Method for solving dendritic crystal morphology and concentration distribution based on arbitrary dimensional phase field model

Abstract

The invention provides a method for solving dendritic crystal morphology and concentration distribution based on a phase field model of any dimension. The method comprises the following steps: establishing a phase field control equation based on any dimension and a solute field control equation based on any dimension; giving physical property parameters required by alloy dendritic crystal growth and numerical simulation; setting an initialized dimensionless concentration and a boundary condition; and performing phase field simulation on the alloy solidification process by using a phase field control equation based on any dimension and a solute field control equation based on any dimension according to the physical property parameters, the initialized dimensionless concentration and the boundary conditions to obtain dendritic crystal morphology and concentration distribution. According to the method, the phase field equation is established by taking any concentration and any temperature as references, the references in the phase field model are unified, the mathematical forms of different phase field equations and the mutual relation between the mathematical forms are determined, and the problem that the phase field equations are not uniform when the dendritic crystal morphology and concentration distribution are solved is solved. And simulation results are difficult to compare due to different reference settings of different phase field models.

Description

Technical Field

[0001] The invention relates to the technical field of microstructure evolution simulation, and in particular to a method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model. Background Art

[0002] In recent years, the phase field method has become increasingly popular in the field of phase change exploration, such as microstructural evolution. Compared with the traditional numerical simulation method using sharp interface models, the phase field method diffuses the solid-liquid interface and does not need to directly track the dynamic evolution of the complex interface. In addition, the phase field method can be easily extended to multiple phases and can also couple random factors such as thermal noise, making it an ideal method for dealing with the evolution of organizational morphology. These advantages make the phase field method widely used in simulating the evolution of microstructures.

[0003] In the process of organizational simulation, the role of phase field variables is to describe the morphological evolution of materials and the change of phase transition position. The phase transition behavior of materials can be obtained by solving the phase field control equation. Concentration field variables are used to describe the distribution of different components in materials. By coupling the solute field control equation and the phase field control equation, the composition change of materials can be characterized. Temperature variables are used to describe the temperature distribution of materials. By coupling the heat conduction equation and the phase field control equation, the temperature evolution process of materials can be described. Physical fields such as solute field and temperature field affect each other, changing the Gibbs free energy distribution of the system, and thus affecting the phase transition behavior of materials. The coupling relationship between phase field variables, concentration field variables and temperature variables can be described by a suitable set of equations. In order to facilitate numerical solutions, these variables usually need to be dimensionless. Dimensionless processing can not only improve the generalization ability of the model, so that the model is not limited to a given alloy system or a specific process condition, but also reduce the number of unknown variables and broaden the scope of application of the model.

[0004] In the prior art, the patent with publication number CN113948158A provides a method for efficiently simulating dendrite growth by phase field method. When the concentration field is dimensionless, the method uses the far field concentration as the concentration reference, and the ratio of latent heat to specific heat is used as the temperature reference in the dimensionless supercooling; the patent with publication number CN108254485A provides a method for simulating the growth of curved dendrites based on phase field method, which quantitatively simulates and predicts the formation of curved dendrites in the solidification process, and dynamically reproduces the formation process of solidification microstructure. Its solute concentration is established based on the far field concentration, and the temperature reference of the characteristic parameter capillary length is the equilibrium solidification interval. The above technical solutions all have common technical problems: the dimensionless standards of the existing phase field models are not unified, and there are differences between the dimension standards of different phase field models; although the physical mechanism is consistent, due to the inconsistent references in different phase field models, the mathematical form of the phase field equation and the relationship between them are not clear; therefore, after obtaining the simulation results, misjudgment is prone to occur when comparing them with each other. Summary of the invention

[0005] In view of the deficiencies in the prior art, the present invention proposes a method for solving the dendrite morphology and concentration distribution based on an arbitrary dimension phase field model to overcome the defects in the prior art.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model comprises the following steps:

[0008] Establish the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions;

[0009] Given the physical property parameters required for the alloy dendrite growth and numerical simulation;

[0010] Set the initial dimensionless concentration and boundary conditions;

[0011] According to the physical parameters, initialized dimensionless concentration and boundary conditions, the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions are used to perform phase field simulation on the alloy solidification process to obtain the dendrite morphology and concentration distribution.

[0012] Furthermore, the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions are constructed in the following manner:

[0013] A phase field order parameter that changes continuously at the phase interface is introduced to characterize the state of the system;

[0014] Construct the total free energy expression of the system and obtain the evolution equations of the phase field and solute field over time;

[0015] The first equilibrium condition and the second equilibrium condition are introduced, and the total free energy expression and the evolution equation of the system are combined to obtain the phase field control equation and the solute field control equation;

[0016] Define dimensionless concentration and dimensionless temperature to link the sharp interface with the phase field model, take any concentration as the concentration basis and any temperature as the temperature basis, and obtain the dimensionless concentration equation of any basis and the dimensionless temperature equation of any basis;

[0017] The dimensionless concentration equation of any reference and the dimensionless temperature equation of any reference are combined with the phase field control equation and the solute field control equation to obtain the phase field control equation based on any dimension and the solute field control equation based on any dimension.

[0018] Furthermore, the total free energy expression of the system is as follows:

[0019]

[0020] In the above formula, φ represents the phase field order parameter, c represents the solute concentration of B, T represents the temperature, V represents the volume, and σ represents the coefficient related to energy; the first term of the integrand is is the interface gradient energy density, and the second term f AB (φ,c,T) is the volume free energy density, A represents the solvent, and B represents the solute.

[0021] Furthermore, the time evolution equation of the phase field is as follows:

[0022]

[0023] The time evolution equation of the solute field is as follows:

[0024]

[0025] In the above formula, φ represents the phase field order parameter, c represents the solute concentration, t represents the time, K φ and K c They represent constants related to phase field variables and concentration field variables, respectively; F represents the total free energy of the system; j at represents the antisolute capture term.

[0026] Furthermore, the phase field control equation is as follows:

[0027]

[0028] The solute field control equation is as follows:

[0029]

[0030] In the above formula, τ = 1 / K φ h,Kφ represents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the coefficient related to energy; R represents the gas constant, T M represents the melting point, k represents the equilibrium distribution coefficient, v0 represents the molar volume and is set as a constant, c represents the solute concentration, T represents the actual external temperature during solidification, m represents the liquidus slope, e represents the natural constant, g(φ) and q(φ) are both interpolation functions, and D is the solute diffusion coefficient in the liquid phase of the alloy.

[0031] Furthermore, the dimensionless concentration equation of the arbitrary benchmark is as follows:

[0032]

[0033] The dimensionless temperature equation for the arbitrary reference is as follows:

[0034]

[0035] In the above formula, U a represents the dimensionless concentration of an arbitrary basis, θ a represents the dimensionless temperature of any reference; e represents the natural constant, k represents the equilibrium distribution coefficient, T represents the actual external temperature during solidification, T M represents the melting point, m represents the liquidus slope, c a Indicates the concentration standard, the value can be any concentration, ΔT a Indicates the temperature reference, and the value can be any temperature.

[0036] Furthermore, the phase field control equation based on arbitrary dimensions is as follows:

[0037]

[0038] In the above formula, τ = 1 / K φ h,K φ represents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the energy-related coefficient, h represents the barrier height; λ represents the coupling constant, θ a represents the dimensionless temperature of any reference, m represents the liquidus slope, c a Indicates the concentration standard, ΔT a represents the temperature reference, k represents the equilibrium distribution coefficient, U a represents the dimensionless concentration of an arbitrary reference; coupling constant λ = -15RT M (1-k)ΔT a / (16v0hm), R represents the gas constant, T Mrepresents the melting point, k represents the equilibrium distribution coefficient, ΔT a represents the temperature reference, v0 represents the molar volume, h represents the potential barrier height, and m represents the liquidus slope;

[0039] The solute field control equation based on arbitrary dimensions is as follows:

[0040]

[0041] In the above formula, k represents the equilibrium distribution coefficient, φ represents the phase field order parameter, U represents the dimensionless concentration, t represents the time, D is the solute diffusion coefficient in the liquid phase of the alloy, and j is at represents the antisolute capture term.

[0042] Furthermore, the physical property parameters required for the alloy dendrite growth and numerical simulation are given, and the physical property parameters include: the solute diffusion coefficient D in the liquid phase of the alloy, the liquidus slope m, the equilibrium distribution coefficient k, the interface Gibbs-Thomson coefficient Г, and the alloy solidification temperature range ΔT0;

[0043] The dimensionless undercooling θ, coupling constant λ, and characteristic parameter capillary length d0 of dendrite growth.

[0044] Furthermore, the initial dimensionless concentrations and boundary conditions are set, including: setting periodic boundary conditions for all variables on all domains, including phase field, solute concentration, and temperature.

[0045] Furthermore, the finite difference method is used to convert the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions from variational equations into algebraic equations through discretization processing.

[0046] It can be seen from the above technical solution that the beneficial technical effects of the present invention are as follows:

[0047] 1. Establish the phase field equation by taking arbitrary concentration and arbitrary temperature as the benchmark, unify the benchmarks in the phase field model, clarify the mathematical forms of different phase field equations and their relationships through mathematical derivation, and solve the problem that the simulation results of different phase field models are difficult to compare due to different benchmark settings.

[0048] 2. A method combining mathematical deduction and phase-field simulation was adopted to ensure the rigor of mathematical and physical mechanisms, realize the numerical simulation of alloy morphology and solute distribution, provide guidance for the further improvement and application of the phase-field model, and broaden the scope of application of the phase-field method, enabling it to simulate solidification behaviors with more complex composition and temperature. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for the specific embodiments or the prior art description. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn according to the actual scale.

[0050] Figure 1 This is a schematic diagram of the method flow of Example 1 of the present invention;

[0051] Figure 2 is a normalized concentration distribution cloud diagram of the Al-3wt.%Cu alloy in Example 1 of the present invention;

[0052] Figure 3 is a normalized concentration distribution cloud diagram of the Mg-6wt.%Gd alloy in Example 1 of the present invention;

[0053] Figure 4 is a dimensionless concentration distribution cloud diagram of the Al-3wt.%Cu alloy in Example 1 of the present invention;

[0054] Figure 5 is a dimensionless concentration distribution cloud diagram of the Mg-6wt.%Gd alloy in Example 1 of the present invention;

[0055] Figure 6 The normalized concentration distribution cloud diagram of the Al-3wt.%Cu alloy in Example 1 of the present invention with different concentrations and temperature references;

[0056] Figure 7 The normalized concentration distribution cloud diagram of the Mg-6wt.%Gd alloy in Example 1 of the present invention with different concentrations and temperature references;

[0057] Figure 8 It is a dimensionless concentration distribution cloud diagram of the Al-3wt.%Cu alloy in Example 1 of the present invention with different concentrations and temperature references;

[0058] Fig. 9 This is a dimensionless concentration distribution cloud diagram of the Mg-6wt.%Gd alloy in Example 1 of the present invention with different concentrations and temperature references. DETAILED DESCRIPTION

[0059] The following embodiments of the technical solution of the present invention are described in detail in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and are therefore only used as examples, and cannot be used to limit the protection scope of the present invention.

[0060] It should be noted that, unless otherwise specified, the technical terms or scientific terms used in this application should have the common meanings understood by those skilled in the art to which the present invention belongs.

[0061] Example 1

[0062] This embodiment provides a method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model, such as Figure 1 As shown, the following steps are included:

[0063] Step 1: Establish an arbitrary dimension phase field model

[0064] For the solidification process of alloys, a phase field order parameter φ that changes continuously at the phase interface is introduced to characterize the state of the system. φ is 1 and -1 in the solid and liquid phases respectively. When φ changes between 1 and -1, it represents the solid-liquid interface. Using an arbitrary concentration and arbitrary temperature as a benchmark, the expression of the phase field equation of the binary alloy dilute solution is derived. According to the Ginzburg-Landau phase transition theory, the total free energy F of the system can be expressed as follows:

[0065]

[0066] In formula (1), φ represents the phase field order parameter, c represents the solute concentration of B, T represents the temperature, V represents the volume, and σ represents the coefficient related to energy; the first term of the integrand is is the interface gradient energy density, and the second term f AB (φ,c,T) is the volume free energy density, A represents the solvent, and B represents the solute.

[0067] According to the Ginzburg-Landau theory, the time evolution equations of the phase field and solute field can be obtained as follows:

[0068]

[0069] In equations (2) and (3), φ represents the phase field order parameter, c represents the solute concentration, t represents the time, and K φ and K c They represent constants related to phase field variables and concentration field variables, respectively; F represents the total free energy of the system; j at represents the antisolute capture term, j at The purpose of is to eliminate the non-equilibrium effect at the diffuse interface caused by assuming that the phase interface has a certain width.

[0070] Introducing the first equilibrium condition δF / δc=μ E and the second equilibrium condition δF / δφ=0, and substituting equation (1) into equation (2) and equation (3), we obtain the phase field control equation (4) and the solute field control equation (5):

[0071]

[0072] In formula (4) and (5), τ = 1 / K φ h,K φ represents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the coefficient related to energy; R represents the gas constant, T M represents the melting point, k represents the equilibrium distribution coefficient, v0 represents the molar volume and is set as a constant, c represents the solute concentration, T represents the actual external temperature during solidification, m represents the liquidus slope, e represents the natural constant, g(φ) and q(φ) are both interpolation functions, and D is the solute diffusion coefficient in the liquid phase of the alloy.

[0073] The definition of dimensionless concentration and dimensionless temperature relates the sharp interface to the phase field model, where the concentration and temperature are based on an arbitrary concentration c a and any temperature ΔT a Expressed as follows:

[0074]

[0075]

[0076] In formula (6) and (7), U a represents the dimensionless concentration of an arbitrary basis, θ a represents the dimensionless temperature of any reference, with the subscript a indicating that the dimensionless concentration or dimensionless temperature is based on any reference; e represents a natural constant, k represents the equilibrium distribution coefficient, T represents the actual external temperature during solidification, and T M represents the melting point, m represents the liquidus slope, c a Indicates the concentration standard, the value can be any concentration, ΔT a Indicates the temperature reference, and the value can be any temperature.

[0077] In formula (6),

[0078] In formula (8), c represents the solute concentration, c a represents the concentration reference, k represents the equilibrium distribution coefficient, and φ represents the phase field order parameter.

[0079] By combining equations (4), (5), (6), (7), and (8), we can obtain the phase field control equation (9) based on arbitrary dimensions and the solute field control equation (10) based on arbitrary dimensions, as follows:

[0080]

[0081] In formula (9), τ = 1 / K φ h,K φrepresents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the energy-related coefficient, h represents the barrier height; λ represents the coupling constant, θ a represents the dimensionless temperature of any reference, m represents the liquidus slope, c a Indicates the concentration standard, ΔT a represents the temperature reference, k represents the equilibrium distribution coefficient, U a Represents a dimensionless concentration on an arbitrary basis.

[0082] Coupling constant λ = -15RT M (1-k)ΔT a / (16v0hm), R represents the gas constant, T M represents the melting point, k represents the equilibrium distribution coefficient, ΔT a represents the temperature reference, v0 represents the molar volume, h represents the barrier height, and m represents the liquidus slope. In this embodiment, a specific reasonable value of the coupling constant is given as an input value to satisfy the dendrite growth.

[0083] In formula (10), k represents the equilibrium distribution coefficient, φ represents the phase field order parameter, U represents the dimensionless concentration, t represents the time, D is the solute diffusion coefficient in the liquid phase of the alloy, and j is at represents the antisolute capture term.

[0084] Step 2: Given the physical parameters required for the alloy dendrite growth and numerical simulation

[0085] According to the needs of the alloy dendrite growth and numerical simulation, the values ​​of the main parameters are first set: the solute diffusion coefficient D in the liquid phase of the alloy, the liquidus slope m, the equilibrium distribution coefficient k, the interface Gibbs-Thomson coefficient Г, and the alloy solidification temperature range ΔT0.

[0086] Then set other parameters: dimensionless undercooling θ of dendrite growth, coupling constant λ, characteristic parameter capillary length d0. d0 is based on the temperature reference, and its value is the ratio of the Gibbs-Thomson coefficient Г to the alloy solidification temperature range ΔT0.

[0087] Step 3: Set initial dimensionless concentration and boundary conditions

[0088] For dimensionless concentration, different dimensionless concentration initialization settings are required for phase field models using different concentration bases. For example, for two special concentration bases, the far-field concentration c ∞ Or liquid side concentration c0 l=c ∞ / k phase field model, corresponding to the two cases of q=1 and q=1 / k in the accompanying drawings of the specification.

[0089] The dimensionless concentration initialization settings are:

[0090]

[0091] In formula (11) and (12), U initial represents the dimensionless concentration initialization value, c represents the solute concentration, c ∞ represents the far-field concentration, k represents the equilibrium distribution coefficient, and φ represents the phase field order parameter.

[0092] For boundary conditions, periodic boundary conditions are set for all variables on all domains, including phase field, solute concentration, and temperature. In a specific embodiment, in the two-dimensional case, the domain size is 409.6×409.6, the minimum grid size is 0.8, and the length unit is the interface size W0. The time and length in the dimensionless phase field model are given by and W0 = λd0 / a1 scaling, a1 = 0.8839, a2 = 0.6267, d0 is the capillary length, D is the solute diffusion coefficient in the liquid phase of the alloy, and λ represents the coupling constant. The seed is initialized at the center of the domain.

[0093] Step 4: According to the physical parameters, initialized dimensionless concentration and boundary conditions, use the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions to perform phase field simulation on the alloy solidification process to obtain the dendrite morphology and concentration distribution.

[0094] To facilitate understanding by those skilled in the art, examples are given below:

[0095] Al-3wt.%Cu and Mg-6wt.%Gd alloys are used, and the solidification process of the alloy is simulated by phase field using equations (9) and (10), and the dendrite morphology and concentration distribution are obtained. In some embodiments, the solution results can be visualized, and the visualization method is not limited and can be implemented in any achievable manner in the prior art, such as using Visit visualization software.

[0096] The results of the visualization are as follows: Figure 2-Figure 9 As shown in the figure, the normalized concentration distribution curve and dimensionless concentration distribution curve are obtained by taking the horizontal midline of the domain in the cloud diagram, where different p and q values ​​represent different concentration and temperature benchmarks, and p = ΔT0 / ΔT a , c a =qc ∞ From the figure, we can see that:

[0097] Al-3wt.%Cu alloy and Mg-6wt.%Gd alloy show typical four-fold symmetry and six-fold symmetry dendrite structures, respectively. The cloud map shows that the phase field models using different dimensions have the same dendrite morphology and concentration range. Taking Al-3wt.%Cu alloy as an example, the c / c∞ range is 0.1394 to 2.128, and the U range is -0.083 to 1.401. For the same alloy, the change in concentration dimension affects the setting of the initialized dimensionless concentration, and then affects the distribution of the dimensionless concentration, but does not affect the simulation result of the normalized concentration, that is, the real concentration remains unchanged. Therefore, it can be proved that through the technical solution of this embodiment, the arbitrary dimension phase field model established can unify the dimensionless standard and ensure that the actual simulation results remain unchanged, and can realize the organizational simulation of the dendrite morphology and concentration distribution of the alloy during the solidification process.

[0098] The technical solution of this embodiment establishes a phase field equation by using arbitrary concentration and arbitrary temperature as a benchmark, unifies the benchmarks in the phase field model, clarifies the mathematical forms of different phase field equations and the relationship between them through mathematical derivation, and solves the problem that simulation results of different phase field models are difficult to compare due to different benchmark settings.

[0099] A method combining mathematical deduction and phase-field simulation was adopted to ensure the rigor of mathematical and physical mechanisms, realize the numerical simulation of alloy morphology and solute distribution, provide guidance for the further improvement and application of the phase-field model, and broaden the scope of application of the phase-field method, enabling it to simulate solidification behavior with more complex composition and temperature.

[0100] Example 2

[0101] Formulas (9) and (10) in Example 1 are variational equations, and there is a divergence operator. Since it is impossible to directly calculate it through algebraic equations in existing computers, it is necessary to discretize the divergence operator into an algebraic equation that is calculated step by step over time or space, and then hand it over to the computer for calculation to improve the calculation efficiency. The specific processing method is as follows:

[0102] The finite difference method is used to discretize the phase field control equation (9) based on arbitrary dimensions and the solute field control equation (10) based on arbitrary dimensions, and equations (9) and (10) are converted from variational equations to algebraic equations.

[0103] When discretizing the phase field control equation, the time discretization is discretized using the forward difference approximate derivative:

[0104]

[0105] Spatial discretization uses central difference approximate derivative for discretization:

[0106]

[0107] In equations (13) and (14), Δt represents the time step, Δx represents the space step, (i, j) represents the node coordinates, φ represents the phase field order parameter, and n represents the number of steps.

[0108] When discretizing the solute field control equation, the finite volume method is used to discretize the divergence term:

[0109]

[0110] In formula (15), JT, JB, JR, and JL are the fluxes of the upper, lower, left, and right surfaces, respectively. Δt represents the time step, Δx represents the space step, t represents the time, and i represents the node coordinate.

[0111] By adopting the technical solution of this embodiment, it is possible to solve the dendrite morphology and concentration distribution based on an arbitrary dimension phase field model in a computer, thereby improving the calculation efficiency.

[0112] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. These modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and specification of the present invention.

Claims

1. A method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model, characterized in that: The following steps are involved: Establish the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions; Given the physical property parameters required for the alloy dendrite growth and numerical simulation; Set the initial dimensionless concentration and boundary conditions; According to the physical parameters, initialized dimensionless concentration and boundary conditions, the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions are used to perform phase field simulation on the alloy solidification process to obtain the dendrite morphology and concentration distribution.

2. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 1, characterized in that: The phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions are constructed in the following manner: A phase field order parameter that changes continuously at the phase interface is introduced to characterize the state of the system; Construct the total free energy expression of the system and obtain the evolution equations of the phase field and solute field over time; The first equilibrium condition and the second equilibrium condition are introduced, and the total free energy expression and the evolution equation of the system are combined to obtain the phase field control equation and the solute field control equation; Define dimensionless concentration and dimensionless temperature to link the sharp interface with the phase field model, take any concentration as the concentration basis and any temperature as the temperature basis, and obtain the dimensionless concentration equation of any basis and the dimensionless temperature equation of any basis; The dimensionless concentration equation of any reference and the dimensionless temperature equation of any reference are combined with the phase field control equation and the solute field control equation to obtain the phase field control equation based on any dimension and the solute field control equation based on any dimension.

3. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 2, characterized in that: The total free energy expression of the system is as follows: In the above formula, φ represents the phase field order parameter, c represents the solute concentration of B, T represents the temperature, V represents the volume, and σ represents the coefficient related to energy; the first term of the integrand is is the interface gradient energy density, and the second term f AB (φ,c,T) is the volume free energy density, A represents the solvent, and B represents the solute.

4. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 2, characterized in that: The time evolution equation of the phase field is as follows: The time evolution equation of the solute field is as follows: In the above formula, φ represents the phase field order parameter, c represents the solute concentration, t represents the time, K φ and K c They represent constants related to phase field variables and concentration field variables, respectively; F represents the total free energy of the system; j at represents the antisolute capture term.

5. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 2, characterized in that: The phase field governing equation is as follows: The solute field control equation is as follows: In the above formula, τ = 1 / K φ h,K φ represents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the coefficient related to energy; R represents the gas constant, T M represents the melting point, k represents the equilibrium distribution coefficient, v0 represents the molar volume and is set as a constant, c represents the solute concentration, T represents the actual external temperature during solidification, m represents the liquidus slope, e represents the natural constant, g(φ) and q(φ) are both interpolation functions, and D is the solute diffusion coefficient in the liquid phase of the alloy.

6. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 2, characterized in that: The dimensionless concentration equation for the arbitrary basis is as follows: The dimensionless temperature equation for the arbitrary reference is as follows: In the above formula, U a represents the dimensionless concentration of an arbitrary basis, θ a represents the dimensionless temperature of any reference; e represents the natural constant, k represents the equilibrium distribution coefficient, T represents the actual external temperature during solidification, T M represents the melting point, m represents the liquidus slope, c a Indicates the concentration standard, the value can be any concentration, ΔT a Indicates the temperature reference, which can be any temperature.

7. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 2, characterized in that: The phase field governing equation based on arbitrary dimensions is as follows: In the above formula, τ = 1 / K φ h,K φ represents the constant related to the phase field variable, h represents the potential barrier height; φ represents the phase field order parameter, t represents the time, σ represents the energy-related coefficient, h represents the barrier height; λ represents the coupling constant, θ a represents the dimensionless temperature of an arbitrary reference, m represents the liquidus slope, c a Indicates the concentration standard, ΔT a represents the temperature reference, k represents the equilibrium distribution coefficient, U a represents the dimensionless concentration of an arbitrary reference; coupling constant λ = -15RT M (1-k)ΔT a / (16v0hm), R represents the gas constant, T M represents the melting point, k represents the equilibrium distribution coefficient, ΔT a represents the temperature reference, v0 represents the molar volume, h represents the potential barrier height, and m represents the liquidus slope; The solute field control equation based on arbitrary dimensions is as follows: In the above formula, k represents the equilibrium distribution coefficient, φ represents the phase field order parameter, U represents the dimensionless concentration, t represents the time, D is the solute diffusion coefficient in the liquid phase of the alloy, and j is at represents the antisolute capture term.

8. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 1, characterized in that: Given the physical property parameters required for the alloy dendrite growth and numerical simulation, the physical property parameters include: solute diffusion coefficient D in the liquid phase of the alloy, liquidus slope m, equilibrium distribution coefficient k, interface Gibbs-Thomson coefficient Г, alloy solidification temperature range ΔT0; The dimensionless undercooling θ, coupling constant λ, and characteristic parameter capillary length d0 of dendrite growth.

9. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 1, characterized in that: Set initial dimensionless concentrations and boundary conditions, including: Setting periodic boundary conditions on all domains for all variables, including phase field, solute concentration, and temperature.

10. The method for solving dendrite morphology and concentration distribution based on an arbitrary dimension phase field model according to claim 7, characterized in that: The finite difference method is used to convert the phase field control equation based on arbitrary dimensions and the solute field control equation based on arbitrary dimensions from variational equations into algebraic equations through discretization.

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