Phase field-lattice Boltzmann method for simulating the growth and motion of large quantities of equiaxed crystals

Abstract

The present invention discloses an efficient and streamlined method for simulating the growth and movement of large quantities of equiaxed crystals using the phase field-lattice Boltzmann method, which solves the key problem that in the multi-field coupled simulation of dendrite growth, as the scale of the computational domain and the number of dendrites increase, the computational resources are extremely consumed and the simulation efficiency is extremely low. The method has been effectively verified in the multi-field coupled simulation of a complete welding molten pool, and is generally applicable to the field of metal solidification. The method proposes a zone-layer hybrid method for a phase field simulation data structure, which replaces the traditional hierarchical data structure to achieve partitioned storage and efficient traversal of the liquid phase and the phase field corresponding to each equiaxed crystal and its derived variables. The efficient and streamlined method proposed in the present invention is used to simulate the solidification of a welding molten pool with 120 base material grains and 276 equiaxed crystals. Compared with the hierarchical data method, the method saves 95% of memory and improves efficiency by 89%, proving that the method can effectively save computing resources and improve simulation efficiency.

Description

Technical Field

[0001] The present invention proposes a phase field-lattice Boltzmann method for simulating the growth and movement of large quantities of equiaxed crystals. The method belongs to the field of multi-field coupled simulation of welding molten pool solidification, and is suitable for the simulation of dendrite growth that couples phase field, solute field, temperature field, flow field and solid phase movement, and is particularly suitable for the simulation of dendrite growth and movement in a complete welding molten pool with a large computational domain and a large number of dendrites. Background Art

[0002] The core control equations for the coupled simulation of phase field, solute field, temperature field, and flow field for dendrite growth and movement in the complete welding molten pool using the phase field-lattice Boltzmann method are shown in equations (1) to (4), which are the Allen-Kahn equation, the Kahn-Hilliard equation, the Fourier heat conduction equation, and the lattice Boltzmann equation, respectively.

[0003]

[0004]

[0005]

[0006]

[0007] Where: φ i is the phase field variable of the ith dendrite, t is the time, is the interfacial energy anisotropy, m is the liquidus slope, T 0 The equilibrium solute concentration of the liquid phase at temperature, T is the transient temperature field distribution, is the concentration C 0 The liquidus temperature under the condition of λ is the phase field-supersaturation field coupling coefficient, U is the dimensionless supersaturation field determined by the solute concentration C, k is the solute equilibrium distribution coefficient, α is the interaction energy coefficient, N is the total number of dendrites, φ 0 is the solid fraction, is the dimensionless solute diffusion coefficient, is the solid phase motion velocity, is the velocity distribution of the flow field; ρ is the density, c is the specific heat capacity, λ con is the thermal conductivity, q is the welding heat flux density; g k is the Boltzmann distribution function, is the position vector, is the discrete velocity, Δt is the discrete time increment, τ is the relaxation time, ω k is the weight coefficient, c is the lattice velocity, v is the kinematic viscosity, h is the Beckermann constant, W 0 is the solid-liquid interface thickness

[0008] The multi-field coupled simulation of dendrite growth based on the phase field method is a computationally intensive problem. Since crystallization occurs at the microscopic scale, the time step and spatial increment of the phase field simulation are extremely small. Compared with the macroscopic scale simulation, the phase field simulation requires large computing resources and low simulation efficiency. In particular, when performing multi-field coupled simulation on the full solidification process of the complete welding molten pool, the computational domain span, field variables to be determined, and the number of dendrites far exceed the pure growth simulation of dendrites in the local molten pool. The computational domain scale often exceeds hundreds of millions of grids and the number of dendrites is tens of thousands. At this time, the huge computing resources required for the simulation, the extremely long simulation time, and the extremely low simulation efficiency have become the most prominent problems.

[0009] According to the phase field control equation (Equation (1)), the array structure of the phase field variables in the two-dimensional simulation of multi-dendrite growth is φ(1:N, 1:n, 1:m), and the overall presentation is that multiple dendrites grow independently in a two-dimensional calculation domain, as shown in Figure 1 (a), where N is the total number of simulated dendrites, m×n is the computational domain size, and the required memory space is N×m×n double-precision real numbers. The size of N has a great impact on the memory space. In addition, there are arrays corresponding to phase field derivative variables, such as the Laplace operator L (1: N, 1: n, 1: m), the phase field variable increment Δφ (1: N, 1: n, 1: m), and the solid phase motion velocity V s (1:N, 1:n, 1:m), which have the same structure as the phase field variable array, which increases the memory space required for simulation exponentially. These four arrays of the form X(1:N, 1:n, 1:m) are actually composed of N two-dimensional arrays, namely X(1, 1:n, 1:m) to X(N, 1:n, 1:m). Storing simulation data in this way is equivalent to storing the phase field and its derived variables of each grain in layers 1 to N. The corresponding spatial hierarchy is as follows Figure 1 As shown in (b), each dendrite is assigned a two-dimensional computational domain. In each computational domain, the liquid phase region occupies most of the space, while the actual liquid phase region is only Figure 1 (a) The area outside the dendrites, so this hierarchical data structure causes a lot of memory redundancy. When the total number of dendrites N is tens of thousands, it will consume huge memory resources and cannot carry out the phase field simulation of the complete welding pool. In addition, in the solution process based on the hierarchical data structure X (1: N, 1: n, 1: m), each time step needs to complete N × m × n traversals and solutions, which will significantly reduce the calculation efficiency, and the simulation efficiency is closely related to the total number of dendrites N.

[0010] In summary, when simulating the growth and movement of large-scale equiaxed crystals in a complete welding pool based on the traditional hierarchical data structure X (1: N, 1: n, 1: m), there are serious defects such as high memory resource consumption, high redundancy, and extremely low simulation efficiency. It is urgent to find a new method to streamline the storage, efficient traversal and solution of phase field simulation data. Summary of the invention

[0011] Based on the characteristic that the maximum number of dendrites that intersect or collide with each other at the same position is no more than six, the present invention proposes a zone-layer hybrid method to simulate the growth and movement of large quantities of equiaxed crystals based on the phase field-lattice Boltzmann method. The specific contents are as follows:

[0012] (1) The entire computational domain is divided into liquid phase and solid phase, such as Figure 2 As shown in (a), (b) and (c), a partition array X(1:n, 1:m) is introduced to replace the traditional hierarchical array X(1:N, 1:n, 1:m), so that the phase field simulation data structure removes the correlation of the total number of dendrites N, realizes the streamlined storage of phase field variables and their derivative variables, greatly saves memory space, and significantly improves the traversal and solution efficiency. The phase field partition array φ(1:n, 1:m) is used to distinguish the solid phase and the liquid phase, φ=-1 is the liquid phase, -1<φ≤1 is the solid-liquid interface and the solid phase;

[0013] (2) The entire solid phase region is further divided into sub-regions occupied by each dendrite, such as Figure 2 As shown in (d), the dendrite number array pnum (1:n, 1:m) and the dendrite name array pname (1:n, 1:m) are introduced to distinguish different dendrites;

[0014] (3) The concept of layers is introduced to deal with the intersection or collision of dendrites due to growth and movement, such as Figure 2 As shown in (e), the partition array X(1:n, 1:m) and the dendrite name array pname(1:n, 1:m) are expanded with the "layer" attribute to form a partition-layer mixed array X(1:Q, 1:n, 1:m) and a dendrite name layered array pname(1:Q, 1:n, 1:m), where Q is the maximum number of dendrite intersections or collisions at the same position, which is a fixed value of 6. At the same time, the number of dendrites that intersect or collide at each node is recorded in the dendrite number array pnum(1:n, 1:m), and the pnum value of the liquid phase node other than the dendrite is set to 0.

[0015] The beneficial effects of the present invention are as follows: the proposed zone-layer hybrid method can save the memory space of 4×N×m×n double-precision real numbers (i.e., variables φ, L, Δφ, V sThe hierarchical array X(1:N, 1:n, 1:m)) is reduced to the memory space required for the zone-level mixed data structure 7×m×n integers plus 4×6×m×n double-precision real numbers (i.e., pnum(1:n, 1:m), pname(1:Q, 1:n, 1:m) and variables φ, L, Δφ, V s The zone-layer hybrid array X(1:Q, 1:n, 1:m)) is used to remove the correlation of the total number of dendrites N in the phase field simulation data structure. When the total number of dendrites N is tens of thousands, the zone-layer hybrid data structure can save a huge amount of memory space and greatly improve the simulation efficiency, making it possible to simulate the phase field of a complete welding pool with a large number of equiaxed crystals. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 Schematic diagram of the hierarchical data structure for multi-dendrite growth simulation

[0017] Figure 2 Schematic diagram of the zone-layer hybrid data structure for multi-dendrite growth simulation

[0018] Figure 3 Simulation results of the solidification process of a complete weld pool with 120 base metal grains and 276 equiaxed grains and a computational domain size of 2048×4096 DETAILED DESCRIPTION

[0019] According to the efficient and streamlined method for simulating the growth and movement of large quantities of equiaxed grains by the phase field-lattice Boltzmann method proposed in the present invention, the solidification process of a complete welding molten pool with 120 base material grains and 276 equiaxed grains and a computational domain size of 2048×4096 was simulated. The following steps were added to the typical process of simulating dendrite growth and movement by the phase field-lattice Boltzmann method:

[0020] Step 1: The phase field variable φ and its derivative variable Laplace operator L, the phase field variable increment Δφ and the solid phase motion velocity V s , introduce the zone-layer hybrid array X(1:6, 1:4096, 1:2048) to replace the traditional layered array X(1:396, 1:4096, 1:2048), store the phase field values ​​of the parent material grains or the equiaxed crystals of the molten pool in the initial state on the first layer of the zone-layer hybrid phase field array, and the values ​​of the other variables are also stored on the first layer of the corresponding array;

[0021] Step 2: Introduce the dendrite number array pnum (1: 4096, 1: 2048) and the dendrite name hierarchical array pname (1: 6, 1: 4096, 1: 2048), store the number of dendrites on each node and the corresponding name, in the initial state, multiple dendrites do not interact or collide with each other, the pnum value of each solid phase node covered by each dendrite is 1, each node corresponds to a unique dendrite name, stored in the first layer of the dendrite name hierarchical array; the pnum value of the liquid phase node not covered by the dendrite is 0, there is no dendrite name, and the pname value is set to 0; during the growth of the dendrite, the liquid phase node at the front of the solid-liquid interface is transformed into the solid phase node covered by each dendrite as the solid-liquid interface advances, and the corresponding pnum value, pname value, phase field variable and its derived variable results also change accordingly, and are stored in the first layer;

[0022] Step 3: When dendrites intersect or collide during their growth and movement, the number and name of the dendrites involved in each intersection or collision are recorded, and the corresponding phase field and derived variable results are calculated and stored in the i-th layer of the zone layer mixing array X(1:6, 1:4096, 1:2048) and the dendrite name layered array pname(1:6, 1:4096, 1:2048), where 1<i<6, until the entire molten pool is completely solidified.

[0023] The above simulation was completed in a computing environment equipped with two AMD EPYC 75422.9GHz 32-core processors and 256G DDR4 memory. The results are as follows Figure 3 As shown. To prove the effectiveness of the method described in this patent, the same number of dendrites and the same computational domain were simulated using the traditional hierarchical data structure under the same computing environment. The memory consumption and simulation time of the two simulations were quantitatively analyzed and it was found that compared with the traditional hierarchical data structure, the zone-layer hybrid data structure proposed in this patent can reduce the memory required for simulation from 209.5G to 10.6G, saving 95% of memory, and can also reduce the simulation time from 531 minutes per 10,000 steps to 51 minutes per 10,000 steps, and improve the simulation efficiency by 89%. The results show that the efficient and streamlined method of simulating the growth and movement of large quantities of equiaxed crystals using the phase field-lattice Boltzmann method proposed in this patent can effectively reduce the consumption of computing resources and greatly improve the simulation efficiency, making the multi-field coupling simulation of the complete welding pool possible.

[0024] The protection scope of the present invention includes but is not limited to the above embodiments. The protection scope of the present invention shall be based on the claims. Any replacement, deformation, and improvement of the present technology that can be easily thought of by those skilled in the art shall fall within the protection scope of the present invention.

Claims

1. An efficient and streamlined method for simulating the growth and movement of large quantities of equiaxed crystals using the phase field-lattice Boltzmann method, characterized in that: It includes the following steps: Step 1: The phase field variable φ and its derived variable Laplace operator L, the phase field variable increment Δφ and the solid phase motion velocity V s , introduce the zone-layer hybrid array X(1:Q, 1:n, 1:m) to replace the traditional layered array X(1:N, 1:n, 1:m), store the phase field values ​​of the parent material grains or the equiaxed crystals in the molten pool in the initial state on the first layer of the zone-layer hybrid phase field array, and the values ​​of the other variables are also stored on the first layer of the corresponding array, where Q is the maximum number of grains where dendrites intersect or collide at a certain position, taking a fixed value of 6, N is the total number of simulated dendrites, m×n is the size of the calculation domain, and the values ​​of N, m, and n change continuously according to the simulation needs; Step 2: Introduce the dendrite number array pnum(1:n, 1:m) and the dendrite name hierarchical array pname(1:Q, 1:n, 1:m) to store the number and corresponding name of the dendrites existing on each node. In the initial state, multiple dendrites do not interact with each other and do not collide. The pnum value of the solid-phase nodes covered by each dendrite is 1, and each node corresponds to a unique dendrite name, which is stored in the first layer of the dendrite name hierarchical array; the pnum value of the liquid-phase nodes not covered by the dendrite is 0, there is no dendrite name, and the pname value is set to 0; during the dendrite growth process, the liquid-phase nodes at the solid-liquid interface front are transformed into the solid-phase nodes covered by each dendrite as the solid-liquid interface advances, and the corresponding results of the pnum value, pname value, phase-field variable and its derivative variables are also changed and stored in the first layer; Step 3: When dendrites intersect or collide during the growth and movement process, record the number and name of the dendrites included in each intersection or collision, and calculate the results of the corresponding phase field and derivative variables, which are respectively stored in the i-th layer of the zone layer mixing array X(1:Q, 1:n, 1:m) and the dendrite name hierarchical array pname(1:Q, 1:n, 1:m), where 1 < i < Q, until the entire molten pool is completely solidified.

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