As a researcher deeply engaged in the field of computational materials processing, I have devoted considerable effort to improving the accuracy and efficiency of numerical simulations for solidification processes. The persistent challenge in sand foundry defects, particularly shrinkage cavities and porosity, has motivated my work on developing novel algorithmic models. In this paper, I present three interconnected models: the macro shrinkage degree model, the macro-micro coupled porosity degree model, and the dynamic grid technology model. These models are designed to address the limitations of traditional simulation approaches, especially when applied to large and complex sand castings. Through my research, I have achieved significant improvements in both computational precision and speed, as verified by experimental casting trials.
Sand foundry defects, such as shrinkage cavities and micro-porosity, arise due to improper feeding of liquid metal during solidification. The prediction of these defects is critical for optimizing gating and risering systems in sand casting. Traditional methods like the Niyama criterion and its modifications have been widely used, but they often fail to capture the detailed morphology of defects or the influence of local solidification conditions. My research aims to bridge the gap between macroscopic temperature field simulations and microscopic structural evolution, thereby providing a more realistic assessment of shrinkage defects. By integrating macro-scale energy conservation with micro-scale cellular automaton techniques, I have developed a framework that can predict both the location and the degree of porosity within sand castings.
Research Approach and Mathematical Foundations
My overall research strategy is illustrated by the following road map: starting from the fundamental heat transfer equations, I discretized the governing equations using the finite difference method. The stability and convergence of the difference schemes were carefully considered. I then introduced three algorithms – macro shrinkage degree, macro-micro porosity degree, and dynamic grid – to enhance the prediction capabilities of the existing TMCAsT simulation platform. The primary goal was to increase both computational precision and efficiency, which are often conflicting requirements in numerical simulation.
The governing equation for three-dimensional transient heat conduction is given by:
$$ \rho c_p \frac{\partial T}{\partial t} = \lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + Q $$
where \(T\) is temperature, \(t\) is time, \(\rho\) is density, \(c_p\) is specific heat, \(\lambda\) is thermal conductivity, and \(Q\) represents the internal heat source, usually the latent heat of solidification. In my finite difference formulation, I used an explicit scheme. For a uniform grid spacing \(\Delta x = \Delta y = \Delta z\), the stability condition is:
$$ \frac{\lambda \Delta t}{\rho c_p (\Delta x)^2} \le \frac{1}{6} $$
This condition ensures that the numerical solution remains stable. In my simulations, I selected the time step \(\Delta t\) based on the most stringent stability requirement, using the liquid-phase heat capacity and the solid thermal conductivity for conservative estimates. This approach guarantees that the temperature field evolution is reliable, which is the foundation for accurate defect prediction.
Latent heat treatment is crucial for capturing the solidification plateau in temperature curves. I adopted the temperature recovery method, which balances the heat loss with the latent heat released during solid phase formation. For alloys with a narrow solidification range, this method is particularly effective. The temperature compensation due to latent heat is computed as:
$$ \Delta T_{recovery} = \frac{L \cdot \Delta f_s}{c_p} $$
where \(L\) is the latent heat, and \(\Delta f_s\) is the incremental solid fraction change during a time step. For the macro-micro coupled model, I refined this approach by separately considering the primary phase and the eutectic phase in an Al-7Si alloy. The solid fraction changes are calculated from nucleation and growth models, which are then fed back to the temperature field.
Macro Shrinkage Degree Model
The original shrinkage cavity prediction algorithm in my code compares the volume of the top pool cell with the cumulative shrinkage volume of the entire molten pool. If the shrinkage volume exceeds the cell volume, the top cell is marked as a cavity. However, this method often over-predicts the cavity depth and produces jagged boundaries because it is highly sensitive to the grid size. To overcome this, I introduced the concept of shrinkage degree – a fractional volume loss within a grid cell due to shrinkage.
In my new algorithm, at each time step, the shrinkage volume generated during that step is calculated and then equally distributed among the top pool cells. Each top cell receives an increment of shrinkage degree, which accumulates over time. When the cumulative shrinkage degree reaches 100%, the cell is converted to a full cavity, and the corresponding volume is subtracted from the total shrinkage. This real-time consumption of shrinkage avoids the artificial situation where cells that have solidified are considered completely full, which is unrealistic. The schematic representation of this algorithm is shown below.
To illustrate the improvement, consider two time steps. In the original algorithm, a shrinkage volume of 3.5 with a top cell volume of 3 would immediately mark the cell as fully void, ignoring the residual 0.5. In my new method, the shrinkage degree is assigned incrementally. For example, at time \(n\), the top pool volume is 3 and the shrinkage increment is 0.5, giving each of the 3 cells a shrinkage degree of 16.7%. At time \(n+1\), the top volume reduces to 1, and the shrinkage becomes 1.6, leading to a 100% degree on that cell, while the remaining cells retain partial degrees. This allows the defect boundary to be fitted more smoothly, independent of the grid size.
I implemented this algorithm in a three-dimensional ductile iron casting simulation. The experimental setup involved a wedge-shaped sand mold, and the material was QT450 ductile iron poured at 1360°C into a graphite mold. The simulation used a grid spacing of 2 mm. The results are summarized in Table 1.
| Algorithm | Height of shrinkage cavity (mm) | Width of shrinkage cavity (mm) | Boundary sharpness |
|---|---|---|---|
| Experimental measurement | 0.5 | 5.0 | Smooth |
| Original algorithm | 2.0 | 6.0 | Jagged |
| Shrinkage degree algorithm | 0.5 | 5.2 | Smooth |
The improved algorithm produced a cavity shape that matched the experimental cross-section much more closely. The shrinkage degree field, which ranges from 0 to 100%, provides a continuous distribution that can be contoured to represent the probable defect morphology. This is a significant advancement for sand foundry defects prediction, as it allows for more precise riser design and quality assessment.
Macro-Micro Coupled Porosity Degree Model
Porosity, or shrinkage micro-porosity, forms in the interdendritic regions when the liquid metal cannot feed the shrinking solid network. The Niyama criterion has been widely used, but it does not account for the actual pressure that drives feeding. I improved this by adopting a static pressure criterion, which compares the local pressure loss with the available metallostatic pressure. The improved criterion is:
$$ \frac{P \sqrt{G}}{\sqrt{R}} < K $$
where \(P\) is the local pressure, \(G\) is the temperature gradient, \(R\) is the cooling rate, and \(K\) is a critical value that depends on the alloy system. This criterion correctly predicts that a region with a high pressure loss but also a high available pressure may not form porosity, because the liquid can still flow.
To further refine porosity prediction, I couple the macroscopic temperature field with a microscopic cellular automaton (CA) model. The key challenge is the synchronization of time steps. The macro time step is typically larger than the micro time step. I introduced an operation order \(N_s\), which is the number of micro steps per macro step. In my implementation, the macro time step was 0.003 s, and the micro time step was 0.001 s, giving \(N_s = 3\). The cooling rate for the micro model is calculated from the macro temperature change over each macro step:
$$ \dot{T}_{micro} = \frac{\Delta T_{macro}}{\Delta t_{micro}} $$
This ensures that the micro simulation follows the real physical cooling path, including the recalescence and the plateau during eutectic solidification.
The micro model uses the Scheil-Gulliver equation for the primary phase solid fraction:
$$ f_{s}^P = 1 – \left( \frac{C_L}{C_0} \right)^{1/(k-1)} $$
where \(C_L\) is the liquid concentration, \(C_0\) is the initial concentration, and \(k\) is the partition coefficient. For eutectic growth, I used the Oldfield continuous nucleation model and the growth law:
$$ V = B \Delta T^2 $$
where \(B\) is a growth constant. The local solid fraction is updated based on the impingement model of Johnson-Mehl-Avrami, which gives:
$$ f_s = 1 – \exp\left( -\frac{4\pi}{3} N_E R^3 \right) $$
where \(N_E\) is the eutectic grain density and \(R\) is the grain radius. By tracking the solid fraction evolution, the latent heat release is computed and fed back to the macro temperature field.
The porosity degree is defined as the volume fraction of pores in a macro cell. I propose that when the pressure loss \(\Delta P\) exceeds the local pressure \(P\), the liquid cannot feed any further, and the subsequent shrinkage contributes directly to porosity. The pressure loss is modeled by Darcy’s law:
$$ u = \frac{K}{\mu} \left( \Delta P + \rho_L g \right) $$
where \(u\) is the interdendritic flow velocity, \(K\) is the permeability, and \(\mu\) is the dynamic viscosity. The velocity due to solidification contraction is related to the cooling rate and temperature gradient:
$$ u = \beta \frac{R}{G} $$
Combining these equations yields the pressure loss:
$$ \Delta P = \frac{\mu \beta R}{K G} $$
When \(\Delta P \ge P\), the porosity degree \(D_p\) accumulates according to:
$$ D_p = \alpha V (1 – f_s) \Delta T + \beta V \Delta f_s $$
where \(\alpha\) and \(\beta\) are the liquid shrinkage coefficient and solidification shrinkage coefficient, respectively.
I validated this model by pouring an Al-7Si cylindrical sample in a sand mold. The temperature curves measured with thermocouples were compared to my simulated curves. Table 2 lists the key temperatures and times of the primary and eutectic plateaus.
| Property | Experimental (s) | Simulation (s) |
|---|---|---|
| Primary plateau start | 12 | 12.5 |
| Primary plateau end | 85 | 88 |
| Eutectic plateau start | 140 | 143 |
| Eutectic plateau end | 230 | 235 |
The close agreement confirms that my temperature recovery model, which couples the micro solid fraction evolution, is accurate. Using this model, I simulated the growth of dendritic and eutectic structures for different numbers of nucleation cores. The porosity degree was found to vary with the number of cores: for 4 cores, the final solid fraction was 0.972 and the porosity degree was 0.028; for 10 cores, the solid fraction was 0.969 and porosity 0.031; for 16 cores, the solid fraction was 0.976 and porosity 0.024. These results indicate that a higher nucleation density generally leads to more dispersed micro-porosity, which is consistent with the physics of interdendritic feeding.
Dynamic Grid Technology
One of the main bottlenecks in high-resolution simulation of sand foundry defects is the computational cost. When the spatial step is halved, the number of cells increases eightfold in three dimensions, and the time step must be reduced by a factor of four due to stability conditions. This results in about a 30-50 times increase in computation time. To mitigate this, I developed a dynamic grid refinement technique. Instead of globally refining the entire mesh, I locally refine only the regions near the solid-liquid interface, where defects are likely to form. The temperature field is first computed on a coarse grid. At each time step, I detect the interface cells and subdivide them into smaller cells. The field values (temperature, solid fraction, etc.) for the new small cells are interpolated from the surrounding coarse cells using a linear interpolation scheme based on the lever rule.
For a one-dimensional case, the interpolated temperature \(T_i\) at a point dividing the segment between two coarse nodes with temperatures \(T_L\) and \(T_R\) into lengths \(a\) and \(b\) is:
$$ T_i = \frac{b T_L + a T_R}{a + b} $$
In three dimensions, I interpolate using the eight surrounding coarse nodes. This method is accurate when the property gradients are smooth, but it is not applied to the outermost cells near the mold boundary because the large property jumps between the casting and the mold could cause overshoots. Since shrinkage defects rarely occur in the first layer adjacent to the mold, this limitation is acceptable.
After local refinement, I perform the defect prediction on the fine cells. The shrinkage degree algorithm and the porosity criterion are applied to these fine cells, allowing for a much more accurate defect shape. After the defect evaluation, the cells that do not contain defects are coarsened back to the original grid, while the defect cells remain fine for storage and visualization. This dynamic process is repeated every time step.
I monitored the number of active cells during the simulation of a cubic casting. The results are shown in Figure 5-5 (not included here, but I present the data). The maximum increase in the number of cells was only about 7% compared to the coarse grid, whereas full global refinement would have increased the cell count by 800%. This demonstrates the efficiency of the method.
| Casting shape | Coarse grid time (s) | Full fine grid time (s) | Dynamic grid time (s) | Speed-up factor vs full fine |
|---|---|---|---|---|
| Cube | 9.72 | 552.62 | 22.53 | 24.5 |
| Step shape | 17.64 | 834.18 | 42.53 | 19.6 |
For the cube casting, the coarse grid (5 mm) took only 9.72 seconds, but the full fine grid (2.5 mm) took 552.62 seconds. With dynamic grid, the computation time was only 22.53 seconds, which is a 24.5-fold speed-up compared to the full fine grid while achieving the same resolution in the defect regions. Similar performance was observed for the step-shaped casting. This dynamic grid technology is therefore highly effective for practical sand foundry defect simulations where computational resources are limited.
Experimental Verification of Shrinkage Defects
To validate the macro shrinkage degree model, I conducted a ductile iron wedge casting experiment. The mold was a sand mold with a cavity width of 120 mm and a height of 60 mm. The charge consisted of 4584 g of pig iron, 1196.5 g of steel scrap, and a nodularizer addition of 104 g. The pouring temperature was 1360°C, and the mold was preheated to 200°C for 20 minutes. After solidification, the casting was sectioned along its centerline, and the shrinkage cavity was revealed by dye penetrant testing.
The comparison between the simulated and experimental results was presented in the previous section. The key finding was that the shrinkage degree algorithm produced a cavity depth of 0.5 mm, which matched the experimental measurement exactly, whereas the original algorithm gave a depth of 2 mm. The width was also closer to the experimental value. Furthermore, the shrinkage degree field allowed me to fit a smooth contour line that represented the actual cavity boundary, rather than a jagged grid-aligned boundary. This is crucial for sand foundry defects assessment because the riser size can be optimized to eliminate such defects.
The macro-micro porosity model was verified using an Al-7Si alloy cylindrical sample. The experimental temperature curves and the simulated curves agreed well, as shown in Table 2. The porosity degree predictions for different nucleation densities were qualitatively consistent with the expectation that a finer grain structure leads to more distributed micro-porosity. However, the model does not yet account for dissolved gas porosity, which is common in aluminum castings. Future work could include hydrogen diffusion and bubble nucleation to provide a more complete prediction.
The dynamic grid technology was tested on two simple shapes (cube and step). The accurate defect shape was obtained with a significantly reduced computational cost. This technology can be easily integrated into existing casting simulation software, making it a valuable tool for foundry engineers who need quick and precise defect predictions for sand castings of any size.
Conclusion
In my research, I have developed and validated three algorithmic models that significantly enhance the numerical simulation of sand foundry defects. The macro shrinkage degree model introduces a continuous defect indicator that improves the prediction of shrinkage cavity morphology and size. The macro-micro porosity degree model couples macroscopic heat transfer with microscopic solidification kinetics, enabling the prediction of micro-porosity distribution and degree. The dynamic grid technology offers a practical solution for reducing computation time while maintaining high accuracy in defect-prone areas.
Through systematic experiments on ductile iron and aluminum alloys, I have demonstrated that these models yield results consistent with real sand castings. The improvements in both accuracy and efficiency make these models suitable for industrial applications, where rapid and reliable defect prediction is essential for optimizing gating and risering systems. I believe that the integration of these models into commercial simulation packages can greatly aid foundry engineers in producing high-quality castings with minimal defects.
Future research directions include extending the macro-micro coupling to three-dimensional full-scale simulations, incorporating gas porosity and shrinkage interaction, and optimizing the dynamic grid refinement criteria based on local gradients. The ultimate goal is to create a fully coupled, high-fidelity simulation framework that can handle the complexities of industrial sand foundry defects with minimal computational expense.