Multicomponent alloy solidification phase field simulation acceleration method based on shallow neural network

Abstract

The invention discloses a multi-component alloy solidification phase field simulation acceleration method based on a shallow neural network. The method comprises the following steps: constructing a parameter space according to characteristics of a target multi-component alloy system; solving a quasi-equilibrium thermodynamic nonlinear equation set based on the equal chemical potential condition and solute conservation to generate a quasi-equilibrium thermodynamic condition data set; constructing an artificial neural network model with a single hidden layer, and training the model by taking the parameter space and the quasi-equilibrium thermodynamic condition data set as input and output of the model respectively; solving an order parameter equation and a solute diffusion equation, inputting the phase components, solute component components and temperature of the current time step into a trained artificial neural network model, and outputting quasi-equilibrium thermodynamic conditions of the current time step; according to quasi-equilibrium thermodynamic conditions output by the model, chemical driving force of the current time step is calculated on the basis of the thermodynamic phase equilibrium principle and serves as initial driving force of the next time step, and loop iteration is conducted till simulation is finished.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of computational materials science and industrial simulation, in particular to a multi-component alloy solidification phase field simulation acceleration method based on a shallow neural network. BACKGROUND

[0002] As one of the most commonly used mesoscale microstructure simulation methods, the phase field method faces severe challenges in the simulation of the microstructure of multi-component engineering alloys. The root cause lies in the fact that the calculation of the chemical driving force of the phase transition process requires real-time acquisition of the quasi-equilibrium composition of the coexisting phase region. The calculation of the quasi-equilibrium composition requires the construction of a complex Gibbs free energy model based on the CALPHAD database, and then a set of nonlinear equations (i.e., quasi-equilibrium thermodynamic conditions) based on the phase equilibrium thermodynamic theory is solved iteratively. The number of nonlinear equations is related to the number of alloy components and phase data, and also depends on the number of calculation grid points of the coexisting phase region, resulting in an exponential increase in the computational complexity of the phase field simulation of multi-component alloys, especially three-dimensional simulation.

[0003] In view of the efficiency of solving the quasi-equilibrium thermodynamic conditions of multi-component alloys, there are currently various methods to improve the efficiency, including thermodynamic extrapolation, table lookup, and machine learning nonlinear equation solver. Although the above methods improve the efficiency of acquiring quasi-equilibrium thermodynamic data in the phase field simulation process, they all have their own limitations, especially most of the methods still need to solve complex nonlinear equations in real time to update the quasi-equilibrium thermodynamic data to meet the needs of the phase field simulation. SUMMARY

[0004] To solve the above technical problems, the purpose of the present application is to provide a multi-component alloy solidification phase field simulation acceleration method based on a shallow neural network.

[0005] The present application provides a multi-component alloy solidification phase field simulation acceleration method based on a shallow neural network, comprising:

[0006] Step 1: According to the characteristics of the target multi-component alloy system, a parameter space containing phase composition, solute component composition and temperature is constructed;

[0007] Step 2: Based on the parameter space, a high-precision numerical solver is used to numerically solve the quasi-equilibrium thermodynamic nonlinear equation set based on the equal chemical potential condition and solute conservation, and generate a quasi-equilibrium thermodynamic condition data set;

[0008] Step 3: An artificial neural network model with a single hidden layer is constructed, the parameter space and the quasi-equilibrium thermodynamic condition data set are respectively taken as the input and output of the model, the artificial neural network model is trained, and the optimal number of hidden layer neurons is determined;

[0009] Step 4: Embed the optimized artificial neural network model into the phase field model, solve the order parameter equation and solute diffusion equation by numerical method, take the phase composition, solute composition and temperature of the current time step as input, and output the quasi-equilibrium thermodynamic condition of the current time step through the artificial neural network model;

[0010] Step 5: According to the quasi-equilibrium thermodynamic condition output by the model, the chemical driving force of the current time step is calculated based on the thermodynamic phase equilibrium principle, which is used as the initial driving force of the next time step, and the iteration is repeated until the simulation is completed.

[0011] The shallow neural network based multi-component alloy solidification phase field simulation acceleration method of the application can predict quasi-equilibrium thermodynamic data by training a lightweight artificial neural network model, replace the nonlinear equation iteration solution in the traditional coupled CALPHAD method, and completely avoid the solution of complex nonlinear equation sets. In the case of Ni-Cr-Al ternary alloy, the simulation time of millimeter level is reduced to 1 / 80 of the traditional method, and the solute distribution error is less than 5%. The method is suitable for microstructure prediction of high-temperature alloy, high-strength steel and other engineering materials, and solves the calculation bottleneck of large-scale phase field simulation. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1 The flowchart of the shallow neural network based multi-component alloy solidification phase field simulation acceleration method of the application;

[0013] Figure 2a The prediction accuracy of the model with different number of hidden layer neurons of the application;

[0014] Figure 2b The training speed of the model with different number of hidden layer neurons of the application;

[0015] Figure 3a The calculation efficiency of the traditional method and the machine learning acceleration method of the application in different size calculation domains;

[0016] Figure 3b The calculation efficiency ratio of the traditional method and the machine learning acceleration method of the application in different size calculation domains;

[0017] Figure 4a The solid-liquid interface morphology of the traditional method and the acceleration method of the application at different times,

[0018] Figure 4b The chemical driving force of the traditional method and the acceleration method of the application at different times;

[0019] Figure 4c The Al element distribution of the traditional method and the acceleration method of the application at different times,

[0020] Figure 4d This is a comparison of Cr element distribution at different times between the traditional method and the accelerated method of this invention;

[0021] Figure 4e The difference between the traditional method and the accelerated method of this invention is the change in the growth rate of the tip of the primary dendrite arm of equiaxed crystal over time.

[0022] Figure 4f The difference between the traditional method and the accelerated method of this invention is the change of Al element composition at the tip of the primary dendrite arm of equiaxed crystal over time;

[0023] Figure 4g The changes in Cr element composition at the tip of the primary dendrite arm of an equiaxed crystal over time are compared between the traditional method and the accelerated method of this invention.

[0024] Figure 5a This invention describes the accelerated simulation of the grain morphology evolution process of a Ni-Cr-Al ternary alloy.

[0025] Figure 5b The accelerated method of this invention simulates the chemical driving force of the Ni-Cr-Al ternary alloy;

[0026] Figure 5c It is the distribution of Al elemental composition;

[0027] Figure 5d It shows the distribution of Cr elemental composition. Detailed Implementation

[0028] The present invention will now be described in detail with reference to specific embodiments.

[0029] like Figure 1 As shown, the present invention provides an accelerated method for simulating the solidification phase field of multi-element alloys based on shallow neural networks, comprising:

[0030] Step 1: Based on the characteristics of the target multi-component alloy system, construct a parameter space including phase composition, solute component composition, and temperature. Specifically:

[0031] Based on the alloy composition and solidification range of the target multi-component alloy system, a parameter space Ω = { is constructed. ,c i , T};wherein For the α phase, the value range is: ∈ [0, 1]; c i The values ​​of each solute component are in the range of c. i ∈ [c i min , c i max ]; T is the absolute thermodynamic temperature, and its range is T ∈ [T min, T max ], c i min c i max T min and T max The results were obtained through equilibrium solidification and Scheil solidification simulations of the target multi-component alloy system.

[0032] Step 2: Based on the parameter space, use a high-precision numerical solver (such as the Levenberg-Marquardt algorithm, residual <10) −12 Numerical solutions were performed on the quasi-equilibrium thermodynamic nonlinear equations based on isochemical potential conditions and solute conservation, generating a quasi-equilibrium thermodynamic condition dataset, specifically:

[0033] According to the principle of phase equilibrium thermodynamics, in a multi-phase coexistence region of a multi-component alloy system, the free energy of the solute component in each phase has the same partial derivative with respect to the composition of each phase, i.e., the chemical potential is equal. Furthermore, the solute component in each phase must satisfy the law of conservation of mass. The quasi-equilibrium thermodynamic conditions are solved using a numerical solver, generating a quasi-equilibrium thermodynamic condition dataset D={D k}, k = 1,2,...,n, where n is the number of samples in the parameter space, and the quasi-equilibrium thermodynamic conditions include the equilibrium composition of solute elements in each phase.

[0034] Taking the Ni-Cr-Al ternary alloy system as an example, the input parameters include: sequence parameter ϕ, solute element composition c. Al and c Cr The range of temperature T, sampling step size, and total number of samples are shown in Table 1. The quasi-equilibrium thermodynamic equations were then solved using the Levenberg-Marquardt algorithm, yielding 39,204 quasi-equilibrium thermodynamic data points.

[0035] Table 1. Range of Input Variables and Sampling Step Size

[0036]

[0037] Step 3: Construct an artificial neural network model with a single hidden layer. Use the parameter space and quasi-equilibrium thermodynamic condition datasets as the model's input and output, respectively. Train the artificial neural network model and determine the optimal number of neurons in the hidden layer. Specifically:

[0038] An artificial neural network model F: Ω→D with a single hidden layer is constructed, where Ω is a parameter space containing phase components, solute component compositions and temperature, and D is quasi-equilibrium thermodynamic data, the number of hidden layer neurons N ∈ [15, 25], the activation function between each layer is a Sigmoid function, and the constructed shallow artificial neural network model is trained, and the optimal number of hidden layer neurons N is determined by the system.

[0039] In this embodiment, the artificial neural network structure with a single hidden layer is constructed as follows: the input layer has 4 nodes corresponding to 4 input parameters (c , c Al , c Cr and T); the output layer contains 4 nodes corresponding to 4 output parameters (c liq Al , c liq Cr , c sol Al and c sol Cr ); the hidden layer contains 1-100 nodes, and the performance of the hidden layer with different numbers of nodes needs to be evaluated through subsequent tests; and the activation function uses a Sigmoid function.

[0040] The data set is divided into a training set and a test set in a ratio of 8:2, 5-fold cross-validation is used in the model training process to avoid overfitting. FIG. 2 is a comparison of the network performance with different numbers of hidden layer neurons of the present application, Figure 2a is the prediction accuracy of the model with different numbers of hidden layer neurons of the present application; Figure 2b is the training speed of the model with different numbers of hidden layer neurons of the present application. After testing, when N < 20, the mean square error MSE > 10 -9 , at this time the prediction model has a larger error, and when N > 20, although the model mean square error is smaller, more computing resources are consumed to optimize the model. When the number of hidden layer neurons exceeds 20, the model prediction error is close to 10 −9 , which is about 4 orders of magnitude higher than the simplest single neuron model. For the solute concentration field variable in the phase field simulation, this is a very small error, and the model training speed is only about 1 / 5 of the most basic model. Therefore, considering the mean square error and the model training speed, the number of hidden layer neurons is set to 20, and this optimized model is used in the subsequent actual phase field simulation, which simplifies the model as much as possible while ensuring the performance and accuracy of the phase field simulation.

[0041] Step 4: Embed the trained artificial neural network model into the phase field model, solve the order parameter equation and solute diffusion equation by numerical method, take the phase composition, solute element composition and temperature at the current time step as input, output the quasi-equilibrium thermodynamic condition at the current time step through the artificial neural network model, specifically:

[0042] Step 4.1: Construct the order parameter equation and solute diffusion equation as follows:

[0043]

[0044]

[0045] wherein, is the volume fraction of the a phase; is the volume fraction of the β phase; is the volume fraction of the γ phase; i is the total composition of element i; is the interfacial mobility of two phases; is a numerical parameter determined according to the distribution of order parameter values of the four neighboring grid points of the calculation grid point during simulation; is the interfacial energy of β and γ phases; is the interfacial energy of α and γ phases; is the Laplace operator; is the gradient operator; is the interfacial width; is the chemical driving force of a to β phase transition; is the molar volume of the alloy; is the diffusion coefficient of element i in the a phase; v is the number of coexisting phases; represents the composition of element i in the a phase; is the anti-solute cut-off phase.

[0046] Step 4.2: Numerically solve the order parameter equation and solute diffusion equation by finite difference method, discretize the calculation domain in space using uniform grid, discretize in time using explicit Euler method, and accelerate the solution process of partial differential equation by means of GPU vectorized parallel computing.

[0047] Step 4.3: Take the solved phase composition, solute element composition and temperature at the current time step as input, output the quasi-equilibrium thermodynamic condition at the current time step through the artificial neural network model.

[0048] Step 5: According to the quasi-equilibrium thermodynamic condition output by the model, calculate the chemical driving force at the current time step based on the principle of thermodynamic phase equilibrium, take it as the initial driving force of the next time step, and iterate until the simulation ends.

[0049] In practice, the chemical driving force and chemical potential are calculated according to the following formulas:

[0050]

[0051]

[0052] in, It is the free energy density of the β phase. It is the free energy density of the α phase. It is the chemical potential of component i. This represents the equilibrium composition of component i in phase β. This represents the equilibrium composition of component i in phase α.

[0053] First, the accuracy and efficiency of this invention are evaluated by simulating the solidification process of a single equiaxed dendrite in a Ni-10 at.% Cr-20 at.% Al alloy. Traditional phase-field simulations require iteratively solving nonlinear equations to obtain quasi-equilibrium thermodynamic data (QETD), but this invention utilizes a machine learning model to predict QETD-ML. As the computational domain increases, the grain size also increases, and the number of mesh points in the solid-liquid interface region increases accordingly. Therefore, the number of nonlinear equations that need to be solved using traditional methods also increases significantly, leading to an exponential increase in computation time with the increase in computational domain size. Figure 3a As shown. This invention fundamentally avoids directly solving the nonlinear quasi-equilibrium equations. It only uses the sequence parameters, composition, and temperature of the current time step as inputs, and employs a machine learning model to predict quasi-equilibrium thermodynamic data. Therefore, the computation time is almost unaffected by the size of the computational domain. Figure 3a As shown. Furthermore, the neural network structure used in this invention is simple, ensuring that the inference of quasi-equilibrium thermodynamic data does not reduce the efficiency of phase-field simulation calculations, such as... Figure 3b As shown, when the computational domain size reaches 1000 μm, the computation time of this invention is approximately 1 / 80 of that of the conventional method.

[0054] Figure 4a-4g This comparison of the computational accuracy of the method of this invention with that of traditional methods shows that the performance of the machine learning model in this invention is optimized, therefore the improvement in computational efficiency has not led to a decrease in the accuracy of the simulation results. Figure 4a As shown, the order parameter distribution indicates that the solid-liquid interface simulated by the two methods at different solidification times ( = = 0.5) Very similar; such as Figure 4b As shown, the chemical driving force error in the tip region of the primary dendrite arm is less than 5%. Furthermore, as... Figure 4c and 4d The distributions of Al and Cr elements simulated by both methods remained consistent at different solidification times; for example... Figure 4eAs shown, the growth rates of equiaxed grain tips of Al and Cr simulated by the two methods at different solidification times are also very close quantitatively; for example... Figure 4f and 4g The quantitative results of the two methods simulating the changes in the tip solute composition of Al and Cr elements with different solidification times are also very close.

[0055] Simulations using a single equiaxed grain phase field (small computational domain, 400 × 400 μm) 2 After verifying the efficiency and accuracy of the invention, it was then used in a larger computational domain (1000 × 1000 μm). 2 The polycrystalline solidification process of a Ni-10 at.%Cr-20 at.%Al ternary alloy was simulated in [the simulation]. Figure 5a As shown, the simulation results reproduce well the free growth and collision growth stages of equiaxed crystals; chemical driving force ( Figure 5b ) and solute element distribution ( Figure 5c and 5d This fully demonstrates the ability of this invention to perform large-scale phase field simulation of the solidification process of multi-component alloys.

[0056] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., 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 method for accelerating the simulation of solidification phase field of multi-element alloys based on shallow neural networks, characterized in that, include: Step 1: Based on the characteristics of the target multi-component alloy system, construct a parameter space including phase composition, solute component composition, and temperature; Step 2: Based on the parameter space, the quasi-equilibrium thermodynamic nonlinear equations based on the isochemical potential condition and solute conservation are numerically solved using a high-precision numerical solver to generate a quasi-equilibrium thermodynamic condition dataset. Step 3: Construct an artificial neural network model with a single hidden layer. Use the parameter space and quasi-equilibrium thermodynamic condition dataset as the input and output of the model, respectively, to train the artificial neural network model and determine the optimal number of neurons in the hidden layer. Step 4: Embed the trained artificial neural network model into the phase-field model, solve the order parameter equation and solute diffusion equation using numerical methods, and take the phase composition, solute component composition and temperature at the current time step as input, and output the quasi-equilibrium thermodynamic conditions at the current time step through the artificial neural network model. Step 5: Based on the quasi-equilibrium thermodynamic conditions output by the model, calculate the chemical driving force of the current time step based on the thermodynamic phase equilibrium principle, use it as the initial driving force for the next time step, and iterate until the simulation ends.

2. The accelerated simulation method for solidification phase field of multi-element alloys based on shallow neural networks according to claim 1, characterized in that, Step 1 specifically involves: Based on the alloy composition and solidification range of the target multi-component alloy system, a parameter space Ω = { is constructed. , c i ,T};where For the α phase, the value range is: ∈ [0, 1]; c i The values ​​of each solute component are in the range of c. i ∈ [c i min , c i max ]; T is the absolute thermodynamic temperature, and its range is T ∈ [T min , T max ], c i min c i max T min and T max The results were obtained through equilibrium solidification and Scheil solidification simulations of the target multi-component alloy system.

3. The accelerated simulation method for solidification phase field of multi-element alloys based on shallow neural networks according to claim 1, characterized in that, Step 2 specifically involves: According to the principle of phase equilibrium thermodynamics, in a multi-phase coexistence region of a multi-component alloy system, the free energy of the solute component in each phase has the same partial derivative with respect to the composition of each phase, i.e., the chemical potential is equal. Furthermore, the solute component in each phase must satisfy the law of conservation of mass. The quasi-equilibrium thermodynamic conditions are solved using a numerical solver, generating a quasi-equilibrium thermodynamic condition dataset D={D k }, k = 1,2,...,n, where n is the number of samples in the parameter space, and the quasi-equilibrium thermodynamic conditions include the equilibrium composition of solute elements in each phase.

4. The accelerated simulation method for solidification phase field of multi-element alloys based on shallow neural networks according to claim 1, characterized in that, Step 3 specifically involves: An artificial neural network model F: Ω → D with a single hidden layer is constructed, where Ω is a parameter space including phase composition, solute component composition and temperature, and D is quasi-equilibrium thermodynamic data. The number of hidden layer neurons N ∈ [15, 25], and the activation function between each layer is the Sigmoid function. The constructed shallow artificial neural network model is trained, and the optimal number of hidden layer neurons N is evaluated and determined.

5. The accelerated simulation method for solidification phase field of multi-element alloys based on shallow neural networks according to claim 1, characterized in that, Step 4 specifically involves: Step 4.1: Construct the following sequence parameter equation and solute diffusion equation: in, It is the volume fraction of the α phase; It is the volume fraction of the β phase; It is the volume fraction of the γ phase; c i The total composition of component i; It is the two-phase interface mobility; These are numerical parameters determined during the simulation process based on the distribution of the order parameter values ​​of the four neighboring grid points of a calculated grid point. It is the interfacial energy between the β and γ phases; It is the interfacial energy between the α and γ phases; It is the Laplace operator; It is a gradient operator; It is the width of the interface; It is the chemical driving force behind the α to β phase transition; It is the molar volume of the alloy; is the diffusion coefficient of component i in the α phase; v is the number of coexisting phases; This indicates the composition of component i in phase α; It is the anti-solute intercepting phase; Step 4.2: Numerically solve the order parameter equation and solute diffusion equation using the finite difference method, spatially discretize the computational domain using a uniform grid, perform temporal discretization using the explicit Euler method, and accelerate the solution process of the partial differential equations by leveraging GPU vectorized parallel computing. Step 4.3: Take the phase composition, solute component composition and temperature of the current time step as input, and output the quasi-equilibrium thermodynamic conditions of the current time step through the artificial neural network model.

6. The accelerated simulation method for solidification phase field of multi-element alloys based on shallow neural networks according to claim 5, characterized in that, In step 5, the chemical driving force and chemical potential are calculated according to the following formulas: in, It is the free energy density of the β phase. It is the free energy density of the α phase. It is the chemical potential of component i. This represents the equilibrium composition of component i in phase β. This represents the equilibrium composition of component i in phase α.

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