This paper presents a systematic study on the structural optimization of metal core bars used in sand casting foundry. The traditional design of metal core bars relies heavily on empirical rules, leading to material waste and over-engineering. To address this issue, I propose a topology optimization method based on the Solid Isotropic Microstructures with Penalization (SIMP) interpolation model. The mathematical model, the Optimality Criteria (OC) solver, and the sensitivity filtering technique are discussed in detail. A custom topology optimization program is developed in C++. To validate the method, sand core specimens with different core bar configurations are tested for bending, compressive, and tensile strengths. The results show that the optimized core bars achieve nearly identical reinforcement effects as conventional cylindrical bars, while reducing the core bar mass by at least 59.9%. Furthermore, the proposed method is applied to two large industrial castings – a crown casting and a guide vane casting. Through casting numerical simulation, the stress conditions on the core bars are obtained and used as inputs for topology optimization. The optimized core bar structures are then implemented in actual production. The successful pouring and defect-free castings demonstrate that the optimized core bars satisfy the strength requirements of the sand cores, with mass reductions of 34.16% and 51.24% for the two cases, respectively. This work provides a new scientific basis for core bar design in sand casting foundry, contributing to lightweight and green manufacturing.
1. Introduction
Sand casting foundry is a fundamental manufacturing process for producing complex metallic components used in aerospace, automotive, energy, and heavy machinery industries. During the casting process, sand molds and cores must withstand various mechanical and thermal loads. In many large and complex castings, the use of metal core bars is an effective way to enhance the strength, permeability, and collapsibility of sand cores. However, the conventional design of metal core bars is often arbitrary and experience-driven. Designers typically use simple cylindrical bars without considering the actual stress distribution, resulting in over-designed structures that waste material and increase production costs.
In recent years, topology optimization has emerged as a powerful tool for determining the optimal material distribution within a given design domain. Unlike size and shape optimization, topology optimization allows the creation of new holes and the evolution of structural connectivity, thus providing the greatest potential for weight reduction. The goal is to find the lightest structure that satisfies specific mechanical performance requirements. Since its introduction by Bendsøe and Kikuchi in 1988, numerous topology optimization approaches have been developed, including homogenization, SIMP, ESO, level-set, moving morphable components, and phase-field methods. Among these, the SIMP method is widely used in engineering applications due to its simplicity, numerical stability, and ease of implementation.
Despite the extensive research on topology optimization, its application to casting process design – especially the optimization of metal core bars – has rarely been reported. The unique working condition of core bars inside sand cores, where the bars are subjected to hydrostatic pressure from molten metal, sand shrinkage, and thermal expansion, makes the optimization problem challenging. Moreover, the coupling between the core bar and the surrounding sand material must be considered. To fill this gap, I propose a metal core bar optimization framework that combines casting numerical simulation with SIMP-based topology optimization. The key idea is to obtain realistic loading conditions through numerical simulation of the filling and solidification process, then use these loads as inputs to topology optimization.
Figure above illustrates typical sand casting parts produced in a modern foundry. The complexity and size of these components necessitate careful design of the sand cores and the embedded metal core bars. In this work, I first present the mathematical foundation of the SIMP approach and the OC optimization algorithm. Then, I describe the development of a custom topology optimization program. Next, I validate the method through a series of mechanical tests on sand core specimens with different core bar configurations. Finally, I demonstrate the practical application of the method on two large industrial castings, where the optimized core bars are successfully used in actual production.
2. Topology Optimization Mathematical Model for Metal Core Bars
The topology optimization problem aims to find the optimal distribution of material within a design domain, denoted by $\Omega$, such that the overall compliance (or flexibility) is minimized under a volume constraint. In the context of metal core bars, the design domain represents the volume occupied by the core bar. The material distribution is described by a design variable $\rho_e$ associated with each finite element $e$, where $\rho_e = 0$ represents a void and $\rho_e = 1$ represents a solid region.
2.1 SIMP Material Interpolation Model
In the SIMP model, the elastic modulus of an element is expressed as a power-law function of the relative density:
$$ E_e = \rho_e^p E_0 $$
where $E_0$ is the elastic modulus of the solid material and $p$ is the penalization factor (typically $p=3$). This penalization drives the density values toward 0 or 1, thus suppressing intermediate densities and ensuring that the final design is physically realizable. The element stiffness matrix can be written as:
$$ \mathbf{k}_e = \rho_e^p \mathbf{k}_0 $$
where $\mathbf{k}_0$ is the element stiffness matrix for a solid element. Using the finite element method, the global stiffness matrix $\mathbf{K}$ is assembled as a sum of the element contributions:
$$ \mathbf{K} = \sum_{e=1}^{N} \rho_e^p \mathbf{k}_e^0 $$
where $N$ is the total number of elements in the design domain.
2.2 Optimization Problem Formulation
The objective is to minimize the structural compliance $C$, which is equivalent to maximizing the stiffness. The compliance is defined as:
$$ C = \mathbf{U}^T \mathbf{F} = \sum_{e=1}^{N} \rho_e^p \mathbf{u}_e^T \mathbf{k}_e^0 \mathbf{u}_e $$
where $\mathbf{U}$ is the global displacement vector, $\mathbf{F}$ is the external force vector, and $\mathbf{u}_e$ is the element displacement vector. The volume constraint limits the total material usage to a fraction of the original volume:
$$ \frac{\sum_{e=1}^{N} \rho_e v_e}{V^*} \le 1 $$
where $v_e$ is the volume of element $e$ and $V^*$ is the allowed volume after optimization. The complete optimization problem is formulated as:
$$ \begin{aligned}
& \underset{\rho}{\text{find}} && \rho_e, \quad e=1,\dots,N \\
& \underset{\rho}{\min} && C = \mathbf{U}^T \mathbf{F} \\
& \text{s.t.} && \mathbf{K} \mathbf{U} = \mathbf{F} \\
& && \sum_{e=1}^{N} \rho_e v_e \le V^* \\
& && 0 \le \rho_{\min} \le \rho_e \le 1
\end{aligned} $$
Here, $\rho_{\min}$ is a small positive value (e.g., $10^{-3}$) to avoid singularities in the stiffness matrix.
2.3 Optimality Criteria (OC) Solver
To solve the optimization problem efficiently, the Optimality Criteria method is employed. The Lagrangian function is constructed by adding the constraint terms with Lagrange multipliers:
$$ \mathcal{L} = C + \lambda_1 \left( \sum_{e=1}^{N} \rho_e v_e – V^* \right) + \lambda_2 (\rho_{\min} – \rho_e) + \lambda_3 (\rho_e – 1) + \mathbf{\lambda}_f^T (\mathbf{K}\mathbf{U} – \mathbf{F}) $$
At the optimum, the Kuhn-Tucker conditions must be satisfied. After simplifications, the update rule for the design variables is given by:
$$ \rho_e^{\text{new}} = \begin{cases}
\max(\rho_{\min}, \rho_e – m) & \text{if } \rho_e B_e^{\eta} \le \max(\rho_{\min}, \rho_e – m) \\
\min(1, \rho_e + m) & \text{if } \rho_e B_e^{\eta} \ge \min(1, \rho_e + m) \\
\rho_e B_e^{\eta} & \text{otherwise}
\end{cases} $$
where $m$ is a move limit (typically 0.2), $\eta$ is a damping factor (typically 0.5), and $B_e$ is derived from the sensitivity of the objective and the constraint:
$$ B_e = \frac{-\frac{\partial C}{\partial \rho_e}}{\lambda \frac{\partial V}{\partial \rho_e}} = \frac{p \rho_e^{p-1} \mathbf{u}_e^T \mathbf{k}_e^0 \mathbf{u}_e}{\lambda v_e} $$
The Lagrange multiplier $\lambda$ is determined by a bisection algorithm to satisfy the volume constraint.
2.4 Sensitivity Filtering
To avoid numerical instabilities such as checkerboard patterns and mesh dependency, a sensitivity filter is applied. The filtered sensitivity of element $e$ is computed as the weighted average of the raw sensitivities over a neighborhood $N_e$ within a radius $r_{\min}$:
$$ \widehat{\frac{\partial C}{\partial \rho_e}} = \frac{1}{\rho_e \sum_{f \in N_e} \hat{H}_f} \sum_{f \in N_e} \hat{H}_f \rho_f \frac{\partial C}{\partial \rho_f} $$
where $\hat{H}_f$ is the weight factor defined by:
$$ \hat{H}_f = \max(0, r_{\min} – \text{dist}(e,f)) $$
where $\text{dist}(e,f)$ is the distance between the centers of elements $e$ and $f$. The filtered sensitivity replaces the raw sensitivity in the OC update rule.
2.5 Topology Optimization Procedure
The overall algorithm is implemented in a custom C++ program. The procedure is as follows:
- Define the design domain and generate a finite element mesh.
- Initialize all element densities to 1 (i.e., fully solid).
- Set boundary conditions: fixed supports and external loads.
- Assemble the global stiffness matrix and solve the equilibrium equation $\mathbf{K}\mathbf{U} = \mathbf{F}$.
- Compute the compliance and the sensitivity of each element.
- Apply sensitivity filtering.
- Update the densities using the OC rule.
- Check convergence (e.g., the change in the objective function between two iterations is less than 0.01%). If not converged, go to step 4.
The program is tested on two simple cylindrical examples. The first example involves a cylindrical bar (diameter 20, length 80) fixed at one end with a downward force at the free end. The volume fraction is set to 0.4. The optimization converges in 26 iterations, and the resulting structure resembles a tapered beam that efficiently transmits the load. The second example uses the same cylinder but with both ends fixed and a central vertical load. After 43 iterations, the optimized structure develops a cross-like internal reinforcement. These examples demonstrate the correctness and stability of the developed program.
3. Experimental Verification of Sand Core Strength
To verify that the topology-optimized metal core bars do not degrade the mechanical performance of sand cores, I designed a series of experiments. The sand cores were made of self-hardening sand, consisting of 50-100 mesh silica sand, an inorganic binder (JNY-F61), and a hardener (JNY-30). The binder-to-sand ratio was 3:100, and the hardener-to-binder ratio was 18:100. Three types of specimens were prepared for each test: (1) no core bar, (2) a conventional cylindrical core bar of diameter 6 mm and length 46 mm, and (3) a topology-optimized core bar. The optimized core bars were manufactured by 3D printing using 316L stainless steel, as the topology results have complex shapes that cannot be easily machined by traditional methods. All specimens were cured for 24 hours before testing. A servo-hydraulic universal testing machine was used with a loading rate of 10 mm/min.
3.1 Bending Strength Test
The bending test was conducted using the three-point bending configuration. The sand core specimens had a length of 160 mm, a width of 20 mm, and a height of 22 mm. The loading condition for the topology optimization was designed as a simply supported beam with a concentrated load at the center, which matches the three-point bending setup. The optimization was performed with a volume constraint of 50%, and the resulting topology is shown in the earlier section. Since the raw optimized shape had sharp ends that could induce stress concentration in the sand, the ends were smoothed in the final design. The bending strength of each specimen was calculated from the measured maximum force. Table 1 summarizes the average bending strengths for the three groups.
| Specimen group | Average bending strength (MPa) | Relative change vs. no core bar |
|---|---|---|
| No core bar | 0.79 | – |
| Conventional core bar | 1.07 | +35.4% |
| Optimized core bar (before smoothing) | 0.93 | +17.7% |
| Optimized core bar (after smoothing) | 1.08 | +36.7% |
The initial optimized core bar reduced the bending strength compared to the conventional bar. After rounding the end surfaces, the bending strength increased to 1.08 MPa, which is even slightly higher than that of the conventional bar. The optimized core bar mass was 3.85 g, whereas the conventional bar weighed 9.62 g, corresponding to a 59.9% mass reduction. Therefore, the optimized core bar satisfies the strength requirement with a significantly lighter weight.
3.2 Compressive Strength Test
The compressive test used cylindrical sand core specimens of 20 mm diameter and 50 mm height. The same three groups were prepared. The average compressive strengths are listed in Table 2.
| Specimen group | Average compressive strength (MPa) |
|---|---|
| No core bar | 1.06 |
| Conventional core bar | 0.93 |
| Optimized core bar | 0.94 |
Both types of core bars caused a small reduction in compressive strength (about 12%). This is because the cracks tend to initiate and propagate along the longitudinal direction, and the presence of a cylindrical bar may act as an internal defect interface. However, the difference between the conventional and optimized bars is negligible, indicating that topology optimization does not impair the axial compressive behavior. In actual casting, the axial load on the core is usually small compared with the radial bending load, so this reduction is acceptable.
3.3 Tensile Strength Test
The tensile test used “8-shaped” standard sand core specimens according to the test standard. The average tensile strengths are given in Table 3.
| Specimen group | Average tensile strength (MPa) |
|---|---|
| No core bar | 0.86 |
| Conventional core bar | 0.84 |
| Optimized core bar | 0.84 |
There is no significant difference among the three groups, because the failure always occurs at the narrowest cross-section of the “8-shaped” specimen, and the core bar is not located in that section. Thus, the addition of core bars does not affect the tensile strength.
3.4 Summary of Experimental Results
The experiments confirm that the topology-optimized core bar provides a reinforcement effect comparable to that of a conventional cylindrical bar in bending, while the axial strengths remain nearly unchanged. The mass reduction of 59.9% is a substantial improvement for resource saving. The problem of sharp edges in the optimized design was solved by a simple post-processing step, which demonstrates the importance of considering manufacturability in topology optimization.
4. Application to Industrial Castings
After the successful validation at the specimen level, I applied the proposed method to real large castings in the sand casting foundry. Two cases are presented: a crown casting and a guide vane casting. Both castings are made of GX4CrNi13-4 stainless steel with a pouring temperature of 1580°C.
4.1 Crown Casting
4.1.1 Casting Design and Numerical Simulation
The crown casting has a maximum diameter of 2200 mm and a relatively uniform wall thickness. However, thick sections at the junction of ribs create hot spots. Using a distance-field algorithm, the hot spots were identified at the central disk area, and three risers were placed there. The core in the lower part of the casting is large and is subject to the weight of the upper part and the impact of the pouring liquid. To reinforce this core, a complex metal core bar framework had originally been designed. The original framework consisted of two circular rings connected by nine vertical bars, with the joints bound by steel wire. Its total mass was 41 kg. Although functional, this design was purely empirical and over-engineered.
To obtain the accurate loading conditions on the core bars, I performed a casting numerical simulation using the commercial software Huazhu CAE. The mesh size was 10 mm, creating 7,057,570 elements, of which 1,057,246 belonged to the casting. The simulation parameters are listed in Table 4.
| Parameter | Value |
|---|---|
| Density (g/cm³) | 7.01 |
| Thermal conductivity (W/(m·K)) | 0.09 |
| Heat capacity (J/K) | 0.44 |
| Latent heat (J/g) | 42.98 |
| Liquidus temperature (°C) | 1480.76 |
| Solidus temperature (°C) | 1411.06 |
The filling simulation revealed that the molten metal impinged mainly on three regions of the core surface, directly below the three risers. The solidification simulation showed a uniform shrinkage deformation of the casting, with a slight inward displacement of the inner cavity surfaces, which exerts additional pressure on the core. From these results, the dominant loads on the core bars were identified as three concentrated forces corresponding to the impingement locations.
4.1.2 Topology Optimization of the Core Bar Framework
Because the original framework was an assembly of separate bars, it was not suitable for direct topology optimization. I simplified the design domain by considering the entire framework envelope as a solid cylinder, thereby converting the discrete bar layout problem into a continuous topology optimization problem. The cylinder had the same overall envelope as the original framework. The three concentrated forces were applied at the load positions on the cylinder surface. The central axis was constrained as a fixed support, representing the central bar that holds the framework together. The volume constraint was set to 40% of the original domain.
The topology optimization converged after 163 iterations. The resulting topology showed three prominent structural members extending from the three load points toward the center, connected by a robust central node. Based on this topology, I designed a new core bar framework with three curved arms meeting at a central hub. The final fabricated structure is shown in the insets of the previous sections. The optimized framework mass was about 27 kg, a reduction of 34.16% compared with the original 41 kg.
4.1.3 Production Verification
The optimized metal core bar framework was placed in the sand core, and the casting was produced with the same process parameters as the baseline. During pouring, the sand core remained stable with no cracks or breakage. The final casting was inspected and found to be sound, meeting all quality requirements. This successful production proved that the topology-optimized core bar provides sufficient strength for the sand core and can be safely implemented in the sand casting foundry.
4.2 Guide Vane Casting
4.2.1 Casting Simulation and Optimization
The guide vane casting is a long prismatic component with a maximum length of 1450 mm. The original process used a riser in the middle and two side metal core bars to strengthen the sand mold. To optimize the core bar layout, I again performed numerical simulation with a mesh size of 3 mm, leading to 5,668,488 elements (casting elements: 1,632,252). The filling simulation indicated that the highest pressure occurred on the two side walls near the riser region. The casting deformation was uniform, with no significant local pressure.
The original core bar distribution consisted of eight separate longitudinal bars with no interconnection. I treated the entire sand core region as a cylindrical design envelope and applied two concentrated loads at the high-pressure zones, while fixing both ends. The topology optimization converged after 53 iterations. The resulting topology suggested a lattice-like structure with diagonal cross-braces. Based on this, I redesigned the core bar framework as a truss-like structure with longitudinal members and connecting diagonals. The final framework weighed 13 kg, compared with 28 kg of the original design, a 51.24% mass reduction.
4.2.2 Production Result
The optimized core bars were installed in the sand mold, and the guide vane casting was poured successfully. The mold and cores did not fail, and the cast product met the dimensional and metallurgical specifications. This further confirms the general applicability of the proposed topology optimization method for core bar design in sand casting foundry.
5. Conclusions and Outlook
In this work, I have developed a novel method for optimizing the structure of metal core bars in sand casting foundry based on topology optimization. The main conclusions are as follows:
- A topology optimization program based on the SIMP material interpolation model and the OC solver was successfully implemented. The program is capable of handling three-dimensional design domains and produces clear, manufacturable topologies.
- Sand core strength experiments demonstrated that optimized core bars, after proper edge smoothing, have a reinforcement effect equivalent to that of conventional cylindrical bars in bending. The compressive and tensile strengths of sand cores are not significantly affected by the optimization. The optimized core bar mass can be reduced by 59.9%.
- Combining casting numerical simulation with topology optimization allows a realistic representation of the loading conditions on core bars. The proposed procedure was applied to two industrial castings, resulting in mass reductions of 34.16% and 51.24% while maintaining the required core strength. Both castings were produced successfully, proving the engineering viability of the method.
Future research can extend the method to consider temperature-dependent material properties of the core bars, since the elastic modulus changes at elevated temperatures. In addition, a more accurate multi-physics simulation of the sand core itself (including heat transfer and stress) would provide even more precise boundary conditions for the topology optimization. Finally, the optimization of very slender core bars (length-to-diameter ratio greater than 20:1) still suffers from numerical instabilities; improvements to the interpolation model or the regularization scheme are needed to address this limitation.
In summary, the integration of topology optimization into the design of metal core bars offers a scientific and efficient approach to lightweight core reinforcement in sand casting foundry. This contributes to reducing material consumption, lowering production costs, and promoting sustainable manufacturing.