I investigate the viscosity behavior of wax used in full mold casting because the dimensional accuracy of the wax pattern directly controls the final quality of aerospace titanium alloy structural components. In the full mold casting route, wax injection forming is not a simple shaping step. It determines filling completeness, cooling uniformity, shrinkage distribution, and final warpage. Numerical simulation is therefore a core technology for optimizing full mold casting wax injection, but its predictive reliability depends strongly on the accuracy of the viscosity model. The wax behaves as a non-Newtonian fluid with clear shear-thinning characteristics. Its apparent viscosity changes with both shear rate and temperature, so a model that captures only one of these effects cannot support high-fidelity full mold casting simulation. The Cross-WLF model is suitable for this task because it combines a Cross shear-thinning term with a Williams-Landel-Ferry temperature-sensitive term. However, commercial material databases often lack the seven Cross-WLF parameters for wax-based materials. Traditional fitting approaches also have limitations. A simple least-squares method relies mainly on first-order gradient information and can fall into local minima when the data are strongly nonlinear. Newton-type methods require the Hessian matrix and its inverse, which increases computational complexity and can become numerically unstable when the Hessian is ill-conditioned. I therefore propose a quasi-Newton method that integrates automatic differentiation and singular value decomposition regularization for high-precision viscosity modeling in full mold casting.
My work focuses on Kc-4017b wax, a typical investment casting wax. I combine rheological measurements, Cross-WLF parameter fitting, Moldflow simulation, actual wax injection trials, and blue-light scanning validation. The objective is to fill the missing viscosity parameter gap for wax-based materials in full mold casting simulation and to provide reliable parameter support for the numerical optimization of wax pattern injection forming. In this article, I use the first-person perspective to describe my methods, results, and conclusions. I also use multiple tables and formulas to summarize the model, the optimization procedure, the fitted parameters, and the validation metrics.
The background of full mold casting is closely related to near-net-shape manufacturing. Titanium alloy structural parts in aerospace applications often have complex geometry, thin walls, variable wall thickness, and strict dimensional tolerances. Investment precision casting is one of the most effective ways to produce such parts. Within this process chain, the wax pattern is a sacrificial precursor. If the wax pattern has dimensional deviation, surface defects, or structural instability, these errors can propagate into the ceramic shell and eventually into the metal casting. Traditional wax pattern accuracy control depends on repeated trial-and-error adjustment of injection temperature, pressure, and holding time. This approach usually requires several rounds of trial production, which lengthens the development cycle and increases cost. Numerical simulation can reduce trial cycles by predicting filling, cooling, shrinkage, and warpage. However, the simulation accuracy is highly sensitive to the viscosity model. For a full mold casting wax pattern, the viscosity is a central material property because it controls flow resistance, shear heating, pressure transmission, and the frozen layer development.
The wax used in my study is a non-Newtonian fluid. Its viscosity decreases rapidly at low shear rates and then becomes more stable at high shear rates. At higher temperatures, the same shear rate produces a lower viscosity. This means that both shear rate and temperature must be included in the viscosity model. The Cross-WLF model is expressed as follows:
$$
\eta(\dot{\gamma},T,p)=\frac{\eta_0(T,p)}{1+\left(\frac{\eta_0(T,p)\dot{\gamma}}{\tau^{*}}\right)^{1-n}}
$$
Here, \(\eta\) is the apparent viscosity, \(\dot{\gamma}\) is the shear rate, \(\eta_0\) is the zero-shear viscosity, \(\tau^{*}\) is a characteristic relaxation parameter, and \(n\) is the non-Newtonian index. When \(n<1\), the material exhibits shear thinning. The zero-shear viscosity is described by the WLF temperature-sensitive equation:
$$
\eta_0(T,p)=D_1 \exp\left[-\frac{A_1(T-T^{*})}{A_2+(T-T^{*})}\right]
$$
The glass transition reference temperature is defined as:
$$
T^{*}=D_2+D_3 p
$$
In these equations, \(D_1\) is the zero-shear viscosity coefficient at the reference temperature, \(D_2\) is the glass transition temperature at low pressure, \(A_1\) and \(A_2\) are WLF parameters, \(T\) is the absolute temperature, \(p\) is pressure, and \(D_3\) is the pressure coefficient. For wax-based small-molecule non-Newtonian fluids, \(D_3\) is often taken as zero. Therefore, the Cross-WLF model uses seven parameters: \(n\), \(\tau^{*}\), \(D_1\), \(D_2\), \(D_3\), \(A_1\), and \(A_2\). In my modeling, I set \(D_3=0\) and optimize the remaining six parameters. This reduces the dimensionality while preserving the physically meaningful temperature and shear dependence required by full mold casting simulation.
The material properties of Kc-4017b wax are summarized in Table 1. These values provide the physical basis for the injection experiments and the numerical model.
| Property | Value | Role in full mold casting |
|---|---|---|
| Density | 0.95 g/cm³ | Controls inertia and pressure response during filling |
| Drop point | 94 °C | Indicates thermal stability limit |
| Softening point | 79 °C | Defines the transition toward low stiffness |
| Rheological type | Non-Newtonian, shear thinning | Requires Cross-WLF viscosity modeling |
| Test temperatures | 60 °C, 62 °C, 64 °C | Provide temperature-dependent viscosity data |
I designed an annular stepped wax pattern to represent the variable wall thickness and complex ring-like features often found in full mold casting for aerospace titanium alloy structural parts. The geometry includes multiple wall thicknesses and step transitions. These features are useful because they create non-uniform flow resistance, non-uniform cooling, and localized shrinkage. The annular stepped wax pattern used in my experiments is shown below.
The injection experiments were performed on an MPI-25 t wax injection machine. The wax temperature was 65 °C, the mold temperature was 25 °C, the holding pressure was 1 MPa, and the holding time was 60 s. The gate diameter was 8 mm. I repeated the injection under the same conditions five times to ensure repeatability. After injection, I used blue-light scanning to record the dimensional changes of the wax pattern. The scanned data were then compared with the Moldflow simulation results to evaluate whether the fitted Cross-WLF parameters could reproduce the actual deformation trend. The experimental and simulation parameters are listed in Table 2.
| Parameter | Value | Purpose |
|---|---|---|
| Wax temperature | 65 °C | Defines initial melt thermal state |
| Mold temperature | 25 °C | Controls heat transfer and frozen layer growth |
| Holding pressure | 1 MPa | Drives packing and compensation |
| Holding time | 60 s | Allows pressure transmission and shrinkage compensation |
| Gate diameter | 8 mm | Sets inlet flow area |
| Number of injection trials | 5 | Ensures repeatability |
| Boundary heat transfer coefficient | 2500 W/(m²·K) | Controls wax-mold heat exchange |
| Mesh edge length | 2 mm | Balances accuracy and computational cost |
| Analysis sequence | Cooling-filling-packing-warpage | Covers the full injection forming chain |
For the numerical simulation, I imported the annular stepped wax pattern into Moldflow and generated a dual-domain mesh. The global edge length was set to 2 mm. I checked mesh quality, including aspect ratio and element overlap, to ensure that the mesh met the accuracy requirements for full mold casting simulation. The analysis sequence was set as cooling-filling-packing-warpage coupling. This sequence allowed me to simulate melt filling, pressure transmission during packing, temperature field evolution during cooling, and final warpage caused by non-uniform shrinkage. The viscosity input used the Cross-WLF shear-thinning model so that the simulation reflected the real rheological behavior of the wax.
The parameter fitting problem is a high-dimensional nonlinear optimization problem. If I write the measured viscosity as \(y_i\) and the model prediction as \(\hat{y}_i(\theta)\), where \(\theta\) is the parameter vector, then the least-squares loss function is:
$$
L(\theta)=\sum_{i=1}^{N}\left[y_i-\hat{y}_i(\theta)\right]^2
$$
The parameter vector for the reduced Cross-WLF model is:
$$
\theta=\left[n,\tau^{*},D_1,D_2,A_1,A_2\right]^{T}
$$
The coefficient of determination is used to measure the fitting quality:
$$
R^2=1-\frac{\sum_{i=1}^{N}\left(y_i-\hat{y}_i\right)^2}{\sum_{i=1}^{N}\left(y_i-\bar{y}\right)^2}
$$
I also used the root-mean-square error and the mean absolute error to evaluate the residual behavior:
$$
\mathrm{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(y_i-\hat{y}_i\right)^2}
$$
$$
\mathrm{MAE}=\frac{1}{N}\sum_{i=1}^{N}\left|y_i-\hat{y}_i\right|
$$
The residual for each data point is:
$$
r_i=y_i-\hat{y}_i
$$
A simple least-squares method minimizes the loss using first-order gradient information. It is easy to implement, but it can become trapped in local minima when the model is strongly nonlinear and the parameters are coupled. In the Cross-WLF model, the shear term and the temperature term are coupled through the zero-shear viscosity. This coupling becomes more significant when the temperature range is wide and the shear rate spans several orders of magnitude. Therefore, a simple least-squares fit may produce parameters that match part of the data but fail in other regions. For full mold casting simulation, such regional failure is unacceptable because the filling front, the frozen layer, and the packing pressure all depend on the local viscosity.
Newton’s method uses both the gradient and the Hessian matrix. The iteration formula is:
$$
\theta_{k+1}=\theta_k-H^{-1}(\theta_k)\nabla L(\theta_k)
$$
Here, \(\nabla L(\theta_k)\) is the gradient and \(H(\theta_k)\) is the Hessian matrix. Newton’s method can capture second-order parameter coupling and may converge faster near the optimum. However, it requires the inverse of the Hessian. The computational complexity of this inversion is typically \(O(n^3)\). For a seven-parameter Cross-WLF model, this cost is significant. More importantly, the Hessian may be singular or ill-conditioned. If the condition number is very large, the inverse may amplify numerical errors and cause oscillation or divergence. This is a serious problem for wax viscosity data because the viscosity changes rapidly with temperature and shear rate.
To overcome these limitations, I use a quasi-Newton method. Instead of computing the Hessian directly, I construct a positive definite approximation of the inverse Hessian. The update formula is:
$$
\theta_{k+1}=\theta_k-\alpha_k B_k \nabla L(\theta_k)
$$
Here, \(B_k\) approximates \(H^{-1}\), and \(\alpha_k\) is the step length. The search direction is:
$$
p_k=-B_k \nabla L(\theta_k)
$$
I update \(B_k\) using the BFGS formula:
$$
B_{k+1}=\left(I-\rho_k s_k y_k^{T}\right)B_k\left(I-\rho_k y_k s_k^{T}\right)+\rho_k s_k s_k^{T}
$$
The quantities in this formula are:
$$
\rho_k=\frac{1}{y_k^{T}s_k}
$$
$$
s_k=\theta_{k+1}-\theta_k
$$
$$
y_k=\nabla L(\theta_{k+1})-\nabla L(\theta_k)
$$
The BFGS update preserves positive definiteness under suitable conditions and reduces the computational complexity to \(O(n^2)\). This is a major advantage for full mold casting viscosity modeling because the parameter fitting can be performed efficiently without constructing the full Hessian.
I also use automatic differentiation to obtain accurate gradients. Manual differentiation of the Cross-WLF model is possible but error-prone because the model contains nested exponentials, powers, and coupled terms. Automatic differentiation provides numerical gradients that are consistent with the implemented model. In addition, I apply singular value decomposition regularization when the approximate Hessian matrix is close to singular. The SVD decomposition is:
$$
A=U\Sigma V^{T}
$$
The inverse is computed as:
$$
A^{-1}=V\Sigma^{-1}U^{T}
$$
When the condition number exceeds a threshold, I truncate small singular values. This prevents numerical explosion and improves stability. The step length is determined by the Armijo condition:
$$
L(\theta_k-\alpha_k p_k)\le L(\theta_k)+c_1\alpha_k\nabla L(\theta_k)^{T}p_k
$$
This condition ensures sufficient decrease in the loss function and supports global convergence behavior. The combination of BFGS, automatic differentiation, SVD regularization, and Armijo line search forms the quasi-Newton method I use for high-precision viscosity modeling in full mold casting.
The algorithmic workflow is summarized in Table 3. Each step is designed to address a specific numerical difficulty in Cross-WLF parameter fitting.
| Step | Operation | Numerical benefit |
|---|---|---|
| 1 | Initialize parameters \(n,\tau^{*},D_1,D_2,A_1,A_2\) | Provides a physically reasonable starting point |
| 2 | Evaluate \(\hat{y}_i(\theta)\) using Cross-WLF equations | Connects parameters to measured viscosity |
| 3 | Compute gradients by automatic differentiation | Improves gradient accuracy and avoids manual derivation |
| 4 | Build or update \(B_k\) using BFGS | Avoids direct Hessian inversion |
| 5 | Check condition number and apply SVD regularization | Prevents instability from ill-conditioning |
| 6 | Determine step length by Armijo condition | Ensures sufficient decrease and stable convergence |
| 7 | Update parameters and repeat until convergence | Produces high-precision Cross-WLF parameters |
The viscosity modeling results show that both the quasi-Newton method and the least-squares method capture the general shear-thinning trend. At low shear rates, the difference between the two methods is small. As the shear rate increases, the deviation of the least-squares method becomes larger. The quasi-Newton method remains close to the measured data over the entire range. This behavior is consistent with the mathematical structure of the Cross-WLF model. The low-shear region is dominated by the zero-shear viscosity and the WLF temperature term. The high-shear region is dominated by the Cross shear-thinning term and the non-Newtonian index. A method that captures second-order coupling between these terms can fit both regions more accurately. The quasi-Newton method achieves this through the BFGS approximation and automatic differentiation.
Table 4 presents the fitted Cross-WLF parameters obtained by the quasi-Newton method and the least-squares method. These parameters were imported into Moldflow for the full mold casting wax injection simulation.
| Parameter | Quasi-Newton method | Least-squares method | Physical meaning |
|---|---|---|---|
| \(n\) | 0.62 | 0.72 | Non-Newtonian index; lower value means stronger shear thinning |
| \(\tau^{*}\) | 0.61 | 0.11 | Characteristic relaxation parameter |
| \(D_1\) | 1.08×10¹⁷ | 1.77×10¹⁷ | Zero-shear viscosity coefficient |
| \(D_2\) | 263.85 | 256.66 | Reference glass transition temperature |
| \(A_1\) | 55.85 | 78.13 | WLF temperature sensitivity parameter |
| \(A_2\) | 51.38 | 99.98 | WLF temperature shift parameter |
| \(D_3\) | 0 | 0 | Pressure coefficient, neglected for wax |
The coefficient of determination provides a direct comparison between the two fitting methods. The quasi-Newton method achieves \(R^2=0.989\), while the least-squares method reaches \(R^2=0.901\). This is a significant improvement. The advantage of the quasi-Newton method becomes more obvious at higher temperatures, where the nonlinear coupling between temperature and shear rate is stronger. Table 5 summarizes the \(R^2\) values at different temperatures.
| Temperature | Quasi-Newton \(R^2\) | Least-squares \(R^2\) | Improvement |
|---|---|---|---|
| 60 °C | 0.991 | 0.902 | 0.089 |
| 62 °C | 0.989 | 0.899 | 0.090 |
| 64 °C | 0.988 | 0.903 | 0.085 |
| Overall | 0.989 | 0.901 | 0.088 |
The residual distribution also supports this conclusion. The quasi-Newton residuals are concentrated in a narrow interval, while the least-squares residuals are more dispersed and contain larger values. I summarize the residual statistics in Table 6.
| Metric | Quasi-Newton method | Least-squares method |
|---|---|---|
| Mean residual | Close to zero | Slightly biased |
| Residual standard deviation | Small | Larger |
| Maximum absolute residual | Lower | Higher |
| RMSE | Lower | Higher |
| MAE | Lower | Higher |
| Residual pattern | Uniform | Localized peaks |
After fitting, I imported both parameter sets into Moldflow and performed the full injection simulation. The filling stage is particularly important for full mold casting because the melt front temperature and viscosity determine whether the cavity fills completely before the frozen layer blocks the flow. In my simulation, the melt front temperature remains at 62 °C, which is consistent with the set temperature. The simulation also shows that the melt near the wall cools rapidly to the mold temperature and its viscosity rises to a solid-like value. The front viscosity predicted by the quasi-Newton parameters is 1.36 Pa·s, while the least-squares parameters predict 0.86 Pa·s. The higher value from the quasi-Newton model is more consistent with the measured viscosity data. This demonstrates that the improved fitting method provides more realistic viscosity input for full mold casting simulation.
Table 7 summarizes the key filling indicators from the Moldflow simulation. These indicators show how the viscosity model affects the filling front and the thermal field.
| Indicator | Quasi-Newton model | Least-squares model | Interpretation |
|---|---|---|---|
| Melt front temperature | 62 °C | 62 °C | Consistent with the imposed thermal boundary |
| Melt front viscosity | 1.36 Pa·s | 0.86 Pa·s | Quasi-Newton value matches measured viscosity better |
| Wall-adjacent viscosity | Rises to solid-like value | Rises to solid-like value | Frozen layer formation is captured |
| Filling pattern | Smooth front advancement | Slightly less stable | Improved rheological consistency |
| Pressure transmission | More realistic | Less realistic | Better packing prediction |
The cooling stage is strongly affected by viscosity because viscosity controls the relaxation of molecular chains and the synchronization between the temperature field and the shrinkage field. The quasi-Newton model predicts a higher viscosity, which corresponds to more tightly entangled molecular chains, especially at lower temperatures. During cooling, the rearrangement and crystallization of these chains proceed more gradually. This slow shrinkage behavior makes the temperature field and the shrinkage field evolve more synchronously. As a result, the temperature gradient in the cross-section is smaller, and the warpage value is slightly lower than that predicted by the least-squares model. This does not mean that warpage disappears. It means that the improved viscosity model provides a more physically consistent description of the cooling and shrinkage process in full mold casting.
The warpage and shrinkage results are particularly relevant to full mold casting because dimensional accuracy is the final criterion. I analyzed the average volumetric shrinkage and the displacement field. The stepped structure creates large differences in volumetric shrinkage between regions. The regions where the shrinkage rate changes abruptly are the high-risk zones for dimensional deviation. In my simulation, the second-step surface and the gate region show abrupt changes in shrinkage. The blue-light scanning results show displacement mutations at the same locations. The physical wax pattern also shows visible shrinkage marks at these positions. This agreement between simulation and experiment validates the reliability of the proposed viscosity modeling method for full mold casting.
Table 8 compares the simulation trends and the experimental observations. The comparison is qualitative but highly informative because it connects the fitted viscosity parameters to actual dimensional behavior.
| Region | Simulation trend | Blue-light scan trend | Physical wax pattern |
|---|---|---|---|
| Second-step surface | Abrupt shrinkage change | Displacement mutation | Visible shrinkage mark |
| Gate region | Abrupt shrinkage change | Displacement mutation | Visible shrinkage mark |
| Overall annular body | Non-uniform shrinkage | Non-uniform displacement | Non-uniform surface quality |
| Thin-wall transition | Localized cooling effect | Localized deviation | Localized deformation |
The experimental validation confirms that the quasi-Newton Cross-WLF parameters can accurately determine the key deformation regions of the wax pattern. This is important for full mold casting because the wax pattern is not a final product. It is a precision precursor. Any error in the wax pattern is transferred to the ceramic shell and then to the metal casting. Therefore, improving the viscosity model for wax injection directly improves the dimensional control of the entire full mold casting chain.
I further evaluated the numerical stability of the proposed method. The least-squares method is sensitive to the initial parameter guess. When the initial values deviate from the physically reasonable range, the fitting error increases and the method may converge slowly. The Newton method requires the Hessian inverse and can become unstable when the Hessian is ill-conditioned. In contrast, the quasi-Newton method with BFGS and SVD regularization maintains stable convergence across different temperatures and shear-rate ranges. The automatic differentiation ensures that the gradient information is accurate, and the Armijo line search prevents overshooting. These features are especially valuable for full mold casting viscosity data because the experimental data contain multiple temperatures and a wide shear-rate range.
Table 9 lists the main numerical advantages of the proposed method compared with the traditional methods.
| Method | Gradient information | Hessian handling | Computational complexity | Stability | Accuracy for full mold casting wax |
|---|---|---|---|---|---|
| Simple least squares | First order | Not used | Low | Can fall into local minima | Limited |
| Newton method | First and second order | Direct inverse | \(O(n^3)\) | Poor when Hessian is ill-conditioned | Potentially high but unstable |
| Proposed quasi-Newton | Automatic differentiation | BFGS approximation and SVD regularization | \(O(n^2)\) | Stable and positive definite | High |
The physical interpretation of the fitted parameters also supports the method. The non-Newtonian index \(n=0.62\) indicates strong shear thinning. The characteristic relaxation parameter \(\tau^{*}=0.61\) is consistent with the transition from the zero-shear plateau to the power-law region. The WLF parameters \(A_1=55.85\) and \(A_2=51.38\) describe a temperature sensitivity that is reasonable for a wax-based material. The reference temperature \(D_2=263.85\) is physically meaningful within the tested range. In contrast, the least-squares parameters produce a higher non-Newtonian index and a lower relaxation parameter, which weakens the shear-thinning description and leads to a poorer match at high shear rates. This is why the least-squares model predicts a lower front viscosity and a slightly different warpage trend.
I also examined the effect of temperature on the fitting quality. At 60 °C, the molecular entanglement is relatively strong, and the viscosity is more sensitive to shear rate. The difference between the two methods is therefore more obvious. At 62 °C and 64 °C, the molecular mobility increases, and the viscosity decreases. The quasi-Newton method still maintains a higher \(R^2\) and a more uniform residual distribution. This demonstrates cross-temperature adaptability. For full mold casting, this adaptability is essential because the wax experiences a temperature gradient from the injection temperature to the mold temperature. A model that works only at one temperature cannot predict the filling and cooling behavior accurately.
The Moldflow simulation also reveals the coupling between viscosity and heat transfer. When the melt contacts the mold wall, its temperature drops rapidly. The viscosity near the wall increases sharply, forming a frozen layer. The flow is then confined to the central region. The front viscosity determines the resistance to flow and the pressure required to fill the cavity. The quasi-Newton model predicts a front viscosity of 1.36 Pa·s, which is higher than the least-squares prediction of 0.86 Pa·s. This higher viscosity is more consistent with the measured data and produces a more realistic pressure history. The pressure history, in turn, affects the packing stage and the final shrinkage distribution.
The packing stage is controlled by pressure transmission through the melt. A more accurate viscosity model allows the simulation to predict how long the pressure can be transmitted and how much shrinkage can be compensated. In full mold casting, insufficient packing can cause sink marks and dimensional deviation. Excessive packing can cause flash or residual stress. Therefore, the viscosity model is not only a material property but also a process control parameter. My results show that the quasi-Newton model provides a more reliable basis for packing analysis.
The warpage stage integrates the effects of filling, cooling, and packing. The final warpage is caused by non-uniform shrinkage and thermal gradients. The quasi-Newton model predicts a slightly lower warpage than the least-squares model because the temperature gradients are smaller and the shrinkage field evolves more uniformly. This result is consistent with the physical expectation that a more accurate viscosity model should produce a more realistic thermo-mechanical history. The blue-light scanning results confirm that the predicted deformation trend matches the actual wax pattern. The key deformation regions are correctly identified.
I summarize the overall validation logic in Table 10. The table connects the modeling steps to the validation evidence.
| Modeling step | Output | Validation evidence | Significance for full mold casting |
|---|---|---|---|
| Rheological testing | Viscosity-shear-temperature data | Measured at 60 °C, 62 °C, 64 °C | Provides the data foundation |
| Cross-WLF fitting | Seven model parameters | \(R^2=0.989\) | Improves viscosity prediction |
| Moldflow simulation | Filling, cooling, packing, warpage | Front viscosity and temperature | Predicts full process behavior |
| Injection experiment | Physical wax patterns | Five repeated trials | Tests repeatability |
| Blue-light scanning | Displacement field | Matches simulation trend | Validates dimensional accuracy |
| Shrinkage mark inspection | Visible defects | Matches predicted high-risk zones | Confirms process reliability |
The proposed method fills an important gap in full mold casting simulation. Commercial software databases often contain rheological data for common thermoplastics but lack parameters for wax and wax-based composites. Without these parameters, full mold casting simulation cannot accurately predict the injection forming of wax patterns. My work provides a systematic procedure for obtaining these parameters from experimental data. The method is not limited to Kc-4017b wax. It can be extended to other wax-based materials used in full mold casting, including filled waxes and modified waxes. The use of automatic differentiation and SVD regularization makes the method robust to different data ranges and initial guesses.
From a practical perspective, the improved viscosity model can reduce the number of trial injections. In traditional full mold casting wax pattern development, engineers often adjust injection parameters by trial and error. This requires multiple rounds of mold testing and manual inspection. A reliable simulation model can guide the selection of injection temperature, holding pressure, holding time, and gate design. The reduction in trial cycles shortens the development period and lowers the manufacturing cost. For aerospace titanium alloy structural components, this is particularly important because the production volume is often low but the quality requirements are extremely high.
The limits of my study should also be noted. The viscosity data were obtained at 60 °C, 62 °C, and 64 °C. The pressure dependence was neglected because \(D_3\) was set to zero. This is reasonable for wax injection at moderate pressures, but future work could include pressure-dependent measurements. The validation used an annular stepped wax pattern. Other geometries with different gate locations and wall thickness ratios could provide additional validation. The numerical method could also be combined with Bayesian uncertainty quantification to estimate confidence intervals for the fitted parameters. These extensions would further strengthen the reliability of full mold casting simulation.
In conclusion, I have developed a high-precision viscosity modeling method for wax injection forming in full mold casting. The method uses a quasi-Newton algorithm that integrates automatic differentiation, BFGS updating, SVD regularization, and Armijo line search. It solves the high-dimensional nonlinear parameter fitting problem of the Cross-WLF model. The fitted parameters produce \(R^2=0.989\), which is significantly higher than the least-squares result of \(R^2=0.901\). The residual distribution is more uniform, and the model remains stable across different temperatures and shear rates. When the fitted parameters are imported into Moldflow, the simulation predicts the filling front temperature, front viscosity, cooling behavior, shrinkage distribution, and warpage trend. The results agree well with actual injection trials and blue-light scanning data. The key deformation regions are accurately identified. This work provides reliable parameter support for the numerical simulation of wax pattern injection forming in full mold casting and establishes a theoretical basis for the precision investment casting of titanium alloy structural components.
I believe that the proposed method can be applied to other non-Newtonian materials used in full mold casting and related precision casting processes. The combination of automatic differentiation and SVD-regularized quasi-Newton optimization is not limited to Cross-WLF fitting. It can be used for other nonlinear rheological models, such as Carreau-Yasuda and modified WLF models. In future work, I plan to extend the method to multi-material wax systems, pressure-dependent viscosity, and full-scale industrial geometries. The ultimate goal is to achieve a digital twin of the full mold casting wax injection process, in which the viscosity model, the thermal boundary, and the process parameters are continuously updated from measured data. Such a digital twin would further reduce trial-and-error, improve dimensional accuracy, and support the rapid development of complex titanium alloy structural components.
Table 11 provides a final summary of the main quantitative outcomes from my study.
| Outcome | Quasi-Newton method | Least-squares method | Interpretation |
|---|---|---|---|
| Overall \(R^2\) | 0.989 | 0.901 | Higher fitting accuracy |
| Front viscosity in simulation | 1.36 Pa·s | 0.86 Pa·s | Closer to measured rheology |
| Residual distribution | Uniform and narrow | Dispersed with peaks | Better error control |
| Temperature adaptability | Stable from 60 °C to 64 °C | Less stable | Reliable for full mold casting |
| Warpage prediction | Consistent with scanning | Less consistent | Accurate deformation trend |
| Shrinkage mark prediction | Matches visible marks | Not validated | Useful for defect control |
The results confirm that the quasi-Newton method is a practical and reliable tool for viscosity modeling in full mold casting. It improves the accuracy of wax injection simulation, supports process optimization, and provides a foundation for the precision manufacturing of titanium alloy structural components. I expect that the method will be useful for both academic research and industrial application in full mold casting and investment precision casting.