In the field of aerospace manufacturing, titanium alloy structural components serve as critical load-bearing and force-transmitting elements. Their dimensional accuracy requirements are extremely strict. Investment precision casting is one of the most important near-net-shape processes for producing such complex components. Within the entire precision casting workflow, the dimensional deviation, surface quality, and structural stability of the wax pattern directly determine the final quality of the titanium alloy casting. Wax pattern preparation is carried out through injection molding. Conventional wax pattern accuracy control relies on an experience-based trial-and-error approach, in which injection temperature, pressure, and other process parameters are repeatedly adjusted. This usually requires three to five rounds of trial production, leading to long development cycles and high costs. Such a workflow cannot easily meet the development requirements of complex titanium alloy components in aerospace applications. To address this problem, numerical simulation technology has been widely used to optimize wax pattern injection molding. By predicting key physical phenomena such as melt filling, cooling shrinkage, and warpage deformation, numerical simulation can reduce the number of trial molds by more than sixty percent and significantly shorten development cycles and costs. However, the accuracy of numerical simulation depends heavily on the accuracy of material thermophysical parameters. Wax is an organic small-molecule material that exhibits shear-thinning behavior and is therefore a typical non-Newtonian fluid. Its viscosity is the core indicator describing the flowability of molten wax and significantly affects filling completeness, shrinkage uniformity, and warpage deformation. Therefore, accurate viscosity modeling is indispensable for reliable precision casting simulation.
Among the models used to describe the viscosity characteristics of non-Newtonian fluids, the Carreau-Yasuda model, the Williams-Landel-Ferry model, and the Cross-Williams-Landel-Ferry model are the most common. The Carreau-Yasuda model can only reflect the influence of shear rate on viscosity and cannot adapt to the rheological behavior of wax over a wide temperature range. The Williams-Landel-Ferry model considers temperature sensitivity but ignores the influence of shear rate, so it has limitations in describing the shear-thinning phenomenon of non-Newtonian fluids. The Cross-Williams-Landel-Ferry model, through the combination of a Cross shear-thinning term and a Williams-Landel-Ferry temperature-sensitive term, can simultaneously and accurately describe the viscosity characteristics of wax under different shear rates and temperatures. It has therefore become the preferred choice for wax rheological characterization. However, commercial databases such as Moldflow contain only rheological data for traditional thermoplastic polymer materials. They lack viscosity model parameters for paraffin wax and wax-based composite materials. These parameters must be obtained through experimental modeling. The modeling process is a high-dimensional nonlinear optimization problem, and its accuracy faces significant challenges. Traditional simple least squares methods have inherent defects in establishing the Cross-Williams-Landel-Ferry model for wax. First, they rely only on first-order gradient information and cannot capture second-order coupling effects among parameters, making them prone to falling into local optima when modeling strongly nonlinear wax data. Second, they are sensitive to initial values. When parameters deviate from a reasonable range, the error increases and adaptability becomes poor, making it difficult to meet the accuracy requirements of wax pattern injection simulation. Newton-type methods use first-order derivative, that is, gradient, and second-order derivative, that is, Hessian matrix, information to construct a quadratic approximation of the objective function. During iteration, the minimum of this quadratic function is solved to update the search direction and step size, thereby gradually approaching the minimum of the objective function. However, each iteration requires computing the inverse of the Hessian matrix. The complexity of matrix inversion is typically \(O(n^3)\), which makes Newton-type methods computationally expensive for high-dimensional problems and results in low computational efficiency. In addition, if the Hessian matrix is singular or ill-conditioned, with a very large condition number, Newton-type methods may fail to compute normally or converge extremely slowly.
To overcome these problems, I propose a quasi-Newton method that dynamically constructs a positive-definite approximation of the Hessian matrix. This method compensates for the deficiencies of simple least squares in accuracy, efficiency, and adaptability, and avoids the high complexity of computing the Hessian matrix and its inverse in Newton-type methods. I apply this method to actual viscosity test data of a wax material dedicated to investment precision casting. By comparing the modeling residuals and the coefficient of determination \(R^2\) of the quasi-Newton method and the traditional least squares method under different temperatures and shear rate intervals, I verify the modeling accuracy and stability of the quasi-Newton method. At the same time, I construct a three-dimensional model of an annular stepped wax pattern that represents typical features of aerospace titanium alloy structural components. I import the Cross-Williams-Landel-Ferry model parameters obtained by the two methods into Moldflow software to carry out full-process injection molding numerical simulation covering cooling, filling, packing, and warpage. I combine this with actual injection experiments and blue light scanning measurement data. From key indicators such as filling time, melt front temperature, and warpage deformation displacement, I quantitatively evaluate the improvement in numerical simulation accuracy achieved by the improved method. This research aims to provide a new method for efficiently and accurately solving the viscosity model parameters of non-Newtonian wax fluids, fill the gap in the Moldflow database regarding rheological parameters of wax-based composite materials, and provide theoretical support and technical reference for wax pattern preparation process optimization and quality control in the investment precision casting of complex titanium alloy components in aerospace. It also helps shorten component development cycles and reduce production costs.
Experimental framework and material behavior
I selected Kc-4017b wax, a typical investment precision casting wax, as the research object. Its density is \(0.95\ \text{g/cm}^3\), its drop point is \(94\ ^\circ\text{C}\), and its softening point is \(79\ ^\circ\text{C}\). I measured viscosity data at \(60\ ^\circ\text{C}\), \(62\ ^\circ\text{C}\), and \(64\ ^\circ\text{C}\). At each temperature, the wax viscosity decreases rapidly with increasing shear rate at first and then gradually levels off. This clearly reflects the shear-thinning characteristics of a non-Newtonian fluid. At higher temperatures, the viscosity at the same shear rate is lower. This indicates that temperature has a significant influence on wax flowability. Therefore, viscosity modeling must consider both shear rate and temperature effects. These experimental data form the basis for fitting the Cross-Williams-Landel-Ferry model parameters.
I designed an annular stepped wax pattern with multiple wall thicknesses to represent the complex annular variable-wall-thickness structures found in aerospace titanium alloy structural components. This geometry allows me to study the influence of viscosity on the injection filling process of variable-wall-thickness structures. The injection experiments were performed on an MPI-25 t wax injection machine. The wax temperature was \(65\ ^\circ\text{C}\), the mold temperature was \(25\ ^\circ\text{C}\), the holding pressure was \(1\ \text{MPa}\), and the holding time was \(60\ \text{s}\). The gate diameter was \(8\ \text{mm}\). To ensure the accuracy of the experimental results, I repeated the injection trials five times under the same process conditions. 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.
I imported the three-dimensional geometric model of the annular stepped wax pattern into Moldflow software and then carried out mesh generation. I used the built-in mesh generation module to create a dual-domain mesh and set the global edge length of the wax pattern mesh to \(2\ \text{mm}\). After mesh generation, I systematically checked the mesh quality to ensure that key indicators such as aspect ratio and element overlap met the accuracy requirements of Moldflow numerical simulation. The simulation parameters were set to be the same as the experimental parameters. The wax temperature was \(65\ ^\circ\text{C}\), the mold temperature was \(25\ ^\circ\text{C}\), the holding pressure was \(1\ \text{MPa}\), the holding time was \(60\ \text{s}\), and the gate diameter was \(8\ \text{mm}\). The boundary heat transfer coefficient between the wax pattern and the mold was \(2500\ \text{W/(m}^2\cdot\text{K)}\). The numerical simulation analysis sequence was set to a coupled cooling-filling-packing-warpage analysis. This sequence covers the full-process physical phenomena of wax pattern injection molding. It sequentially simulates melt filling flow in the mold cavity, pressure transmission during the packing stage, temperature field evolution during cooling and solidification, and final warpage deformation caused by non-uniform shrinkage. For viscosity settings, I used the Cross-Williams-Landel-Ferry shear-thinning viscosity model, which matches the true viscosity characteristics of wax. I imported the fitted viscosity model parameters into the Moldflow software database to ensure numerical simulation accuracy.
| Parameter | Value | Unit |
|---|---|---|
| Wax temperature | 65 | °C |
| Mold temperature | 25 | °C |
| Holding pressure | 1 | MPa |
| Holding time | 60 | s |
| Gate diameter | 8 | mm |
| Boundary heat transfer coefficient | 2500 | W/(m²·K) |
| Mesh edge length | 2 | mm |
| Analysis sequence | Cooling-filling-packing-warpage | — |
Cross-Williams-Landel-Ferry viscosity model
The Cross-Williams-Landel-Ferry model is formed by combining the Cross shear-thinning model and the Williams-Landel-Ferry temperature-sensitive equation. It can fully describe the quantitative relationship between viscosity, shear rate, and temperature. The viscosity-shear rate relationship is expressed by the Cross model 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 in \(\text{Pa}\cdot\text{s}\), \(\eta_0\) is the zero-shear viscosity in \(\text{Pa}\cdot\text{s}\), which is the viscosity when the shear rate approaches zero, \(\dot{\gamma}\) is the shear rate in \(\text{s}^{-1}\), \(\tau^*\) is the characteristic relaxation parameter in \(\text{Pa}\), which reflects the molecular chain relaxation time, and \(n\) is the non-Newtonian index, dimensionless. When \(n < 1\), the material exhibits shear thinning. The smaller the value of \(n\), the more significant the shear-thinning effect.
The relationship between zero-shear viscosity and temperature is described by the Williams-Landel-Ferry equation:
$$\eta_0(T,p) = D_1 \exp \left( -\frac{A_1 (T – T^*)}{A_2 + (T – T^*)} \right)$$
The glass transition temperature \(T^*\) is defined as:
$$T^* = D_2 + D_3 p$$
In these equations, \(D_1\) is the zero-shear viscosity coefficient at the reference temperature in \(\text{Pa}\cdot\text{s}\), \(D_2\) is the glass transition temperature at low pressure in \(\text{K}\), which determines the critical temperature for the transition from the glassy state to the high-elastic state, \(A_1\) and \(A_2\) are Williams-Landel-Ferry equation parameters, dimensionless and in \(\text{K}\), respectively, which reflect temperature sensitivity and temperature shift characteristics, \(T\) is the thermodynamic temperature in \(\text{K}\), \(T^*\) is the glass transition temperature in \(\text{K}\), \(p\) is the pressure in \(\text{Pa}\), and \(D_3\) is the pressure coefficient. For paraffin wax, a small-molecule non-Newtonian fluid, \(D_3\) is generally taken as zero. Combining the Cross model and the Williams-Landel-Ferry equation, the Cross-Williams-Landel-Ferry model uses seven parameters, namely \(n\), \(\tau^*\), \(D_1\), \(D_2\), \(D_3\), \(A_1\), and \(A_2\), to fully describe the nonlinear variation of wax viscosity.
Parameter fitting methodology
Most existing studies use simple least squares for Cross-Williams-Landel-Ferry model parameter fitting. However, least squares is only suitable for linear models. For the strongly nonlinear and strongly coupled parameter model of wax viscosity, it produces large errors and poor adaptability, making it difficult to meet the accuracy requirements of wax pattern injection numerical simulation. Newton-type methods are typical second-order optimization methods. For a function \(f(x)\), 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. By using first-order derivative, that is, gradient, and second-order derivative, that is, Hessian matrix, information to construct a quadratic model, Newton-type methods can accurately capture nonlinear coupling relationships among parameters and theoretically converge to the global optimum faster. They show better accuracy potential than least squares in high-dimensional nonlinear parameter fitting problems. However, Newton-type methods have significant limitations in practical applications. On one hand, the complexity of directly computing the Hessian matrix and its inverse is as high as \(O(n^3)\), where \(n\) is the dimension of the parameters to be fitted. For the seven key parameters of the Cross-Williams-Landel-Ferry model, this reduces parameter solving efficiency. On the other hand, when the Hessian matrix becomes singular, with a condition number \(\kappa(H) > 10^6\), not only does the inverse matrix solution fail, but the iteration process may also oscillate or even diverge. This makes it difficult to adapt to the strongly nonlinear scenarios in which wax viscosity data change dramatically with temperature and shear rate.
On the basis of traditional least squares and Newton-type methods, I propose a quasi-Newton method that integrates automatic differentiation and singular value decomposition regularization. It does not directly compute the Hessian matrix. Instead, it constructs a positive-definite approximation of the Hessian matrix. Through the gradient change information of the objective function in each iteration and automatic differentiation technology, it improves computational accuracy. At the same time, it uses singular value decomposition 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 Broyden-Fletcher-Goldfarb-Shanno formula is used to dynamically update the inverse Hessian approximation, avoiding the high complexity of directly computing the Hessian matrix. The computational complexity is reduced to \(O(n^2)\). This improved scheme compensates for the deficiencies of simple least squares in accuracy, efficiency, and adaptability, as well as the high complexity and possible numerical instability of Newton-type methods when computing the Hessian matrix and its inverse. The specific mathematical model is as follows.
The core of the quasi-Newton method is to construct a positive-definite approximation of the Hessian matrix. Through the gradient information of the objective function during iteration, the approximate value of the inverse Hessian matrix is recursively updated using the Broyden-Fletcher-Goldfarb-Shanno iteration formula. In each iteration \(k\), the difference between the model prediction \(\hat{y}_i^{(k)}\) and the actual data \(y_i\) is calculated. I use the loss function \(L(\theta^{(k)}) = \sum_i (y_i – \hat{y}_i^{(k)})^2\) to measure this difference. The iterative update formula is:
$$\theta_{k+1} = \theta_k – \alpha_k H_k^{-1} \nabla L(\theta_k)$$
Here, \(k\) is the iteration number, \(\alpha_k\) is the step size, \(H_k^{-1}\) is the inverse Hessian matrix updated by the Broyden-Fletcher-Goldfarb-Shanno 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 Broyden-Fletcher-Goldfarb-Shanno formula:
$$H_{k+1}^{-1} = \left( I – \rho_k s_k y_k^T \right) H_k^{-1} \left( I – \rho_k y_k s_k^T \right) + \rho_k s_k s_k^T$$
In this formula, \(s_k = \theta_{k+1} – \theta_k\) is the parameter update amount, \(y_k = \nabla L(\theta_{k+1}) – \nabla L(\theta_k)\) is the gradient update amount, and \(\rho_k = 1/(y_k^T s_k)\). This formula ensures the positive definiteness of \(H_k^{-1}\). The step size \(\alpha_k\) is determined by the Armijo condition, which is also called the Armijo rule or sufficient decrease condition. It 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 sufficient decrease in the function value, thereby guaranteeing the global convergence of the optimization algorithm. The Armijo condition is expressed as:
$$L(\theta_k + \alpha_k p_k) \le L(\theta_k) + c_1 \alpha_k \nabla L(\theta_k)^T p_k$$
Here, \(p_k = -H_k^{-1} \nabla L(\theta_k)\) is the search direction and \(c_1 = 0.01\). When the matrix \(A\), which is the Hessian approximation matrix constructed during quasi-Newton iteration, is close to singular, I use singular value decomposition to decompose it into the product of three matrices:
$$A = U \Sigma V^T$$
Here, \(U\) and \(V\) are orthogonal matrices, and \(\Sigma\) is a diagonal matrix whose diagonal elements are the singular values of \(A\). The inverse is computed as:
$$A^{-1} = V \Sigma^{-1} U^T$$
This is more stable than direct inversion. To fully explore the advantages of the quasi-Newton method, I separately used simple least squares and the quasi-Newton method for viscosity parameter fitting and comprehensively compared their differences.
| Method | Gradient information | Hessian handling | Computational complexity | Numerical stability | \(R^2\) |
|---|---|---|---|---|---|
| Simple least squares | First order only | Not used | \(O(n)\) | Prone to local optima | 0.901 |
| Newton-type method | First and second order | Exact Hessian and inverse | \(O(n^3)\) | Unstable if singular | Not adopted |
| Quasi-Newton with automatic differentiation and SVD regularization | First and second order via BFGS and automatic differentiation | Approximate inverse with SVD regularization | \(O(n^2)\) | Stable | 0.989 |
Viscosity modeling results
I obtained viscosity modeling results at \(60\ ^\circ\text{C}\), \(62\ ^\circ\text{C}\), and \(64\ ^\circ\text{C}\) using both least squares and the quasi-Newton method. The modeling curves of both methods exhibit shear-thinning rheological behavior as the shear rate increases. At higher temperatures, the viscosity at the same shear rate is lower. This is consistent with the law that the thermal motion of wax molecules intensifies with increasing temperature and the intermolecular forces weaken. The modeling curve of the quasi-Newton method shows better overall agreement with the experimental data. In the low shear rate stage, the deviation between the modeling data and the experimental data is small for both methods. However, as the shear rate increases, the modeling deviation of least squares gradually increases, while the quasi-Newton method still fits the experimental data well. The quasi-Newton method accurately obtains gradients and Hessian matrix information through automatic differentiation technology, ensures numerical stability through singular value decomposition regularization, and dynamically updates the inverse Hessian matrix using the Broyden-Fletcher-Goldfarb-Shanno algorithm. It effectively captures the strongly nonlinear coupling relationship between wax viscosity, shear rate, and temperature. It avoids the defect of least squares, 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 stably converges to a better parameter combination, thereby more accurately reflecting the rheological characteristics of wax.
I further compared the modeling performance at different temperatures. At \(60\ ^\circ\text{C}\), because the wax molecules are relatively tightly entangled, the viscosity is more sensitive to changes in shear rate. The difference in modeling deviation between the two methods is relatively easier to observe, and the advantage of the quasi-Newton method is more prominent at this temperature. As the temperature increases to \(62\ ^\circ\text{C}\) and \(64\ ^\circ\text{C}\), the activity of wax molecules increases, the overall viscosity decreases, and the sensitivity of viscosity to shear rate decreases slightly. However, the quasi-Newton method always maintains higher modeling accuracy than least squares. This indicates that the quasi-Newton method not only performs well under a single temperature condition but also has stable adaptability across temperature intervals. It provides more reliable viscosity model parameter support for subsequent Moldflow numerical simulation of the full wax pattern injection molding process, including filling, shrinkage, and warpage. This helps improve the accuracy of the simulation results.
| Temperature (°C) | Quasi-Newton \(R^2\) | Least squares \(R^2\) | Improvement |
|---|---|---|---|
| 60 | 0.989 | 0.901 | +0.088 |
| 62 | 0.989 | 0.901 | +0.088 |
| 64 | 0.989 | 0.901 | +0.088 |
| Overall | 0.989 | 0.901 | +0.088 |
The scatter plots of measured viscosity versus modeled viscosity at different temperatures show that the quasi-Newton method produces points that are more tightly distributed near the line where measured viscosity equals modeled viscosity. This clearly indicates better consistency between modeled and measured values. The scatter points of least squares are more dispersed and deviate more from this line. The coefficient of determination \(R^2\) is calculated as:
$$R^2 = 1 – \frac{\sum_{i=1}^{N} (y_i – \hat{y}_i)^2}{\sum_{i=1}^{N} (y_i – \bar{y})^2}$$
The \(R^2\) values of the quasi-Newton method at \(60\ ^\circ\text{C}\), \(62\ ^\circ\text{C}\), and \(64\ ^\circ\text{C}\) are all 0.989, higher than the 0.901 of least squares. As the temperature increases, the advantage of the quasi-Newton method in \(R^2\) becomes more significant. The core of the Cross-Williams-Landel-Ferry model is to describe viscosity through the coupling of the shear term, that is, the Cross equation, and the temperature term, that is, the Williams-Landel-Ferry equation. When the temperature increases, the nonlinear relationship between temperature and shear rate becomes more obvious. This further proves that the quasi-Newton method, by capturing the second-order coupling effect among parameters, has better accuracy and stability than least squares, which relies only on first-order gradients, when modeling 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 parameter support for subsequent Moldflow numerical simulation of wax pattern injection molding.
I also analyzed the residual distributions of the two methods. The three-dimensional residual surface of the quasi-Newton method is generally flatter, and the color changes are relatively gentle. This indicates that the residual distribution is more uniform and the residual values are smaller. The residual surface of least squares has obvious raised regions. This means that under certain combinations of shear rate and temperature, the residuals are large and the fluctuations are more severe. The residual histogram of the quasi-Newton method is more concentrated, mainly clustered in smaller residual intervals. The residual histogram of least squares is more dispersed, with more large residuals. This further proves that the quasi-Newton method controls residuals more effectively by more accurately modeling wax viscosity.
| Residual feature | Quasi-Newton method | Least squares method |
|---|---|---|
| Overall distribution | Uniform and concentrated | Scattered with several large residuals |
| Residual surface | Flat, smooth transitions | Raised regions, stronger fluctuations |
| Histogram peak | Narrow, near zero | Broad, shifted toward larger residuals |
| Maximum residual | Smaller | Larger |
| Temperature stability | Stable across 60–64 °C | Accuracy loss at high shear rates |
The Cross-Williams-Landel-Ferry model parameters obtained by the quasi-Newton method and least squares are listed in the following table. I imported them into Moldflow software for injection molding numerical simulation.
| Parameter | Quasi-Newton method | Least squares method | Unit |
|---|---|---|---|
| \(n\) | 0.62 | 0.72 | — |
| \(\tau^*\) | 0.61 | 0.11 | Pa |
| \(D_1\) | \(1.08 \times 10^{17}\) | \(1.77 \times 10^{17}\) | Pa·s |
| \(D_2\) | 263.85 | 256.66 | K |
| \(D_3\) | 0 | 0 | K/Pa |
| \(A_1\) | 55.85 | 78.13 | — |
| \(A_2\) | 51.38 | 99.98 | K |
Moldflow simulation based on the new viscosity modeling parameters
I analyzed the temperature and viscosity of the wax during the filling process. During filling, the melt temperature rapidly decreases to the mold temperature after contact with the wall. Only the melt at the front of the interface maintains a relatively high temperature and advances forward. The melt front temperature is \(62\ ^\circ\text{C}\), which is the same as the set temperature. This indicates that the software simulation accuracy is high. At the same time, because the melt contacting the wall experiences a sudden temperature drop, its viscosity also rises to a solid-like value. The viscosity at the interface front is relatively low. The viscosity value obtained with the quasi-Newton method parameters is \(1.36\ \text{Pa}\cdot\text{s}\), while the value obtained with least squares parameters is \(0.86\ \text{Pa}\cdot\text{s}\). This is consistent with the modeled viscosity values and is closer to the actual viscosity value. This proves the stability and feasibility of the viscosity modeling method.
| Location | Quasi-Newton viscosity (Pa·s) | Least squares viscosity (Pa·s) | Observed trend |
|---|---|---|---|
| Melt front | 1.36 | 0.86 | Quasi-Newton closer to actual |
| Wall-adjacent melt | Solid-like high viscosity | Solid-like high viscosity | Rapid cooling at wall |
| Interior melt | Moderate viscosity | Lower viscosity | Quasi-Newton captures coupling |
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, which is more significant at low temperatures. During the cooling stage, the rearrangement and crystallization speed of molecular chains are more gradual. This slow shrinkage characteristic makes the temperature field and shrinkage field evolution inside the mold more synchronized. As shown in the cross-sectional temperature field results, the temperature gradient in each region is smaller. Therefore, the warpage value is slightly smaller than that obtained with the least squares modeling parameters. This indicates that the quasi-Newton method provides a more physically consistent description of the coupled thermal and rheological behavior during precision casting wax injection.
| Simulation output | Quasi-Newton parameters | Least squares parameters | Physical interpretation |
|---|---|---|---|
| Melt front temperature | \(62\ ^\circ\text{C}\) | \(62\ ^\circ\text{C}\) | Same as set temperature |
| Melt front viscosity | 1.36 Pa·s | 0.86 Pa·s | Quasi-Newton closer to measured rheology |
| Cross-sectional temperature gradient | Smaller | Larger | More synchronous shrinkage |
| Warpage | Slightly smaller | Slightly larger | High viscosity slows molecular rearrangement |
| Shrinkage uniformity | Better | Less uniform | Improved thermal-rheological coupling |
Actual injection experiment validation
To further verify the accuracy of the viscosity modeling parameters obtained by the quasi-Newton method, I conducted actual injection experiments using the simulation parameters. After the holding stage, I removed the wax pattern and let it stand. After the wax pattern cooled, I used blue light scanning to record its displacement values for comparison with the numerical simulation results. The average volume shrinkage numerical simulation result based on the quasi-Newton viscosity modeling parameters shows that, due to the stepped structure, the volume shrinkage of different parts of the wax pattern varies greatly. In particular, regions where the volume shrinkage changes abruptly are high-risk areas for dimensional deviation. In the simulation results, the color changes abruptly on the second-step surface and at the gate position, indicating sudden changes in volume shrinkage. Compared with the actual scanning results, displacement mutations also occur at the corresponding positions, and the dimensional deviation trend is consistent with the numerical simulation results. Shrinkage marks were also found at the corresponding positions on the actual wax pattern. This further verifies the reliability of the new viscosity modeling method.
| Validation metric | Numerical simulation with quasi-Newton parameters | Blue light scanning measurement | Agreement |
|---|---|---|---|
| Deformation trend | Matches stepped geometry effects | Matches stepped geometry effects | High |
| Shrinkage mutation location | Second-step surface and gate | Second-step surface and gate | High |
| Shrink marks | Predicted at abrupt shrinkage zones | Observed at same zones | High |
| Displacement direction | Consistent with cooling contraction | Consistent with measured displacement | High |
| Overall dimensional deviation trend | Captures key risk regions | Confirms key risk regions | High |
Discussion
The results show that the proposed quasi-Newton method solves the high-dimensional nonlinear parameter fitting problem of the Cross-Williams-Landel-Ferry model for wax. The method combines automatic differentiation and singular value decomposition regularization. Automatic differentiation provides accurate gradients, which improves the reliability of the Broyden-Fletcher-Goldfarb-Shanno update. Singular value decomposition regularization handles ill-conditioned Hessian approximations. When the condition number exceeds \(10^6\), small singular values are truncated to prevent numerical explosion. This is particularly important for wax viscosity data, which change strongly with temperature and shear rate. The Armijo condition ensures sufficient decrease in the objective function and helps maintain global convergence. The Broyden-Fletcher-Goldfarb-Shanno formula dynamically updates the inverse Hessian approximation and reduces the computational complexity from \(O(n^3)\) to \(O(n^2)\). This makes the method efficient for the seven-parameter Cross-Williams-Landel-Ferry model.
The coefficient of determination \(R^2 = 0.989\) obtained by the quasi-Newton method is significantly higher than the \(R^2 = 0.901\) obtained by least squares. The residual distribution is also more uniform. This improvement is not merely numerical. It has a direct effect on the reliability of precision casting simulation. The viscosity model parameters determine the predicted filling pattern, pressure transmission, cooling rate, shrinkage, and warpage. Inaccurate parameters can lead to incorrect predictions of defect locations and dimensional deviations. In my validation, the Moldflow simulation results corresponding to the quasi-Newton parameters show a high degree of consistency with the displacement trend measured by blue light scanning. The key deformation regions can be accurately identified. This demonstrates that the new method provides reliable parameter support for the numerical simulation of wax pattern injection molding in titanium alloy investment precision casting.
The method also fills a gap in the Moldflow database. Moldflow contains rheological data for many thermoplastics but lacks viscosity model parameters for paraffin wax and wax-based composite materials. By experimentally measuring viscosity and fitting the Cross-Williams-Landel-Ferry model with the quasi-Newton method, I generated a reliable parameter set that can be imported into Moldflow. This enables engineers to simulate wax injection molding with better confidence. It also supports process optimization for complex aerospace titanium alloy structural components. The use of numerical simulation can reduce trial-and-error iterations and shorten development cycles. This is especially valuable in precision casting, where dimensional tolerances are tight and production costs are high.
The quasi-Newton method also shows stable adaptability across different temperatures and shear rate intervals. It accurately characterizes the coupling effect between wax viscosity, shear rate, and temperature. The fitted parameters are more consistent with actual rheological laws. This improves the prediction accuracy of numerical simulation and provides a solid theoretical foundation for non-Newtonian fluid filling processes. In contrast, least squares relies only on first-order gradient information. It cannot capture second-order coupling effects among parameters. This leads to larger errors at high shear rates and higher temperatures. Newton-type methods can capture second-order information, but they require direct computation of the Hessian matrix and its inverse. This is computationally expensive and numerically fragile when the Hessian is singular or ill-conditioned. The proposed quasi-Newton method avoids these problems by constructing a positive-definite approximation and updating it recursively.
| Comparison item | Simple least squares | Newton-type method | Proposed quasi-Newton method |
|---|---|---|---|
| Gradient order | First order | First and second order | First and second order via BFGS and automatic differentiation |
| Hessian computation | Not used | Exact Hessian and inverse | Approximate inverse with SVD regularization |
| Complexity | \(O(n)\) | \(O(n^3)\) | \(O(n^2)\) |
| Local optima risk | High | Lower but unstable | Low |
| Numerical stability | Moderate | Poor when Hessian is singular | High due to SVD regularization |
| \(R^2\) for wax viscosity | 0.901 | Not adopted | 0.989 |
| Residual uniformity | Scattered | — | Uniform |
| Application to precision casting simulation | Limited | Impractical for high-dimensional fitting | Reliable and efficient |
There are also limitations to this study. The pressure coefficient \(D_3\) was set to zero for paraffin wax because it is a small-molecule non-Newtonian fluid. In future work, I plan to investigate pressure-dependent viscosity effects under higher packing pressures. The current validation used one wax material and one annular stepped geometry. More wax materials and more complex geometries should be tested to generalize the method. In addition, the computational efficiency of the quasi-Newton method can be further improved by parallelizing the automatic differentiation and optimizing the Broyden-Fletcher-Goldfarb-Shanno update for large data sets. The method can also be integrated with machine learning surrogate models to accelerate the optimization of injection process parameters in precision casting.
Conclusions
I studied Kc-4017b wax, a dedicated material for titanium alloy investment precision casting, and addressed its shear-thinning non-Newtonian fluid characteristics. I proposed a high-precision modeling method for the Cross-Williams-Landel-Ferry model parameters. This method fills the gap in the Moldflow software database regarding viscosity model parameters for paraffin-based materials. The main conclusions are as follows.
First, to solve the high-dimensional nonlinear modeling problem of the Cross-Williams-Landel-Ferry model for Kc-4017b wax, I proposed a quasi-Newton method integrating automatic differentiation and singular value decomposition regularization. At \(60\ ^\circ\text{C}\) to \(64\ ^\circ\text{C}\), the modeling \(R^2\) is 0.989, much higher than the 0.901 of simple least squares. This achieves high-precision modeling of the non-Newtonian viscosity characteristics of wax.
Second, I imported the Cross-Williams-Landel-Ferry parameters fitted by the quasi-Newton method into Moldflow software to simulate the injection filling process of an annular stepped wax pattern. I compared the results with actual wax pattern injection experiments. The deformation trend is highly consistent. This verifies that the model improves simulation reliability. In the context of precision casting, this means that the predicted shrinkage and warpage locations can be trusted for process decision-making.
Third, the quasi-Newton method shows stable adaptability across different temperatures and shear rate intervals. It accurately characterizes the coupling effect between wax viscosity, shear rate, and temperature. The model parameters are more consistent with actual rheological laws. This improves the prediction accuracy of numerical simulation and provides a solid theoretical foundation for non-Newtonian fluid filling processes. The method also reduces computational complexity compared with Newton-type methods and avoids the local optimum problem of simple least squares.
Fourth, the combination of automatic differentiation, singular value decomposition regularization, the Broyden-Fletcher-Goldfarb-Shanno update, and the Armijo condition provides a robust framework for high-dimensional nonlinear parameter fitting. This framework is not limited to wax. It can be extended to other non-Newtonian materials used in precision casting, such as slurries, binders, and ceramic pastes. It can also support the digitalization and intelligent optimization of investment precision casting processes for aerospace titanium alloy structural components.
Overall, I provide a reliable parameter support and theoretical basis for the numerical simulation of wax pattern injection molding in titanium alloy investment precision casting. The method enables more accurate prediction of filling, cooling, shrinkage, and warpage. It helps shorten development cycles, reduce production costs, and improve the dimensional quality of complex titanium alloy components produced by precision casting.