a commonly used optimization method for solving various complex multivariate optimization problems.\u003c/p\u003e \u003cp\u003eThe schematic representation of the sequential steps involved in the implementation of the SAO optimization method is visually depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, showcasing the systematic flow and intricate details of this approach.\u003c/p\u003e \u003cp\u003e(Step 1) Initialize with number of sequences\u0026thinsp;=\u0026thinsp;1.\u003c/p\u003e \u003cp\u003e(Step 2) Confirm the optimization objective, constraints, and design variable ranges.\u003c/p\u003e \u003cp\u003e(Step 3) A Latin hypercube design (LHD) is used to generate some initial sampling points.\u003c/p\u003e \u003cp\u003e(Step 4) Perform numerical simulation using Moldex3D software to numerically calculate the objective function for the sampling points.\u003c/p\u003e \u003cp\u003e(Step 5) Approximate all the functions as RBF networks. Here, the approximated objective function is expressed as\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{f}_{i}\\left(x\\right)\\le\\:0\\:\\:i=\\text{1,2},\\dots\\:,k$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e(Step 6) If convergence is satisfied, terminate the SAO algorithm and output the result. Otherwise, the number of sequences is added by one and the Pareto optimal solution will be added as a new sampling point and return to step 3.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Radial Basis Function neural network\u003c/h2\u003e \u003cp\u003eOwing to their outstanding performance, RBF (Radial Basis Function) neural networks have been extensively employed in a diverse range of fields, including but not limited to pattern recognition, function approximation, system identification, and control. One of the major advantages of RBF neural networks is their fast-learning speed, which is achieved through a simple structure that requires fewer iterations to converge than other types of neural networks. Another advantage of RBF neural networks is their excellent approximation performance, which enables them to approximate complex functions with high accuracy. In addition, RBF neural networks have good generalization ability, which allows them to perform well on new, unseen data. A radial basis function is utilized in the hidden layer of RBF neural networks to transfer the input space into a higher dimensional space where linear separation is achievable. This enables the network to record nonlinear interactions between inputs and outputs, which is frequently required for difficult problem solving. The Gaussian function, which has a bell-shaped curve and is defined by a center and a width parameter, is the most often used radial basis function. Overall, the RBF neural network is an useful tool for handling difficult problems with nonlinear input-output relationships. Its straightforward structure, quick learning speed, high approximation performance, and generalization capabilities make it a popular choice for a wide range of applications in industry and academics. The topology of a radial-basis neural network with numerous inputs and multiple outputs is depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e below.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe complete interpolation method requires the interpolation function to pass through each sample point, i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:F\\left({X}^{n}\\right)={d}^{n}\\)\u003c/span\u003e\u003c/span\u003e. There is a total of k sample points.\u003c/p\u003e \u003cp\u003eThe RBF method is to choose k basis functions, each corresponding to a training data, each of which is of the form \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\phi\\:(\\parallel\\:X-{X}^{k}\\parallel\\:)\\)\u003c/span\u003e\u003c/span\u003e, which is called radial basis function since the distance is radially homogeneous. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\parallel\\:X-{X}^{k}\\parallel\\:\\)\u003c/span\u003e\u003c/span\u003e denotes the modulus of the difference vector, or called the 2-parametric number.\u003c/p\u003e \u003cp\u003eThe interpolation function based on the radial basis function is\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:F\\left(x\\right)=\\sum\\:_{k=1}^{n}{\\omega\\:}_{k}{\\phi\\:}_{k}\\parallel\\:X-{X}^{k}\\parallel\\:={\\omega\\:}_{1}{\\phi\\:}_{1}\\parallel\\:X-{X}^{1}\\parallel\\:+\\:{\\omega\\:}_{2}{\\phi\\:}_{2}\\parallel\\:X-{X}^{2}\\parallel\\:+\\dots\\:{\\omega\\:}_{n}{\\phi\\:}_{n}\\parallel\\:X-{X}^{n}\\parallel\\:$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe following Gaussian kernel is generally used as the basis function: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:hi\\left(x\\right)=\\text{e}\\text{x}\\text{p}(-\\frac{{\\left(x-xi\\right)}^{T}(x-xi)}{{r}^{2}j})\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003ewhere xi is the i-th sampling point and ri is the width of the i-th basis function. The response yi is computed at the sampling point xi. The learning of RBF networks is usually done by solving.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the horizontal coordinate is the design variable dimension, and the vertical coordinate is the solution vector dimension. For each design variable, the high precision numerical solution corresponds to a unique solution vector, as shown by the solid line in the figure. At each time when a new design variable *X needs to be calculated, the data of certain sample points, as shown by the solid dots in the figure, have been obtained in the previous period. Based on these sample points, the variation of the solution vector with the design variables can be predicted by approximate modeling, and the next sample point is sampled based on this prediction, as shown by the star-shaped point in the figure. The results in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e demonstrate that, when the iteration format is left unchanged and the optimization is carried out, the distance between the predicted value and the converged value is significantly smaller than the distance between the initial value and the converged value, indicating that using the approximate model acceleration significantly decreases the total amount of iterative steps of the high-precision simulation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 Case study","content":"\u003cp\u003eA company\u0026apos;s selection of liquid silicone lenses for a car lamp is shown in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e. The object has a maximum flesh thickness of approximately 22.483 mm, a minimum flesh thickness of approximately 0.101 mm, and a volume of approximately 53,972.35 mm\u003csup\u003e3\u003c/sup\u003e. We model it using modeling software, and the model that is created will be used in the test\u0026apos;s mold flow analysis.\u003c/p\u003e\n\u003cp\u003eThe essential parameters linked to injection molding allows users to set in this inquiry are shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. Based on the literature analysis, five process factors were chosen and experimentally analyzed: melt temperature (\u0026deg;C), mold temperature (\u0026deg;C), filling time (s), maturation time (s), and maturation pressure (MPa), all of which are substantially connected with the density of residual stress values.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eExperimental optimization parameters for injection molding of liquid silicone lenses for automotive lights\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"3\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eUnit\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eRange\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMelt temperature\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e℃\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u0026thinsp;~\u0026thinsp;30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMold temperature\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e℃\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e150\u0026thinsp;~\u0026thinsp;170\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCooling temperature\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e℃\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCooling time\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003es\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFilling time\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003es\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.8\u0026thinsp;~\u0026thinsp;2.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCuring time\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003es\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026thinsp;~\u0026thinsp;20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCuring pressure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u0026thinsp;~\u0026thinsp;40\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eAmong them, the residual stress detection methods for injection molded parts can be basically categorized as lossy testing and nondestructive testing. The principle of lossy testing is to destroy the original structure of the product, break the original stress equilibrium state to make the product appear deformation or appearance changes, so as to calculate the size and distribution of residual stress. The common methods of lossy stress detection include drilling, layer peeling, solvent method, etc. Non-destructive testing is the use of instrumentation to detect changes in the physical properties of the product due to the presence of stress, and thus calculate the stress distribution. The common polarized stress test is one in which, according to the law of stress optics, the refractive index of a transparent plastic material changes when the material is subjected to stress. This is reflected under the stress polarization equipment by the color distribution presented on the product. Through the color distribution state, it is possible to simply observe the residual stress distribution on the product. It is characterized by a simple method, easy operation and inexpensive equipment, and is a widely used nondestructive stress detection method. The disadvantage is that the results are generally measured qualitatively and cannot be calibrated in quantitative terms.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eInput-output parameters (Experimental data) in this study.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"11\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003eGroup\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003eMelt temperature (℃)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003eFilling time (s)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 6.9083%;\"\u003e\n \u003cp\u003eCuring time (s)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 19.2027%;\"\u003eCuring pressure (MPa)\u003cbr\u003e\u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 14.5191%;\"\u003eMold temperature (℃)\u003cbr\u003e\u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 12.1773%;\"\u003eAverageVon-Mises thermal stress (Mpa)\u003cbr\u003e\u003c/th\u003e\n \u003cth align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003eVolume Shrinkage(%)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e23.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e19.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e20.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e154.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e21.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e4.97\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e29.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e15.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e33.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e153.70\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e21.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.75\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e22.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e15.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n 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style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e152.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e18.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.74\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e20.73\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e16.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e29.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e165.83\u003c/p\u003e\n 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style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e4.96\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e23.97\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e17.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e30.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e168.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e21.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e4.04\u003c/p\u003e\n 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style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e24.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e17.21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e17.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e155.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e22.64\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e5.14\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n 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style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e20.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e19.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e37.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e166.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e19.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.59\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e25.75\u003c/p\u003e\n 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\u003cp\u003e16.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e28.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e158.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e22.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e4.19\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e27.84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e18.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e16.73\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e151.72\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e22.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e5.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e22.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e19.82\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e39.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e156.88\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e18.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.46\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e2.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e26.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e18.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e32.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e160.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e19.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e4.00\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e24.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e15.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e33.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e167.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e22.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.66\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" style=\"width: 6.4399%;\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 11.826%;\"\u003e\n \u003cp\u003e1.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 6.3228%;\"\u003e\n \u003cp\u003e25.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 16.6267%;\"\u003e\n \u003cp\u003e18.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 19.2027%;\"\u003e\n \u003cp\u003e37.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 14.5191%;\"\u003e\n \u003cp\u003e162.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 12.1773%;\"\u003e\n \u003cp\u003e19.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" style=\"width: 13.3482%;\"\u003e\n \u003cp\u003e3.52\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e demonstrates the injection molding experimental data, i.e., the effect of variations of five factors on residual stress and volume shrinkage. These data contain a large amount of feature and label information that can be used to train and validate the performance of the model. The data need to be pre-processed and cleaned to ensure the accuracy and consistency of the data before proceeding to modeling, and finally they will be used for the building of RBF networks for LSR lens injection molding. The pre-processed and cleaned data will serve as the foundation for constructing RBF networks for LSR lens injection molding, which constitutes a crucial step in our experimental process.\u003c/p\u003e\n\u003cp\u003eA multilayer radial basis neural network with three network topologies (i.e., input, hidden, and output) was used to map the injection molding process of liquid silicone rubber lenses. The mapping process was performed using MATLAB 2020 software and a neural network toolbox. The hidden layer neurons use Gaussian activation functions, while the output layer neurons use pure linear functions.\u003c/p\u003e\n\u003cp\u003eThe input layer of the neural network consists of five neurons corresponding to the number of independent variables, while the output layer has two neurons matching the number of dependent variables. Figure \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e shows the multilayer LSR neural network developed in the MATLAB 2020 environment, which was used to predict process parameters and evaluate the quality of the molded LSR lenses.\u003c/p\u003e"},{"header":"5 Result and discussion","content":"\u003cp\u003eThe energy used in mass production cannot be overlooked, despite the fact that the cost of energy consumption during injection molding is very modest. Up to 30\u0026nbsp;billion kWh of energy are used annually by the global injection molding industry, which accounts for 10% of all energy used globally and generates a significant amount of carbon emissions. Thus, it is essential to maximize energy efficiency to ensure product quality. The main piece of machinery used in the injection molding process is the injection molding machine. Using high-tech tools like servo motors with high precision and energy-efficient injection molding machines can drastically save energy usage. Using low-energy LSR materials can also significantly reduce the amount of energy needed to heat plastic in the injection molding machine, leading to a more environmentally friendly and energy-efficient manufacturing process. Consequently, it is possible to increase energy efficiency and reduce emissions in the injection molding process while maintaining the quality of the final product by upgrading equipment and employing high-quality materials.\u003c/p\u003e \u003cp\u003eWhen a substance undergoes cooling from a state of high temperature and high pressure to a state of low temperature and low pressure, its volume changes. This change is known as volume shrinkage, which is defined as the percentage difference in volume before and after cooling. Volume shrinkage is positive when the substance undergoes a reduction in volume and negative when the substance expands due to excessive holding pressure.\u003c/p\u003e \u003cp\u003eThe shrinkage rate of the entire plastic part differs from that of its profile, leading to the formation of internal residual stresses that mimic the effects of external forces. The distribution of residual stress is closely related to the optical properties of the product, and the presence of residual stress can significantly impact its optical performance. Areas of the product with higher residual stress tend to exhibit poorer optical performance than areas with lower residual stress. If the volume shrinkage of a product is uneven, it can compromise not only the dimensional accuracy of the product but also indirectly impact its optical performance. This is due to the potential irregularities in the product's shape caused by the uneven shrinkage, which may result in deviations on the surface of optical components and ultimately affect their optical performance.\u003c/p\u003e \u003cp\u003eLSR material is characterized by low shrinkage and warpage, which makes it an ideal material for applications where dimensional accuracy and optical performance are critical. The volume change of a product before and after V/P switching is a reliable indicator of the injection molding process's carbon footprint. Failure to adjust the process parameters correctly can result in higher shrinkage and a subsequent increase in the carbon footprint of the product.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo gain a better understanding of the interaction between different factors, response surface plots were employed to visualize the effect of each factor on the residual stress in the plastic part. In this study, five influential factors were considered, including melt temperature, maturation pressure, maturation time, filling time, and mold temperature.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e displays a 3D surface that demonstrates how different factors influence the residual stress in the plastic part. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(a) shows the interaction between melt temperature and maturation pressure on the residual stress values. The results reveal that the residual stress value of the lens is at its minimum at a melt temperature of 25\u0026ndash;30\u0026deg;C and a maturation pressure of approximately 40 MPa. Similarly, Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(b) illustrates the relationship between maturation time and maturation pressure on the residual stress values. The results demonstrate that the residual stress values decrease as the maturation time increases at a constant maturation pressure. Figure\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(c) shows the effect of filling time and maturation pressure on the residual stresses, where the effect of filling time is comparatively minor compared to the significant impact of maturation pressure on the residual stresses. Furthermore, Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(d) displays the response surface plots of filling time and maturation time on the residual stress values. The results show that the residual stress values of the lenses gradually decrease with increasing curing time. In Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(e), the interaction between melt temperature and maturation time on the residual stress values is demonstrated. The results indicate that the residual stress values of the lenses are at their minimum when the maturation time is 18\u0026ndash;20 seconds, and the melt temperature is higher. Finally, Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e(f) presents the effect of melt temperature and mold temperature on residual stresses. The results show that the lenses have the smallest residual stress values when the mold temperature is 150\u0026deg;C and the melt temperature is approximately 30\u0026deg;C. Overall, these response surface plots provide valuable insight into the effects of different factors on residual stress values in plastic parts.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eⅠ, Ⅱ and Ⅲ point injection-molding parameter results.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eItem\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFilling time (s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMelt temperature (℃)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCuring pressure (MPa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eCuring time (s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMold temperature (℃)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eAverageVon-Mises thermal stress (Mpa)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eVolume Shrinkage(%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eⅠ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e26.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e33.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e19.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e150.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e18.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.78\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eⅡ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e29.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e39.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e19.49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e150.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e17.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e3.75\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eⅢ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e27.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e28.79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e20.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e150.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e16.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e4.14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTo optimize these parameters and improve the efficiency and quality of injection molding, the optimal Latin hypercube sampling method was used to obtain samples in this study. We selected five input parameters as the input layer of the model, namely melt temperature, mold temperature, holding time, holding pressure and filling time, while a radial basis (RBF) neural network model was constructed with minimum volume shrinkage and minimum residual stress as the output layer. The model has strong nonlinear approximation capability and good accuracy and can effectively perform prediction and optimization with a large amount of data.\u003c/p\u003e \u003cp\u003eThrough numerical calculations, we determined the Pareto boundary between residual stress and volume shrinkage to obtain the best combination of parameters. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e, we compared three different points on the Pareto boundary and found that point I had a larger value of residual stress and a smaller volume shrinkage, while point II had more average values of residual stress and volume shrinkage. The distribution of stress values at point III was improved and was similar to points I and II. By analyzing the results in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, we conclude that by constructing the RBF neural network model, the best combination of parameters can be obtained, thus reducing the cost of production trial and error and saving production time and material.\u003c/p\u003e \u003cp\u003eIt is important to note that there must be some trade-offs between the optimization objectives. In practice, optimizing one objective may lead to the sacrifice of another, so the best balance needs to be found. In injection molding, the balance between the two objectives, minimum residual stress and minimum volume shrinkage, requires careful trade-offs and considerations. In actual production, choosing the best injection molding parameters can not only improve production efficiency and quality, but also significantly reduce production costs.\u003c/p\u003e"},{"header":"6 Conclusion","content":"\u003cp\u003eBased on polymer injection molding theory, this paper aims to investigate the molding process parameters of optical liquid silicone injection molded products, focusing on their residual stress and volume shrinkage, to improve the quality of the injection molded products of headlight lenses. After analysis, the main parameters that affect the product quality include filling time, melting temperature, curing time, curing pressure and mold temperature. Combined with the mold flow analysis software Moldex-3D and intelligent optimization algorithms, the optimal process parameters for optical plastic lenses were studied in depth, aiming to achieve the dual target optimization of minimum residual stress and optimal volume shrinkage. Usually, numerical simulations of injection molding are intensive and can be optimized using radial basis function (RBF) networks and SAO optimization methods. The research results are as follows:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eA multi-objective optimization of the maturation pressure distribution and process parameters was carried out to reduce residual stresses and density. The ripening pressure distribution, which varies during the ripening stage, was used, and optimized. In addition to the ripening pressure distribution, melt temperature, injection time, holding pressure, and holding time were used as process parameters for optimization. the intensity of the PIM numerical simulation was high, so the sequential approximation optimization method with radial basis functions was used. The Pareto boundary between residual stress and density was determined by numerical calculations. In order to verify the validity of the numerical calculation results, experiments were conducted. The experimental results are in good agreement with the numerical calculation results. The effectiveness of the method to reduce the residual stresses and density was verified by numerical calculations and experimental results.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWhen it comes to optimization of injection molding process parameters, the sequential approximation optimization method has many significant advantages. First, the sequential approximate optimization method can find the optimal solution more quickly and efficiently than the traditional trial-and-error or empirical methods. Second, the method can obtain sufficient data with fewer trials, thus reducing the cost and time of the trials. In addition, the sequential approximate optimization method can optimize multiple parameters simultaneously, thus achieving diversified process parameter optimization. Finally, the method can be updated and adjusted to suit the needs of process parameter optimization as the optimization process is continuously performed.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWe propose a new optimization approach for RBF neural networks that combines the concepts of sequential approximation optimization with multi-objective optimization. The original multi-objective optimization problem is simplified into a single-objective optimization problem using this method, making the optimization process more efficient and effective. A series of experiments were carried out to examine the feasibility and effectiveness of the suggested strategy. The results show that the suggested optimization method considerably enhances plastic part production by lowering residual stress values and volume shrinkage by 27.4% and 10.3%, respectively. Additionally, the suggested method minimizes the time required to establish the ideal process parameters, resulting in a more efficient production process.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe proposed optimization method can be applied to a wide range of industries that utilize plastic injection molding, as it offers a powerful tool for improving the quality and productivity of plastic parts. Moreover, the method can also be extended to other optimization problems in manufacturing, providing a valuable tool for optimizing complex systems and processes.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthor Contributions:\u003c/strong\u003e Conceptualization, H.C.; Data curation, Y.L.; Methodology, S.L.; Project administration, Y.S.; Writing—original draft, H. C. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u0026nbsp;\u003c/strong\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement:\u003c/strong\u003e The authors declare that the data supporting the results of this study are available in the paper. If any raw data files in other formats are required, they can be obtained from the corresponding author upon reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments:\u0026nbsp;\u003c/strong\u003eThis research was funded by the 2023 Guangdong Province Science and Technology Special Fund Project—the Guangdong Taiwan Normal University Excellent Project and Technical Support by Xuying Biomedicine Co., Ltd., and Software Support by CoreTech System Co., Ltd., which are gratefully acknowledged.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of Interest:\u0026nbsp;\u003c/strong\u003eThe authors declare no conflict of interest.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eChen, W.-C.; Nguyen, M.-H.; Chiu, W.-H.; Chen, T.-N.; Tai, P.-H., Optimization of the plastic injection molding process using the Taguchi method, RSM, and hybrid GA-PSO. \u003cem\u003eThe International Journal of Advanced Manufacturing Technology \u003c/em\u003e\u003cstrong\u003e2015,\u003c/strong\u003e 83, (9-12), 1873-1886.\u003c/li\u003e\n\u003cli\u003eLin, W. 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In the optimization process, the radial basis function network replaces the finite element reanalysis and allows the construction of an approximate functional relationship between quality and process conditions. In this study, injection molding of objects was simulated and analyzed while varying the filling time, melt temperature, mold temperature, curing pressure, and curing time schemes to better understand the aspects affecting the optimization process. Using the automobile optical liquid silicone lens as an example, the Pareto boundary is used to determine the residual stress and volume shrinkage, as well as the deviation function and radial basis function network. Because numerical simulations are time-consuming, the radial basis function sequential approximation optimization method is applied. The product had the highest quality when the filling time was 1.57s, the melt temperature was 27.18\u0026deg;C, the mold temperature was 150\u0026deg;C, the curing time was 20.02s, and the curing pressure was 28.79 MPa, according to numerical results. Experiments were carried out to test the efficacy of the proposed approach. Nondestructive analysis is used to determine the target values (residual stress and volume shrinkage). Because nondestructive testing does not damage materials, workpieces, or buildings, the inspection rate of items can be quite high following nondestructive testing. Furthermore, numerical and experimental data demonstrate that the technique effectively reduces residual stress and volume shrinkage.\u003c/p\u003e","manuscriptTitle":"Quality optimization of liquid silicon lenses based on sequential approximation optimization and radial basis function networks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-17 14:48:00","doi":"10.21203/rs.3.rs-4898818/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-10-07T05:11:11+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-28T16:45:20+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"68693936939801322456712775266069383923","date":"2024-09-17T02:27:50+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-16T22:37:04+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"224805484644183864770463691481771028601","date":"2024-09-16T22:08:15+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"250167629631437416313631322826581750139","date":"2024-09-16T20:22:45+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-09-16T20:17:58+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-09-16T20:16:47+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-08-23T02:23:40+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-08-20T12:39:14+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-08-12T08:32:13+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d1bbda04-7f29-4a4d-b43a-3e38d23b2522","owner":[],"postedDate":"September 17th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":37725244,"name":"Physical sciences/Optics and photonics/Optical physics/Micro optics"},{"id":37725245,"name":"Physical sciences/Optics and photonics/Optical physics/Nonlinear optics"},{"id":37725246,"name":"Physical sciences/Physics/Fluid dynamics"},{"id":37725247,"name":"Physical sciences/Physics/Optical physics"},{"id":37725248,"name":"Physical sciences/Physics/Statistical physics thermodynamics and nonlinear dynamics"},{"id":37725249,"name":"Physical sciences/Mathematics and computing/Scientific data"}],"tags":[],"updatedAt":"2025-02-10T16:07:04+00:00","versionOfRecord":{"articleIdentity":"rs-4898818","link":"https://doi.org/10.1038/s41598-025-87753-7","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-02-03 15:57:18","publishedOnDateReadable":"February 3rd, 2025"},"versionCreatedAt":"2024-09-17 14:48:00","video":"","vorDoi":"10.1038/s41598-025-87753-7","vorDoiUrl":"https://doi.org/10.1038/s41598-025-87753-7","workflowStages":[]},"version":"v1","identity":"rs-4898818","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4898818","identity":"rs-4898818","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}