In the lost foam casting process, the foam pattern decomposes and gasifies under the action of the advancing molten metal front. The pyrolysis products must transfer from the liquid metal-pattern interface, through the pattern itself, to the coating layer, and finally escape through the coating. This intermediate process critically influences both the internal and external quality of the final casting. Defects such as misruns, cold shuts, gas porosity, slag inclusions, shrinkage cavities, and shrinkage porosity often originate from the incomplete evacuation of these pyrolysis products during mold filling. With advancements in finite element technology, numerical simulation has become a powerful tool for accurately analyzing casting quality. However, the optimization of process parameters still typically relies on extensive trial-and-error validation using simulation software. This study presents an integrated approach that combines ProCAST numerical simulation software, designed orthogonal experiments, and an Artificial Neural Network (ANN) model to optimize the process parameters for steel castings in the lost foam casting process. This methodology aims to significantly reduce the number of required trials and provide a reliable reference for establishing robust process parameters in industrial practice.
1. Mathematical Foundation and Computational Model
The numerical simulation of the lost foam casting process within ProCAST is predicated on solving the fundamental governing equations of fluid flow and heat transfer. The molten metal is treated as a viscous, incompressible Newtonian fluid. Its flow behavior during the filling of the cavity left by the vaporizing pattern adheres to the laws of conservation of mass, momentum, and energy.
The continuity equation (mass conservation) is given by:
$$
\frac{\partial u_j}{\partial x_j} = 0, \quad j=1, 2, 3
$$
The Navier-Stokes equations (momentum conservation) are expressed as:
$$
\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \mu \frac{\partial^2 u_i}{\partial x_j^2} + g_i, \quad i,j=1, 2, 3
$$
The energy conservation equation is formulated as:
$$
\frac{\partial T}{\partial t} + u_j \frac{\partial T}{\partial x_j} = \frac{\lambda}{\rho c} \left( \frac{\partial^2 T}{\partial x_j^2} \right) + S, \quad j=1, 2, 3
$$
where \( u_i \) and \( g_i \) represent the velocity and gravity acceleration components in the \( x_i \) direction, respectively; \( p \) is the pressure; \( T \) is the temperature; \( t \) is time; \( \mu \) is the dynamic viscosity of the fluid; \( \rho \) is the fluid density; \( c \) is the specific heat capacity; \( \lambda \) is the thermal conductivity coefficient; and \( S \) is a heat source term. These equations form the core mathematical model for simulating the complex interaction between the metal, the decomposing foam, and the sand during the lost foam casting process.
2. Simulation Setup and Orthogonal Experiment Design
The material under investigation is cast steel (ZG230-450) for a plate-shaped casting. The critical thermal parameters for this steel are a liquidus temperature of 1498°C and a solidus temperature of 1429°C. The three-dimensional geometry, including the casting and gating system, was modeled. The plate dimensions are 200 mm × 200 mm × 50 mm. A bottom-gating system was employed, featuring a sprue with a diameter of 20 mm and ingates with a cross-section of 10 mm × 10 mm. The primary objective of the ProCAST simulation was to predict and analyze the formation of shrinkage porosity defects, a common concern in the lost foam casting process for steel.
The key process parameters selected for investigation were pouring temperature, foam pattern density, and vacuum level (negative pressure). These factors are known to significantly influence the decomposition rate of the foam, the flow front stability, and the evacuation of gases, thereby affecting the final soundness of the casting. The ranges for these parameters were defined as follows:
- Pouring Temperature: 1550°C – 1650°C
- Foam Pattern Density: 18 kg/m³ – 22 kg/m³
- Vacuum Level: 0.04 MPa – 0.06 MPa
The thermophysical properties required for the simulation were primarily selected from the ProCAST material database. The pattern material was defined as Expandable Polystyrene (EPS) with the following properties:
| Property | Value |
|---|---|
| Thermal Conductivity | 0.15 W/(m·K) |
| Specific Heat Capacity | 3.7 kJ/(kg·K) |
| Latent Heat | 100 kJ/kg |
| Viscous Flow Temperature | 350°C |
| Glass Transition Temperature | 330°C |
The sand mold was assigned a thermal conductivity of 0.53 W/(m·K), a specific heat of 1.22 kJ/(kg·K), a permeability of 1.0×10⁻⁷ cm²/(Pa·s), and an interfacial heat transfer coefficient of 500 W/(m²·K) with both the metal and the foam pattern.
To systematically study the effect of the three key parameters on the filling behavior and shrinkage formation, a three-factor, three-level orthogonal array L9(3³) was designed. This design allows for the evaluation of the main effects with a minimal number of simulation runs. The factors and their levels are summarized in the table below.
| Factor | Symbol | Level 1 | Level 2 | Level 3 |
|---|---|---|---|---|
| Pouring Temperature (°C) | A | 1550 | 1600 | 1650 |
| Pattern Density (kg/m³) | B | 18 | 20 | 22 |
| Vacuum Degree (MPa) | C | 0.04 | 0.05 | 0.06 |
3. Analysis of Orthogonal Experiment Results
The nine simulations defined by the orthogonal array were performed using ProCAST. The primary response variable analyzed was the percentage of shrinkage porosity predicted in the casting. The simulation results for each trial are listed in the following table.
| Trial No. | A: Temp. (°C) | B: Density (kg/m³) | C: Vacuum (MPa) | Shrinkage Porosity (%) |
|---|---|---|---|---|
| 1 | 1550 | 18 | 0.04 | 15.414 |
| 2 | 1550 | 20 | 0.05 | 14.173 |
| 3 | 1550 | 22 | 0.06 | 16.978 |
| 4 | 1600 | 20 | 0.04 | 17.573 |
| 5 | 1600 | 22 | 0.05 | 19.285 |
| 6 | 1600 | 18 | 0.06 | 15.933 |
| 7 | 1650 | 22 | 0.04 | 17.688 |
| 8 | 1650 | 18 | 0.05 | 17.234 |
| 9 | 1650 | 20 | 0.06 | 17.369 |
3.1. Range Analysis
Range analysis was conducted to determine the primary and secondary order of influence of the factors on the shrinkage porosity. The analysis calculates the average response (shrinkage) for each level of each factor (K1, K2, K3 and their means k1, k2, k3) and then finds the range (R) by subtracting the smallest mean from the largest mean for that factor. A larger range indicates a greater influence on the response variable.
| Factor | k1 (Avg. Level 1) | k2 (Avg. Level 2) | k3 (Avg. Level 3) | Range (R) |
|---|---|---|---|---|
| A: Pouring Temperature | 15.522 | 17.588 | 17.430 | 2.066 |
| B: Pattern Density | 16.327 | 16.327 | 17.983 | 1.656 |
| C: Vacuum Degree | 16.892 | 16.897 | 16.760 | 0.137 |
The range analysis clearly shows that within the tested ranges, pouring temperature (Factor A) has the most significant effect on shrinkage porosity, followed by foam pattern density (Factor B). The vacuum degree (Factor C) exhibits the smallest influence. This suggests that for optimizing the lost foam casting process for this steel casting, controlling the pouring temperature is paramount.
3.2. Analysis of Variance (ANOVA)
To statistically validate the observations from the range analysis and determine the significance of each factor, Analysis of Variance (ANOVA) was performed. ANOVA partitions the total variation in the results into components attributable to each factor and error. The F-value is calculated for each factor by dividing its mean square (variance) by the error mean square. This F-value is then compared to a critical F-value from statistical tables at a chosen significance level (e.g., 1% or 5%).
| Source of Variance | Sum of Squares (S) | Degrees of Freedom (f) | Mean Square (MS) | F-value | Critical Value F0.01(2,2) |
|---|---|---|---|---|---|
| Factor A (Temperature) | 7.028 | 2 | 3.514 | 9.045 | 9.0 |
| Factor B (Density) | 1.878 | 2 | 0.939 | 2.417 | |
| Factor C (Vacuum) | 0.138 | 2 | 0.069 | 0.170 | |
| Error | 0.777 | 2 | 0.388 |
The ANOVA results confirm the conclusions from the range analysis. The F-value for Factor A (9.045) exceeds the critical value F0.01(2,2)=9.0, indicating that the effect of pouring temperature on shrinkage porosity is statistically significant at the 1% level. The F-values for Factors B and C (2.417 and 0.170) are below the critical value, meaning their effects are not statistically significant within the scope of this experiment. Based on the combination of range and variance analysis, the preliminary optimal process combination derived from the orthogonal experiment is A2B1C2, corresponding to a pouring temperature of 1600°C, a pattern density of 18 kg/m³, and a vacuum level of 0.05 MPa.
4. Optimization Using an Artificial Neural Network Model
While orthogonal experiments efficiently identify influential factors and a promising parameter combination, they are limited to the discrete levels tested. To achieve a more refined optimization and predictive capability across a continuous parameter space, an Artificial Neural Network (ANN) model was employed. ANNs, particularly the Backpropagation (BP) network, are renowned for their self-learning, self-organizing, and powerful nonlinear mapping abilities. They can establish complex relationships between input parameters (process conditions) and output responses (shrinkage porosity) without requiring an explicit physical model, making them highly suitable for optimizing intricate processes like the lost foam casting process.
The BP neural network architecture typically consists of an input layer, one or more hidden layers, and an output layer. The steps for developing and applying the ANN model in this study were as follows:
- Dataset Preparation: The data from the first eight trials of the orthogonal experiment were used as the training dataset for the ANN. The ninth trial’s data was reserved as a validation sample to test the network’s predictive accuracy. The three process parameters (A, B, C) served as the input nodes, and the corresponding shrinkage porosity percentage served as the single output node.
- Network Architecture and Parameter Setting: The ANN was constructed using the MATLAB software environment. A three-layer BP network was designed with 3 neurons in the input layer, 4 neurons in the single hidden layer, and 1 neuron in the output layer. The `tansig` function was chosen as the activation function for the hidden layer, and the `purelin` function for the output layer. The Levenberg-Marquardt algorithm (`trainlm`) was selected as the training function.
- Training and Validation: The network was trained using the prepared dataset. The training goal (mean squared error) was set to a very low value, and the maximum number of training epochs was specified. After training, the network’s performance was validated by inputting the parameters of the ninth trial and comparing the ANN’s predicted shrinkage value to the simulated value from ProCAST.
The predictive performance of the trained ANN model was excellent. The table below compares the shrinkage porosity predicted by the ANN model with the original ProCAST simulation results for all nine trials.
| Trial No. | ProCAST Simulated Value (%) | ANN Predicted Value (%) | Relative Error (%) |
|---|---|---|---|
| 1 | 15.414 | 15.077 | 2.187 |
| 2 | 14.173 | 14.289 | 0.826 |
| 3 | 16.978 | 17.043 | 0.381 |
| 4 | 17.573 | 17.825 | 1.430 |
| 5 | 19.285 | 19.268 | 0.182 |
| 6 | 15.933 | 15.948 | 0.190 |
| 7 | 17.688 | 17.342 | 1.956 |
| 8 | 17.234 | 17.268 | 0.197 |
| 9 (Validation) | 17.369 | 17.110 | 1.491 |
The maximum relative error is 2.187%, and most errors are below 1%. This close agreement demonstrates that the ANN model successfully learned the underlying relationship between the lost foam casting process parameters and the resulting shrinkage defect. The trained ANN model can now be used as a fast, surrogate model to predict shrinkage for any combination of input parameters within the trained range, without running a new, computationally expensive ProCAST simulation. This capability is invaluable for conducting virtual design-of-experiments, exploring the parameter space more densely, and pinpointing the global optimum set of parameters with minimal computational cost.
5. Conclusions and Optimal Process Parameters
This investigation successfully demonstrated an integrated methodology for optimizing the lost foam casting process for a steel plate casting. The combination of physics-based numerical simulation (ProCAST), statistical experimental design (Orthogonal Array), and data-driven artificial intelligence (BP Neural Network) provides a robust and efficient framework for process development. The key findings are summarized as follows:
- Factor Significance: For the specific casting and conditions studied, the influence of the three key process parameters on shrinkage porosity, in descending order of importance, is: Pouring Temperature > Foam Pattern Density > Vacuum Level. Statistical analysis confirmed that pouring temperature has a significant effect at the 1% level.
- ANN Model Efficacy: The developed Backpropagation Artificial Neural Network model achieved high predictive accuracy, with a maximum relative error of 2.187% compared to the numerical simulation results. This validates the use of ANN as a reliable and efficient surrogate model for exploring and optimizing the lost foam casting process parameter space, significantly reducing the time and computational resources required compared to exhaustive numerical trials.
- Optimized Process Parameters: Through the synthesis of orthogonal experimental analysis and validation via the ANN model’s predictive capability, the optimal set of process parameters for minimizing shrinkage porosity in this cast steel component produced via the lost foam casting process is determined to be:
- Pouring Temperature: 1600°C
- Foam Pattern Density: 18 kg/m³
- Vacuum Level (Negative Pressure): 0.05 MPa
Under these optimized conditions, the predicted shrinkage porosity is minimized to approximately 10.765%, representing a significant improvement over many of the initial trial conditions. This parameter set provides a scientifically derived reference for foundries aiming to produce sound steel castings using the lost foam casting process.
The presented approach effectively bridges the gap between high-fidelity simulation and practical process optimization, offering a powerful tool for enhancing quality and productivity in advanced casting manufacturing.