CALPHAD Phase-Field Simulation Coupling Method and Apparatus Based on Tensor Decomposition

Abstract

This invention discloses a CALPHAD phase-field simulation coupling method and apparatus based on tensor decomposition. The coupling method includes the following steps: acquiring thermodynamic data from the CALPHAD thermodynamic database; separating the composition-related logarithmic terms from the Gibbs free energy expression, with the remaining portion as the data to be processed; constructing the data to be processed into a multidimensional tensor and performing tensor regularization to obtain a factor matrix and coefficients; constructing a low-dimensional function expression based on the factor matrix and coefficients; recombining the low-dimensional function expression with the separated logarithmic terms and coupling it into a multiphase-field model to calculate the thermodynamic data of the target system in real time; and predicting the performance of the target material by numerically solving the governing equations and simulating the evolution of the target microstructure. This invention improves the computational efficiency of phase-field simulation by transforming complex thermodynamic functions into polynomial operations through tensor decomposition; it also improves numerical stability by separating the logarithmic terms, exhibiting good system versatility.

Description

Technical Field

[0001] This invention relates to the field of materials computational design and simulation technology, and in particular to a CALPHAD phase-field simulation coupling method and apparatus based on tensor decomposition. Background Technology

[0002] With the development of advanced manufacturing, the demand for high-performance structural and functional materials is constantly increasing. The macroscopic properties of materials largely depend on their microstructure. Therefore, accurately predicting and controlling the evolution of microstructure is crucial for optimizing material properties. Predicting the evolution of material microstructure through theoretical simulation methods and guiding material design has become an important means of modern materials research and development.

[0003] Phase-field methods are powerful tools for simulating the evolution of material microstructures, and their accuracy is highly dependent on the input thermodynamic data. The CALPHAD method can obtain phase equilibrium information of multi-component systems by establishing thermodynamic models, and coupling the CALPHAD database with phase-field models has become an important direction for predicting the microstructure of materials.

[0004] However, the existing technologies described above have the following problems: (1) Directly calling the CALPHAD database involves a complex iterative solution process, which is computationally intensive and severely limits the time and space scales of phase-field simulation, resulting in extremely low efficiency; (2) The Gibbs free energy expression in the CALPHAD database usually contains logarithmic terms related to the composition (such as the ideal mixing entropy term). When the composition is close to pure elements, the logarithmic terms and their derivatives will exhibit numerical singularities, leading to unstable or even divergent solutions to the phase-field equations.

[0005] Furthermore, while existing technologies have attempted to approximate thermodynamic functions using polynomial fitting, high-order fitting is often required in high-dimensional systems to ensure accuracy, resulting in too many fitting coefficients, which increases the computational burden and fails to meet the application requirements. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art by providing a CALPHAD phase-field simulation coupling method and device based on tensor decomposition, which can maintain the accuracy of the CALPHAD database, improve computational efficiency, and enhance numerical stability.

[0007] To achieve the above objectives, the technical solution adopted in this invention is: a CALPHAD phase-field simulation coupling method based on tensor decomposition.

[0008] Step 1: Based on the CALPHAD thermodynamic database of the target system, obtain the thermodynamic data of the system as sample data, separate the logarithmic terms related to composition in the Gibbs free energy expression, and use the remaining part as the data to be processed.

[0009] Step 2: Construct the data to be processed into a multidimensional tensor;

[0010] Step 3: Perform tensor regularization on the multidimensional tensor to obtain the factor matrix and corresponding coefficients;

[0011] Step 4: Construct a low-dimensional functional expression for the thermodynamic quantity based on the factor matrix and the corresponding coefficients;

[0012] Step 5: Recombine the low-dimensional function expression with the logarithmic terms separated in Step 1 and couple them into the multiphase field model. Calculate the thermodynamic data of the target system in real time based on the multiphase field model.

[0013] Step 6: Simulate the dynamic evolution of the target material's microstructure by numerically solving the governing equations of the multiphase field model;

[0014] Step 7: Predict the microstructure and mechanical properties of the target material based on the simulation results.

[0015] Optionally, the tensor regularization decomposition adopts CP decomposition, the expression of which is:

[0016]

[0017] in For a tensor containing multiple factors, To decompose the rank, This is the factor vector for the corresponding dimension.

[0018] Optionally, the decomposition rank of the tensor regularization... The value range is 3 to 10.

[0019] Optionally, the tensor decomposition optimizes the decomposition rank by minimizing the least squares cost function. and factor matrix

[0020] Optionally, the target system is an Nb-C-Ni-Co quaternary system; the thermodynamic data include Gibbs free energy, chemical potential, and diffusion potential derivative.

[0021] Optionally, in the Nb-C-Ni-Co quaternary system, the mass ratio of Ni to Co is 0.3 to 3.0.

[0022] Optionally, the multiphase field model includes the Allen-Cahn equation and the Cahn-Hilliard equation.

[0023] Optionally, the numerical solution employs a hybrid approach combining spatial discretization and time progression.

[0024] Optionally, the dynamic evolution process of the microstructure includes grain growth and grain coarsening processes.

[0025] A coupling device for implementing the method, comprising:

[0026] The data acquisition module is used to obtain the thermodynamic data of the target system from the CALPHAD thermodynamic database as sample data, separate the logarithmic terms related to composition in the Gibbs free energy expression, and use the remaining part as data to be processed.

[0027] The tensor construction module is used to construct the data to be processed into a multidimensional tensor, wherein the dimensions of the multidimensional tensor correspond to the mole fractions of the independent components in the system.

[0028] The tensor decomposition module is used to perform tensor regularization decomposition on the multidimensional tensor to obtain the factor matrix and corresponding coefficients.

[0029] The thermodynamic function reconstruction module is used to construct a low-dimensional functional expression of thermodynamic quantities based on the factor matrix and corresponding coefficients. The low-dimensional functional expression is used to perform polynomial approximation of CALPHAD-type thermodynamic functions.

[0030] The phase-field coupling module is used to recombine the low-dimensional function expression with the logarithmic terms separated in step 1 and couple them to the multiphase field model, and calculate the thermodynamic data of the target system in real time based on the multiphase field model;

[0031] The numerical solution module is used to simulate the dynamic evolution process of the target material's microstructure by numerically solving the governing equations of the multiphase field model.

[0032] The results analysis module is used to predict the microstructure characteristics and mechanical properties of the target material based on the simulation results.

[0033] Due to the application of the above technical solution, the present invention has the following advantages compared with the prior art:

[0034] This invention transforms the complex CALPHAD function into a weighted sum of multiple one-dimensional function products through tensor canonical decomposition. This simplifies the calculation of thermodynamic quantities at each time step and grid point in phase field simulation from a complex iterative solution process to a simple polynomial evaluation, reducing computation time and enabling large-scale, long-term microstructure evolution simulation.

[0035] Meanwhile, tensor regularization can achieve a lower decomposition rank. Capture the main characteristics of high-dimensional thermodynamic data by adjusting The value of can be flexibly balanced between computational efficiency and fitting accuracy. While ensuring computational speed, it can still maintain high thermodynamic data fitting accuracy, keeping the deviation between the simulation results and the original CALPHAD database within an acceptable range.

[0036] Secondly, by separating the logarithmic terms related to the composition in the Gibbs free energy and preserving their analytical form without participating in tensor decomposition, the numerical singularity of the logarithmic terms and their derivatives when the composition is close to pure components is avoided, thereby improving the robustness of solving the phase field equations, allowing for larger time steps, and reducing the risk of computational divergence.

[0037] This invention performs tensor decomposition and low-dimensional reconstruction of the polynomial part of thermodynamic functions. It does not depend on the CALPHAD model form of a specific material system, has good portability and scalability, and is highly practical. Attached Figure Description

[0038] The accompanying drawings are provided to further illustrate the present disclosure and form part of the specification. They are used together with the following detailed description to explain the present disclosure, but do not constitute a limitation thereof. In the drawings:

[0039] Figure 1 This is a flowchart illustrating the coupling preparation process of an embodiment of the invention;

[0040] Figure 2 This is a schematic diagram of the construction of the thermodynamic data tensor of the quaternary system in an embodiment of the present invention. (a) and (b) represent different sampling step sizes. A schematic diagram of tensor construction below;

[0041] Figure 3 This is a schematic diagram of the CP decomposition structure of tensors in an embodiment of the invention;

[0042] Figure 4 The figures shown are related to the microstructure of NbC-Ni-Co cermets simulated using the method of the present invention in the embodiments of the present invention, wherein (a) represents the different microstructures of NbC-15Ni-5Co (wt.%) alloy. Microstructure diagrams under different Ni / Co ratios, (b) is a comparison of the simulated and experimental microstructure morphology of NbC-Ni-Co cermets with different Ni / Co ratios;

[0043] Figure 5 This is a comparison diagram of the phase composition of alloys with different Ni / Co ratios in the embodiments of the present invention, showing the comparison between the simulation prediction results of the method of the present invention, the experimental measurement results, and the CALPHAD thermodynamic calculation (TC) results. Detailed Implementation

[0044] To enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present application, and not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative effort should fall within the scope of protection of the present application.

[0045] This disclosure provides a CALPHAD phase-field simulation coupling method based on tensor decomposition to solve the technical problems in the prior art where directly calling the CALPHAD database is extremely inefficient and easily leads to unstable or even divergent solutions to the phase-field equations.

[0046] For ease of understanding, the specific process in the embodiments of this application is described below.

[0047] Example 1: Phase field simulation of microstructure evolution during liquid-phase sintering of NbC-Ni-Co cermets.

[0048] A CALPHAD phase-field simulation coupling method based on tensor decomposition includes the following steps:

[0049] Step 1: Select the evaluated and optimized CALPHAD thermodynamic database for the Nb-C-Ni-Co quaternary system. Using MATLAB software and its Thermo-Calc toolbox, within a set temperature range (e.g., 1450 °C), traverse the mole fraction space of each component (with step sizes of 0.05 and 0.001) to extract the Gibbs free energy of the system. Chemical potential Isothermodynamic quantities. In this process, the Gibbs free energy expression is broken down into two parts:

[0050]

[0051] in, For the logarithmic term related to the composition, i.e., the ideal mixture entropy part, The remaining portion includes the Gibbs free energy contribution from the reference surface and the excess Gibbs free energy contribution. Step 1 only extracts... The dataset;

[0052] Step 2: Construct the data to be processed into a multidimensional tensor, wherein the dimensions of the multidimensional tensor correspond to the mole fractions of the independent components in the system;

[0053] Specifically, such as Figure 2 As shown, the extracted The data is constructed as a four-dimensional tensor Since the sum of the mole fractions of the four components in the quaternary system is 1, and only three are independent, the first three dimensions of the tensor correspond to the mole fractions of Nb, Ni, and Co, respectively; among them, the mole fraction of C is given by the equation... The fourth dimension can be used to store different thermodynamic quantities (such as...) (itself or its derivative), that is ,like Figure 2 A dataset consisting of tetrahedral green spheres.

[0054] in, This indicates the sampling step size for the mole fraction; the smaller the value, the larger the dataset of thermodynamic quantities, such as... Figure 2 As shown in (a) and (b) in the figure.

[0055] Step 3: Perform tensor regularization on the multidimensional tensor to obtain the factor matrix and corresponding coefficients;

[0056] like Figure 3 As shown, specifically, for the constructed four-dimensional tensor Perform CP decomposition and set the decomposition rank. The value is 3 to 10, and in this embodiment, the rank decomposition is preferred. Then, the following optimization problem is solved by minimizing the least squares cost function:

[0057]

[0058] Obtain the factor matrix and weight vector .in It is the first ( The first dimension A factor vector.

[0059] Step 4: Construct a low-dimensional functional expression for the thermodynamic quantity based on the factor matrix and the corresponding coefficients;

[0060] In this step, based on the factor matrix and coefficients obtained from the decomposition, a low-dimensional functional expression for the thermodynamic quantity is constructed:

[0061]

[0062] This expression is essentially a polynomial approximation of the original CALPHAD thermodynamic function (with the logarithmic part removed), replacing the complex original CALPHAD function with this polynomial.

[0063] in, By factor vector A continuous function obtained by interpolation (such as spline interpolation). This expression is simply the sum of products of polynomials or simple functions, has an analytical form, and is therefore extremely fast to calculate.

[0064] Step 5: Recombine the low-dimensional function expression with the logarithmic terms separated in Step 1 and couple them into the multiphase field model. Then, calculate the thermodynamic data of the target system in real time based on the multiphase field model.

[0065] Specifically, a multiphase field model is established, and order parameters are defined. Indicates the first Orientation and concentration field of individual NbC grains This represents the concentration distribution of each component. The total free energy functional of the system. Includes chemical free energy density and interfacial energy density:

[0066]

[0067] Among them, chemical free energy density It consists of a reconstructed low-dimensional function and logarithmic terms:

[0068]

[0069] In the formula, Indicates different phases, Indicates a liquid matrix. These represent solid-phase grains with different orientations. The composition field is also included. correspond Each independent component.

[0070] The interfacial free energy density is expressed as:

[0071]

[0072] The chemical potential is derived using the reconstructed chemical free density (equation (5)). The diffusion potential, obtained through a linear combination of chemical potentials, is substituted into the Cahn-Hilliard equation, which primarily describes how the components evolve over time (diffusion, phase separation):

[0073]

[0074] And the Allen-Cahn equation, which mainly describes how the structure (grains, phase domains) evolves over time (interface migration, grain growth):

[0075]

[0076] in For atomic diffusion mobility, For interface dynamics coefficients.

[0077] Step 6: Simulate the dynamic evolution of the target material's microstructure by numerically solving the governing equations of the multiphase field model. The numerical solution adopts a hybrid mode combining spatial discretization and time-progression. Spatial discretization can use the finite difference method, finite element method, spectral method, or a combination thereof, while time-progression can use explicit, implicit, or semi-implicit schemes.

[0078] Specifically, spatial discretization employs the finite difference method to discretize the space, with the grid size set to [value missing]. The simulation was performed using a semi-implicit Fourier spectroscopy method or an explicit Euler method, with initial conditions set as randomly distributed NbC nuclei and a uniform binder phase (Ni-Co alloy), and periodic boundary conditions. Numerical simulations were conducted using the phase-field simulation program developed in this invention, recording the evolution data of field variables sequentially according to the simulation time step, thus completely reproducing the growth and Oswald ripening (coarsening) behavior of NbC grains throughout the simulation process.

[0079] Step 7: Predict the microstructure and mechanical properties of the target material based on the simulation results.

[0080] After the simulation, the microstructure morphology at different simulation times was analyzed to observe grain shape and continuity; the grain size distribution histogram, the relationship between average grain diameter and time, the mean free path of the binder phase, and the volume fraction of each phase were statistically analyzed; the microstructures obtained under different ratios and different Ni / Co ratios are shown below. Figure 4 As shown.

[0081] Specifically, see Figure 4 (a) In Phase-field simulation microstructure morphology of NbC-15Ni-5Co cermet with ratios of 1.0, 1.3, 1.5, and 2.0. This allows for clear capture of different microstructures. The subtle changes in grain morphology and contact state under the ratio verified that the fast coupled phase-field simulation method based on tensor decomposition and CALPHAD database of the present invention can accurately characterize the influence of interface energy related parameters on the microstructure evolution of NbC-Ni-Co cermets, and has reliable simulation accuracy and practicality.

[0082] See Figure 4(b) This method sets multiple Ni / Co ratio variables: A1-15Ni5Co, A2-10Ni10Co, and A3-5Ni15Co. For each ratio, it simultaneously presents the microstructure morphology obtained from phase-field simulation and the microstructure morphology observed in actual experiments, achieving a direct comparison between simulation and experimental results. The figure shows the differences in grain coarsening and distribution caused by changes in the wettability of the binder phase under different Ni / Co ratios. Furthermore, the simulated morphology and the experimental morphology show good consistency, verifying the accuracy of the simulation method of this invention.

[0083] Finally, the simulation results of this invention were experimentally verified. In this embodiment, three groups of NbC-Ni-Co cermets with different Ni / Co ratios (A1-15Ni5Co, A2-10Ni10Co, and A3-5Ni15Co) were selected as the research objects, and the corresponding results were obtained... Figure 5 (a), 5(b), and 5(c) are three sets of phase composition comparison diagrams. The phase composition results obtained by phase field simulation of the present invention, the experimentally measured phase composition results, and the phase composition results calculated by CALPHAD thermodynamics (TC) are compared and analyzed side by side.

[0084] Figure 5 The experimentally measured phase composition data are labeled with solid symbols, the simulated phase composition data are labeled with hollow symbols, and the phase composition data of the phase field simulation method of this invention are labeled with TC / CALPHAD and presented as a curve, which are the thermodynamically calculated phase composition data.

[0085] The results show that, under the mole fraction dimensions of Nb, Ni, Co, and C, the simulation results of this invention are in high agreement with the experimental results, with solid and hollow symbols corresponding and overlapping, and the simulation data are all within the error range of the experimental measurements. Furthermore, the differences in NbC grain coarsening characteristics and spatial distribution caused by variations in the wettability of the binder phase under different Ni / Co ratios can be accurately simulated using the method of this invention, and the simulated microstructure morphology is highly consistent with the experimentally measured morphology. These results fully verify the accuracy and reliability of the simulation method described in this invention and also demonstrate its practicality in the field of metal-ceramic material process optimization.

[0086] The above-described embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A CALPHAD phase-field simulation coupling method based on tensor decomposition, characterized in that: Step 1: Based on the CALPHAD thermodynamic database of the target system, obtain the thermodynamic data of the system as sample data, separate the logarithmic terms related to composition in the Gibbs free energy expression, and use the remaining part as the data to be processed. Step 2: Construct the data to be processed into a multidimensional tensor; Step 3: Perform tensor regularization on the multidimensional tensor to obtain the factor matrix and corresponding coefficients; Step 4: Construct a low-dimensional functional expression for the thermodynamic quantity based on the factor matrix and the corresponding coefficients; Step 5: Recombine the low-dimensional function expression with the logarithmic terms separated in Step 1 and couple them into the multiphase field model. Calculate the thermodynamic data of the target system in real time based on the multiphase field model. Step 6: Simulate the dynamic evolution of the target material's microstructure by numerically solving the governing equations of the multiphase field model; Step 7: Predict the microstructure and mechanical properties of the target material based on the simulation results.

2. The tensor decomposition based CALPHAD phase field simulation coupling method of claim 1, wherein: The tensor regularization uses CP decomposition, and its expression is: in For a tensor containing multiple factors, To decompose the rank, This is the factor vector for the corresponding dimension.

3. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 2, characterized in that: The decomposition rank of the tensor regularization The value range is 3 to 10.

4. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 2, characterized in that: The tensor decomposition optimizes the decomposition rank by minimizing the least squares cost function. and factor matrix.

5. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 1, characterized in that: The target system is a Nb-C-Ni-Co quaternary system; the thermodynamic data include Gibbs free energy, chemical potential, and diffusion potential derivative.

6. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 5, characterized in that: In the Nb-C-Ni-Co quaternary system, the mass ratio of Ni to Co is 0.3 to 3.

0.

7. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 1, characterized in that: The multiphase field model includes the Allen-Cahn equation and the Cahn-Hilliard equation.

8. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 1, characterized in that: The numerical solution employs a hybrid approach combining spatial discretization and time progression.

9. The CALPHAD phase-field simulation coupling method based on tensor decomposition as described in claim 1, characterized in that: The dynamic evolution process of the microstructure includes grain growth and grain coarsening.

10. A coupling device for implementing the method according to any one of claims 1, characterized in that, include: The data acquisition module is used to obtain the thermodynamic data of the target system from the CALPHAD thermodynamic database as sample data, separate the logarithmic terms related to composition in the Gibbs free energy expression, and use the remaining part as data to be processed. The tensor construction module is used to construct the data to be processed into a multidimensional tensor, wherein the dimensions of the multidimensional tensor correspond to the mole fractions of the independent components in the system. The tensor decomposition module is used to perform tensor regularization decomposition on the multidimensional tensor to obtain the factor matrix and corresponding coefficients. The thermodynamic function reconstruction module is used to construct a low-dimensional functional expression of thermodynamic quantities based on the factor matrix and corresponding coefficients. The low-dimensional functional expression is used to perform polynomial approximation of CALPHAD-type thermodynamic functions. The phase-field coupling module is used to recombine the low-dimensional function expression with the logarithmic terms separated in step 1 and couple them to the multiphase field model, and calculate the thermodynamic data of the target system in real time based on the multiphase field model; The numerical solution module is used to simulate the dynamic evolution process of the target material's microstructure by numerically solving the governing equations of the multiphase field model. The results analysis module is used to predict the microstructure characteristics and mechanical properties of the target material based on the simulation results.

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