A multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing

Abstract

The application discloses a multi-scale coupling organization evolution prediction method for titanium alloy additive manufacturing, relates to the technical field of additive manufacturing, and solves the problems that in the existing numerical simulation of the titanium alloy additive manufacturing organization, grain growth and phase change simulation lack effective coupling, it is difficult to carry out unified and comprehensive analysis on organization evolution, and the pertinence and usability of process parameter optimization and organization regulation are limited. Through macroscopic temperature field finite element solving, the full-time temperature field history is obtained; the temperature history is cross-scale mapped, the space mapping relationship of the macroscopic finite element grid to the mesoscopic cellular automaton and the microscopic phase field model grid is established based on the bilinear interpolation, the mesoscopic cellular automaton simulates the beta grain growth, and the beta grain characteristics are output through initialization, remelting epitaxy, nucleation judgment and other steps; the total free energy functional is constructed in the microscopic phase field model, the alpha phase multivariant sequence parameter set is defined, the sequence parameter evolution equation is solved, and the titanium alloy beta to alpha phase change dynamics simulation and variant selection behavior prediction are realized.

Description

Technical Field

[0001] This invention relates to the field of additive manufacturing technology, specifically to a multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing. Background Technology

[0002] Titanium alloys possess high specific strength, corrosion resistance, and good high-temperature resistance, making them a commonly used material for critical load-bearing components in aerospace and other fields. Additive manufacturing can achieve near-net-shape forming of complex titanium alloy components, but the heat input during the forming process is significantly transient and localized. The material undergoes rapid heating and cooling within a very short time, forming a complex thermal cycle. This thermal cycle directly controls the growth behavior of primary β grains and the subsequent β→α phase transformation process, thus determining the microstructure, size, and spatial distribution of the α phase. Since microstructure characteristics play a decisive role in the strength, plasticity, and anisotropy of titanium alloys, numerous studies have attempted to predict and optimize the microstructure of titanium alloys under additive manufacturing processes through numerical simulation.

[0003] In existing numerical simulations of microstructures, the temperature variation over time during the process is difficult to accurately measure across the entire field and process. Temperature field solutions are typically obtained using the finite element method (FEM) or finite volume method, with the temperature history used as the thermal driving input for microstructure evolution. At the mesoscale, cellular automata, due to their high computational efficiency and clear evolutionary rules, are commonly used to simulate grain nucleation and competitive growth processes. They can output information such as grain size, grain boundary morphology, and grain region division, making them suitable for describing the formation and growth trend of primary β grains, as shown in CN102750425B. At the microscale, the phase field method can describe phase interface migration and phase transformation competition. It can introduce terms such as chemical free energy, interfacial energy, and elastic strain energy within the free energy framework to simulate the phase distribution evolution and morphological characteristics in the β→α phase transformation of titanium alloys, as shown in CN117150813A.

[0004] However, the aforementioned models still struggle to provide a unified and comprehensive analysis of the microstructure evolution during titanium alloy additive manufacturing in practical applications. While both grain growth simulation and phase deformation morphology simulation use temperature field and temperature history as driving variables, they typically establish separate computational domains, parameter systems, and solution processes: the former focuses on obtaining structural information such as the morphology, size, and spatial distribution of primary β grains, while the latter focuses on describing the evolution of phase distribution and morphology over time during the β→α phase transformation. Due to the lack of effective coupling under the same temperature history conditions, the temperature input format, time step, and spatial resolution of the two models are often inconsistent, making it difficult to analyze the two types of results in the same coordinate system and time series. Therefore, single models or independent analyses cannot provide a continuous description of the microstructure evolution process, limiting the relevance and usability of their results for process parameter optimization and microstructure control. Summary of the Invention

[0005] In view of the shortcomings and deficiencies of the prior art, the present invention provides a multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing, which uses temperature history as a unified driver and can effectively couple temperature field solution, grain evolution and phase transformation microstructure evolution.

[0006] To achieve the above objectives, the present invention provides a multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing, comprising the following steps: S1. Solve the macroscopic temperature field using finite element method, obtain and set the process parameters and geometric parameters, establish a finite element heat transfer model of the deposition process, and solve for the temperature field history of the entire deposition time history. Based on the temperature field history, the liquidus temperature threshold criterion is used to determine the molten pool region, solidification front region and remelting region at each time, and the movement trajectory of the solidification front and key heat over time are extracted. S2. Temperature history cross-scale coupling mapping: construct the mesoscale cellular automata computational domain and the microscale phase field computational domain, and establish the spatial alignment relationship between the two. Based on bilinear interpolation, a spatial mapping relationship is established from macroscopic finite element mesh to mesoscopic cellular automata and microscopic phase field model mesh, and the finite element time series is synchronized so that the historical data of macroscopic temperature field drives the evolution of mesoscopic and microscopic models with a unified time step. S3. Mesoscopic cellular automaton grain growth simulation: Initialize cellular automaton phase state and grain state variables; calculate solid-liquid interface propagation speed and growth tendency based on temperature gradient, cooling rate and local undercooling; determine the preferred occupation according to preset competitive selection criteria and output β grain morphology and size. S4. Simulation of multivariate phase transition in microscopic phase field: Construct the total free energy functional in the microscopic phase field model, define the set of multivariate order parameters for the α phase, and introduce the characteristic strain matrix that characterizes the crystallographic orientation relationship to establish the eigenstrain field of the multivariate; Driven by the temperature history in step S1, solve the order parameter evolution equation to realize the dynamic simulation of β→α phase transition of titanium alloy and the prediction of variant selection behavior.

[0007] Preferably, in step S1, the process parameters and geometric parameters include the geometric dimensions of the substrate and deposition channels, the number of layers and channels, the scanning path and reciprocating strategy, the interlayer cooling time, the power and efficiency of the heat source, the moving speed, the shape parameters of the heat source, as well as the environmental heat transfer and radiation boundary conditions, and the thermal and mechanical constitutive parameters of the material.

[0008] Preferably, the specific process of establishing the finite element heat transfer model of the deposition process includes: using the transient heat conduction equation as the governing equation, adopting convection and radiation equivalent heat dissipation methods for the boundary conditions, and selecting the Goldak double ellipsoid heat source, Gaussian distribution heat source, or oscillating laser-arc composite heat source with equivalent oscillation trajectory processing according to the heat source type. After obtaining the full-time temperature field history, the cross-sectional node coordinates and the corresponding temperature-time series are output as CSV or other structured files as input data for cross-scale coupling.

[0009] Preferably, in step S2, a mesoscale cellular automata computational domain and a microscale phase-field computational domain are constructed. The grid size, time step, and boundary conditions are set according to the target resolution to establish the spatial alignment between the mesoscale cellular automata computational domain and the microscale phase-field computational domain. The specific process of bilinear interpolation is as follows: For any cell center coordinate in the mesoscopic or microscopic computational domain, find the temperatures of the four corner nodes of the macroscopic finite element element in which it is located, and calculate the temperature value of the cell center according to geometric weights to ensure the spatial continuity of the temperature field. When synchronizing the finite element time series, piecewise cubic Hermite interpolation is used to make the temperature series of each spatial point continuous. After constructing the continuous temperature history, the input is resampled according to the time step required by the mesoscopic cellular automata model and the microscopic phase field model.

[0010] Preferably, in step S3, the mesoscopic cellular automata grain growth simulation specifically includes the following steps: S31. Initialization of the computational domain for grain growth simulation: Initialize the phase state and grain state variables in the mesoscopic grain growth model, and characterize the phase state with the solidity ratio. The phase state includes liquid phase, solid phase, and interface cell where solid and liquid phases coexist. The substrate region uses Voronoi or random seeding to generate equiaxed initial grains and assigns them random orientations. The deposition region is initially in liquid phase or in a state to be solidified. S32. Remelting Criterion and Resolidation Epitaxial Inheritance: The liquidus temperature is used as the criterion for remelting. When the temperature of any cell in the mesoscopic cellular automaton is greater than or equal to the criterion, the cell is marked as liquid. When the temperature of the liquid cell is in the solidification range and there are solidified cells in its neighborhood, the epitaxial inheritance rule is adopted so that the liquid cell inherits the grain number and grain orientation angle of its neighboring solidified cells. A new grain nucleation event is triggered only when there are no solidified cells available for inheritance in the neighborhood of the liquid cell. S33. The nucleation model adopts the Gaussian distribution probability nucleation model. In each time step, the nucleation probability is calculated for candidate cells that meet the solidification conditions, and the nucleation event is determined by combining random numbers. S34. Grain growth rate and solid fraction update: The solid-liquid interface propagation rate is calculated based on temperature gradient, cooling rate and local undercooling, and the solid fraction of the interface cell is updated in combination with the interface propagation rate. S35. Capture and competitive growth rules: Based on the capture probability determination rule and the best occupation rule, after completing the grain evolution simulation, output the ID distribution, grain boundary profile and solidification front evolution information of β grains over time.

[0011] Preferably, in step S35, the interface cell traverses candidate liquid phase cells in its neighborhood and calculates the capture probability. The capture probability consists of a propulsion distance probability term, an orientation factor, and a neighbor type correction term, specifically: (1) Advancement distance probability term: The greater the advancement distance of the interface cell, the greater the corresponding capture probability; (2) Orientation factor: The grain orientation angle of the interface cell is mapped to the unit vector of the main growth direction. The cosine of the angle between the unit vector and the direction of the candidate liquid phase cell is calculated and the direction weight is formed. (3) Neighbor type correction term: Different weight coefficients are assigned to orthogonal neighbors and diagonal neighbors to suppress grid pseudo bias. When multiple grain fronts can capture the same liquid phase cell at the same time, the preferred occupancy rule is adopted.

[0012] Preferably, in step S4, the simulation of multivariate phase transitions in the microscopic phase field specifically includes the following steps: S41. The micro-phase field computational domain is used as a node in the macro-temperature field. The temperature in the computational domain is approximately uniform at the same time. The macro-temperature history is mapped to the time step of the phase field solution through time interpolation, thereby realizing the coupling of the macro-micro scale temperature history. S42. Set up the set of α-phase sequence parameters and β-phase sequence parameters, and construct the total free energy functional, including chemical free energy, interfacial energy and elastic strain energy terms; S43. Introduce the characteristic strain matrix that characterizes the crystallographic orientation relationship, and combine it with the microelasticity theory to couple the stress-free transformation strain of each α phase variant with the order parameter to form a spatially distributed multivariate eigenstrain field. S44. The α-phase nucleation process is simulated using the probability distribution of Poisson. The expected number of nuclei in the unit volume is calculated to determine whether nucleation occurs. When nucleation occurs, the α-phase variant sequence parameter is assigned to the new nucleus. S45. Driven by the temperature history in step S1, solve the sequence parameter evolution equation to realize the dynamic simulation of the β→α phase transformation of titanium alloy and the prediction of variant selection behavior.

[0013] Preferably, in step S44, the phase field nucleation determination is implemented using the Poisson distribution probability formula. The specific process is as follows: within each time step, the nucleation rate is calculated based on the local temperature, undercooling, phase fraction, and defect state of each grid cell in the phase field calculation domain. Based on the nucleation rate, the grid cell volume, and the time step, the expected number of nuclei in the grid cell volume is calculated. Based on this, it is determined whether the grid cell has nucleated. When it is determined that the grid cell has nucleated, the variant sequence parameter is assigned to the new nucleus in the α phase variant according to the preset rule.

[0014] Preferably, in step S45, the specific process of solving the order parameter evolution is as follows: using the variational derivative of the free energy with respect to the order parameter as the driving force, the Allen-Cahn equation is used as the order parameter evolution equation; the Fourier spectrum method is used to numerically solve the order parameter evolution equation, and a semi-implicit linear term and an explicit nonlinear term update strategy is adopted in the solution process to improve the stability of the numerical solution and allow for a larger time step. An explicit nucleation strategy is introduced, and the nucleation rate is set based on classical nucleation theory to make the nucleation events follow a Poisson distribution. This triggers nucleation differences in different spatial regions at the point where thermal cycling conditions change, thus realistically describing the spatiotemporal distribution differences of α-phase nucleation caused by changes in cooling conditions.

[0015] The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing proposed in this invention has the following beneficial effects: (1) The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing of the present invention unifies macroscopic finite element heat transfer, mesoscopic cellular automata grain evolution and microscopic phase field multivariate phase transformation in the same framework. With temperature history as the unified driving quantity, the spatial mapping relationship from macroscopic to mesoscopic and microscopic is established by bilinear interpolation. Combined with piecewise cubic Hermite interpolation to achieve time series synchronization, a stable cross-scale coupling interface is constructed. This solves the problems of independent simulation of grain growth and phase transformation in existing simulations, inconsistent temperature input form, time step and spatial resolution. It allows the prediction of the solidified grain structure of the molten pool and the microstructure evolution of the solid β→α phase transformation to be completed in the same coordinate and time series, realizing the continuous description of the microstructure evolution process of titanium alloy additive manufacturing, and effectively improving the consistency and accuracy of microstructure prediction under deposition thermal cycling.

[0016] (2) The multi-scale coupled microstructure evolution prediction method of the present invention has both good scalability and engineering practicality. It can achieve efficient solution while maintaining the key physical mechanism, and can also flexibly calibrate parameters according to different titanium alloy material systems and additive manufacturing process windows. It provides a reliable theoretical basis and technical support for the optimization of process parameters and the collaborative design of microstructure and properties in titanium alloy additive manufacturing, and significantly improves the pertinence and usability of simulation results for actual production. Attached Figure Description

[0017] Figure 1 This is a flowchart of the multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing in this invention; Figure 2 This is a temperature variation diagram of additive titanium alloy deposition at the macroscopic scale according to the present invention; Figure 3 This is a grain growth evolution diagram of additively manufactured titanium alloys at the mesoscale according to the present invention; Figure 4 This is a diagram showing the β→α phase transformation evolution of additively manufactured titanium alloys at the microscale according to the present invention. Detailed Implementation

[0018] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0019] like Figure 1 As shown, the multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing of the present invention takes the single-layer, single-pass forming of titanium alloy TC4 in additive manufacturing as the prediction object, and constructs a multi-scale coupled microstructure evolution prediction process driven by temperature history, including the following steps: S1. Finite element solution of macroscopic temperature field: Obtain and set process and geometric parameters, including matrix and deposition channel geometry, number of layers and channels, scanning path and reciprocating strategy, interlayer cooling time, heat source power and efficiency, moving speed, heat source shape parameters, as well as environmental heat transfer and radiation boundary conditions, material thermal properties and mechanical constitutive parameters. Provide a unified input for subsequent finite element solution and cross-scale mapping.

[0020] A finite element heat transfer model of the deposition process was established to obtain the temperature field history throughout the entire deposition timeline. The specific process included: using ABAQUS simulation software, the transient heat conduction equation was used as the governing equation, and convection and radiation equivalent heat dissipation were adopted as boundary conditions. The convection effect within the molten pool was neglected, and its influence was represented by boundary heat transfer equivalents to improve the efficiency and stability of engineering calculations. Regarding heat source loading, a Goldak double ellipsoidal heat source could be used for the electric arc, and a Gaussian distributed heat source could be used for the laser. When using an oscillating laser-electric arc composite heat source, an equivalent oscillation trajectory treatment could be introduced based on the Gaussian heat source. That is, when the laser completes one or more oscillations within a time step, the heat flux density distribution along the oscillation direction within that time step is approximated as uniform or an equivalent distribution obtained by integrating the trajectory, to simplify loading and ensure energy conservation. Based on the temperature field history, the liquidus temperature threshold criterion was used to determine the molten pool region, solidification front region, and remelting region at each time point, and the movement trajectory and key heat of the solidification front over time were extracted.

[0021] The full-time temperature field history can be obtained by solving the problem. Then, the cross-sectional node coordinates and the corresponding temperature-time series are output as CSV or other structured files as input data for cross-scale coupling.

[0022] Figure 2 The temperature field distribution throughout the deposition time history was obtained by solving the macroscopic finite element heat transfer model. The figure shows that at different time points, the area near the heat source center consistently maintains the highest temperature zone. The high-temperature zone moves synchronously forward along the scanning direction with the heat source, forming a continuous heat-affected zone behind it. The temperature drop is most significant near the molten pool boundary and the upper surface of the substrate, with a large spatial temperature gradient, indicating that this region is the main sensitive area for subsequent solidification front movement and microstructure transformation. In contrast, the temperature distribution in the high-temperature core area inside the molten pool is relatively gentle, corresponding to the material being fully melted or near a liquid phase. Therefore, based on the spatial temperature distribution at different time points, the molten pool profile, remelting range, and liquidus position can be further determined, and key thermal history parameters such as temperature gradient and cooling rate can be extracted, providing a unified thermal driving input for subsequent cellular automata grain growth simulation and phase field phase transition simulation.

[0023] S2. Temperature history cross-scale mapping: Construct mesoscale cellular automata computational domain and microscale phase field computational domain. Set grid size, time step and boundary conditions according to target resolution to establish spatial alignment relationship between mesoscale cellular automata computational domain and microscale phase field computational domain.

[0024] A spatial mapping is established from macroscopic meshes to mesoscopic and microscopic meshes. Mesoscopic cellular automata meshes and microscopic phase-field meshes are typically at micrometer-level resolution, while macroscopic finite element meshes are at millimeter-level resolution. Therefore, a spatial mapping relationship is established from macroscopic finite element meshes to mesoscopic cellular automata and microscopic phase-field model meshes based on bilinear interpolation. The specific process of bilinear interpolation is as follows: For any cell center coordinate in the mesoscopic or microscopic computational domain... Find the temperatures of the four corner nodes of the macroscopic finite element element in which it resides. , , , The temperature value at the center of the cell is calculated using geometric weights. This ensures the spatial continuity of the temperature field and reduces numerical oscillations.

[0025] For the synchronization and continuity of temperature history, since grain solidification and phase transformation evolution require higher time resolution, the time step of the finite element output frame is often insufficient to directly drive the other two models. When synchronizing the finite element time series, piecewise cubic Hermite interpolation is used to make the temperature series of each spatial point continuous, thus constructing a continuous temperature history. Then, according to the time steps required by the mesoscopic cellular automata model and the microscopic phase-field model. , By resampling the input, the historical data of the macroscopic temperature field can drive the evolution of mesoscopic and microscopic models with a unified time step.

[0026] S3. Mesoscopic cellular automata grain growth simulation: Initialize the cellular automata phase state and grain state variables; calculate the solid-liquid interface propagation speed and growth tendency based on temperature gradient, cooling rate, and local undercooling; determine the preferred occupying grain and output the β-grain morphology and size according to a preset competitive selection criterion; specifically, the following steps are included: S31. Initialization of the computational domain for grain growth simulation: Divide the mesoscopic cellular automata computational domain into a cellular mesh, and initialize the phase state and grain state variables, including solidity, in the mesoscopic grain growth model. Phase identifier (liquid / solid-liquid / solid), grain ID, grain orientation angle Equivalent solute concentration The phase state is characterized by the solid fraction: Indicates liquid phase, Indicates solid phase, This represents the interface cell. The substrate region uses Voronoi or random seeding to generate equiaxed initial grains with random orientation, and the deposition region is initially in a liquid phase or in a state to be solidified.

[0027] S32, Remelting Criterion and Resolidation Epitaxial Inheritance, Based on Liquidus Temperature (or the user-defined remelting threshold) serves as the basis for remelting determination: when the temperature of a certain cell meets... When the temperature drops back to the solidification range and there are already solidified cells in the neighborhood of the liquid phase cell, the epitaxial inheritance rule is preferentially adopted: the grain ID and orientation of the neighboring solid phase cells are inherited. This enables continuous epitaxial growth across multiple layers of deposition. New nucleation events are triggered only when no inheritable solid-phase cells exist in the neighborhood, thus avoiding excessive random nucleation that could weaken the texture and columnar crystal continuity.

[0028] S33. The nucleation model adopts a Gaussian probability distribution nucleation model, with supercooling... As the independent variable, the nucleation density follows a Gaussian distribution with respect to supercooling, with the maximum nucleation density... Peak subcooling with standard deviation This is an adjustable parameter. Within each time step, the nucleation probability is calculated for candidate cells that meet the solidification conditions. The system uses random numbers to determine whether a nucleation event has occurred. If nucleation is successful, the solid fraction of the cell is set to a very small initial value (e.g., 0.001), a new grain ID and a random orientation angle are assigned, and the growth stage begins.

[0029] S34. Grain growth rate and solid fraction update: The solid-liquid interface propagation rate is calculated based on temperature gradient, cooling rate and local undercooling, and the solid fraction of the interface cell is updated in combination with the interface propagation rate.

[0030] The total undercooling can be obtained by superimposing thermal undercooling, solute undercooling, and curvature undercooling:

[0031] Solute supercooling can be achieved through equivalent composition correction related to solid fraction, while curvature supercooling can be obtained from the solid-liquid interface curvature based on the Gibbs-Thomson effect. The growth rate can be expressed using a KGT model to reduce the complexity of cell-by-cell solution. At each time step, the interface advancement distance is calculated based on the growth rate, and the solid fraction increment is obtained by normalizing it according to the cell characteristic length. Update the solid phase index of the interface cell. If the cell It then transforms into a solid phase and locks in the grain ID and orientation information.

[0032] S35. Capture and Competitive Growth Rules: For interface cells, candidate liquid phase cells are traversed within their neighborhoods, and the capture probability is calculated based on the capture probability determination rule and the optimal occupancy rule. The capture probability consists of a propulsion distance probability term, an orientation factor, and a neighbor type correction term, specifically: (1) Propulsion: The greater the propulsion distance, the greater the capture probability.

[0033] (2) Orientation factor: the orientation angle carried by the interface cell The vector is mapped to the unit vector of the main growth direction, and the cosine of the angle between the vector and the candidate neighbor direction is calculated to form the direction weight. A lower limit is set to retain randomness and avoid numerical anisotropy.

[0034] (3) Neighbor type: Orthogonal neighbors and diagonal neighbors are assigned different weights (e.g., orthogonal 1.0, diagonal 0.7) to suppress mesh pseudo-bias. The winning grain is determined by a combination of interface advancement speed advantage and orientation advantage; if the ranking is the same, a fixed priority or random elimination is used to ensure numerical stability. In order to obtain a morphology that is closer to the "bent grain boundary" of real columnar crystals, "weak gradient bias + related perturbation" can be introduced into the selection rule: that is, the growth direction is affected by both the grain's own orientation and the local thermal gradient direction, and a spatially related low-frequency perturbation is superimposed on the thermal gradient direction to be used for equivalent molten pool convection and thermal flow line fluctuation, thereby avoiding grain boundaries that are too straight and parallel.

[0035] After completing the grain evolution simulation, the ID distribution, grain boundary profile, and solidification front evolution information of the β grains over time are output.

[0036] Figure 3This diagram illustrates the grain growth evolution of additively manufactured titanium alloys at the mesoscale. As shown, in the initial deposition stage, the region near the substrate and melt pool boundary solidifies first. Existing substrate grains epitaxially inherit from the newly formed β grains, forming a columnar grain initiation zone that preferentially grows along the direction of maximum temperature gradient. As the solidification front continues to advance into and above the melt pool, grains with different orientations begin to compete for growth. Grains more aligned with the local heat flow direction and with faster interface advancement preferentially occupy liquid phase cells, while grains with significantly deviated growth directions are gradually suppressed or even terminated. Further observation at later time points reveals that some grain boundaries continuously migrate and merge, and the columnar β grains gradually elongate and coarsen along the deposition direction. The grain boundary morphology evolves from initially dispersed fine nuclei into a continuous grain structure with a clear directionality.

[0037] S4. Simulation of multivariable phase transitions in the microscopic phase field, specifically including the following steps: S41. Since the computational domain of the phase-field model is at a microscale, with a representative volume element size of 32μm×32μm×32μm, its spatial scale is much smaller than the characteristic change scale of the macroscopic temperature field during additive manufacturing. Without significantly introducing errors, the microscopic phase-field computational domain is used as a node in the macroscopic temperature field. The temperature within this computational domain is approximately uniform at any given moment and evolves over time according to the temperature history of that material point. The temperature history can be obtained from macroscopic thermal analysis and mapped to the time step of the phase-field solution through time interpolation, thus achieving macro-microscale temperature history coupling.

[0038] S42. For the β→α phase transition of titanium alloys, a set of α-phase sequence parameters and β-phase sequence parameters are set. This invention sets 13 sequence parameters: 12 α-phase sequence parameters and 1 β-phase sequence parameter, satisfying that each sequence parameter is non-negative and sums to 1. A total free energy functional is constructed, including chemical free energy, interfacial energy, and elastic strain energy terms.

[0039] The chemical free energy is constructed using Landau polynomials to create a potential well-barrier structure and reflects multivariate mutual repulsion. Its coefficients can be obtained from the Gibbs free energy difference between the α / β phases or its fitting, and converted to unit volume energy density through molar volume conversion, thus unifying it with the dimensions of interface energy and elastic energy. The interface energy is characterized by the order parameter gradient term to represent the interface thickness, and an anisotropic tensor can be introduced to reflect orientation-related interface energy differences.

[0040] S43. Introducing a characteristic strain matrix representing crystallographic orientation relationships, and employing microelasticity theory, the stress-free transformation strain of each α variant is coupled with the order parameter to form a spatially distributed characteristic strain field. Under the homogeneous elastic approximation, the elastic interaction is written as a frequency-domain convolution, and a fast Fourier transform is used for efficient solution. This step can be used to capture the influence of variant selection, variant grouping, and long-range elastic interactions on phase deformation morphology, improving the physical consistency of predictions.

[0041] S44. Simulation of the α-phase nucleation process is carried out using a Poisson distribution probability. Specifically, at each time step... Within the grid cell, the nucleation rate is calculated based on information such as local temperature T, undercooling ΔT, phase fraction, and defect state. (unit:· ), and obtain the desired number of nuclei within the unit volume. Subsequently, based on the Poisson distribution... The number of nucleation events k is obtained through sampling; when k ≥ 1, the cell is determined to have nucleated. To improve numerical stability and repeatability, the nucleation criterion is simplified to... The result is compared with a uniformly random number in the range of [0,1] to determine whether nucleation has occurred. When nucleation occurs, the variant order parameter of the new nucleus is assigned to one of the 12 α variants according to a preset rule.

[0042] S45. Driven by the temperature history of step S1, the Allen-Cahn equation is used as the order parameter evolution equation, with the variational derivative of free energy with respect to the order parameter as the driving force. The Fourier spectral method is used to numerically solve the order parameter evolution equation. During the solution process, a semi-implicit linear term and an explicit nonlinear term update strategy are adopted to improve the stability of the numerical solution and allow for a larger time step. Explicit nucleation is introduced, and the nucleation rate is set based on classical nucleation theory. The nucleation events follow a Poisson distribution, triggering nucleation differences in different spatial regions at the point of change in thermal cycling conditions, so as to more realistically describe the spatiotemporal distribution differences of α-phase nucleation caused by changes in cooling conditions. This achieves the simulation of the β→α phase transformation dynamics of titanium alloys and the prediction of variant selection behavior.

[0043] Figure 4 The diagram illustrates the β→α phase transformation evolution of additively manufactured titanium alloys at the microscale. The drawing shows that the β-to-α phase transformation exhibits a clear temporal and selective pattern at different time points. In the initial stage of the transformation, a small amount of α phase precipitates first in regions with more favorable local thermal histories, preferentially appearing near β grain boundaries or in areas with higher strain energy, forming discrete initial α phase nuclei. As the cooling process continues, the nucleated α phase continues to grow along the preferred variant direction that satisfies the Burgers orientation relationship. Lamellar or acicular α phases gradually elongate and thicken, while different variants compete for selection in local regions. At subsequent time points, the volume fraction of the α phase further increases, adjacent α lamellae contact each other and form grouped distribution characteristics, while the untransformed β phase is gradually compressed into residual regions between lamellae or near grain boundaries.

[0044] This invention presents a multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing. It unifies macroscopic finite element heat transfer, mesoscopic cellular automata grain evolution, and microscopic phase field multivariate phase transitions within the same framework. Using temperature history as a unified driving variable, it achieves a continuous description of the microstructure evolution process in titanium alloy additive manufacturing, effectively improving the consistency and accuracy of microstructure prediction under deposition thermal cycling. Furthermore, this multi-scale coupled microstructure evolution prediction method can efficiently solve problems while preserving key physical mechanisms. It can also flexibly calibrate parameters according to different titanium alloy material systems and additive manufacturing process windows, providing reliable theoretical basis and technical support for optimizing process parameters and co-designing microstructure and properties in titanium alloy additive manufacturing. This significantly enhances the relevance and usability of simulation results for actual production.

[0045] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementations should not be considered beyond the scope of this invention.

[0046] Finally, although this specification describes embodiments, not every embodiment contains only one independent technical solution. This way of describing the specification is only for clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing, characterized in that, The steps include the following: S1. Finite element solution of macroscopic temperature field: Obtain and set process parameters and geometric parameters, establish a finite element heat transfer model of the deposition process, and solve for the temperature field history of the entire deposition time. Based on the temperature field history, the liquidus temperature threshold criterion is used to determine the molten pool region, solidification front region and remelting region at each time, and the movement trajectory and thermal history parameters of the solidification front over time are extracted. S2. Temperature history cross-scale coupling mapping: Construct a mesoscale cellular automata computational domain and a microscale phase field computational domain, and establish their spatial alignment relationship; Based on bilinear interpolation, a spatial mapping relationship is established from macroscopic finite element mesh to mesoscopic cellular automata and microscopic phase field model mesh, and the finite element time series is synchronized so that the historical data of macroscopic temperature field drives the evolution of mesoscopic and microscopic models with a unified time step. S3. Mesoscopic cellular automaton grain growth simulation: Initialize cellular automaton phase state and grain state variables, calculate solid-liquid interface propagation speed and growth tendency based on thermal history parameters, determine the preferred occupation according to preset competitive selection criteria, and output β grain morphology and size. S4. Simulation of multivariate phase transition in microscopic phase field: Construct the total free energy functional in the microscopic phase field model, define the set of multivariate order parameters for the α phase, and introduce the characteristic strain matrix that characterizes the crystallographic orientation relationship to establish the eigenstrain field of the multivariate; Driven by the temperature history in step S1, solve the order parameter evolution equation to realize the dynamic simulation of β→α phase transition of titanium alloy and the prediction of variant selection behavior.

2. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 1, characterized in that, In step S1, the process parameters and geometric parameters include the geometric dimensions of the substrate and deposition channels, the number of layers and channels, the scanning path and reciprocating strategy, the interlayer cooling time, the power and efficiency of the heat source, the moving speed, the shape parameters of the heat source, as well as the environmental heat transfer and radiation boundary conditions, and the thermal and mechanical constitutive parameters of the material.

3. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 2, characterized in that, The specific process of establishing a finite element heat transfer model for the deposition process includes: using the transient heat conduction equation as the governing equation, adopting convection and radiation equivalent heat dissipation as the boundary conditions, and selecting Goldak double ellipsoidal heat source, Gaussian distribution heat source, or oscillating laser-arc composite heat source with equivalent oscillation trajectory processing according to the heat source type. After obtaining the full-time temperature field history, the cross-sectional node coordinates and the corresponding temperature-time series are output as CSV or other structured files as input data for cross-scale coupling.

4. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 1, characterized in that, In step S2, a mesoscale cellular automata computational domain and a microscale phase-field computational domain are constructed. The mesh size, time step, and boundary conditions are set according to the target resolution, and the spatial alignment relationship between the mesoscale cellular automata computational domain and the microscale phase-field computational domain is established. The specific process of bilinear interpolation is as follows: For any cell center coordinate in the mesoscopic or microscopic computational domain, find the temperatures of the four corner nodes of the macroscopic finite element element in which it is located, and calculate the temperature value of the cell center according to geometric weights to ensure the spatial continuity of the temperature field. When synchronizing the finite element time series, piecewise cubic Hermite interpolation is used to make the temperature series of each spatial point continuous. After constructing the continuous temperature history, the input is resampled according to the time step required by the mesoscopic cellular automata model and the microscopic phase field model.

5. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 1, characterized in that, In step S3, the mesoscopic cellular automata grain growth simulation specifically includes the following steps: S31. Initialization of the computational domain for grain growth simulation: Initialize the phase state and grain state variables in the mesoscopic grain growth model, and characterize the phase state with the solidity ratio. The phase state includes liquid phase, solid phase, and interface cell where solid and liquid phases coexist. The substrate region uses Voronoi or random seeding to generate equiaxed initial grains and assigns them random orientations. The deposition region is initially in liquid phase or in a state to be solidified. S32. Remelting Criterion and Resolidation Epitaxial Inheritance: The liquidus temperature is used as the criterion for remelting. When the temperature of any cell in the mesoscopic cellular automaton is greater than or equal to the criterion, the cell is marked as liquid. When the temperature of the liquid cell is in the solid-liquid solidification range and there are solidified cells in its neighborhood, the epitaxial inheritance rule is adopted so that the liquid cell inherits the grain number and grain orientation angle of its neighboring solidified cells. A new grain nucleation event is triggered only when there are no available solidified cells to inherit in the neighborhood of the liquid cell. S33. The nucleation model adopts the Gaussian distribution probability nucleation model. In each time step, the nucleation probability is calculated for candidate cells that meet the solidification conditions, and the nucleation event is determined by combining random numbers. S34. Grain growth rate and solid fraction update: The solid-liquid interface propagation rate is calculated based on temperature gradient, cooling rate and local undercooling, and the solid fraction of the interface cell is updated in combination with the interface propagation rate. S35. Capture and competitive growth rules: Based on the capture probability determination rule and the best occupation rule, after completing the grain evolution simulation, output the ID distribution, grain boundary profile and solidification front evolution information of β grains over time.

6. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 5, characterized in that, In step S35, the interface cell traverses candidate liquid phase cells in its neighborhood and calculates the capture probability. The capture probability consists of a propulsion distance probability term, an orientation factor, and a neighbor type correction term, specifically: (1) Advancement distance probability term: The greater the advancement distance of the interface cell, the greater the corresponding capture probability; (2) Orientation factor: The grain orientation angle of the interface cell is mapped to the unit vector of the main growth direction. The cosine of the angle between the unit vector and the direction of the candidate liquid phase cell is calculated and the direction weight is formed. (3) Neighbor type correction term: Different weight coefficients are assigned to orthogonal neighbors and diagonal neighbors to suppress grid pseudo bias. When multiple grain fronts can capture the same liquid phase cell at the same time, the preferred occupancy rule is adopted.

7. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 1, characterized in that, Step S4, the simulation of multivariate phase transitions in the microscopic phase field, specifically includes the following steps: S41. The micro-phase field computational domain is used as a node in the macro-temperature field. The temperature in the computational domain is approximately uniform at the same time. The macro-temperature history is mapped to the time step of the phase field solution through time interpolation, thereby realizing the coupling of the macro-micro scale temperature history. S42. Set up a set of α-phase multivariate sequence parameters and β-phase sequence parameters, and construct a total free energy functional, including chemical free energy, interfacial energy and elastic strain energy terms; S43. Introduce the characteristic strain matrix that characterizes the crystallographic orientation relationship, and combine it with the microelasticity theory to couple the stress-free transformation strain of each α phase variant with the order parameter to form a spatially distributed multivariate eigenstrain field. S44. The α-phase nucleation process is simulated using the probability distribution of Poisson. The expected number of nuclei in the unit volume is calculated to determine whether nucleation occurs. When nucleation occurs, the α-phase variant sequence parameter is assigned to the new nucleus. S45. Driven by the temperature history in step S1, solve the sequence parameter evolution equation to realize the dynamic simulation of the β→α phase transformation of titanium alloy and the prediction of variant selection behavior.

8. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 7, characterized in that, In step S44, the phase field nucleation determination is implemented using the Poisson distribution probability formula. The specific process is as follows: within each time step, the nucleation rate is calculated based on the local temperature, undercooling, phase fraction and defect state of each grid cell in the phase field calculation domain. Based on the nucleation rate, grid cell volume and time step, the expected number of nuclei in the grid cell volume is calculated. Based on this, it is determined whether the grid cell has nucleated. When it is determined that the grid cell has nucleated, the variant order parameter is assigned to the new crystal nucleus in the α phase variant according to the preset rule.

9. The multi-scale coupled microstructure evolution prediction method for titanium alloy additive manufacturing according to claim 7, characterized in that, In step S45, the specific process of solving the order parameter evolution is as follows: using the variational derivative of the free energy with respect to the order parameter as the driving force, the Allen-Cahn equation is used as the order parameter evolution equation; the Fourier spectrum method is used to numerically solve the order parameter evolution equation, and a semi-implicit linear term and an explicit nonlinear term update strategy is adopted in the solution process to improve the stability of the numerical solution and allow for a larger time step. An explicit nucleation strategy is introduced, and the nucleation rate is set based on classical nucleation theory to make the nucleation events follow a Poisson distribution. This triggers nucleation differences in different spatial regions at the point where thermal cycling conditions change, thus realistically describing the spatiotemporal distribution differences of α-phase nucleation caused by changes in cooling conditions.

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