A high-precision numerical simulation method, system and product for a wax mold filling process of a lost wax casting

Abstract

This invention belongs to the field of numerical simulation of investment casting, and relates to a high-precision numerical simulation method, system, and product for the wax pattern filling process of investment casting. The method includes updating boundary conditions and determining the time step Δt; solving for the flow field, temperature field, and updating the viscosity based on the known field quantities of the current time step; the known field quantities include the flow field and temperature field obtained in the previous time step; performing interface tracking based on the flow field to obtain the free surface between the wax and air, thereby distinguishing between filled and unfilled areas; determining whether to continue time progression according to a preset filling ratio; if time progression is stopped, outputting the results of the flow field, temperature field, viscosity field, and the change of the free surface over time during the entire wax pattern filling process.

Description

Technical Field

[0001] This invention belongs to the field of numerical simulation of investment casting, and more specifically, relates to a high-precision numerical simulation method for the wax pattern filling process of investment casting. Background Technology

[0002] Investment casting is a crucial precision forming process capable of manufacturing complex structures, high dimensional accuracy, and high surface quality parts, with wide applications in aerospace, energy, automotive manufacturing, and high-end equipment. As a key intermediate product in the investment casting process, the quality of the wax pattern directly affects subsequent shell making, dewaxing, and the dimensional accuracy and surface quality of the final casting. Therefore, the stability and integrity of the wax pattern filling process are among the most important factors influencing the overall quality of investment casting.

[0003] In the wax pattern preparation process, molten wax enters the mold cavity under pressure and completes its flow, filling, and cooling under the constraints of complex geometric boundaries. This process is a typical transient, non-isothermal filling process, involving not only the velocity distribution and pressure transmission of the wax in the cavity, but also heat exchange between the wax and the mold wall, changes in flowability caused by local temperature drops, and the evolution of the flow state due to the continuous advancement of the free surface during filling. Especially for wax pattern structures with thin walls, narrow grooves, locally thick areas, and complex turning channels, problems such as local stagnation, underfilling, uneven filling, and uneven temperature distribution are prone to occur during the filling process, thus affecting the quality of the wax pattern.

[0004] Currently, numerical simulation methods for the filling process are widely used in injection molding, polymer flow, and general fluid filling analysis. These methods typically calculate and analyze the flow and heat transfer behavior of fluids in the mold cavity by establishing mass conservation equations, momentum conservation equations, and energy conservation equations. However, most existing methods are geared towards general injection molding or ordinary filling problems, and when directly applied to the wax pattern filling process of investment casting, the following shortcomings still exist: First, most existing methods are designed to establish general models for general polymer melts or common filling objects, which are not specific enough for wax pattern filling in investment casting. They are difficult to fully reflect the filling characteristics of wax material under the combined effects of temperature changes, shear state and boundary constraints in complex cavities, resulting in poor object adaptability. Secondly, existing methods often employ a first-order time-progression scheme for transient solutions. While the first-order scheme is simple to implement, it exhibits significant time discretization errors in problems with pronounced transient evolution characteristics, such as wax model filling. To maintain simulation accuracy, smaller time steps are often required, leading to an increase in the overall number of computational steps and a decrease in computational efficiency, making it difficult to balance accuracy and efficiency. Third, the description of wax viscosity in existing methods is often too simplified. Some methods only consider constant viscosity or only consider the influence of a single factor, and fail to fully reflect the variation law of wax viscosity under the combined influence of temperature, shear state and pressure. As a result, the prediction accuracy of flow stagnation, local underfilling and uneven filling is insufficient in complex cavities, thin-walled areas and local cooling sensitive areas. Fourth, for the wax mold filling process where changes in flow field, temperature field and related viscosity field work together, if the traditional strongly coupled fully implicit solution method is used, it is easy to lead to a large scale of equations, high solution complexity and difficulty in engineering implementation; if a simple low-order separate solution method is used, it is easy to have insufficient accuracy in complex boundaries and long-term progress.

[0005] With the increasing demands on the forming quality of complex structural parts in investment casting technology, how to design a high-precision numerical simulation method for the wax pattern filling process of investment casting, so that it can not only improve the simulation adaptability for the specific object of wax pattern filling, but also achieve a good balance between time progression accuracy, viscosity description accuracy, numerical stability and engineering feasibility, has become a key technical problem that urgently needs to be solved in the field of numerical simulation of investment casting. Summary of the Invention

[0006] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a high-precision numerical simulation method, system, and product for the wax pattern filling process in investment casting. The aim is to resolve the shortcomings in accuracy of existing numerical simulation methods for wax pattern filling in investment casting, particularly the low time discretization accuracy and inaccurate description of wax rheological behavior in traditional methods during transient filling processes. This results in limited accuracy in predicting the velocity field, pressure field, temperature field, and the evolution of the filling state during the wax pattern filling process.

[0007] To achieve the above objectives, according to one aspect of the present invention, a high-precision numerical simulation method for the wax pattern filling process of investment casting is provided, comprising the following model and steps: A three-dimensional incompressible flow control model for the wax pattern filling process includes: a continuity equation to ensure fluid mass conservation; a momentum conservation equation to describe fluid momentum transport; and an energy conservation equation to describe the heat transfer process. Among them, the viscous term in the momentum conservation equation is expressed in the form of a generalized Newtonian fluid to allow viscosity to be a variable that varies in space and time; the energy conservation equation is used to solve the temperature field of the wax. Wax viscosity field model: Based on the zero-shear viscosity and shear thinning laws, the viscosity of wax is expressed as a function of shear rate, temperature and pressure; BDF2 time discretization of the governing equations: The time derivatives of velocity and temperature are discretized using the BDF2 scheme. In the case of equal time steps, the unknowns of the current time step are constructed by the historical information of the previous time step, thus obtaining second-order time accuracy. In the discretization process, the nonlinear convection term is handled by second-order explicit extrapolation, while the viscous diffusion term determined by the freezing viscosity and the thermal diffusion term in the temperature field are handled implicitly. Step S1: Update boundary conditions and determine time step Δt; boundary conditions include the inlet flow rate of the wax model, the wall boundary between the wax and the mold, and the free surface between the wax and the air; Step S2: Based on the known field quantities of the current time step, first update the viscosity based on the wax viscosity field model, then solve the flow field using the three-dimensional incompressible flow control model based on the updated viscosity field, and then solve the temperature field using the BDF2 time discretization of the control equation; the known field quantities include the flow field and temperature field obtained in the previous time step. Step S3: Based on the flow field, interface tracking is performed to obtain the free surface between the wax and the air, thereby distinguishing between filled and unfilled areas; Step S4: Determine whether to continue time progression based on the preset filling ratio. If the preset filling ratio is not reached, proceed to step S1; otherwise, end time progression and output the flow field, temperature field, viscosity field, and the results of the free surface changes over time during the entire wax mold filling process.

[0008] Furthermore, in the three-dimensional incompressible flow control model: (1) Continuity equation: Under the assumption of incompressibility, the flow field satisfies the continuity equation: , Where u=(u,v,w) is the fluid velocity vector, and u,v,w represent the velocity components of the wax in the x, y, and z coordinate directions, respectively. This represents the divergence operator, used to characterize the degree of volume expansion or compression of a velocity vector field; (2) Momentum conservation equation: When the viscosity of the wax is not considered constant, but varies with temperature, pressure, and shear rate, the momentum conservation equation for an incompressible generalized Newtonian fluid is as follows: , in, t represents the density of the wax; t represents time. This represents the partial derivative of velocity with respect to time. For convection terms; This represents the gradient operator; P is the pressure; μ is the dynamic viscosity of the wax. Represents the Laplace operator; This is the volume force term, used to represent the external force acting on a unit volume of fluid; Generalized viscosity is used to represent the viscous characteristics of waxes under non-Newtonian conditions as state parameters change; is the strain rate tensor, used to characterize the local deformation rate of the velocity field; This indicates that divergence calculation is performed on a tensor field; (3) Energy conservation equation: For the non-isothermal wax mold filling process, the energy conservation equation is established as follows: , in, T represents the specific heat capacity of the wax; T represents the temperature. It represents the partial derivative of temperature with respect to time, used to characterize local transient changes in temperature; For temperature convection; k is the thermal conductivity. Q is the heat conduction term; Q is the volumetric heat source term.

[0009] Furthermore, the strain rate tensor Defined as: , in, Represents the velocity gradient tensor Transpose of; This represents the symmetric part of the velocity gradient tensor.

[0010] Furthermore, the expression for the wax viscosity field model is as follows: , in, For wax materials at temperature T and shear rate Generalized viscosity under pressure P; Zero shear viscosity; Where is the shear rate; P is the pressure; This is the critical stress parameter; i It is a power-law exponent used to characterize the degree of pseudoplasticity of a material.

[0011] Furthermore, zero shear viscosity The calculation is as follows: , , , in, For material viscosity scale parameters; This represents an exponential function with the natural constant e as its base. These are the fitting parameters for temperature sensitivity; The reference transition temperature is a characteristic temperature used to characterize changes in the viscosity of a material. Temperature correction parameters related to pressure; These are the fitting parameters under normal pressure conditions; This is the pressure correlation coefficient, used to characterize the effect of pressure on viscosity model parameters; This is the reference transition temperature under low pressure or reference pressure conditions.

[0012] Furthermore, for the velocity field, under the condition of equal time steps, the BDF2 time discretization form of its time derivative is: , in, , and These represent the fluid velocity vectors at times n+1, n, and n-1, respectively; for the nonlinear convection term in the momentum conservation equation... A second-order explicit extrapolation is used to incorporate it into the right-hand side of the discrete equation, thus avoiding nonlinear iterative solutions. .

[0013] Further, step S2 includes: S21: Viscosity Field Update Based on the velocity field, pressure field, and temperature field obtained in the previous time step, the shear rate is calculated and the viscosity field of the wax is updated. The updated viscosity field will be used to solve the subsequent flow field and temperature field, realizing the sequential coupling between multiple physics fields. S22: Solving the flow field Within each time step, a variable coefficient diffusion operator is first constructed using the viscosity field updated in the current time step, and the velocity prediction equation is solved to obtain the intermediate velocity field. Then, the pressure Poisson equation is established to perform pressure correction on the intermediate velocity field to obtain a velocity field that satisfies the incompressibility condition, and the pressure field is updated synchronously. S23: Temperature Field Solution After obtaining the velocity field at the current time step, the updated velocity field is substituted into the energy conservation equation. The convective transport term in the energy conservation equation is calculated explicitly, while the thermal diffusion term is processed implicitly under the BDF2 framework to obtain the temperature field distribution at the current time step. The temperature field result is used as the input for updating the viscosity field at the next time step to reflect the influence of the cooling process on the viscosity change of the wax.

[0014] Furthermore, the free surface update process in step S3 is as follows: Define the volume fraction function F, where 0 ≤ F ≤ 1, to characterize the volume occupancy of the wax material within the computational cell: F = 1 indicates that the cell is completely filled with wax material, F = 0 indicates that the cell is empty, and 0 < F < 1 indicates that the cell contains a free surface; the governing equation for the transport of the volume fraction function F with the flow field is: , After obtaining the velocity field at each time step, the F-field is updated by solving this governing equation to determine the instantaneous position of the free surface of the wax material and the distribution state of the filling region.

[0015] According to another aspect of the present invention, there is provided a computer system, including a memory, a processor, and a computer program stored on the memory, wherein the processor executes the computer program to implement a high-precision numerical simulation method for the investment casting wax pattern filling process as described in any one of the preceding items.

[0016] According to another aspect of the present invention, there is provided a computer program product, including a computer program, which implements a high-precision numerical simulation method for the investment casting wax pattern filling process as described in any one of the preceding items when executed by a processor.

[0017] Generally speaking, compared with the prior art, the above technical solutions conceived by the present invention can achieve the following beneficial effects: 1. Instead of directly applying general injection molding or ordinary filling analysis methods to the wax pattern filling process, the present invention specifically establishes a high-precision numerical simulation framework for the investment casting wax pattern filling process. Compared with traditional general simulation methods, the present invention can better conform to the actual physical characteristics of the wax material during the wax pattern filling process in multiple dimensions, thereby improving the description accuracy of the entire wax pattern filling process and enhancing the consistency between the numerical simulation results and the actual filling process.

[0018] 2. The present invention uses a second-order time advancement strategy to perform transient numerical simulation on the wax pattern filling process. Compared with traditional first-order time advancement methods, second-order time advancement can utilize more historical time step information to construct the discrete results of the current time step, thereby effectively reducing the truncation error in time discretization and more accurately describing the variation laws of the velocity field, pressure field, and temperature field of the wax material with time during the filling process. Therefore, the "high precision" of the present invention is first reflected in the improvement of time discretization accuracy.

[0019] 3. This invention employs a viscosity model that more accurately reflects the true rheological properties of wax. Compared to constant viscosity models or simplified models that only consider the influence of a single factor, the viscosity model used in this invention can more fully reflect the combined effects of temperature changes and shear forces on the flowability of wax, thereby improving the accuracy of characterizing the rheological behavior of wax and further enhancing the accuracy of predicting the evolution of the filling state during the wax mold filling process. Therefore, the "high accuracy" mentioned in this invention is reflected not only in the time progression aspect but also in the description of the wax viscosity field. Attached Figure Description

[0020] Figure 1 This is an overall flowchart of the high-precision numerical simulation method for the wax mold filling process in a preferred embodiment of the present invention; Figure 2 This is a diagram of the high-precision time discretization and explicit / implicit combination solution structure in a preferred embodiment of the present invention. Figure 3 The preferred embodiment of the present invention is a weakly coupled progression diagram of the flow field—temperature field—viscosity field sequence. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0022] This invention provides a preferred high-precision numerical simulation method for the wax pattern filling process in investment casting, such as... Figure 1 As shown, the main models and steps include the following: 1. Mathematical Model Construction: A Three-Dimensional Incompressible Flow Control Model for Wax Pattern Filling Process A three-dimensional incompressible flow control model was established to address the flow and heat transfer behavior of molten wax within the mold cavity during the wax mold filling process. The control model includes: a continuity equation to ensure fluid mass conservation; a momentum conservation equation to describe fluid momentum transport; and an energy conservation equation to describe the heat transfer process.

[0023] (1) Continuity equation: To ensure the mass conservation of the wax material during the filling process, under the assumption of incompressibility, the flow field satisfies the continuity equation: , Where u=(u,v,w) is the fluid velocity vector, and u,v,w represent the velocity components of the wax in the x, y, and z coordinate directions, respectively. The divergence operator is used to characterize the degree of volume expansion or compression of the velocity vector field. This equation shows that, under incompressible conditions, the wax element does not have a volume source term or mass defect during flow, and its local inflow and outflow remain in equilibrium.

[0024] (2) Momentum conservation equation: To describe the flow behavior of wax material within the mold cavity under pressure, viscous resistance, and external forces, the momentum conservation equation is used as follows: , in, t represents the density of the wax; t represents time. The partial derivative of velocity with respect to time is used to characterize the local variation of velocity over time. The convection term is used to characterize the momentum transport of fluid particles caused by changes in their spatial position. This represents the gradient operator; P is the pressure; μ is the dynamic viscosity of the wax. This represents the Laplace operator, used to characterize the second-order spatial diffusion effect of the velocity field; This is a volume force term, used to represent the external force acting on a unit volume of fluid.

[0025] This formula reflects the momentum balance relationship of the wax material during the filling process, namely the balance between the inertial term, pressure term, viscosity term and external force term.

[0026] When the viscosity of the wax is no longer considered constant, but varies with temperature, pressure, and shear rate, the viscosity term in the above equation can no longer be simplified to a constant coefficient form. At this point, for an incompressible generalized Newtonian fluid, the momentum conservation equation can be further written as: , in, Generalized viscosity is used to represent the viscous characteristics of waxes under non-Newtonian conditions as state parameters change; Let be the strain rate tensor, used to characterize the local deformation rate of the velocity field; where This indicates that divergence calculations are performed on the tensor field to obtain the contribution of viscous stress to the momentum equation.

[0027] This formula can more accurately reflect the non-Newtonian rheological behavior of molten wax during the filling process, thereby improving the ability of numerical simulation to characterize the actual filling process.

[0028] The strain rate tensor Defined as: , in, Represents the velocity gradient tensor Transpose of; This represents the symmetric part of the velocity gradient tensor.

[0029] This formula is used to calculate the shear and tensile deformation rates of fluid elements during flow, and is the basis for establishing the generalized Newtonian fluid viscous stress model.

[0030] In this embodiment, the viscosity η of the wax is preferably determined by the Cross-WLF model, and its specific expression and discrete solution method can be found in the following implementation steps.

[0031] (3) Energy conservation equation: For the non-isothermal wax mold filling process, in order to describe the evolution of the wax temperature field under the combined effects of convection and conduction, the following energy conservation equation is established: , in, T represents the specific heat capacity of the wax; T represents the temperature. It represents the partial derivative of temperature with respect to time, used to characterize local transient changes in temperature; is the temperature convection term, used to characterize heat transport caused by fluid flow; k is the thermal conductivity. Q is the thermal conductivity term, used to characterize the diffusion and transfer of heat in space; Q is the volumetric heat source term, used to characterize the generation or absorption of internal heat per unit volume.

[0032] This formula is used to solve the transient temperature distribution of wax during the filling process and provides temperature field input for viscosity updates, solidification determination and subsequent flow calculations.

[0033] 2. Establishment of the viscosity field model of wax Preferably, this embodiment uses the Cross-WLF non-Newtonian viscosity model to describe the viscosity of the wax. Based on zero-shear viscosity and shear thinning, this model expresses the viscosity of the wax as a function of shear rate, temperature, and pressure, thereby characterizing the viscosity changes of the wax in different filling stages and different local regions.

[0034] The model expression is: , in, For wax materials at temperature T and shear rate Generalized viscosity under pressure P; Zero shear viscosity is used to represent the viscosity of a material when the shear rate approaches zero. is the shear rate, used to characterize the speed of local shear deformation in a fluid; P is the pressure. It is a critical stress parameter used to characterize the characteristic stress of a material as it transitions from a low shear viscosity plateau region to a shear-thinning region. i It is a power-law exponent used to characterize the degree of pseudoplasticity of a material.

[0035] This formula is used to describe the coupling relationship between the viscosity of wax and the changes in temperature, pressure and shear rate.

[0036] Among them, zero shear viscosity Given by the WLF equation: , , , in, For material viscosity scale parameters; This represents an exponential function with the natural constant e as its base. These are the fitting parameters for temperature sensitivity; The reference transition temperature is a characteristic temperature used to characterize changes in the viscosity of a material. Temperature correction parameters related to pressure; These are the fitting parameters under normal pressure conditions; This is the pressure correlation coefficient, used to characterize the effect of pressure on viscosity model parameters; This is the reference transition temperature under low pressure or reference pressure conditions.

[0037] In this embodiment, , i , , , , and These are all material model parameters, which can be obtained by fitting wax rheological experimental data.

[0038] During the numerical solution process, the shear rate and viscosity values ​​are calculated based on the velocity field, temperature field, and pressure field obtained in the previous or current time step. The updated viscosity field is then used in the next flow field solution.

[0039] 3. Time Discreteness of the BDF2 of the Governing Equations In this embodiment, such as Figure 2 As shown, to improve the time accuracy of the numerical simulation of the filling process, the time term of the governing equations is discretized using a second-order backward difference scheme (BDF2). For the velocity and temperature fields, under the condition of equal time steps, the discretized form of their time derivatives is as follows: , in , and These represent the fluid velocity vectors at times n+1, n, and n-1, respectively.

[0040] For the nonlinear convection term in the momentum conservation equation A second-order explicit extrapolation is used to incorporate it into the right-hand side of the discrete equation, thus avoiding nonlinear iterative solutions. , The convection term in the temperature equation also adopts the same second-order explicit extrapolation form.

[0041] To address the influence of shear rate, temperature, and pressure on wax viscosity, a viscosity freezing strategy is employed during time progression. This involves calculating the viscosity field at the current time step (time n) using the field variables from the previous time step (time n-1). Within the current time step, it is considered as a known coefficient: , in, These represent the pressure, temperature, and shear rate at time n-1, respectively.

[0042] Based on the above discretization process, the velocity prediction step discrete equation of the momentum equation can be constructed as a system of linear equations containing a variable coefficient diffusion operator, with the following specific form: , Where Δt is the time step. This approach places both the viscous diffusion term and the thermal diffusion term in the temperature field on the left-hand side of the equation for implicit solution, which ensures the stability of the numerical calculation while also taking into account computational efficiency through explicit extrapolation of the convection term.

[0043] Based on the above model, the main steps of the numerical simulation method of the present invention are as follows: Step S1: Update boundary conditions and determine time step Δt; boundary conditions include the inlet flow rate of the wax model, the wall boundary between the wax and the mold, and the free surface between the wax and the air; Step S2: Based on the known field quantities of the current time step, complete the flow field solution, temperature field solution, and viscosity update; the known field quantities include the flow field and temperature field obtained in the previous time step; Step S3: Based on the flow field, interface tracking is performed to obtain the free surface between the wax and the air, thereby distinguishing between filled and unfilled areas; Step S4: Determine whether to continue time progression based on the preset filling ratio. If the preset filling ratio is not reached, proceed to step S1; otherwise, end time progression and output the flow field, temperature field, viscosity field, and the results of the free surface changes over time during the entire wax mold filling process.

[0044] Preferably, such as Figure 3 As shown, step S2 includes the following sub-steps: S21: Viscosity Field Update In the current time step, based on the flow field and temperature field obtained in the previous time step, the shear rate is calculated using the wax viscosity field model and the viscosity field of the wax is updated. Among them, the flow field is the velocity field and the pressure field.

[0045] The updated viscosity field will be used to solve the flow field and temperature field in the next time step, realizing the sequential coupling between multiple physical fields.

[0046] S22: Solving the flow field In each time step, first, a variable coefficient diffusion operator is constructed using the viscosity field updated in the current time step and the pressure term is ignored. , and the intermediate velocity field is obtained by solving the velocity prediction equation: , where I is the identity matrix, is the intermediate velocity field, represents the variable coefficient viscous operator defined by the frozen viscosity field.

[0047] After obtaining the intermediate velocity field , the pressure correction is carried out using the projection method. First, the pressure increment that satisfies the Poisson equation is solved: , Then, the intermediate velocity is corrected using the pressure increment to obtain the correct velocity field: , Correspondingly, the pressure field is updated to: , The updated flow field will be used for subsequent temperature field solving and free surface updating.

[0048] S23: Solving the temperature field After obtaining the velocity field of the current time step, it is substituted into the energy conservation equation; the convective transport term in the energy conservation equation is calculated explicitly, and the heat diffusion term is treated implicitly under the BDF2 framework to obtain the temperature field distribution of the current time step.

[0049] The result of the temperature field is used as the input for viscosity field updating to reflect the influence of the cooling process on the viscosity change of the wax.

[0050] S3: Free surface updating In this embodiment, the volume of fluid (VOF) method is used to track the free surface during the wax filling process. The volume fraction function F is defined, and its value range is 0 ≤ F ≤ 1, which is used to characterize the volume occupancy of the wax in the computational cell: F = 1 means the cell is completely filled with wax, F = 0 means the cell is empty, and 0 < F < 1 means there is a free surface in the cell.

[0051] The governing equation for the volume fraction function F as a function of flow field transport is: , After obtaining the velocity field at each time step, the instantaneous position of the wax front and the distribution of the filling region can be determined by solving the above transport equation to update the F field.

[0052] S4: Time Step Progression, Result Output and Application The above process is executed cyclically in the order of time progression until all time steps are calculated. After all time steps are calculated, the velocity field, pressure field, temperature field, viscosity field, and the changes of the free surface over time during the entire wax mold filling process are output.

[0053] Based on the above results, the design of gates and runners, filling time, local cooling sensitive areas, potential underfill areas, and process parameter settings can be evaluated, thereby providing a quantitative basis for the optimization design of wax mold structure and the determination of process parameters.

[0054] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A high-precision numerical simulation method for a wax pattern molding process for investment casting, characterized by, The model and steps include the following: A three-dimensional incompressible flow control model for the wax pattern filling process includes: a continuity equation to ensure fluid mass conservation; a momentum conservation equation to describe fluid momentum transport; and an energy conservation equation to describe the heat transfer process. Among them, the viscous term in the momentum conservation equation is expressed in the form of a generalized Newtonian fluid to allow viscosity to be a variable that varies in space and time; the energy conservation equation is used to solve the temperature field of the wax. Wax viscosity field model: Based on the zero-shear viscosity and shear thinning laws, the viscosity of wax is expressed as a function of shear rate, temperature and pressure; BDF2 time discretization of the governing equations: The time derivatives of velocity and temperature are discretized using the BDF2 scheme. In the case of equal time steps, the unknowns of the current time step are constructed by the historical information of the previous time step, thus obtaining second-order time accuracy. In the discretization process, the nonlinear convection term is handled by second-order explicit extrapolation, while the viscous diffusion term determined by the freezing viscosity and the thermal diffusion term in the temperature field are handled implicitly. Step S1: Update boundary conditions and determine time step Δt; boundary conditions include the inlet flow rate of the wax model, the wall boundary between the wax and the mold, and the free surface between the wax and the air; Step S2: Based on the known field quantities of the current time step, first update the viscosity based on the wax viscosity field model, then solve the flow field using the three-dimensional incompressible flow control model based on the updated viscosity field, and then solve the temperature field using the BDF2 time discretization of the control equation; the known field quantities include the flow field and temperature field obtained in the previous time step. Step S3: Based on the flow field, interface tracking is performed to obtain the free surface between the wax and the air, thereby distinguishing between filled and unfilled areas; Step S4: Determine whether to continue time progression based on the preset filling ratio. If the preset filling ratio is not reached, proceed to step S1; otherwise, end time progression and output the flow field, temperature field, viscosity field, and the results of the free surface changes over time during the entire wax mold filling process.

2. The high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 1, characterized in that, In the three-dimensional incompressible flow control model: (1) Continuity equation: Under the assumption of incompressibility, the flow field satisfies the continuity equation: Where u=(u,v,w) is the fluid velocity vector, and u,v,w represent the velocity components of the wax in the x, y, and z coordinate directions, respectively. This represents the divergence operator, used to characterize the degree of volume expansion or compression of a velocity vector field; (2) Momentum conservation equation: When the viscosity of the wax is not considered constant, but varies with temperature, pressure, and shear rate, the momentum conservation equation for an incompressible generalized Newtonian fluid is as follows: in, t represents the density of the wax; t represents time. This represents the partial derivative of velocity with respect to time. For convection terms; This represents the gradient operator; P is the pressure. This is the volume force term, used to represent the external force acting on a unit volume of fluid; Generalized viscosity is used to represent the viscous characteristics of waxes under non-Newtonian conditions as state parameters change; is the strain rate tensor, used to characterize the local deformation rate of the velocity field; This indicates that divergence calculation is performed on a tensor field; (3) Energy conservation equation: For the non-isothermal wax mold filling process, the energy conservation equation is established as follows: in, T represents the specific heat capacity of the wax; T represents the temperature. It represents the partial derivative of temperature with respect to time, used to characterize local transient changes in temperature; For temperature convection; k is the thermal conductivity. Q is the heat conduction term; Q is the volumetric heat source term.

3. The high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 2, characterized in that, The strain rate tensor Defined as: in, Represents the velocity gradient tensor Transpose of; This represents the symmetric part of the velocity gradient tensor.

4. The high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 1, characterized in that, The expression for the viscosity field model of wax is: in, For wax materials at temperature T and shear rate Generalized viscosity under pressure P; Zero shear viscosity; Where is the shear rate; P is the pressure; This is the critical stress parameter; i It is a power-law exponent used to characterize the degree of pseudoplasticity of a material.

5. A high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 4, characterized in that, Zero shear viscosity The calculation is as follows: in, For material viscosity scale parameters; This represents an exponential function with the natural constant e as its base. These are the fitting parameters for temperature sensitivity; The reference transition temperature is a characteristic temperature used to characterize changes in the viscosity of a material. Temperature correction parameters related to pressure; These are the fitting parameters under normal pressure conditions; This is the pressure correlation coefficient, used to characterize the effect of pressure on viscosity model parameters; This is the reference transition temperature under low pressure or reference pressure conditions.

6. A high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 2, characterized in that, For the velocity field, under the condition of equal time steps, the BDF2 time discretization form of its time derivative is: in, , and These represent the fluid velocity vectors at times n+1, n, and n-1, respectively; for the nonlinear convection term in the momentum conservation equation... A second-order explicit extrapolation is used to incorporate it into the right-hand side of the discrete equation, thus avoiding nonlinear iterative solutions. 。 7. A high-precision numerical simulation method for the wax pattern filling process of investment casting according to any one of claims 1 to 6, characterized in that, Step S2 includes: S21: Viscosity Field Update Based on the velocity field, pressure field, and temperature field obtained in the previous time step, the shear rate is calculated and the viscosity field of the wax is updated. The updated viscosity field will be used to solve the subsequent flow field and temperature field, realizing the sequential coupling between multiple physics fields. S22: Solving the flow field At each time step, first, a variable-coefficient diffusion operator is constructed using the viscosity field updated at the current time step, and the velocity prediction equation is solved to obtain the intermediate velocity field. Subsequently, a pressure Poisson equation is established to correct the pressure of the intermediate velocity field, obtaining a velocity field that satisfies the incompressible condition, and the pressure field is updated synchronously. S23: Temperature field solution After obtaining the velocity field at the current time step, the updated velocity field is substituted into the energy conservation equation. The convective transport term in the energy conservation equation is calculated explicitly, and the heat diffusion term is treated implicitly under the BDF2 framework to obtain the temperature field distribution at the current time step. The temperature field result is used as the input for updating the viscosity field at the next time step to reflect the influence of the cooling process on the viscosity change of the wax material.

8. A high-precision numerical simulation method for the wax pattern filling process of investment casting according to claim 1, characterized in that, The free surface update process of step S3 is as follows: Define the volume fraction function F, where 0 ≤ F ≤ 1, to characterize the volume occupancy of the wax material in the computational cell: F = 1 indicates that the cell is completely filled with wax material, F = 0 indicates that the cell is empty, and 0 < F < 1 indicates that the cell contains a free surface. The control equation for the transport of the volume fraction function F with the flow field is: After obtaining the velocity field at each time step, the F field is updated by solving this control equation to determine the instantaneous position of the free surface of the wax material and the distribution state of the filling region.

9. A computer system comprising a memory, a processor, and a computer program stored in the memory, characterized in that, The processor executes the computer program to implement a high-precision numerical simulation method for the investment casting wax mold filling process according to any one of claims 1 to 8.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements a high-precision numerical simulation method for the investment casting wax mold filling process according to any one of claims 1 to 8.

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