Application of Fuzzy Set Theory to Quantitative Analysis of Correctness of the Mathematical Model Based on the ADI Method during Solidification

The explicit finite difference (EFD) method is used to calculate the casting temperature field during the solidification process. Because of its limited time step, the computational efficiency of the EFD method is lower than that of the alternating direction implicit (ADI) method. A model based on the equivalent specific heat method and the ADI method that improves computational efficiency is established.The error of temperature field simulation comes frommodel simplification, the acceptable hypotheses and calculation errors caused by different time steps, and the different mesh numbers that are involved in the process of numerical simulation.This paper quantitatively analyzes the degree of similarity between simulated and experimental results by the hamming distance (HD). For a thick-walled position, the time step influences the simulation results of the temperature field and the number of casting meshes has little influence on the simulation results of temperature field. For a thin-walled position, the time step has minimal influence on the simulation results of the temperature field and the number of casting meshes has a larger influence on the simulation results of temperature field.


Introduction
The 3D heat transfer equation of the temperature field during the solidification process is as follows [1][2][3]: where  is the temperature,  is the time,  is the average density of the liquid phase and the solid phase,   is the specific heat,  is the convectional parameter, Q is the inner heat source, and  is the latent heat,   is the solid phase fraction.
The energy conservation equation is usually solved by the EFD method, and computational efficiency is lower due to its limited time step [4][5][6].
The critical time step Δ in the EFD method can be taken as follows [6][7][8]: where Δ, Δ, and Δ are the mesh sizes in the , , and  directions, respectively.In this study, the equivalent specific heat method is adopted to describe the latent heat and the high-order ADI method that is fourth order in space and second order in time.This high-order mathematical model is based on the equivalent specific heat method, and the high-order ADI method is more accurate than the EFD method [7][8][9][10][11][12].
The error of temperature field simulation comes from model simplification, the acceptable hypotheses and calculation errors of the different time steps, and the different mesh numbers involved in the process of numerical simulation.

Method
Truncation errors This new high-order mathematical model Fourth order in space and second order in time The EFD method Second order in space and first order in time The degree of similarity between the simulation and the experimental results is quantitatively analyzed using the hamming distance [13][14][15].

Mathematical Model
The energy conservation equation can be given as the following: where   is the temperature of the liquid phase and   is the temperature of the solid phase.With the equivalent specific heat method [8]: The discretization equations of this high-order mathematical model based on the equivalent specific heat method and the high-order ADI method can be given as the following: where  = (/   );  2  ,  2  , and  2  are the second-order central difference operators.
Finally,  +1  can be obtained from (6).Each step has a tridiagonal system of equations that can be quickly calculated using the Thomas algorithm [16].
The calculation speed of this high-order mathematical model is faster because it is unconditionally stable.Table 1  shows that this high-order mathematical model is more accurate than the EFD method.
According to the HD, the degree of similarity between sets  and  can be evaluated by the function (, ): Equation ( 7) is used to quantitatively analyze the degree of similarity between the simulation results and the experimental results.

Experimental Results and Discussion
The 3D model is shown in Figure 1; the geometric figure of the casting is shown in Figure 2; the casting mould is 200 mm × 100 mm × 100 mm; the pouring speed is 0.35 m/s; and the pouring temperature is 670 ∘ C. The necessary physical parameters are shown in Table 2.The size of the mesh is 1.0 mm × 1.0 mm × 1.0 mm and the number of meshes is 2,000,000.
All of the thermocouples are connected by coaxial cables to a data logger and interfaced with a computer.The temperature data are automatically acquired.A schematic representation of the experimental setup, which is connected  to the data acquisition and analysis system, is shown in Figure 3.The experimental results are shown in Figure 4.

Temperature Simulation of Point A.
In this section, the high-order mathematical model, which is based on the equivalent specific heat method and the high-order ADI method, is used to compute the energy conservation equation.the temperature derived from the simulation method.These include the different time steps and the different mesh numbers.
The fuzzy set  can be described as follows: The mesh number remains constant.
(1) The time step is Δ and the number of casting meshes is 124031.The fuzzy set  can be described as follows: According to (7), the degree of similarity between sets  and  can be evaluated: (2) The time step is 20Δ and the number of casting meshes is 124031.The fuzzy set  can be described as follows: According to (7), the degree of similarity between sets  and  can be evaluated: (3) The time step is 200Δ and the number of casting meshes is 124031.The fuzzy set  can be described as follows: According to (7), the degree of similarity between sets  and  can be evaluated: The time step remains constant.
(1) The time step is 20Δ and the number of casting meshes is 6825.The fuzzy set  can be described as follows: According to (7), the degree of similarity between sets  and  can be evaluated: (2) The time step is 20Δ and the number of casting meshes is 672963.The fuzzy set  can be described as follows: According to (7), the degree of similarity between sets  and  can be evaluated:  The hardware environment: microcomputer.
The error of temperature field simulation comes from model simplification, the acceptable hypotheses and calculation errors that can be caused by the different time steps, and the different mesh numbers that are involved in the process of numerical simulation.Because the heat transfer model is based on the energy conservation equation (see (1)) and the governing equations ( 6), the loss of accuracy comes from calculation error that can be caused by the different time steps and the different mesh numbers.
The conclusions of the analysis and computations can be described as follows.
(1) The number of casting meshes remains constant and the different time steps are adopted to compute the temperature simulation, with great changes in the degrees of similarity between the simulation results and the experimental results: (2) The time step remains constant and the different mesh numbers are adopted to compute the temperature simulation, with the degrees of similarity between the simulation results and the experimental results changing slightly: ( (,  4 ) = 0.986140,  (,  2 ) = 0.988659,  (,  5 ) = 0.995946) .
In short, this high-order mathematical model is based on the equivalent specific heat method and the high-order ADI method, which can be used to calculate the temperature field.For the thick-walled position (see point A), the time step has a large influence on the simulation results of the temperature field and the number of casting meshes has little influence on the simulation results of temperature field.
In Figure 5, for the thick-walled position, the same conclusions hold: (a) the number of casting meshes remains constant and the changes of the time steps change the simulation results of the temperature field; (b) the time step remains constant and change in the mesh numbers brings little change in the simulation results of the temperature field.These are given to illustrate the validity of the analysis method that uses the hamming distance.
The simulation results and the experimental results can only be qualitatively analyzed by the figure analysis.For the first time, this study analyzes the hamming distance to quantitatively ascertain the degree of similarity between the simulation results and the experimental results.The quantitative analysis is based on hamming distance and it is more accurate than qualitative analysis based on the figure analysis.

Temperature Simulation of Point B.
The analysis method is similar and its steps are as follows.First, the number of casting meshes is 124031 and this number remains constant.The different time steps (Δ = 0.000026 s, 20Δ = 0.00052 s, and 200Δ = 0.0052 s) are adopted to compute the temperature simulation of point B. Second, the time step is 20Δ and this remains constant.The different mesh numbers (6825, 124031, and 672963) are adopted to compute the temperature simulation of point B. The hamming distance can evaluate the degrees of similarity between the simulation results and the experimental results.Figure 6 illustrates the validity of the analysis method of the hamming distance.
The conclusions are that for the thin-walled position (see point B); the time step has little influence on the simulation results of the temperature field and the number of casting meshes has a large influence on the simulation results of the temperature field.
This high-order mathematical model, which is based on the equivalent specific heat method and the high-order ADI method, is superior to the explicit finite difference method.In this section, the number of casting meshes remains constant and the calculation time between the explicit finite difference method and the high-order mathematical model is shown in Table 3.

Conclusions
(1) The high-order mathematical model based on the equivalent specific heat method and the high-order ADI method can be used to effectively compute the temperature simulation.Because this mathematical model is unconditionally stable, the different time steps can be chosen with quick calculation.
(2) For the first time, this paper demonstrates how the analysis method of the hamming distance can be used to quantitatively analyze the degree of similarity between the simulation results of the temperature field and the experimental results of the temperature field.
(3) For the thick-walled position (see point A), the time step has a large influence on the simulation results of the temperature field and the number of casting meshes has little influence on the simulation results of the temperature field.For the thin-walled position

Experiment results
The number of casting meshes is 124031 The number of casting meshes is 6825 The number of casting meshes is 672963 The time step is 0.00052 s.The number of casting meshes is 124031 The number of casting meshes is 6825 The number of casting meshes is 672963 The time step is 0.00052 s. (see point B), the time step has little influence on the simulation results of the temperature field and the number of casting meshes has a sizable influence on the simulation results of the temperature field.

Table 2 :
Physical parameters of casting and mold.

Table 3 :
The comparison results of calculation time.