In modern manufacturing, the lost foam casting process has emerged as a pivotal technique for producing complex metal components, particularly in the automotive industry for aluminum alloy parts. This process offers significant advantages over traditional sand casting, including superior dimensional accuracy, reduced post-processing, and lower defect rates. However, the mechanical properties of castings, such as tensile strength and elongation, are highly sensitive to various process parameters in the lost foam casting process. As a researcher focused on enhancing casting quality, I aimed to optimize these parameters to achieve optimal performance in ZL102 aluminum-silicon alloy castings. The selection of parameters like modifier dosage, casting wall thickness, and pouring temperature critically influences microstructural evolution and defect formation, thereby affecting final properties. To address this multi-objective optimization challenge, I employed the grey correlation method, a robust tool within grey system theory for handling limited data and uncertain systems. This study integrates orthogonal experimental design with grey relational analysis to systematically evaluate and optimize the lost foam casting process, ensuring that both tensile strength and elongation are maximized simultaneously.
The lost foam casting process involves using a foam pattern that vaporizes upon metal pouring, leaving a precise cavity for the molten metal. This technique is renowned for its flexibility and efficiency, especially for high-volume production of intricate shapes. Key parameters in the lost foam casting process include the amount of modifier used to refine microstructure, the wall thickness of the casting which affects cooling rates, and the pouring temperature that influences fluidity and solidification. In this work, I focused on ZL102 alloy, a common aluminum-silicon alloy used in automotive applications due to its good castability and mechanical properties. The goal was to determine the optimal combination of these parameters to enhance tensile strength and elongation, which are critical indicators of durability and performance in service conditions.
To achieve this, I designed an experimental framework based on the L9(3^3) orthogonal array, which efficiently explores three factors at three levels with only nine trials. The factors were modifier dosage (2.5%, 3.0%, 3.5%), casting wall thickness (5 mm, 9 mm, 13 mm), and pouring temperature (720°C, 760°C, 800°C). Each trial involved preparing expandable polystyrene (EPS) patterns with specified dimensions, coating them with a refractory coating, and assembling them in a sand mold without binders. The浇注 system was designed as an open type to ensure proper filling and minimize turbulence. After pouring the molten ZL102 alloy, castings were extracted and machined into standard tensile specimens according to GB/T 228.1-2010. Tensile tests were conducted on a universal testing machine at a constant rate, and the results for tensile strength and elongation were recorded as the response variables. This orthogonal design allowed for a comprehensive assessment of the lost foam casting process while minimizing experimental effort.
The grey correlation method, developed by Deng Julong, is particularly suited for multi-objective optimization where relationships between factors and responses are complex or nonlinear. It analyzes the geometric proximity between a reference sequence (ideal performance) and comparative sequences (experimental results) to compute relational degrees. In this study, I applied this method to the orthogonal experimental data, treating tensile strength and elongation as the target functions. Both are “larger-the-better” characteristics, meaning higher values indicate better mechanical performance. The grey relational analysis involves several steps: data normalization to eliminate unit differences, calculation of grey relational coefficients for each response, and determination of overall grey relational degrees by weighting the coefficients. This approach transforms multi-response optimization into a single objective problem, facilitating the identification of optimal parameter settings in the lost foam casting process.
The mathematical formulation of the grey correlation method begins with data preprocessing. For “larger-the-better” responses, the original data are normalized using the following formula to obtain dimensionless values:
$$ x_i(k) = \frac{x_i^0(k) – \min x_i^0(k)}{\max x_i^0(k) – \min x_i^0(k)} $$
where \( x_i^0(k) \) is the original value for the \( i \)-th trial and \( k \)-th response, \( \min x_i^0(k) \) and \( \max x_i^0(k) \) are the minimum and maximum values across all trials for that response, and \( x_i(k) \) is the normalized value. This ensures all data fall within [0, 1], with 1 representing the best performance. Next, the grey relational coefficient \( \xi_i(k) \) is computed to measure the similarity between the normalized sequence and the ideal sequence (which is set to 1 for all responses in this case). The coefficient is given by:
$$ \xi_i(k) = \frac{\Delta_{\min} + \varphi \Delta_{\max}}{\Delta_{0i}(k) + \varphi \Delta_{\max}} $$
where \( \Delta_{0i}(k) = |1 – x_i(k)| \) is the absolute difference between the ideal and normalized values, \( \Delta_{\min} = \min_i \min_k \Delta_{0i}(k) \) and \( \Delta_{\max} = \max_i \max_k \Delta_{0i}(k) \) are the global minimum and maximum differences, and \( \varphi \) is the distinguishing coefficient, typically set to 0.5 to balance influence. Finally, the overall grey relational degree \( \gamma_i \) for each trial is calculated as the weighted average of the coefficients:
$$ \gamma_i = \frac{1}{n} \sum_{k=1}^{n} \lambda_k \xi_i(k) $$
where \( n \) is the number of responses (2 in this study), and \( \lambda_k \) is the weight for each response. Since both tensile strength and elongation are equally important for mechanical properties, I assigned equal weights: \( \lambda_1 = \lambda_2 = 0.5 \). A higher \( \gamma_i \) indicates that the trial’s parameter combination yields responses closer to the ideal, thus representing better overall performance in the lost foam casting process.
The orthogonal experimental results are summarized in Table 1, which lists the parameter combinations and the corresponding tensile strength and elongation values for ZL102 alloy castings. These data form the basis for the grey relational analysis. From the table, it is evident that the responses vary significantly across different parameter settings, highlighting the sensitivity of the lost foam casting process to these factors. For instance, higher tensile strengths are observed at certain modifier dosages and pouring temperatures, while elongation shows different trends. This variability underscores the need for a systematic optimization approach to balance both objectives.
| Trial No. | Modifier Dosage (%) | Casting Wall Thickness (mm) | Pouring Temperature (°C) | Tensile Strength (MPa) | Elongation (%) |
|---|---|---|---|---|---|
| 1 | 2.5 | 5 | 720 | 127.66 | 2.88 |
| 2 | 2.5 | 9 | 760 | 106.83 | 1.99 |
| 3 | 2.5 | 13 | 800 | 92.41 | 1.31 |
| 4 | 3.0 | 5 | 760 | 145.66 | 2.86 |
| 5 | 3.0 | 9 | 800 | 105.83 | 1.42 |
| 6 | 3.0 | 13 | 720 | 90.45 | 1.93 |
| 7 | 3.5 | 5 | 800 | 159.41 | 2.23 |
| 8 | 3.5 | 9 | 720 | 113.54 | 1.91 |
| 9 | 3.5 | 13 | 760 | 127.23 | 2.32 |
Using the grey correlation method, I normalized the tensile strength and elongation data from Table 1. The normalization process transforms raw values into comparable scales, as shown in the calculations below. For tensile strength, the maximum value is 159.41 MPa (Trial 7) and the minimum is 90.45 MPa (Trial 6). Thus, the normalized tensile strength for Trial 1 is computed as:
$$ x_1(1) = \frac{127.66 – 90.45}{159.41 – 90.45} = \frac{37.21}{68.96} \approx 0.540 $$
Similarly, for elongation, the maximum is 2.88% (Trial 1) and the minimum is 1.31% (Trial 3), so the normalized elongation for Trial 1 is:
$$ x_1(2) = \frac{2.88 – 1.31}{2.88 – 1.31} = \frac{1.57}{1.57} = 1.000 $$
This process is repeated for all trials, resulting in normalized values that range from 0 to 1. The ideal sequence is defined as (1, 1) for both responses, representing the best possible performance. Next, the grey relational coefficients are calculated using the formula with \( \varphi = 0.5 \). For Trial 1, the absolute differences are \( \Delta_{01}(1) = |1 – 0.540| = 0.460 \) and \( \Delta_{01}(2) = |1 – 1.000| = 0.000 \). The global minimum difference \( \Delta_{\min} \) is 0.000 (from Trial 1’s elongation), and the global maximum difference \( \Delta_{\max} \) is determined from all trials; for instance, Trial 3 has a high difference in tensile strength. After computing \( \Delta_{\max} \), the grey relational coefficient for Trial 1’s tensile strength is:
$$ \xi_1(1) = \frac{0.000 + 0.5 \times \Delta_{\max}}{0.460 + 0.5 \times \Delta_{\max}} $$
and for elongation:
$$ \xi_1(2) = \frac{0.000 + 0.5 \times \Delta_{\max}}{0.000 + 0.5 \times \Delta_{\max}} = 1.000 $$
The exact value of \( \Delta_{\max} \) depends on all trials; from the data, the maximum absolute difference occurs for Trial 3’s tensile strength: \( \Delta_{03}(1) = |1 – 0.000| = 1.000 \) after normalization (since Trial 3 has the lowest tensile strength, its normalized value is 0). Thus, \( \Delta_{\max} = 1.000 \). Then, \( \xi_1(1) = \frac{0.0 + 0.5 \times 1.0}{0.460 + 0.5 \times 1.0} = \frac{0.5}{0.960} \approx 0.521 \). The overall grey relational degree for Trial 1 is \( \gamma_1 = 0.5 \times 0.521 + 0.5 \times 1.000 = 0.761 \). This procedure is applied to all trials, and the results are presented in Table 2.
| Trial No. | Modifier Dosage (%) | Casting Wall Thickness (mm) | Pouring Temperature (°C) | Grey Relational Coefficient (Tensile Strength), \( \xi_1 \) | Grey Relational Coefficient (Elongation), \( \xi_2 \) | Grey Relational Degree, \( \gamma \) |
|---|---|---|---|---|---|---|
| 1 | 2.5 | 5 | 720 | 0.521 | 1.000 | 0.761 |
| 2 | 2.5 | 9 | 760 | 0.396 | 0.469 | 0.432 |
| 3 | 2.5 | 13 | 800 | 0.340 | 0.333 | 0.337 |
| 4 | 3.0 | 5 | 760 | 0.715 | 0.975 | 0.845 |
| 5 | 3.0 | 9 | 800 | 0.392 | 0.350 | 0.371 |
| 6 | 3.0 | 13 | 720 | 0.333 | 0.452 | 0.393 |
| 7 | 3.5 | 5 | 800 | 1.000 | 0.547 | 0.774 |
| 8 | 3.5 | 9 | 720 | 0.429 | 0.447 | 0.438 |
| 9 | 3.5 | 13 | 760 | 0.517 | 0.584 | 0.550 |
From Table 2, the grey relational degree \( \gamma \) serves as a comprehensive performance index for each parameter combination in the lost foam casting process. Trial 4 exhibits the highest \( \gamma \) value of 0.845, indicating that with a modifier dosage of 3.0%, casting wall thickness of 5 mm, and pouring temperature of 760°C, the responses are closest to the ideal. However, to identify the optimal level for each factor, I performed a mean grey relational degree analysis. This involves calculating the average \( \gamma \) for each level of the three factors, as shown in Table 3. For example, for modifier dosage, Level 1 (2.5%) includes Trials 1, 2, and 3, with \( \gamma \) values of 0.761, 0.432, and 0.337, respectively. The mean is \( (0.761 + 0.432 + 0.337)/3 = 0.510 \). Similarly, means are computed for other levels and factors.
| Process Parameter | Mean Grey Relational Degree (Level 1) | Mean Grey Relational Degree (Level 2) | Mean Grey Relational Degree (Level 3) | Range (Max – Min) |
|---|---|---|---|---|
| Modifier Dosage | 0.510 | 0.536 | 0.587 | 0.077 |
| Casting Wall Thickness | 0.793 | 0.414 | 0.427 | 0.379 |
| Pouring Temperature | 0.531 | 0.609 | 0.494 | 0.115 |
The range values in Table 3 represent the difference between the maximum and minimum mean grey relational degrees for each factor. A larger range implies that the factor has a more significant influence on the overall performance in the lost foam casting process. Here, casting wall thickness has the largest range of 0.379, followed by pouring temperature at 0.115, and modifier dosage at 0.077. Therefore, the order of influence on the mechanical properties of ZL102 alloy castings is: casting wall thickness > pouring temperature > modifier dosage. This finding is crucial for practitioners of the lost foam casting process, as it highlights that controlling wall thickness is paramount for achieving desired tensile strength and elongation. The optimal level for each factor is determined by selecting the level with the highest mean grey relational degree: modifier dosage at 3.5% (Level 3, mean 0.587), casting wall thickness at 5 mm (Level 1, mean 0.793), and pouring temperature at 760°C (Level 2, mean 0.609). Thus, the theoretically optimal parameter combination for the lost foam casting process is modifier dosage of 3.5%, casting wall thickness of 5 mm, and pouring temperature of 760°C.
To validate this optimization, I conducted a confirmation experiment using the optimal parameters derived from the grey correlation method. The lost foam casting process was executed under these conditions, and the resulting ZL102 alloy castings were tested for tensile strength and elongation. The results, presented in Table 4, show a tensile strength of 163.5 MPa and an elongation of 3.0%. These values exceed the best outcomes from the orthogonal trials (e.g., Trial 7 had 159.41 MPa tensile strength and Trial 1 had 2.88% elongation), demonstrating the effectiveness of the grey correlation method in optimizing the lost foam casting process. The improvement can be attributed to the synergistic effects of the parameters: a higher modifier dosage refines the silicon phase and reduces porosity, a thinner wall thickness promotes faster cooling and finer microstructure, and an intermediate pouring temperature balances fluidity and gas evolution. This validation underscores the practical utility of integrating grey relational analysis with orthogonal design for multi-objective optimization in manufacturing processes like lost foam casting.
| Modifier Dosage (%) | Casting Wall Thickness (mm) | Pouring Temperature (°C) | Tensile Strength (MPa) | Elongation (%) |
|---|---|---|---|---|
| 3.5 | 5 | 760 | 163.5 | 3.0 |
The success of this optimization hinges on the underlying mechanisms of the lost foam casting process. Modifier dosage, typically involving sodium-based compounds, alters the solidification behavior by inhibiting the growth of primary silicon and promoting a finer eutectic structure. In ZL102 alloy, this leads to enhanced strength and ductility. Casting wall thickness directly affects the cooling rate; thinner walls result in rapid solidification, which refines grain structure and reduces defects like shrinkage pores. However, excessively thin walls may increase turbulence during pouring, so an optimal thickness is essential. Pouring temperature influences the viscosity of the molten metal and the decomposition of the foam pattern. Lower temperatures may cause incomplete filling, while higher temperatures can lead to excessive gas generation and porosity. The optimal temperature of 760°C ensures adequate fluidity without compromising integrity. By leveraging the grey correlation method, these complex interactions are quantified, enabling a balanced optimization that maximizes both tensile strength and elongation in the lost foam casting process.
In broader contexts, the grey correlation method offers several advantages for optimizing the lost foam casting process. It is particularly suited for situations with limited experimental data, as it does not require large sample sizes or assumptions about data distribution. The method’s ability to handle multiple responses simultaneously makes it ideal for real-world manufacturing where quality is multi-faceted. Moreover, the computational steps are straightforward and can be automated, facilitating integration into process control systems. For the lost foam casting process, this means that manufacturers can quickly identify optimal settings for new alloys or component geometries, reducing trial-and-error and improving efficiency. The use of orthogonal design further enhances this by providing a structured exploration of the parameter space, ensuring that all interactions are considered without exhaustive testing.
To further illustrate the application of the grey correlation method, consider the general framework for optimizing any lost foam casting process. Suppose there are \( m \) factors (e.g., \( A_1, A_2, \dots, A_m \)) each at \( l \) levels, and \( n \) responses (e.g., \( Y_1, Y_2, \dots, Y_n \)). An orthogonal array like \( L_a(b^c) \) can be used, where \( a \) is the number of trials, \( b \) is the number of levels, and \( c \) is the number of factors. After conducting experiments, the responses are normalized based on their characteristics. For “larger-the-better” responses, the normalization formula is as above; for “smaller-the-better” responses, such as defect count, the formula is:
$$ x_i(k) = \frac{\max x_i^0(k) – x_i^0(k)}{\max x_i^0(k) – \min x_i^0(k)} $$
Then, the grey relational coefficients are computed using the same formula with \( \varphi \) often set to 0.5 for neutrality. The overall grey relational degree is calculated as a weighted sum, where weights can be assigned based on engineering priorities. For instance, if tensile strength is deemed twice as important as elongation in the lost foam casting process, one might set \( \lambda_1 = 0.667 \) and \( \lambda_2 = 0.333 \). The optimal factor levels are those that maximize the mean grey relational degree for each factor, and the significance of factors is assessed via range analysis or analysis of variance (ANOVA) if data permits. This systematic approach ensures robust optimization across diverse scenarios in the lost foam casting process.
In conclusion, this study demonstrates the effective use of the grey correlation method to optimize key parameters in the lost foam casting process for ZL102 aluminum alloy. Through orthogonal experimentation and grey relational analysis, the optimal combination was identified as modifier dosage of 3.5%, casting wall thickness of 5 mm, and pouring temperature of 760°C. This setting yielded superior tensile strength and elongation compared to all orthogonal trials, validating the method’s efficacy. The analysis also revealed that casting wall thickness has the greatest influence on mechanical properties, followed by pouring temperature and modifier dosage. These insights provide valuable guidance for engineers and practitioners seeking to enhance the quality and performance of castings produced via the lost foam casting process. Future work could explore additional factors such as coating permeability, vibration assistance, or alloy composition, and apply advanced grey models like fuzzy grey correlation for even more nuanced optimization. Ultimately, the integration of grey system theory with traditional experimental design offers a powerful toolkit for advancing the lost foam casting process in industrial applications.
The lost foam casting process continues to evolve with advancements in materials and technology. By employing optimization techniques like the grey correlation method, manufacturers can achieve higher efficiency, reduced waste, and improved product quality. This study contributes to that evolution by providing a concrete methodology for parameter optimization, emphasizing the importance of a holistic approach that considers multiple performance metrics. As industries increasingly adopt data-driven methods, the principles outlined here can be extended to other casting processes and materials, fostering innovation and sustainability in manufacturing. The lost foam casting process, with its unique advantages, stands to benefit greatly from such systematic optimization efforts, ensuring its continued relevance in producing high-integrity components for demanding applications.