A method and apparatus for numerical simulation of solidification process of steel castings with large time step

Abstract

The present application belongs to the technical field of numerical simulation of casting process, and discloses a large time step numerical simulation method and equipment for the solidification process of steel castings, which comprises the following steps: (1) using MAC staggered grid to discretize the calculation domain corresponding to the solidification process of the steel castings to obtain a grid; constructing a multi-physical field coupling model; (2) using an implicit and explicit time advancing method based on BDF2 and AB2 to discretize the momentum conservation equation in the flow field contained in the multi-physical field coupling model; (3) in each time step, first, updating the solid phase fraction according to the current temperature field and solute field; then, determining the flow region according to the updated solid phase fraction; subsequently, using the projection method to solve the incompressible flow in the determined flow region to obtain the velocity field and pressure field; then, solving the temperature field and solute field based on the updated velocity field; (4) dynamically updating the physical property parameters, thereby completing the simulation. The present application improves the calculation efficiency.

Description

Technical Field

[0001] This invention belongs to the technical field of numerical simulation of casting processes, and more specifically, relates to a numerical simulation method and equipment for the solidification process of steel castings with a large time step. Background Technology

[0002] In the casting industry, especially in the production of large steel castings (such as heavy machine tool bases and hydroelectric turbine runners), macroscopic segregation and other defects are prone to occur during the solidification process, seriously affecting the mechanical properties and service safety of the components. Traditional trial-and-error methods, relying on experience, are costly and time-consuming, and struggle to deeply reveal the defect formation mechanisms. With the development of computational mechanics, numerical simulation of the casting solidification process has become a key tool for predicting defects and optimizing processes. By solving the energy equation coupled with the latent heat of phase change, it reproduces the temperature field evolution and thus predicts the solidification sequence and defect risk, providing significant guidance for process design.

[0003] Currently, explicit time-progression schemes, especially explicit Euler methods, are widely used in commercial casting simulation software and most self-developed programs due to their simplicity, lack of iteration requirements, and ease of implementation. However, the explicit Euler method is a conditionally stable algorithm, and its stability is constrained by strict CFL conditions. For heat conduction problems, these CFL conditions require the time step to be less than a critical value proportional to the square of the mesh size. When performing detailed simulations of large castings, the spatial mesh is enormous and the mesh size is extremely small in order to distinguish geometric details and temperature boundary layers, resulting in a severe limitation on the maximum allowable stable time step. Simulating solidification processes that last for hours or even tens of hours requires executing millions or even tens of millions of time steps, resulting in enormous computational loads and staggering time consumption, severely restricting simulation efficiency, especially in applications requiring multi-parameter studies and rapid process evaluation, becoming a prominent bottleneck.

[0004] Therefore, developing a time discretization scheme that combines high-order accuracy, good numerical stability (such as unconditional stability or a wide stability domain), and feasible computational efficiency is a key research direction for improving the practicality of casting solidification simulation. Among numerous time discretization methods, the backward difference formula series, especially the second-order BDF scheme, shows the potential to resolve this contradiction. The BDF2 scheme is an implicit linear multistep method with second-order time accuracy, theoretically providing a more accurate transient process approximation than the first-order explicit Euler method for the same number of time steps. More importantly, the BDF2 scheme has superior numerical stability compared to explicit methods, typically allowing for much larger time steps, thus potentially improving overall computational efficiency by significantly reducing the total number of time steps, providing a new algorithmic approach for efficient and high-fidelity simulation of large castings. Although the BDF2 scheme requires additional steps at startup and incurs some single-step computational overhead from solving implicit equations within each time step, its advantage of trading "large time steps" for overall efficiency improvement is significant in long-term transient simulations.

[0005] In summary, existing solidification simulation methods, such as the explicit Euler method, suffer from stability limitations and low computational efficiency, making it difficult to meet the engineering requirements for rapid and high-precision simulation of large castings. Developing novel solution frameworks based on higher-order implicit or semi-implicit schemes to achieve breakthroughs in both stability and efficiency has significant theoretical value and promising engineering application prospects. Summary of the Invention

[0006] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a numerical simulation method and apparatus for the solidification process of steel castings with a large time step, which aims to solve the problems of limited time step and low computational efficiency caused by existing explicit time-progression schemes.

[0007] To achieve the above objectives, according to one aspect of the present invention, a numerical simulation method for the solidification process of steel castings with a large time step is provided, comprising the following steps: (1) The computational domain corresponding to the solidification process of steel casting is spatially discretized by MAC staggered grid to obtain the grid, wherein pressure, temperature, total solute concentration and solid fraction are stored in the center of the grid cell, and velocity components are stored at the grid cell interface; at the same time, a multi-physics coupling model of the solidification process of steel casting including flow field, temperature field, solute field and phase transformation field is constructed. The multi-physics coupling model uses solid fraction to realize the coupling between the physical fields. (2) The momentum conservation equation in the flow field contained in the multiphysics coupling model is discretized by an implicit and explicit time-progression method based on the second-order backward difference formula BDF2 and the second-order Adams-Bashforth scheme AB2, so that the variables of the momentum conservation equation can be represented by points on the grid; wherein: the unsteady term of the momentum conservation equation is implicitly discretized by BDF2, the convection term is explicitly extrapolated by AB2, and the diffusion term and the drag term of the smear region are implicitly processed; (3) In each time step, the solid fraction is first updated based on the current temperature field and solute field; then the flow region is determined based on the updated solid fraction; then the incompressible flow is solved by projection method in the determined flow region to obtain the velocity field and pressure field; then the temperature field and solute field are solved based on the updated velocity field, and the corresponding scalar distribution is updated. (4) Based on the solid fraction, material type and solidification state, the flow resistance parameters, heat transfer parameters and solute transport parameters in the computational domain are dynamically updated, thereby obtaining the temperature field, velocity field, pressure field, solid fraction field, solute field and material distribution results during the solidification process of steel castings.

[0008] Furthermore, the flow field is used to describe the motion behavior of liquid metal during solidification, the temperature field is used to describe the heat transfer process, the solute field is used to describe the migration and distribution behavior of alloying elements, and the phase transformation field is used to describe the evolution process of the material from the liquid phase to the solid-liquid two-phase region and the solid phase region.

[0009] Furthermore, the phase transition field is characterized by a solid fraction, which is updated based on the coupling of the temperature field and the solute field.

[0010] Furthermore, the solid fraction is used to simultaneously characterize the local solidification state and serves as a coupling variable for flow region renewal, paste region resistance calculation, and solute distribution calculation, thereby achieving unified coupling between the flow field, temperature field, solute field, and phase change field.

[0011] Furthermore, the momentum conservation equation is discretized to form a Helmholtz-type linear equation system, which is used to solve the intermediate velocity field within the time step; a pressure Poisson equation is constructed based on the intermediate velocity field, and the intermediate velocity field is projected and corrected based on the pressure Poisson equation to obtain the velocity field and pressure field that satisfy the continuity equation.

[0012] Furthermore, when the solid fraction reaches a preset threshold, the corresponding region is switched from a flowing region to a non-flowing region; the flow resistance in the mushy region is dynamically adjusted according to the solid fraction.

[0013] Furthermore, dynamic updates of regional attributes are used in conjunction with mobile region masks.

[0014] Furthermore, after obtaining the new solid-phase fractional field, a region mask is constructed to limit the solution range for different physical fields, including the following region masks: A flow region mask is used to define the solution domain for the velocity and pressure fields. Temperature solution region mask, used to limit the update range of the temperature field; The solute solution region mask is used to limit the update range of the solute field.

[0015] The present invention also provides a numerical simulation system for the solidification process of steel castings with a large time step. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it performs the numerical simulation method for the solidification process of steel castings with a large time step as described above.

[0016] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the numerical simulation method for the solidification process of steel castings with a large time step as described above.

[0017] In summary, compared with the prior art, the numerical simulation method and equipment for the solidification process of steel castings with a large time step provided by the present invention have the following advantages: 1. This invention employs an implicit-explicit time discretization strategy based on BDF2 and AB2 for the momentum conservation equation. In this strategy, rigid terms such as diffusion and sludge resistance are implicitly handled, while convection terms are explicitly extrapolated. This effectively weakens the stability constraints caused by the CFL condition in traditional explicit methods, freeing the time step from strict constraints. Under the premise of ensuring numerical stability, the acceptable time step is significantly increased, thereby greatly reducing the total number of time steps and improving the overall computational efficiency.

[0018] 2. This invention couples the flow field, temperature field, solute field, and phase transition field, and combines them with a sequential coupling solution strategy, so that the physical fields can transmit information through shared variables (solid fraction), thereby avoiding the numerical oscillation and convergence difficulties that are prone to occur in traditional strong coupling solutions, and improving the stability of multi-physics coupling calculations.

[0019] 3. This invention constructs a dynamic region masking mechanism based on the solid fraction and updates the flow region, temperature solution region, and solute solution region in real time according to the change of solid fraction. This enables the computational domain to evolve dynamically with the solidification process, thereby realistically reflecting processes such as liquid phase channel contraction, pasty region expansion, and the formation of a fully solidified region, and improving the characterization ability of the regional evolution characteristics of the solidification process.

[0020] 4. This invention establishes a dynamic updating mechanism for the resistance parameters and key physical properties of the mushy region with the solid fraction, enabling accurate description of the flow attenuation behavior, thermal property evolution, and solute redistribution characteristics within the mushy region during solidification. This mechanism effectively improves the prediction accuracy of macrosegregation, thus providing more reliable numerical basis for optimizing cooling rates and controlling solidification defects in casting processes, and enhancing the pertinence of engineering analysis and process optimization. Attached Figure Description

[0021] Figure 1 This is a flowchart of a numerical simulation method for the solidification process of steel castings with a large time step, provided in an embodiment of the present invention. Figure 2 These are detailed diagrams illustrating the advancement process according to embodiments of the present invention; Figure 3 This is a schematic diagram of the MAC grid variable arrangement; Figure 4 This is a schematic diagram illustrating the application of the BDF2-AB2 scheme in the momentum equation. Detailed Implementation

[0022] 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.

[0023] Numerical simulations of the solidification process of steel castings typically involve multi-physics coupled processes such as flow, heat transfer, solute transport, and phase transformation. The momentum conservation equations are interdependent, highly nonlinear, and exhibit significant rigidity. Especially in high spatial resolution or large-size casting simulations, if explicit time-progression methods are used, the time step must meet strict stability conditions, leading to a significant increase in the number of time steps required for computation. This results in high computational costs and low solution efficiency. Furthermore, under conditions such as the evolution of mushy regions, regional flow blockage, and dynamic changes in material parameters, coupled computational instability or decreased convergence performance can easily occur.

[0024] This invention provides a numerical simulation method for the solidification process of steel castings with a large time step. The numerical simulation method is based on the BDF2-AB2 implicit and explicit time discretization scheme. By splitting rigid and non-rigid terms, the method increases the usable time step while ensuring numerical stability and calculation accuracy, thereby improving the computational efficiency of steel casting solidification simulation.

[0025] The numerical simulation method mainly includes the following steps: S1. The computational domain corresponding to the solidification process of steel casting is spatially discretized using a MAC staggered grid to obtain a grid. Pressure, temperature, total solute concentration and solid fraction are stored in the center of the grid cell, and velocity components are stored at the grid cell interface. At the same time, a multi-physics coupling model of the solidification process of steel casting is constructed, which includes flow field, temperature field, solute field and phase transformation field. The multi-physics coupling model uses solid fraction to realize the coupling between various physical fields.

[0026] Specifically, scalars such as pressure, temperature, total solute concentration, and solid fraction are stored at the center of the grid cells, while the velocity components in the three coordinate directions are stored on the corresponding grid cell surfaces to improve the pressure-velocity coupling stability in the incompressible flow solution process.

[0027] The computational domain can be established based on the actual geometry of the casting, mold, insulation sleeve, and air area, and material distribution information can be imported through external voxel files or geometric data files to form a multi-material computational domain.

[0028] Imported material identifiers can be mapped to flow regions, thermally conductive solid regions, insulating regions, and inactive regions for subsequent flow solutions, thermal conductivity calculations, and boundary treatment.

[0029] The flow field describes the motion behavior of liquid metal during solidification, the temperature field describes the heat transfer process, the solute field describes the migration and distribution of alloying elements, and the phase transformation field describes the evolution of the material from the liquid phase to the solid-liquid two-phase region and the solid phase region. The phase transformation field is characterized by the solid fraction, which is updated based on the coupling of the temperature field and the solute field.

[0030] The solid fraction is used to simultaneously characterize the local solidification state and serves as a coupling variable for flow region renewal, paste region resistance calculation, and solute distribution calculation, thereby achieving unified coupling between the flow field, temperature field, solute field, and phase change field.

[0031] S2 employs an implicit / explicit time-progression method based on the second-order backward difference formula BDF2 and the second-order Adams-Bashforth scheme AB2 to discretize the momentum conservation equations in the flow field contained in the multiphysics coupling model, so as to use points on the grid to represent the variables of the momentum conservation equations; wherein: the unsteady-state terms of the momentum conservation equations are implicitly discretized using BDF2, the convection terms are explicitly extrapolated using AB2, and the diffusion terms and the drag terms in the smear region are implicitly processed.

[0032] Specifically, the momentum conservation equation is discretized using the BDF2-AB2 implicit-explicit time-margining method: the momentum conservation equation in the flow field is discretized using an implicit-explicit time-margining method based on the second-order backward difference formula BDF2 and the second-order Adams-Bashforth scheme AB2. Unsteady-state terms are implicitly discretized using the BDF2 scheme, convection terms are explicitly extrapolated using the AB2 scheme, and diffusion terms and drag terms in the sludge region are implicitly handled. By implicitly discretizing rigid terms and explicitly discretizing non-rigid terms in the momentum conservation equation, the time-margining stability can be improved while maintaining the accuracy of second-order time discretization. The discretized momentum conservation equation forms a Helmholtz-type linear equation system, used to solve for the intermediate velocity field within the time step. A pressure Poisson equation is constructed based on the intermediate velocity field, and the intermediate velocity field is projected and corrected based on the pressure Poisson equation to obtain the velocity and pressure fields that satisfy the continuity equation.

[0033] S3, at each time step, first update the solid fraction based on the current temperature field and solute field; then determine the flow region based on the updated solid fraction; then solve the incompressible flow in the determined flow region using the projection method to obtain the velocity field and pressure field; then solve the temperature field and solute field based on the updated velocity field, and update the corresponding scalar distribution.

[0034] When the solid fraction reaches a preset threshold, the corresponding region can be switched from a flowing region to a non-flowing region to characterize the dynamic contraction of the flow channel during solidification and the inhibitory effect of the fully solidified region on flow. The flow resistance in the mushy region is dynamically adjusted according to the solid fraction to reflect the obstructive effect of the dendritic framework on the flow of liquid metal. The temperature field and solute field can be advanced using an implicit / explicit time discretization method; the convection term is explicitly processed, and the diffusion term is implicitly processed to balance stability and computational efficiency. The solute field can be updated using the total solute mass conservation form, and the liquid and solid phase concentrations are calculated in conjunction with the solid fraction to characterize the solute redistribution behavior during solidification.

[0035] S4 dynamically updates the flow resistance parameters, heat transfer parameters, and solute transport parameters in the computational domain based on the solid fraction, material type, and solidification state.

[0036] Different material regions can correspond to different thermal conductivity, thermal diffusivity, and solute transport coefficients. Flow resistance parameters can increase with increasing solid fraction to reflect the physical process of gradually decreasing flow capacity in the mushy region. Dynamic updates of region attributes can be used in conjunction with flow region masks to improve the physical consistency of solidification process simulation.

[0037] Material properties are dynamically updated based on the solid fraction, where: 1) The viscosity changes continuously between the liquid phase viscosity and the high viscosity of the solid phase; 2) Thermal conductivity is updated based on material type and phase. 3) The solute transport coefficient is adjusted according to the change in liquid phase fraction.

[0038] S5, repeat steps S2 to S4 until the preset simulation time or solidification end criterion is reached, output the temperature field, velocity field, pressure field, solid fraction field, solute field and material distribution results during the solidification process of steel castings, which are used for solidification behavior analysis, defect prediction and process optimization of steel castings.

[0039] The output results may also include liquid phase concentration field, solid phase concentration field and different region type identification results to support segregation analysis, solidification front evolution analysis and subsequent visualization processing.

[0040] The following combination Figures 1 to 4 The specific embodiments of the present invention will be described in further detail below.

[0041] This invention provides a multiphysics coupled numerical simulation method for the solidification process of steel castings based on the BDF2-AB2 implicit-explicit time discretization scheme. It is applicable to the numerical analysis of flow, heat transfer, solute transport, and phase transformation behavior of steel castings during solidification, and is particularly suitable for efficient simulation of large-size castings and high spatial resolution mesh conditions. Specifically: (1) Computational domain establishment and spatial discretization In this embodiment, a three-dimensional computational domain for the solidification process of the steel casting is first established. This computational domain may include multiple material regions such as the casting region, mold region, insulation sleeve region, and air region. Depending on the actual modeling requirements, the computational domain can be generated directly based on regular geometric parameters, or spatial distribution information of each region can be imported from an external geometric model, voxelized data file, or material distribution file to establish a multi-material computational domain.

[0042] Preferably, material distribution information can be imported via an external voxel file, and the original material identifiers can be mapped to flowing metal regions, thermally conductive solid regions, insulating regions, and inactive regions for subsequent flow solving, heat conduction calculation, and boundary region identification. The corresponding material identifier mapping and region partitioning logic can be supported by the material import module within the program.

[0043] In terms of spatial discretization, this implementation uses a MAC staggered grid to discretize the computational domain. Scalars such as pressure, temperature, total solute concentration, and solid fraction are defined at the center of each grid cell; the velocity components in the three coordinate directions are defined at the center of the corresponding grid cell surface. This staggered arrangement effectively reduces the pseudo-coupling problem between the pressure and velocity fields in incompressible flow solutions, improving numerical stability and solution accuracy.

[0044] Furthermore, let the number of grid nodes in the three-dimensional computational domain in the three coordinate directions be Nx, Ny, and Nz, respectively, forming a structured three-dimensional mesh. For the case of a regular mesh, the grid spacing in each direction can be constant; for the case of a non-uniform mesh, the grid scale in each direction can also be set according to the step size information provided in the externally imported file.

[0045] (2) Multiphysics initialization After establishing the computational domain and completing spatial discretization, the various physical fields involved in the calculation of the solidification process of steel castings are initialized, including the temperature field, velocity field, pressure field, total solute concentration field, solid fraction field, and material distribution field.

[0046] The initial temperature of the metal region can be set according to the pouring temperature, while the initial temperatures of the mold region, the insulation region, and the air region can be set according to the ambient temperature or the preheating temperature. The initial velocity field is preferably set to a zero velocity field, but a non-zero initial flow can also be set according to the initial pouring conditions. The initial pressure field can be set to a uniform distribution. The initial total solute concentration field is set according to the nominal alloy composition of the molten steel. The initial solid fraction field is calculated based on the initial temperature field and the initial solute concentration field combined with the phase diagram relationship.

[0047] Furthermore, when using the BDF2-AB2 implicit-explicit time discretization scheme, it is necessary to store the variable values ​​of at least one historical time layer for the construction of subsequent BDF2 time terms and AB2 convection terms extrapolation. For the initial calculation phase, a first-order time progression scheme can be used to obtain the first historical time layer data, and then the process can be switched to the BDF2-AB2 scheme for formal time progression.

[0048] (3) Establishment of multiphysics coupling relationship In this embodiment, the solidification process of steel castings is described by a multi-physics field coupling model of flow field, temperature field, solute field and phase transformation field.

[0049] Among them, the flow field is used to describe the flow behavior of liquid metal under the action of gravity, pressure gradient and local resistance; the temperature field is used to describe the heat conduction and convection heat transfer process in the molten steel, the mold and the surrounding area; the solute field is used to describe the transport and redistribution behavior of alloying elements in the liquid phase region and the solid-liquid two-phase region; and the phase transformation field is used to describe the solidification evolution process of the material from the liquid phase to the paste region and the solid phase region.

[0050] In this embodiment, the phase transition field is characterized by the solid fraction. The solid fraction value ranges from 0 to 1. When the solid fraction is 0, it corresponds to the completely liquid phase region; when the solid fraction is between 0 and 1, it corresponds to the paste region; and when the solid fraction is 1, it corresponds to the completely solidified region. The solid fraction is updated according to the local temperature and solute concentration and serves as the core coupling variable between the flow field, temperature field, solute field, and phase transition field.

[0051] Furthermore, the solid fraction is not only used to characterize the local solidification state, but also to control the opening and closing of the flow region, the resistance intensity of the mushy region, and the distribution relationship of the solute between the liquid and solid phases, thereby achieving unified multi-field coupling in the solidification process.

[0052] (4) Overall process of time advancement like Figure 1 and Figure 2 As shown, in this embodiment, the entire solidification process of the steel casting is solved by a time-step approach. Let the time step be Δt and the total simulation time be temp. Starting from the initial moment, the simulation proceeds step-by-step according to the preset time step until the total simulation time is reached or the solidification termination condition is met.

[0053] Within each time step, the multiphysics coupling solution process is executed in the following order: 1) Update the solid fraction field based on the current temperature field and solute field; 2) Determine the flow region, temperature solution region, and solute solution region based on the updated solid fraction field; 3) Discretize the momentum conservation equation in the flow field using the BDF2-AB2 implicit-explicit time-progression scheme to solve for the intermediate velocity field; 4) Construct the pressure Poisson equation and apply pressure correction to the intermediate velocity field to obtain the velocity field and pressure field that satisfy the continuity equation; 5) Solve for the temperature field based on the updated velocity field; 6) Solve for the solute field based on the updated velocity field and phase transition information; 7) Dynamically update flow resistance parameters, heat transfer parameters, and solute transport parameters based on solid fraction, material type, and solidification state; 8) When the output time is reached, output the relevant physical quantity field.

[0054] By using the above sequential coupling process, stable coupling solutions between various physical fields can be achieved during the solidification of steel castings.

[0055] (5) Solid phase fraction update and region mask reconstruction.

[0056] At the beginning of each time step, the solid fraction of each grid cell is first calculated based on the current temperature field and the total solute concentration field. The solid fraction can be updated according to a preset phase diagram relationship, the temperature range relationship between the solidus and liquidus lines, or a temperature-concentration coupling function.

[0057] For regions above the liquidus line, the solid fraction is low or zero; for regions below the solidus line, the solid fraction is high or close to one; for regions between the liquidus and solidus lines, a paste-like region with a solid fraction that evolves continuously with temperature and solute changes is formed.

[0058] After obtaining the new solid-phase fractional field, a region mask is further constructed to limit the solution range for different physical fields. Preferably, it includes the following region masks: A flow region mask is used to define the solution domain for the velocity and pressure fields. Temperature solution region mask, used to limit the update range of the temperature field; The solute solution region mask is used to limit the update range of the solute field.

[0059] The flow region mask is determined based on the material type and a solid fraction threshold. When a grid cell is located in a metallic region and its solid fraction is below a preset threshold, it can be identified as a flowable region; when its solid fraction reaches or exceeds the threshold, it can be identified as a non-flowable region or a weakly flowable region. This mechanism dynamically reflects the contraction and closure of liquid phase channels during solidification.

[0060] (6) Solving the flow field In this embodiment, the flow field is solved using an incompressible flow model, and combined with... Figure 4 The BDF2-AB2 implicit-explicit time propagation strategy shown addresses the momentum conservation equation.

[0061] Specifically, the unsteady-state terms in the momentum conservation equation are implicitly discretized using the second-order backward difference formula BDF2, the convection terms are explicitly extrapolated using the second-order Adams-Bashforth scheme AB2, and the diffusion and sludge drag terms are implicitly handled. This splitting method improves numerical stability under large time step conditions while maintaining second-order time accuracy.

[0062] The discretized momentum conservation equations can be constructed as a Helmholtz-type linear system of equations to solve for the intermediate velocity field in three directions. Since the intermediate velocity field typically does not strictly satisfy the incompressible continuity equation, a projection method is further used to correct for pressure in the intermediate velocity field.

[0063] Specifically, a pressure Poisson equation is constructed based on the intermediate velocity field, and the source terms of the pressure Poisson equation are determined by the divergence of the intermediate velocity field. After obtaining the pressure field based on the pressure Poisson equation, the intermediate velocity field is projected and corrected using the pressure field to obtain the final velocity field and pressure field that satisfy the mass conservation condition.

[0064] Furthermore, after the velocity field projection is completed, the velocity in the non-flow region is forcibly set to zero or limited to a minimum value to reflect the inhibitory effect of the fully solidified region or the highly stagnant region on the flow.

[0065] (7) Resistance treatment of the pasty area In this embodiment, to reflect the hindering effect of the dendritic framework on the flow of liquid metal during solidification, a mushy region resistance term based on the solid fraction is introduced into the momentum conservation equation. This mushy region resistance term increases with increasing solid fraction, thereby simulating the physical characteristic of gradually decreasing flow capacity during the transition from the liquid phase to the mushy region and then to the fully solidified region.

[0066] Preferably, the resistance term in the mushy region participates implicitly in the momentum conservation equation discretization to enhance numerical stability under high solid fraction conditions. The resistance term in the mushy region can also be used in conjunction with a flow region mask to further weaken the flow solution intensity when the region is close to complete solidification.

[0067] (8) Solving the temperature field After solving the flow field, the temperature field is derived based on the updated velocity field. The governing equations of the temperature field include unsteady-state terms, convection terms, and diffusion terms, and can include latent heat effects, boundary heat transfer terms, and differences in heat transfer between different material regions as needed.

[0068] Preferably, the temperature field is solved using an implicit / explicit time discretization method, where the convection term is explicitly handled and the diffusion term is implicitly handled, to balance stability and computational efficiency. For different material regions, such as the molten steel region, the mold region, the insulation jacket region, and the air region, different thermal conductivity or thermal diffusivity parameters can be set, thereby achieving joint solution of the heat conduction process of multiple materials within a unified computational domain.

[0069] Once the temperature field is updated, the results will be used for calculating the solid fraction and determining the region evolution in the next time step.

[0070] (9) Solving the solute field Following the temperature field update, the solute field is further advanced. The solute field is used to describe the migration and redistribution behavior of alloying elements during solidification.

[0071] Preferably, the solute field is discretized and solved using the total solute mass conservation form. Its governing equations include unsteady-state terms, convection terms, and diffusion terms. The convection term reflects the influence of the flow field on solute migration, and the diffusion term reflects the transport process driven by the solute concentration gradient.

[0072] After the total solute concentration field is updated, the liquid and solid concentration fields can be further calculated based on the current solid fraction to describe the redistribution of solute between different phases during solidification. Furthermore, solute transport parameters can be dynamically adjusted according to material type and solidification state to more realistically reflect changes in solute migration capacity in the liquid and paste regions.

[0073] (10) Dynamic updating of physical property parameters and regional attributes At each time step, after solving for the flow field, temperature field, and solute field, the physical properties and regional attributes of the local elements are dynamically updated.

[0074] The dynamic update may include at least the following: 1) Update the flow resistance parameters based on the solid fraction; 2) Update heat transfer parameters according to material type; 3) Update solute transport parameters according to material type and solidification state; 4) Update the flow region mask and related region identifiers based on the solid fraction threshold.

[0075] Different thermal conductivity, thermal diffusivity, and solute transport coefficients can be used for different material regions; the flow resistance parameters in the mushy region increase with the increase of solid fraction; for the fully solidified region, the flow resistance parameters can be removed from the flow solution domain. Through the above dynamic updating method, the flow capacity, heat transfer capacity, and solute migration capacity of each region during solidification can be synchronized with the local solidification state, improving the physical consistency of the simulation results.

[0076] (11) Initial conditions and boundary conditions In this implementation, the corresponding initial conditions and boundary conditions need to be set before the calculation begins.

[0077] Boundary conditions can be set according to the actual casting process, for example: Apply adiabatic or convective heat transfer boundary conditions to the outer boundary of the computational domain. Apply no-slip boundary, free-slip boundary, or closed boundary conditions to the velocity boundary; Apply zero flux boundary conditions or specified concentration boundary conditions to the solute boundary; Apply a reference pressure boundary or a local isobaric boundary condition to the pressure field.

[0078] Different combinations of initial and boundary conditions can be used under different casting process conditions.

[0079] (12) Results output and post-processing After the preset output time is reached, multiple physical quantities during the solidification process of the steel casting are output, including: 1) Velocity field; 2) Pressure field; 3) Temperature field; 4) Solid-phase fractional field; 5) Total solute concentration field; 6) Liquid phase concentration field; 7) Solid phase concentration field; 8) Material distribution area or area identification area.

[0080] By performing post-processing analysis on the above results, information such as solidification front evolution characteristics, local flow attenuation behavior, temperature gradient distribution, segregation tendency, and potential defect formation areas can be obtained, thus providing a basis for steel casting feeding design, cooling process optimization, segregation control, and quality prediction.

[0081] (13) Effect description Compared with the traditional explicit time-stepping method, this implementation adopts the BDF2-AB2 implicit-explicit time discretization scheme to split the rigid and non-rigid terms in the momentum conservation equation, so that the time step is no longer strictly limited by the explicit stability condition, thereby significantly reducing the total number of computation steps and improving the computational efficiency of solidification simulation of large steel castings.

[0082] Meanwhile, this implementation uses the solid fraction as a unified coupling variable among the flow field, temperature field, solute field, and phase transition field. Combined with dynamic updates to the flow region, enhanced resistance in the mushy region, and dynamic adjustment mechanisms for multiple material parameters, it provides a more reasonable characterization of the evolution processes in the liquid, mushy, and solid regions. This improves the stability, accuracy, and engineering applicability of the multiphysics coupling solution. The present invention also provides a numerical simulation system for the solidification process of steel castings with a large time step. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it performs the numerical simulation method for the solidification process of steel castings with a large time step as described above.

[0083] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the numerical simulation method for the solidification process of steel castings with a large time step as described above.

[0084] 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 numerical simulation method for the solidification process of steel castings with a large time step, characterized in that, The steps are as follows: (1) The computational domain corresponding to the solidification process of steel casting is spatially discretized by MAC staggered grid to obtain the grid, wherein pressure, temperature, total solute concentration and solid fraction are stored in the center of the grid cell, and velocity components are stored at the grid cell interface; at the same time, a multi-physics coupling model of the solidification process of steel casting including flow field, temperature field, solute field and phase transformation field is constructed. The multi-physics coupling model uses solid fraction to realize the coupling between the physical fields. (2) The momentum conservation equation in the flow field contained in the multiphysics coupling model is discretized by an implicit and explicit time-progression method based on the second-order backward difference formula BDF2 and the second-order Adams-Bashforth scheme AB2, so that the variables of the momentum conservation equation can be represented by points on the grid; wherein: the unsteady term of the momentum conservation equation is implicitly discretized by BDF2, the convection term is explicitly extrapolated by AB2, and the diffusion term and the drag term of the smear region are implicitly processed; (3) In each time step, the solid fraction is first updated according to the current temperature field and solute field; then the flow region is determined according to the updated solid fraction; then the incompressible flow is solved by projection method in the determined flow region to obtain the velocity field and pressure field; then the temperature field and solute field are solved respectively based on the updated velocity field. (4) Based on the solid fraction, material type and solidification state, the flow resistance parameters, heat transfer parameters and solute transport parameters in the computational domain are dynamically updated, thereby obtaining the temperature field, velocity field, pressure field, solid fraction field, solute field and material distribution results during the solidification process of steel castings.

2. The numerical simulation method for the solidification process of steel castings with a large time step as described in claim 1, characterized in that: The flow field describes the motion behavior of liquid metal during solidification, the temperature field describes the heat transfer process, the solute field describes the migration and distribution behavior of alloying elements, and the phase transformation field describes the evolution of materials from the liquid phase to the solid-liquid two-phase region and the solid phase region.

3. The numerical simulation method for the solidification process of steel castings with a large time step as described in claim 2, characterized in that: The phase transition field is characterized by the solid fraction, which is updated based on the coupling of the temperature field and the solute field.

4. The numerical simulation method for the solidification process of steel castings with a large time step as described in claim 3, characterized in that: The solid fraction is used to simultaneously characterize the local solidification state and serves as a coupling variable for flow region renewal, paste region resistance calculation, and solute distribution calculation, thereby achieving unified coupling between the flow field, temperature field, solute field, and phase change field.

5. The numerical simulation method for the solidification process of steel castings with a large time step as described in claim 1, characterized in that: The momentum conservation equation is discretized to form a Helmholtz-type linear equation system, which is used to solve the intermediate velocity field within the time step. Based on the intermediate velocity field, the pressure Poisson equation is constructed, and the intermediate velocity field is projected and corrected based on the pressure Poisson equation to obtain the velocity field and pressure field that satisfy the continuity equation.

6. The numerical simulation method for the solidification process of steel castings with a large time step as described in any one of claims 1-5, characterized in that: When the solid fraction reaches a preset threshold, the corresponding region is switched from a flowing region to a non-flowing region; the flow resistance in the mushy region is dynamically adjusted according to the solid fraction.

7. The numerical simulation method for the solidification process of steel castings with a large time step as described in any one of claims 1-5, characterized in that: Dynamic updates of region attributes are used in conjunction with mobile region masks.

8. The numerical simulation method for the solidification process of steel castings with a large time step as described in claim 7, characterized in that: After obtaining the new solid-phase fractional field, a region mask is further constructed to limit the solution range for different physical fields, including the following region masks: A flow region mask is used to define the solution domain for the velocity and pressure fields. Temperature solution region mask, used to limit the update range of the temperature field; The solute solution region mask is used to limit the update range of the solute field.

9. A numerical simulation system for the solidification process of steel castings with a large time step, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the numerical simulation method for the solidification process of steel castings with a large time step as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the numerical simulation method for the solidification process of steel castings with a large time step as described in any one of claims 1-8.

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