My research focuses on developing a systematic methodology for optimizing metal core bar structures used in sand casting. The motivation arises from the fact that traditional metal core bar design relies almost entirely on empirical experience and often employs simple cylindrical geometries that, while functional, lead to material overuse and increased production costs. In modern sand casting, complex castings demand sand molds and cores with varying strength, collapsibility, and gas generation requirements across different sections. Metal core bars are instrumental in reinforcing sand cores, enhancing permeability, and improving collapsibility. However, without a scientific basis for material distribution, the design of core bars in intricate mold assemblies remains suboptimal. To address this gap, I introduced topology optimization—a method that seeks optimal material distribution under given mechanical performance constraints—into the realm of metal core bar design. This integration aims to reduce material consumption while maintaining or even improving the mechanical performance of sand cores, ultimately mitigating sand casting defects such as mold wall movement, core deformation, and cracking during pouring. My work encompasses mathematical modeling, algorithm development, experimental validation, and real-world industrial applications, as detailed in the following sections.
1. Introduction and Research Background
Sand casting remains one of the most versatile and widely adopted manufacturing processes, particularly for producing large and complex components in industries such as automotive, aerospace, energy, and heavy machinery. The process involves creating a mold from sand, pouring molten metal into the cavity, and allowing it to solidify. While sand casting offers significant advantages—low cost, adaptability to various alloys, and the ability to form intricate geometries—it also presents considerable challenges. Sand molds and cores must withstand thermal, mechanical, and chemical loads during pouring and solidification. Any failure in the mold or core can directly lead to sand casting defects including sand wash, erosion, mold penetration, and core shift, which compromise the quality and integrity of the final casting.
One of the critical strategies to reinforce sand cores is the incorporation of metal core bars. These bars enhance the structural integrity of cores, improve gas permeability, and facilitate easier collapsibility after solidification. Traditional core bar design, however, has seen little innovation. Designers typically resort to simple, uniform cylindrical bars, positioning them based on intuition rather than rigorous analysis. For complex and large-scale castings, this approach often results in significant material waste and, paradoxically, may not achieve the optimal reinforcement required for specific loading scenarios. The core bar structure might be over-engineered in some regions while under-performing in others, the latter potentially giving rise to sand casting defects.
The advent of structural optimization techniques, particularly topology optimization, offers a promising avenue to overcome these limitations. Topology optimization is a mathematical approach that optimizes material layout within a given design space, for a given set of loads, boundary conditions, and constraints, with the goal of maximizing the system’s performance. Since its inception in the 1980s, topology optimization has evolved significantly, branching into various methods such as homogenization, density-based approaches like SIMP (Solid Isotropic Material with Penalization), evolutionary strategies, level-set methods, and moving morphable components. Among these, the SIMP method has gained widespread acceptance in engineering due to its computational efficiency, ease of implementation, and robustness. By applying topology optimization to metal core bar design, it becomes possible to generate structures that exactly match the strength requirements of sand cores, thereby reducing weight and material consumption, while maintaining the necessary mechanical support to prevent sand casting defects.
In this research, I propose a comprehensive method that integrates topology optimization with casting process simulation. The workflow begins by obtaining the stress conditions on sand cores during the casting process via numerical simulation. These conditions serve as the input for topology optimization, which then determines the optimal distribution of metal core bar material. The optimized designs are subsequently validated through mechanical strength tests and actual casting trials. This methodology not only provides a theoretical basis for core bar design but also offers a new perspective on reducing sand casting defects in practical production.
2. Topology Optimization Mathematical Model
Topology optimization for metal core bars is fundamentally a discretized problem of determining whether or not material should exist at each point within a design domain. However, solving such 0-1 discrete problems at scale is computationally prohibitive. The density-based approach circumvents this by relaxing the design variables to continuous values within the range [0, 1], thereby transforming the problem into a continuous optimization task that can be solved using standard mathematical programming algorithms. In the SIMP method, a material interpolation scheme relates the material properties of each element to its relative density.
The core of the SIMP model is the relationship between the Young’s modulus of an element, E(x), and its relative density, ρ(x), expressed as:
$$E(x) = \rho(x)^p E_0$$
where E₀ is the material’s actual Young’s modulus and p is the penalization factor. The penalization factor, which is typically greater than or equal to 3, serves to discourage intermediate densities and drive the solution towards a clear 0-1 distribution, thus aligning the continuous optimization result with the practical discrete requirement. Based on this, the elemental stiffness matrix is scaled similarly:
$$k_e(\rho_e) = \rho_e^p k_0$$
where k₀ is the stiffness matrix of a fully solid element.
The objective of the topology optimization problem is to minimize the overall compliance (or maximize the stiffness) of the metal core bar structure. This is subject to a resource constraint, typically the total volume or mass of the structure. The optimization problem statement is given by:
Minimize:
$$C(\rho) = \mathbf{U}^T\mathbf{F} = \sum_{e=1}^{N} (\rho_e)^p \mathbf{u}_e^T \mathbf{k}_0 \mathbf{u}_e$$
Subject to:
$$\mathbf{K}(\rho)\mathbf{U} = \mathbf{F}$$
$$V(\rho) = \sum_{e=1}^{N} \rho_e v_e \leq V^*$$
$$0 < \rho_{min} \leq \rho_e \leq 1$$
Here, C(ρ) is the objective function representing compliance, U is the global displacement vector, F is the applied force vector, K is the global stiffness matrix assembled from the elemental stiffness matrices, uₑ is the elemental displacement vector, vₑ is the elemental volume, V* is the prescribed volume constraint, and ρ_min is a small positive number to avoid singularity in the stiffness matrix.
For the solution of this optimization problem, I employed the Optimality Criteria (OC) method, which is well-suited for problems with a single constraint and simple objective functions. The OC method derives heuristic update rules for the design variables based on the stationary conditions of the Lagrangian function. The Lagrangian for the minimization problem is formulated as:
$$L = C(\rho) + \lambda_1(V – V^*) + \lambda_2^T(\mathbf{K}\mathbf{U} – \mathbf{F}) + \sum_{e=1}^{N} \lambda_3^e(\rho_{min} – \rho_e) + \sum_{e=1}^{N} \lambda_4^e(\rho_e – 1)$$
where λ₁, λ₂, λ₃, and λ₄ are Lagrange multipliers for the respective constraints. At the optimum, the derivative of the Lagrangian with respect to the design variable should vanish. Solving these equations yields the optimality condition:
$$B_e = \frac{-\partial C / \partial \rho_e}{\partial V / \partial \rho_e} = \lambda_1$$
This condition implies that the ratio of sensitivity of compliance to volume is constant for all elements. In practice, a heuristic scheme updates the densities based on this condition. The update formula is:
$$\rho_e^{new} = \begin{cases} \max(\rho_{min}, \rho_e – m) & \text{if } \rho_e B_e^{\eta} \leq \max(\rho_{min}, \rho_e – m) \\ \rho_e B_e^{\eta} & \text{if } \max(\rho_{min}, \rho_e – m) < \rho_e B_e^{\eta} < \min(1, \rho_e + m) \\ \min(1, \rho_e + m) & \text{if } \rho_e B_e^{\eta} \geq \min(1, \rho_e + m) \end{cases}$$
where m is the move limit, which restricts the change in density in each iteration to ensure stability, and η is a numerical damping factor, typically set to 0.5.
One of the well-known issues with density-based topology optimization is the appearance of numerical instabilities, most notably checkerboard patterns and mesh dependency, which render the optimized results unsuitable for manufacturing. To overcome these issues, I adopted a mesh-independency filtering technique, specifically the sensitivity filter. The sensitivity of the compliance with respect to the density variable is modified by a weighted average of the sensitivities in a fixed neighborhood. The filtered sensitivity is given by:
$$\widehat{\frac{\partial C}{\partial \rho_e}} = \frac{1}{\rho_e \sum_{f=1}^{N} \hat{H}_f} \sum_{f=1}^{N} \hat{H}_f \rho_f \frac{\partial C}{\partial \rho_f}$$
where the convolution operator, \hat{H}_f, is defined as:
$$\hat{H}_f = \max(0, r_{min} – \text{dist}(e,f))$$
Here, r_min is the filter radius and dist(e,f) is the distance between the centers of elements e and f. This filtering process smooths the sensitivity field, effectively eliminating checkerboard patterns and reducing mesh dependency.
Table 1 provides a comparison of various topology optimization methodologies, highlighting the selection rationale for the SIMP method in this work.
| Method | Computational Efficiency | Stability | Optimized Result | Application |
|---|---|---|---|---|
| Homogenization | High | Good | Contains many micro-structures | Composite materials |
| Density-based (SIMP) | High | Fair | Checkerboards possible; zigzag boundaries | Widely used in engineering |
| ESO | Low | Poor | Zigzag boundaries | Static and dynamic problems |
| ICM | Low | Poor | Zigzag boundaries | Static and dynamic problems |
| MMC | High | Good | Smooth boundaries | Rarely applied |
| Level-set | Low | Good | Smooth boundaries | Rarely applied |
Based on this comparison, the SIMP method was selected for its balance of computational efficiency, ease of implementation, and robust performance across various applications. The entire topology optimization procedure developed for metal core bar design is summarized in the algorithm flow presented in Table 2.
| Step | Action |
|---|---|
| Initialization | Assign density values to all elements; define design domain, boundary conditions, and loads. |
| FEA Solve | Assemble global stiffness matrix, solve for displacements and strains. |
| Objective | Calculate the total compliance of the metal core bar structure. |
| Sensitivity | Compute compliance sensitivity with respect to each element’s density. |
| Filter | Apply the sensitivity filter to suppress numerical instabilities. |
| Update | Use the Optimality Criteria algorithm to update element densities. |
| Convergence | Check if the change in objective or design variables falls below a defined threshold. If not, return to Step 2. |
Through this algorithm, I was able to perform topology optimization for various core bar geometries. For the initial verification, an elementary cylindrical model with a diameter of 20 units and length of 80 units was used. The cylinder was subjected to fixed constraints at one end and a concentrated vertical load at the other end. The volume constraint was set to 40% of the original volume. The optimization converged after 26 iterations, producing a lightweight structure that maintained the required stiffness. This simple case demonstrated the capability of the program and laid the groundwork for more complex optimization tasks.
3. Experimental Verification of Optimized Core Bars
To ascertain the practical applicability of the optimized metal core bars, I conducted a series of mechanical strength tests on sand cores. The objective was to compare the performance of sand cores with no core bars, with traditional cylindrical core bars, and with topology-optimized core bars. I designed three distinct experiments: a three-point bending test for radial strength, a compressive test for axial strength, and a tensile test for axial strength. In all experiments, the core bars were fabricated using 316L stainless steel via 3D printing to achieve the complex geometries defined by the topology optimization. The sand used was a self-hardening sand mixture, consisting of 50-100 mesh silica sand, an inorganic binder (JNY-F61), and a curing agent (JNY-30), with a mass ratio of 100:3:0.54. The dimensions of the test specimens varied according to the test type. For each test group, a sufficient number of samples (ranging from 20 to 30) were fabricated to ensure statistical reliability.
3.1 Three-Point Bending Test
The three-point bending test was used to evaluate the radial strength of the sand cores. The specimen was a rectangular bar with dimensions 160 mm × 20 mm × 22 mm. The test setup is illustrated in Figure 1. The loading was applied at a constant speed of 10 mm/min using a servo-controlled universal testing machine.
For the topology optimization of the core bar used in this test, I simplified the problem to a beam with fixed supports at both ends and a concentrated load at its center. After 57 iterations, the optimization yielded a structure that tapered towards the ends and thickened at the center. While this shape was highly efficient in terms of material usage, the initial optimized core bar had sharp edges at its ends, which acted as stress concentrators within the sand core, leading to premature failure. This was apparent in the initial experiment results, where the sand cores with the optimized core bars (before edge smoothing) showed a lower average bending strength of 0.93 MPa, compared to 1.07 MPa for those with traditional cylindrical bars. To address this, I performed a post-processing step to round the sharp edges of the optimized core bar. After this edge smoothing, the bending strength of the sand cores with the optimized core bars increased significantly to 1.08 MPa. This value was comparable to, and even slightly better than, the strength provided by traditional core bars. The comparative results are summarized in Table 3.
| Specimen Group | Average Bending Strength (MPa) | Relative Change vs. No Bar |
|---|---|---|
| No core bar | 0.79 | – |
| Traditional core bar | 1.07 | +33.58% |
| Optimized core bar (before smoothing) | 0.93 | +16.27% |
| Optimized core bar (after smoothing) | 1.08 | +34.94% |
The most significant finding was that the optimized core bar, after edge smoothing, had a mass of 3.85 grams, compared to 9.62 grams for the traditional core bar—a reduction of 59.9%. This demonstrates that a significant material saving can be achieved without any loss in mechanical performance, effectively preventing any performance-induced sand casting defects.
3.2 Compressive Test
The compressive test was designed to evaluate the axial strength of the sand cores. Cylindrical sand core specimens with a diameter of 20 mm and height of 50 mm were prepared. The core bar used inside was identical to the one used in the bending test. The compressive test involved 30 samples per group. The results, presented in Table 4, showed an interesting phenomenon.
| Specimen Group | Average Compressive Strength (MPa) |
|---|---|
| No core bar | 1.06 |
| Traditional core bar | 0.93 |
| Optimized core bar | 0.94 |
Contrary to the bending test, the addition of metal core bars, whether traditional or optimized, led to a slight decrease in compressive strength (~12.3% and ~11.3%, respectively). This is attributed to the fact that the core bars align parallel to the direction of the compressive load, potentially acting as initiation sites for vertical cracks. Nevertheless, the decrease was consistent for both types of bars, reaffirming that the optimized core bar did not compromise axial strength beyond what is already expected from the traditional design. In real casting scenarios, the axial loads on cores are substantially smaller than radial loads, and thus this minor reduction is acceptable and does not contribute to sand casting defects in practice.
3.3 Tensile Test
The tensile test utilized standard “8-shaped” sand core specimens to evaluate bonding strength under axial tension. Similar to the compressive test, the results, as shown in Table 5, revealed that the addition of core bars had a negligible effect on tensile strength.
| Specimen Group | Average Tensile Strength (MPa) |
|---|---|
| No core bar | 0.86 |
| Traditional core bar | 0.84 |
| Optimized core bar | 0.84 |
The fracture in the tensile specimens consistently occurred at the narrowest cross-section of the “8-shaped” specimen, perpendicular to the core bar direction. Since the core bars did not bridge this critical region, they had minimal influence on the tensile strength. This result confirms that the topology-optimized core bars behave exactly like the traditional bars regarding axial tensile performance.
4. Application in Large Castings
Having validated the proposed method in standard laboratory tests, I proceeded to apply it to actual industrial castings. The first case study involved a crown casting, while the second involved a guide vane casting. Both castings required robust metal core bar structures to support their sand cores during pouring. A critical challenge in this application was determining the realistic loading conditions on the metal core bars. To achieve this, I employed the commercial casting simulation software “InteCAST” to simulate the mold filling and solidification processes. The simulation outcomes, including fluid pressure distributions and casting deformation tendencies, were used to define the boundary conditions for the topology optimization of the core bars.
4.1 Case Study: Crown Casting
The crown casting had a maximum diameter of 2200 mm. The original process used a core bar structure consisting of two rings and nine columns, weighing 41 kg. Although this design was functional, it was over-engineered and assembled from discrete rods tied together, which limited its overall structural efficiency. The casting material was GX4CrNi13-4 stainless steel, poured at 1580°C. The chemical composition is given in Table 6.
| C | Si | Mn | S | P | Cr | Ni | Mo | P+S |
|---|---|---|---|---|---|---|---|---|
| 0.06 | 1.0 | 1.0 | 0.025 | 0.035 | 12.0-13.5 | 3.5-5.0 | 0.7 | 0.045 |
The simulation, performed with a mesh size of 10 mm, resulted in over 7 million elements and provided detailed insights into the filling behavior. The results indicated that the primary impact of the molten metal on the mold floor occurred directly beneath the three risers, positioned at the vertices of a triangle. This localised impact was identified as the dominant loading case for the core bars. The material properties used in the simulation are listed in Table 7.
| Density (g/cm³) | Thermal Conductivity (W/(m·K)) | Specific Heat (J/K) | Latent Heat (J/g) | Liquidus (℃) | Solidus (℃) |
|---|---|---|---|---|---|
| 7.01 | 0.09 | 0.44 | 42.98 | 1480.76 | 1411.06 |
To perform topology optimization on the core bar distribution, I converted the original frame structure concept into a single solid cylinder that represented the entire design space. The concentrated impact forces from the molten metal were applied at points on this cylinder corresponding to the riser positions. A fixed constraint was applied along the central axis to represent the fastening effect of the original structure. After 163 optimization iterations, the algorithm yielded a clear material distribution that emphasized a triangular support structure connecting the three loaded regions to the central constraint. Based on this result, I designed a new, integrated core bar structure. The actual casting was then produced using this new structure. The optimization achieved a weight reduction of 34.16%, bringing the core bar mass down to approximately 27 kg. The pouring process proceeded smoothly, and the final casting was complete and defect-free. This confirmed that the optimized core bars adequately supported the sand cores, effectively preventing sand casting defects such as core shift and mold deformation.
4.2 Case Study: Guide Vane Casting
In the second case, a guide vane casting with a maximum length of 1450 mm was studied. The original design utilized eight slender cylindrical rods to reinforce the sand core, weighing a total of 28 kg. The simulation of this casting, using a 3 mm mesh size, revealed that the pressure on the side walls was most concentrated near the risers located in the middle of the casting. The entire core bar frame was transformed into a columnar design domain. The fill-induced pressures were applied to the corresponding areas, and fixed supports were added at both ends. After 53 iterations, the topology optimization produced a truss-like structure that efficiently transferred the loads to the supports. The resulting design, which was lighter by 51.24% (weighing only 13 kg), maintained sufficient strength, as evidenced by a successful casting run without any signs of sand core failure. The comparison of the original and optimized core bars for both castings is summarized in Table 8.
| Casting | Original Weight (kg) | Optimized Weight (kg) | Weight Reduction (%) | Outcome |
|---|---|---|---|---|
| Crown | 41 | 27 | 34.16% | Successful pour |
| Guide Vane | 28 | 13 | 51.24% | Successful pour |
These industrial validations not only demonstrate the practicality of the topology optimization method for metal core bars but also highlight a significant step forward in minimizing sand casting defects by ensuring that the core structure is optimally designed to withstand the specific loads it is subjected to, thereby enhancing the overall quality and reliability of the casting process.
5. Conclusions and Future Perspectives
In this research, I have successfully developed and validated a systematic method for the structural optimization of metal core bars in sand casting, based on topology optimization. The main conclusions are drawn as follows:
First, I established a topology optimization framework using the SIMP material interpolation model and the OC solution algorithm. I developed a dedicated C++ program capable of generating efficient core bar structures. The numerical instabilities inherent in density-based methods, such as checkerboard patterns, were effectively mitigated through the implementation of sensitivity filtering.
Second, the mechanical performance of sand cores containing the optimized core bars was thoroughly evaluated. Through bending, compressive, and tensile tests, I confirmed that the optimized core bars, after appropriate edge smoothing, provide strength enhancement equivalent to traditional cylindrical bars. Crucially, a material savings of up to 59.9% was achieved, which was highly promising for cost and resource efficiency.
Third, the integrated methodology, combining casting numerical simulation with topology optimization, was successfully applied to two large-scale industrial castings. The optimized core bar structures, which were designed to withstand the specific pressures from the molten metal during pouring, proved to be effective in production. The weight reductions of 34.16% and 51.24% for the crown and guide vane castings, respectively, clearly demonstrate the potential of this method to reduce waste and lower production costs.
Looking forward, there are several promising avenues for future research. The current topology optimization models assume a constant material elastic modulus. However, in casting, the high temperatures significantly affect material properties. Future work could integrate temperature-dependent material models into the optimization algorithm to achieve even higher accuracy. Additionally, as numerical simulation techniques for sand cores become more mature, obtaining the actual stress states on the core bars directly from detailed sand core simulations will enable more precise topology optimization conditions. Finally, extending the optimization to very slender core bars, where the current algorithm can be unstable, represents an important technical challenge to address.
In summary, this research introduces a scientific approach to metal core bar design, replacing empirical methods with an engineering-driven optimization process. The demonstrated success in both experimental and industrial settings underscores the value of this approach in improving casting quality, reducing material waste, and effectively mitigating sand casting defects. I believe this methodology will serve as a valuable reference for future process optimization in the foundry industry, promoting more sustainable and intelligent manufacturing practices.