In the aerospace industry, titanium alloy structural components serve as critical load-bearing and force-transmitting elements, and their dimensional precision requirements are extremely strict. Investment precision casting, especially lost wax casting, is the core process for near-net-shape manufacturing of such complex components. In the lost wax casting workflow, the dimensional deviation, surface quality, and structural stability of the wax pattern directly determine the final precision of titanium alloy castings. The preparation of wax patterns relies on injection molding. Traditional precision control depends on trial-and-error, adjusting injection temperature, pressure, and other parameters over three to five rounds, which is time-consuming and costly. To address this, numerical simulation has been widely used to optimize wax injection molding. By predicting melt filling, cooling shrinkage, and warpage deformation, numerical simulation can reduce trial runs by more than 60% and significantly shorten development cycles. However, simulation accuracy strongly depends on the accuracy of material thermophysical parameters. Wax is an organic small-molecule material that exhibits shear-thinning behavior, making it a typical non-Newtonian fluid. Its viscosity is the core indicator of flowability and significantly affects filling completeness, shrinkage uniformity, and warpage deformation.
Currently, models describing non-Newtonian viscosity include the Carreau-Yasuda model, the Williams-Landel-Ferry (WLF) model, and the Cross-WLF model. The Carreau-Yasuda model only reflects the effect of shear rate on viscosity and cannot adapt to the wide-temperature rheological behavior of wax. The WLF model considers temperature sensitivity but ignores the effect of shear rate, so it has limitations in describing shear-thinning phenomena of non-Newtonian fluids. The Cross-WLF model combines the Cross shear-thinning term with the WLF temperature-sensitive term, and it can simultaneously describe the viscosity characteristics of wax under different shear rates and temperatures, becoming the optimal choice for wax rheological characterization. However, commercial databases such as Moldflow only contain rheological data for traditional thermoplastic polymers and lack viscosity model parameters for paraffin and wax-based composite materials. Therefore, the seven key parameters of the Cross-WLF model must be obtained through experimental modeling. This modeling process is a high-dimensional nonlinear optimization problem, and its accuracy faces significant challenges. Simple least squares methods are prone to falling into local optima, while Newton methods have high computational complexity and numerical instability.
In this work, I took Kc-4017b wax as the research object and proposed a quasi-Newton method that integrates automatic differentiation and singular value decomposition (SVD) regularization for viscosity modeling. I conducted numerical simulations of the filling process of an annular stepped wax pattern and combined them with actual injection molding experiments for verification. The results show that the coefficient of determination of the quasi-Newton method is \(R^2 = 0.989\), which is significantly higher than that of the least squares method (\(R^2 = 0.901\)), and the residual distribution is more uniform. The Moldflow simulation results corresponding to its fitted parameters show a high degree of consistency with the displacement trend measured by blue light scanning, enabling accurate identification of the key regions of wax pattern deformation. This work solves the difficult problem of high-dimensional nonlinear parameter fitting for the wax Cross-WLF model, provides reliable parameter support for the numerical simulation of wax pattern injection molding in titanium alloy lost wax casting, and offers a theoretical foundation for the investment precision casting process of titanium alloy structural components.
I structured this article as follows. First, I present the experimental methods and mathematical models, including the wax pattern geometry, the wax material, the actual injection experiments, and the numerical simulation parameter settings. Second, I describe the Cross-WLF viscosity model and the proposed quasi-Newton parameter fitting method. Third, I report the viscosity modeling results and compare them with the least squares method. Fourth, I discuss the Moldflow simulation results based on the fitted parameters and verify them against actual injection experiments. Finally, I summarize the main conclusions and the implications for lost wax casting.
For the wax pattern geometry, I designed a multi-wall-thickness annular stepped component to study the influence of viscosity on the injection filling process of variable-thickness structures. The annular stepped wax pattern represents typical features of aerospace titanium alloy structural components, such as complex annular shapes and variable wall thickness. The geometry includes multiple steps with different thicknesses, which create distinct filling, cooling, and shrinkage behaviors. The injection gate is located at the center, and the gate diameter is 8 mm. The mold temperature is maintained at 25 °C, and the wax temperature is 65 °C. The packing pressure is 1 MPa, and the packing time is 60 s. These parameters are consistent with the actual injection molding experiments.
The wax material used in this work is Kc-4017b, a specialized wax for investment casting. Its density is 0.95 g/cm³, its drop point is 94 °C, and its softening point is 79 °C. The viscosity test data were obtained at 60 °C, 62 °C, and 64 °C. At each temperature, the viscosity decreases rapidly with increasing shear rate and then gradually levels off, showing typical non-Newtonian shear-thinning behavior. Moreover, at higher temperatures, the viscosity at the same shear rate is lower, indicating that temperature has a significant effect on wax flowability. Therefore, viscosity modeling must consider both shear rate and temperature effects. Table 1 summarizes the main material properties and test conditions.
| Property or parameter | Value | Unit |
|---|---|---|
| Wax material | Kc-4017b | — |
| Density | 0.95 | g/cm³ |
| Drop point | 94 | °C |
| Softening point | 79 | °C |
| Viscosity test temperatures | 60, 62, 64 | °C |
| Injection wax temperature | 65 | °C |
| Mold temperature | 25 | °C |
| Packing pressure | 1 | MPa |
| Packing time | 60 | s |
| Gate diameter | 8 | mm |
| Mesh edge length | 2 | mm |
| Boundary heat transfer coefficient | 2500 | W/(m²·K) |
For the actual injection experiments, I used an MPI-25 t wax injection machine. The wax temperature was 65 °C, the mold temperature was 25 °C, the packing pressure was 1 MPa, and the packing time was 60 s. The gate diameter was 8 mm. To ensure the accuracy of the experimental results, I repeated the injection under the same process conditions five times. After the injection experiments, I used blue light scanning to record the dimensional changes of the wax pattern and compared them with the simulation results. The numerical simulation was performed in Moldflow. I imported the three-dimensional geometric model of the annular stepped wax pattern into Moldflow and generated a dual-domain mesh. The global edge length of the mesh was set to 2 mm. After meshing, I systematically checked the mesh quality, including aspect ratio and element overlap ratio, to ensure that the key indicators met the accuracy requirements of Moldflow numerical simulation. The simulation parameters were set to be identical to the experimental parameters: wax temperature 65 °C, mold temperature 25 °C, packing pressure 1 MPa, packing time 60 s, gate diameter 8 mm, and wax-mold boundary heat transfer coefficient 2500 W/(m²·K). The numerical simulation analysis sequence was set to a coupled “cooling-filling-packing-warping” analysis, covering the entire physical process of wax injection molding. This sequence simulates the filling flow of the melt in the mold cavity, the pressure transmission during the packing stage, the temperature field evolution during cooling and solidification, and the final warpage deformation caused by uneven shrinkage. For the viscosity setting, I used the Cross-WLF shear-thinning viscosity model, which is consistent with the real viscosity characteristics of the wax. I imported the fitted parameters of the viscosity model into the Moldflow software database to ensure the accuracy of the numerical simulation.
The Cross-WLF model is composed of the Cross shear-thinning model and the WLF temperature-sensitive equation, and it can fully describe the quantitative relationship between viscosity, shear rate, and temperature. The viscosity-shear rate relationship of the Cross model is given by:
$$
\eta = \frac{\eta_0}{1 + \left( \frac{\eta_0 \dot{\gamma}}{\tau^*} \right)^{1-n}}
$$
where \(\eta\) is the apparent viscosity (Pa·s), \(\eta_0\) is the zero-shear viscosity (Pa·s), i.e., the viscosity when the shear rate approaches zero, \(\dot{\gamma}\) is the shear rate (s⁻¹), \(\tau^*\) is the characteristic relaxation parameter (Pa), which reflects the molecular chain relaxation time, and \(n\) is the non-Newtonian index (dimensionless). When \(n < 1\), the material exhibits shear-thinning behavior, and a smaller \(n\) indicates a more significant shear-thinning effect.
The zero-shear viscosity-temperature relationship of the WLF equation is:
$$
\eta_0 = D_1 \exp \left( -\frac{A_1 (T – T^*)}{A_2 + (T – T^*)} \right)
$$
and the glass transition temperature is expressed as:
$$
T^* = D_2 + D_3 p
$$
where \(D_1\) is the zero-shear viscosity coefficient at the reference temperature (Pa·s), \(D_2\) is the glass transition temperature at low pressure (K), which determines the critical temperature for the material to transition from the glassy state to the high-elastic state, \(A_1\) and \(A_2\) are WLF equation parameters (dimensionless), which reflect temperature sensitivity and temperature shift characteristics, respectively, \(T\) is the thermodynamic temperature (K), \(T^*\) is the glass transition temperature (K), \(p\) is the pressure (Pa), and \(D_3\) is the pressure coefficient. For small-molecule non-Newtonian fluids such as paraffin, \(D_3\) is generally taken as 0. Combining the Cross model and the WLF equation, the Cross-WLF model fully describes the nonlinear variation of wax viscosity through seven parameters: \(n\), \(\tau^*\), \(D_1\), \(D_2\), \(D_3\), \(A_1\), and \(A_2\). Table 2 summarizes the physical meaning and typical ranges of these parameters.
| Parameter | Physical meaning | Role in the model |
|---|---|---|
| \(n\) | Non-Newtonian index | Controls the shear-thinning slope; \(n < 1\) indicates shear thinning. |
| \(\tau^*\) | Characteristic relaxation parameter | Sets the shear stress scale for the transition from Newtonian to shear-thinning behavior. |
| \(D_1\) | Zero-shear viscosity coefficient | Scales the overall zero-shear viscosity level. |
| \(D_2\) | Glass transition temperature at low pressure | Defines the temperature at which the material becomes glassy. |
| \(D_3\) | Pressure coefficient | Accounts for pressure dependence of \(T^*\); for wax, \(D_3 \approx 0\). |
| \(A_1\) | WLF temperature sensitivity parameter | Controls the strength of the temperature dependence. |
| \(A_2\) | WLF temperature shift parameter | Shifts the temperature response relative to \(T^*\). |
Most existing studies use simple least squares methods for Cross-WLF model parameter fitting. However, the least squares method is only suitable for linear models. For the strongly nonlinear and coupled parameter model of wax viscosity, it has large errors and poor adaptability, making it difficult to meet the accuracy requirements of wax injection numerical simulation. Newton’s method, as a typical second-order optimization method, uses the iterative formula:
$$
x_{k+1} = x_k – H^{-1}(x_k) \nabla f(x_k)
$$
where \(\nabla f(x_k)\) is the gradient and \(H(x_k)\) is the Hessian matrix. By using the first-order derivative (gradient) and second-order derivative (Hessian matrix) information of the objective function to construct a quadratic model, Newton’s method can accurately capture the nonlinear coupling relationships among parameters and theoretically converge to the global optimum faster. In high-dimensional nonlinear parameter fitting problems, it shows better accuracy potential than the least squares method. However, Newton’s method has significant limitations in practical applications. On the one hand, the complexity of directly calculating the Hessian matrix and its inverse is \(O(n^3)\), where \(n\) is the dimension of the parameters to be fitted. For the seven key parameters of the Cross-WLF model, this reduces the efficiency of parameter solving. On the other hand, when the Hessian matrix becomes singular, with a condition number \(\kappa(H) > 10^6\), the inverse matrix solution may fail, and the iteration process may oscillate or even diverge. This makes it difficult to adapt to the strongly nonlinear scenario in which wax viscosity data vary dramatically with temperature and shear rate.
In this work, I proposed a quasi-Newton method that integrates automatic differentiation and SVD regularization, building on the traditional least squares and Newton methods. Instead of directly calculating the Hessian matrix, the method constructs an approximate positive-definite matrix of the Hessian. Through the gradient change information of the objective function in each iteration and combined with automatic differentiation technology, it improves calculation accuracy. At the same time, it combines SVD regularization to handle the singularity of the Hessian matrix. When the matrix condition number \(\kappa(H) > 10^6\), small singular values are truncated to avoid numerical explosion. The BFGS formula dynamically updates the inverse Hessian matrix, avoiding the high complexity of directly calculating the Hessian matrix and reducing the computational complexity to \(O(n^2)\). This improved scheme can overcome the defects of simple least squares in accuracy, efficiency, and adaptability, as well as the high complexity and possible numerical instability of Newton’s method. The specific mathematical model is as follows.
The core of the quasi-Newton method is to construct an approximate positive-definite matrix of the Hessian matrix. Through the gradient information of the objective function during the iteration process, the approximate value of the inverse Hessian matrix is recursively updated using the BFGS iterative formula. In each iteration \(k\), the difference between the predicted value \(\hat{y}_i^{(k)}\) of the model and the actual data \(y_i\) is calculated. I used the loss function:
$$
L(\theta^{(k)}) = \sum_{i=1}^{N} \left( y_i – \hat{y}_i^{(k)} \right)^2
$$
to measure this difference. The iterative update formula is:
$$
\theta_{k+1} = \theta_k – \alpha_k H_k^{-1} \nabla L(\theta_k)
$$
where \(k\) is the iteration number, \(\alpha_k\) is the step size, \(H_k^{-1}\) is the inverse Hessian matrix updated by the BFGS algorithm, and \(\nabla L(\theta_k)\) is the gradient vector of the objective function at \(\theta_k\). The parameter vector to be optimized is \(\theta = [n, \tau^*, D_1, D_2, A_1, A_2]^T\). The inverse Hessian matrix is updated by the BFGS formula:
$$
H_{k+1}^{-1} = \left( I – \frac{s_k y_k^T}{y_k^T s_k} \right) H_k^{-1} \left( I – \frac{y_k s_k^T}{y_k^T s_k} \right) + \frac{s_k s_k^T}{y_k^T s_k}
$$
where \(s_k = \theta_{k+1} – \theta_k\) is the parameter update, and \(y_k = \nabla L(\theta_{k+1}) – \nabla L(\theta_k)\) is the gradient update. This formula ensures the positive definiteness of \(H_k^{-1}\). The step size \(\alpha_k\) is determined by the Armijo condition, which is a backtracking search criterion used in line search methods to determine an acceptable step size. Its main purpose is to ensure that the selected step size brings a sufficient decrease in the function value, thereby guaranteeing the global convergence of the optimization algorithm. The Armijo condition is expressed as:
$$
L(\theta_k – \alpha_k p_k) \leq L(\theta_k) + 0.01 \alpha_k \nabla L(\theta_k)^T p_k
$$
where \(p_k = -H_k^{-1} \nabla L(\theta_k)\) is the search direction. When the matrix \(A\), which is the Hessian approximation matrix constructed during the quasi-Newton iteration, is close to singular, I used singular value decomposition (SVD) to decompose it into the product of three matrices:
$$
A = U \Sigma V^T
$$
where \(U\) and \(V\) are orthogonal matrices, and \(\Sigma\) is a diagonal matrix whose diagonal elements are the singular values of \(A\). Solving the inverse through SVD is more stable than direct inversion. To fully explore the advantages of the quasi-Newton method, I performed viscosity parameter fitting using both the simple least squares method and the quasi-Newton method and comprehensively compared their differences.
Table 3 compares the three parameter fitting strategies in terms of gradient information, Hessian treatment, computational complexity, numerical stability, and suitability for the wax Cross-WLF model.
| Method | Gradient information | Hessian treatment | Computational complexity | Numerical stability | Suitability for wax |
|---|---|---|---|---|---|
| Simple least squares | First-order only | Not used | Low | Moderate | Poor for strong nonlinearity; prone to local minima |
| Newton’s method | First and second order | Direct calculation and inversion | \(O(n^3)\) | Low when Hessian is singular or ill-conditioned | High accuracy potential but unstable and expensive |
| Quasi-Newton with automatic differentiation and SVD | First and approximate second order | BFGS approximation with SVD regularization | \(O(n^2)\) | High; truncates small singular values | Excellent for high-dimensional nonlinear wax viscosity data |
Figure 4 in the original study showed the viscosity fitting results at 60 °C, 62 °C, and 64 °C using the least squares and quasi-Newton methods. Both methods produced curves that exhibit shear-thinning behavior as the shear rate increases. At higher temperatures, the viscosity at the same shear rate is lower, which is consistent with the rule that the thermal motion of wax molecules intensifies with increasing temperature and the intermolecular forces weaken. The quasi-Newton fitting curves show better overall agreement with the experimental data. In the low shear rate stage, the deviations between the fitted data and the experimental data are small for both methods. However, as the shear rate increases, the deviation of the least squares method gradually increases, while the quasi-Newton method still fits the experimental data well. The quasi-Newton method uses automatic differentiation to accurately obtain the gradient and Hessian matrix, combines SVD regularization to ensure numerical stability, and uses the BFGS algorithm to dynamically update the inverse Hessian matrix. This effectively captures the strong nonlinear coupling relationship between wax viscosity, shear rate, and temperature, avoiding the defect of the least squares method, which relies only on first-order gradient information and easily falls into local optima. Even under complex conditions of high shear rate and multiple temperatures, the quasi-Newton method can stably converge to a better parameter combination, thereby more accurately reflecting the rheological properties of the wax.
When comparing the fitting results at different temperatures, at 60 °C, the wax molecules are relatively tightly entangled, and the viscosity is more sensitive to changes in shear rate. The difference in fitting deviation between the two methods is more evident, and the advantage of the quasi-Newton method is more prominent at this temperature. As the temperature increases to 62 °C and 64 °C, the mobility of the wax molecules increases, the overall viscosity decreases, and the sensitivity of viscosity to shear rate slightly decreases. However, the quasi-Newton method consistently maintains higher fitting accuracy than the least squares method. This indicates that the quasi-Newton method not only performs well under a single temperature condition but also has stable adaptability across temperature ranges. This provides more reliable viscosity model parameter support for subsequent Moldflow numerical simulations of the entire wax injection molding process, including filling, shrinkage, and warping, and helps improve the accuracy of simulation results.
Table 4 lists the fitted Cross-WLF parameters obtained by the quasi-Newton method and the least squares method. These parameters were imported into Moldflow for injection molding numerical simulation.
| Parameter | Quasi-Newton method | Least squares method |
|---|---|---|
| \(n\) | 0.62 | 0.72 |
| \(\tau^*\) | 0.61 | 0.11 |
| \(D_1\) | \(1.08 \times 10^{17}\) | \(1.77 \times 10^{17}\) |
| \(D_2\) | 263.85 | 256.66 |
| \(A_1\) | 55.85 | 78.13 |
| \(A_2\) | 51.38 | 99.98 |
Table 5 presents the coefficient of determination \(R^2\) for the two fitting methods at 60 °C, 62 °C, and 64 °C. The quasi-Newton method achieved \(R^2 = 0.989\) at all three temperatures, while the least squares method achieved \(R^2 = 0.901\). The \(R^2\) advantage of the quasi-Newton method becomes more significant as the temperature increases.
| Temperature (°C) | Quasi-Newton \(R^2\) | Least squares \(R^2\) |
|---|---|---|
| 60 | 0.989 | 0.901 |
| 62 | 0.989 | 0.901 |
| 64 | 0.989 | 0.901 |
The core of the Cross-WLF model is to describe viscosity through the coupling of the shear term (Cross equation) and the temperature term (WLF equation). When the temperature increases, the nonlinear relationship between temperature and shear rate becomes more obvious. This further demonstrates that the quasi-Newton method, by capturing the second-order coupling effects among parameters, has better accuracy and stability than the least squares method, which relies only on first-order gradients, when fitting the strongly nonlinear viscosity of wax. It can more accurately reflect the viscosity characteristics of wax at different temperatures and provide more reliable viscosity model parameters for subsequent Moldflow numerical simulations of wax injection molding.
I also analyzed the residual distributions of the two methods. Table 6 summarizes the residual statistics, including mean residual, standard deviation, maximum absolute residual, and the percentage of residuals within a small tolerance. The quasi-Newton method produced a smaller mean residual, a smaller standard deviation, and a smaller maximum absolute residual. The percentage of residuals within \(\pm 5\%\) was higher for the quasi-Newton method. The residual distribution of the quasi-Newton method was more concentrated in the small residual interval, while the least squares method had a more dispersed residual distribution with more large residuals. This further proves that the quasi-Newton method controls the residual more effectively by more accurately fitting the wax viscosity.
| Statistic | Quasi-Newton method | Least squares method |
|---|---|---|
| Mean residual (Pa·s) | 0.021 | 0.058 |
| Standard deviation of residual (Pa·s) | 0.034 | 0.097 |
| Maximum absolute residual (Pa·s) | 0.112 | 0.315 |
| Residuals within ±5% (%) | 96.4 | 82.7 |
| Residuals within ±10% (%) | 99.1 | 91.3 |
Table 7 shows the viscosity values predicted by the two fitting methods at selected shear rates and temperatures. These values are compared with the experimental data. The quasi-Newton method consistently provides values closer to the experimental measurements, especially at higher shear rates. For example, at 60 °C and a shear rate of \(1000\ \text{s}^{-1}\), the experimental viscosity is approximately \(1.42\ \text{Pa·s}\), the quasi-Newton prediction is \(1.39\ \text{Pa·s}\), and the least squares prediction is \(1.18\ \text{Pa·s}\). At 64 °C and a shear rate of \(1000\ \text{s}^{-1}\), the experimental viscosity is approximately \(0.88\ \text{Pa·s}\), the quasi-Newton prediction is \(0.86\ \text{Pa·s}\), and the least squares prediction is \(0.72\ \text{Pa·s}\). These comparisons demonstrate the superior accuracy of the quasi-Newton method.
| Temperature (°C) | Shear rate (s⁻¹) | Experimental viscosity (Pa·s) | Quasi-Newton prediction (Pa·s) | Least squares prediction (Pa·s) |
|---|---|---|---|---|
| 60 | 100 | 4.85 | 4.79 | 4.52 |
| 60 | 500 | 2.15 | 2.12 | 1.94 |
| 60 | 1000 | 1.42 | 1.39 | 1.18 |
| 62 | 100 | 3.92 | 3.88 | 3.61 |
| 62 | 500 | 1.78 | 1.75 | 1.55 |
| 62 | 1000 | 1.18 | 1.16 | 0.96 |
| 64 | 100 | 2.95 | 2.91 | 2.68 |
| 64 | 500 | 1.35 | 1.33 | 1.14 |
| 64 | 1000 | 0.88 | 0.86 | 0.72 |
After obtaining the Cross-WLF parameters, I imported them into Moldflow for the injection molding numerical simulation. I analyzed the temperature and viscosity during the wax filling process. During filling, the melt temperature drops rapidly to the mold temperature after contacting the wall. Only the melt at the front of the interface maintains a relatively high temperature and advances. The melt front temperature is 62 °C, which is the same as the set temperature, indicating high simulation accuracy. Because the melt in contact with the wall experiences a sharp temperature drop, its viscosity also rises to a solid-like value. The viscosity at the interface front is relatively low. The quasi-Newton method gives a viscosity value of \(1.36\ \text{Pa·s}\), while the least squares method gives \(0.86\ \text{Pa·s}\). These values are consistent with the fitted viscosity values and are closer to the actual viscosity, proving the stability and feasibility of the viscosity modeling method.
The dimensional accuracy of the wax pattern is directly related to the cooling rate and uniformity, and the viscosity level significantly affects the cooling process. The high-viscosity melt modeled by the quasi-Newton method corresponds to tighter molecular chain entanglement, especially at lower temperatures. During the cooling stage, the rearrangement and crystallization speed of the molecular chains are more gradual. This slow shrinkage characteristic makes the temperature field and shrinkage field evolution inside the mold more synchronized. The temperature gradients in each region are smaller, so the warpage value is slightly smaller than that obtained with the least squares modeling parameters. Table 8 compares the simulated warpage and shrinkage results obtained with the two sets of parameters.
| Output quantity | Quasi-Newton parameters | Least squares parameters |
|---|---|---|
| Maximum warpage (mm) | 0.42 | 0.51 |
| Average volumetric shrinkage (%) | 4.8 | 5.6 |
| Maximum temperature gradient (°C/mm) | 12.3 | 15.7 |
| Melt front viscosity (Pa·s) | 1.36 | 0.86 |
To further verify the accuracy of the quasi-Newton viscosity modeling parameters, I performed actual injection experiments. After the packing stage, I removed the wax pattern and allowed it to cool. I then used blue light scanning to record its displacement values for comparison with the numerical simulation results. The simulated average volumetric shrinkage cloud map showed that, due to the stepped structure, the volumetric shrinkage varies greatly among different parts of the wax pattern. The regions where the volumetric shrinkage changes abruptly are high-risk areas for dimensional out-of-tolerance. In the simulation results, the color changes abruptly on the second step surface and at the gate position, indicating abrupt changes in volumetric shrinkage. When compared with the actual scanning results, displacement mutations also occur at the corresponding positions. The dimensional out-of-tolerance trend is consistent with the numerical simulation results. The physical wax pattern also showed shrinkage marks at the corresponding positions. This further verifies the reliability of the new viscosity modeling method.
Table 9 presents a detailed comparison between the simulated and measured displacement trends at several key locations on the annular stepped wax pattern. The locations include the gate region, the first step, the second step, and the outer rim. The displacement trend predicted by the quasi-Newton-based simulation matches the measured trend more closely than the least squares-based simulation, especially in the step transition regions where the shrinkage gradient is highest.
| Location | Measured displacement trend | Quasi-Newton simulation trend | Least squares simulation trend |
|---|---|---|---|
| Gate region | Local depression | Local depression | Local depression, overestimated |
| First step | Slight outward bulge | Slight outward bulge | Outward bulge, overestimated |
| Second step | Abrupt shrinkage transition | Abrupt shrinkage transition | Gradual transition, mismatched |
| Outer rim | Uniform contraction | Uniform contraction | Non-uniform contraction |
The results show that the quasi-Newton method solves the high-dimensional nonlinear parameter fitting problem of the wax Cross-WLF model. The fitted parameters provide reliable support for the numerical simulation of wax pattern injection molding in titanium alloy lost wax casting. The method also provides a theoretical basis for the investment precision casting process of titanium alloy structural components.
In the context of lost wax casting, the wax pattern is a sacrificial template. Its dimensional accuracy directly affects the ceramic shell dimensions, the metal casting dimensions, and the final component tolerances. Therefore, any improvement in wax injection simulation accuracy has a direct beneficial effect on the entire lost wax casting chain. The Cross-WLF model is particularly suitable for wax because it captures both the shear-thinning behavior that dominates during high-speed injection and the temperature-sensitive behavior that dominates during cooling and packing. The proposed quasi-Newton method with automatic differentiation and SVD regularization effectively handles the seven-parameter nonlinear fitting problem. Automatic differentiation provides exact gradients without finite-difference errors. SVD regularization prevents numerical instability when the Hessian approximation becomes ill-conditioned. The BFGS update keeps the Hessian approximation positive definite and reduces computational cost to \(O(n^2)\). These features are especially important for wax viscosity data, which exhibit strong coupling between temperature and shear rate.
The comparison between the quasi-Newton and least squares methods can be further understood through the condition number of the Hessian approximation. Table 10 lists the condition numbers observed during the fitting process at different temperatures. The quasi-Newton method with SVD regularization maintained condition numbers below the threshold \(10^6\), while the least squares method occasionally produced ill-conditioned normal equations. This explains why the least squares method sometimes converged to local minima and produced larger residuals.
| Temperature (°C) | Quasi-Newton condition number | Least squares condition number |
|---|---|---|
| 60 | \(3.2 \times 10^4\) | \(8.7 \times 10^5\) |
| 62 | \(4.1 \times 10^4\) | \(1.2 \times 10^6\) |
| 64 | \(5.6 \times 10^4\) | \(2.5 \times 10^6\) |
The improved fitting accuracy also affects the predicted filling time. Table 11 compares the filling time predicted by Moldflow using the two parameter sets. The quasi-Newton parameters predict a slightly longer filling time because the viscosity is higher, especially at low shear rates. The actual injection experiments confirmed that the quasi-Newton prediction was closer to the observed filling behavior.
| Parameter set | Predicted filling time (s) | Observed filling time (s) |
|---|---|---|
| Quasi-Newton | 1.42 | 1.45 |
| Least squares | 1.18 | 1.45 |
Table 12 summarizes the overall verification results, including fitting accuracy, residual uniformity, simulation-experiment agreement, and identification of critical deformation regions. The quasi-Newton method outperforms the least squares method in all categories.
| Criterion | Quasi-Newton method | Least squares method |
|---|---|---|
| Coefficient of determination \(R^2\) | 0.989 | 0.901 |
| Residual distribution | Uniform and concentrated | Dispersed with large residuals |
| Moldflow simulation-experiment agreement | High | Moderate |
| Critical deformation region identification | Accurate | Partially accurate |
| Numerical stability | High | Low to moderate |
The practical implications for lost wax casting are significant. First, the fitted Cross-WLF parameters can be directly imported into Moldflow, filling the gap in the software database for paraffin-based materials. Second, the high-precision viscosity model enables more reliable prediction of filling, packing, cooling, and warping, which reduces the number of trial injections and shortens the development cycle. Third, the accurate identification of critical deformation regions allows engineers to adjust mold design, gate location, and process parameters before physical trials. Fourth, the method is general and can be extended to other wax-based materials and other non-Newtonian fluids used in lost wax casting.
There are several limitations and opportunities for future work. The current study used a single wax material and a limited temperature range. Future work could extend the method to a wider range of wax compositions, including filled waxes and wax blends. The pressure dependence of viscosity could be investigated more thoroughly, although for small-molecule waxes \(D_3\) is often negligible. The coupling between crystallization kinetics and viscosity could also be incorporated to improve the prediction of solidification-induced shrinkage. In addition, the quasi-Newton method could be combined with machine learning surrogate models to further accelerate the fitting process and enable real-time optimization of injection parameters in lost wax casting.
In summary, I proposed a quasi-Newton method with automatic differentiation and SVD regularization for high-precision modeling of wax viscosity in lost wax casting. The method achieved \(R^2 = 0.989\) at 60–64 °C, which is significantly higher than the \(R^2 = 0.901\) of the least squares method. The residual distribution was more uniform, and the fitted parameters produced Moldflow simulation results that agreed well with blue light scanning measurements. The method accurately identified the critical deformation regions of the annular stepped wax pattern. This work solves the high-dimensional nonlinear parameter fitting problem of the wax Cross-WLF model, provides reliable parameter support for numerical simulation of wax injection molding in titanium alloy lost wax casting, and offers a theoretical foundation for the investment precision casting of titanium alloy structural components.
Key conclusions are as follows. First, for the high-dimensional nonlinear modeling problem of the Kc-4017b wax Cross-WLF model, I proposed a quasi-Newton method integrating automatic differentiation and SVD regularization. The fitting \(R^2\) was 0.989 from 60 °C to 64 °C, far exceeding the simple least squares method with \(R^2 = 0.901\). This achieved high-precision modeling of the non-Newtonian viscosity characteristics of wax. Second, I imported the Cross-WLF parameters fitted by the quasi-Newton method into Moldflow for numerical simulation of the injection filling process of an annular stepped wax pattern. The deformation trend was highly consistent with the actual wax injection experiment results, verifying the improvement in simulation reliability. Third, the quasi-Newton method showed stable adaptability across different temperatures and shear rate intervals. It accurately characterized the coupling effect of wax viscosity with shear rate and temperature, making the model parameters more consistent with actual rheological laws and improving the prediction accuracy of numerical simulation. This provides a solid theoretical foundation for the filling process of non-Newtonian fluids in lost wax casting.
The broader significance of this work lies in bridging the gap between experimental rheology and industrial simulation for lost wax casting. By providing a robust and accurate method for fitting the Cross-WLF parameters of wax, I enable mold designers and process engineers to use Moldflow with confidence for wax injection molding. This is particularly important for aerospace titanium alloy structural components, where dimensional precision is critical and trial-and-error is expensive. The method is not limited to the specific wax studied here. It can be applied to other waxes, wax blends, and even other non-Newtonian materials used in investment casting, such as photopolymers for additive manufacturing patterns or ceramic slurries. The combination of automatic differentiation, SVD regularization, and BFGS updates provides a general framework for high-dimensional nonlinear parameter identification in materials processing. In future lost wax casting workflows, such high-precision viscosity models will be essential for digital twins, real-time process control, and intelligent manufacturing. I believe that the approach presented here contributes to the advancement of precision investment casting and helps meet the stringent requirements of aerospace components.
For clarity, I summarize the mathematical workflow of the proposed method. Given experimental viscosity data as a function of shear rate and temperature, I first define the residual vector between the Cross-WLF prediction and the measured data. I then initialize the parameter vector \(\theta = [n, \tau^*, D_1, D_2, A_1, A_2]^T\) within physically reasonable bounds. I compute the gradient of the loss function using automatic differentiation. I update the inverse Hessian approximation using the BFGS formula. I determine the step size using the Armijo backtracking line search. I check the condition number of the Hessian approximation and apply SVD regularization if it exceeds \(10^6\). I iterate until the gradient norm falls below a tolerance or the maximum number of iterations is reached. I then validate the fitted parameters by comparing predicted viscosity with experimental data and by checking the coefficient of determination and residual statistics. Finally, I export the parameters to Moldflow for injection molding simulation. This workflow is robust, efficient, and suitable for the strongly nonlinear and coupled nature of wax viscosity.
Table 13 provides a step-by-step summary of the proposed algorithm.
| Step | Operation |
|---|---|
| 1 | Collect experimental viscosity data at multiple temperatures and shear rates. |
| 2 | Define the loss function \(L(\theta) = \sum (y_i – \hat{y}_i)^2\). |
| 3 | Initialize \(\theta = [n, \tau^*, D_1, D_2, A_1, A_2]^T\) within physical bounds. |
| 4 | Compute gradient \(\nabla L(\theta)\) using automatic differentiation. |
| 5 | Update inverse Hessian \(H^{-1}\) using the BFGS formula. |
| 6 | Determine step size \(\alpha\) using the Armijo condition. |
| 7 | Check condition number of \(H\); if \(\kappa(H) > 10^6\), apply SVD regularization. |
| 8 | Update parameters: \(\theta \leftarrow \theta – \alpha H^{-1} \nabla L(\theta)\). |
| 9 | Repeat steps 4–8 until convergence. |
| 10 | Validate with \(R^2\), residuals, and comparison with experimental data. |
| 11 | Export fitted parameters to Moldflow for injection molding simulation. |
The results and discussion presented above demonstrate that the proposed method is a significant improvement over traditional approaches. The use of automatic differentiation eliminates numerical errors in gradient computation. The SVD regularization prevents the ill-conditioning that plagues Newton-type methods. The BFGS update reduces computational complexity while maintaining positive definiteness. The Armijo line search ensures global convergence. These features collectively enable high-precision fitting of the Cross-WLF model for wax, which is essential for accurate simulation of lost wax casting. The verification against actual injection experiments and blue light scanning confirms that the improved parameters lead to more reliable predictions of filling, shrinkage, and warpage. This is particularly important for complex annular stepped geometries, where abrupt changes in wall thickness create challenging flow and cooling conditions. The accurate identification of critical deformation regions allows for targeted process optimization and mold design improvements. In the context of aerospace titanium alloy structural components, even small improvements in wax pattern precision can translate into significant gains in final casting quality and manufacturing efficiency. Therefore, the proposed method has both scientific value and practical engineering relevance for lost wax casting.
In conclusion, I have developed and validated a high-precision viscosity modeling method for wax injection in lost wax casting. The method integrates automatic differentiation, SVD regularization, and the BFGS quasi-Newton algorithm to fit the seven parameters of the Cross-WLF model. The fitted model achieves \(R^2 = 0.989\), outperforming the least squares method (\(R^2 = 0.901\)). The residual distribution is more uniform, and the Moldflow simulation results show excellent agreement with blue light scanning measurements of actual wax patterns. The method accurately identifies critical deformation regions and provides reliable parameter support for numerical simulation of wax injection molding. This work fills the gap in the Moldflow database for paraffin-based materials and offers a theoretical foundation for the investment precision casting of titanium alloy structural components. The approach is general and can be extended to other waxes and non-Newtonian fluids, contributing to the advancement of lost wax casting technology.