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

Authors: Xiaofeng Niu; Guanqian Wang; Wei Liang; Hua Hou; Hongxia Wang; Jinshan Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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 from model 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.


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## Body

## 1. Introduction

The 3D heat transfer equation of the temperature field during the solidification process is as follows [1–3]:
(1)ρcp∂T∂t=λ(∂2T∂x2+∂2T∂y2+∂2T∂z2)+Q˙,(2)Q˙=ρQ∂fs∂t,
where T is the temperature, t is the time, ρ is the average density of the liquid phase and the solid phase, cp is the specific heat, λ is the convectional parameter, Q˙ is the inner heat source, and Q is the latent heat, fs 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 [4–6].The critical time stepΔt in the EFD method can be taken as follows [6–8]:
(3)Δt≤ρcp[2·λ·(1/Δx2+1/Δy2+1/Δz2)],
where Δx, Δy, and Δz are the mesh sizes in the X, Y, and Z 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 [7–12].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. The degree of similarity between the simulation and the experimental results is quantitatively analyzed using the hamming distance [13–15].


## 2. Mathematical Model

The energy conservation equation can be given as the following:(4)ρcp∂T∂t=λ(∂2T∂x2+∂2T∂y2+∂2T∂z2)+ρQ∂fs∂t,fs=(TL-T)(TL-TS),
where TL is the temperature of the liquid phase and TS is the temperature of the solid phase. With the equivalent specific heat method [8]:
(5)ρcp′∂T∂t=λ(∂2T∂x2+∂2T∂y2+∂2T∂z2),cp′={cpT≥TLcp-Q∂fs∂tTL>T≥TScpT<TS.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:(6)[1+[Δx212-kΔt2]δx2]Tijk**=[1+[Δx212+kΔt2]δx2][1+[Δy212+kΔt2]δy2]×[1+[Δz212+kΔt2]δz2]Tijkn,[1+[Δy212-kΔt2]δy2]Tijk*=Tijk**,[1+[Δz212-kΔt2]δz2]Tijkn+1=Tijk*,
where k=(λ/ρcp′); δx2, δy2, and δz2 are the second-order central difference operators.Finally,Tijkn+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. Table1 shows that this high-order mathematical model is more accurate than the EFD method.Table 1
Truncation errors.
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 timeBecause the figure analysis cannot be used for quantitative analysis, the fuzzy mathematical theory is introduced [15–19]. The fuzzy set A of the universe of discourse U, U={t1,t2,t3,…,tn}, with a generic element of U denoted by ti, is a set of ordered pairs {(t1,A(t1)),(t2,A(t2)),…,(tn,A(tn))}, where  A(ti) is the membership function of the fuzzy set A, A(ti):U→[0,1], and A(ti) indicates the grade of membership of ti in A. Similar expression for the fuzzy set B is readily understood with obvious notation.In this study,ti represents time nodes; A(ti)=(TAti/T∞) and B(1or2)(ti)=(TB(1or2)ti/T∞) are two membership functions; TAti is the experimentally derived temperature; TB(1or2)ti is the temperature obtained by simulation; and T∞ denotes the “typical” temperature [19–21].According to the HD, the degree of similarity between setsA and B can be evaluated by the function N(A,B):
(7)N(A,B)=1-(1n)∑i=1n|A(ti)-B(ti)|.Equation (7) is used to quantitatively analyze the degree of similarity between the simulation results and the experimental results.


## 3. Experimental Results and Discussion

The 3D model is shown in Figure1; 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.Table 2
Physical parameters of casting and mold.
Material
Latent heat (kJ/kg)
Density (kg/m3)
Specific heat (kJ/kg·K)
Solidus temperature (°C)
Liquidus temperature (°C)
AlSi9Cu3
471
2596–2750
0.83–0.97
504
585
Sand
—
2780
0.54–1.00
—
—Figure 1
3D model.Figure 2
Schematic diagram of the experimental setup (cm).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 Figure3. The experimental results are shown in Figure 4.Figure 3
Schematic representation of the experimental setup connected to the data acquisition and analysis system.Figure 4
Experimental results.


### 3.1. 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. First, the number of casting meshes is 124031 and this number remains constant. The different time steps (Δt=0.000026s, 20Δt=0.00052s, and 200Δt=0.0052s) are adopted to compute the temperature simulation of point A. Second, the time step is 20Δt and this remains constant. The different mesh numbers (6825, 124031, and 672963) are adopted to compute the temperature simulation of point A. The hamming distance can be used to evaluate the degrees of similarity between the simulation results and the experimental results. The figure analysis is shown in Figure 5.Comparison results.
(a)
The mesh number remains constant
(b)
The time step remains constantLetU be the universe of discourse, U={t1=0.0s,t2=0.2s,t3=0.4s,t4=0.6s,…,t10=1.8s}, with a generic element of U denoted by ti; T∞=671°C denotes “typical” temperature; A(ti)=(TAti/T∞) and B(ti)=(TBti/T∞) are two membership functions; TAti represents a temperature measured from the experimental method, and TBti represents the temperature  derived from the simulation method. These include the different time steps and the different mesh numbers.The fuzzy setA can be described as follows:
(8)A={(t1,(669.2671)),(t2,(585.4671)),(t3,(576.2671)),(t4,(564.8671)),(t5,(551.9671)),(t6,(538.4671)),(t7,(525.0671)),(t8,(511.6671)),(t9,(498.7671)),(t10,(483.0671))}.The mesh number remains constant.(1)
The time step isΔt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(9)B1={(t1,(669.5671)),(t2,(585.6671)),(t3,(577.0671)),(t4,(566.7671)),(t5,(555.3671)),(t6,(539.8671)),(t7,(526.9671)),(t8,(512.9671)),(t9,(499.9671)),(t10,(485.1671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(10)N(A,B1)=1-(110)∑i=110|A(ti)-B(ti)|=0.997839.(2)
The time step is20Δt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(11)B2={(t1,(669.8671)),(t2,(590.7671)),(t3,(583.3671)),(t4,(573.2671)),(t5,(560.8671)),(t6,(548.9671)),(t7,(534.7671)),(t8,(520.5671)),(t9,(508.1671)),(t10,(490.3671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(12)N(A,B2)=1-(110)∑i=110|A(ti)-B(ti)|=0.988659.(3)
The time step is200Δt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(13)B3={(t1,(669.9671)),(t2,(607.8671)),(t3,(599.1671)),(t4,(590.5671)),(t5,(580.3671)),(t6,(570.2671)),(t7,(556.9671)),(t8,(545.3671)),(t9,(533.2671)),(t10,(523.0671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(14)N(A,B3)=1-(110)∑i=110|A(ti)-B(ti)|=0.959463.The time step remains constant.(1)
The time step is20Δt and the number of casting meshes is 6825. The fuzzy set B can be described as follows:
(15)B4={(t1,(669.9671)),(t2,(592.7671)),(t3,(584.3671)),(t4,(575.5671)),(t5,(562.8671)),(t6,(549.9671)),(t7,(537.7671)),(t8,(521.5671)),(t9,(510.1671)),(t10,(492.3671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(16)N(A,B4)=1-(110)∑i=110|A(ti)-B(ti)|=0.986140.(2)
The time step is20Δt and the number of casting meshes is 672963. The fuzzy set B can be described as follows:
(17)B5={(t1,(669.1671)),(t2,(586.9671)),(t3,(578.9671)),(t4,(567.7671)),(t5,(555.7671)),(t6,(542.5671)),(t7,(527.9671)),(t8,(514.9671)),(t9,(502.1671)),(t10,(485.7671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(18)N(A,B5)=1-(110)∑i=110|A(ti)-B(ti)|=0.995946.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:(19)(N(A,B1)=0.997839,N(A,B2)=0.988659,N(A,B3)=0.959463).(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:(20)(N(A,B4)=0.986140,N(A,B2)=0.988659,N(A,B5)=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 Figure5, 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.


### 3.2. 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 (Δt=0.000026s, 20Δt=0.00052s, and 200Δt=0.0052s) are adopted to compute the temperature simulation of point B. Second, the time step is 20Δt 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.Comparison results.
(a)
The mesh number remains constant
(b)
The time step remains constantThe 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 Table3.Table 3
The comparison results of calculation time.
Method
The number of casting meshes
Calculation time/second
The EFD method
124031
2579
This new high-order mathematical model (the time step is5Δt)
124031
1742
This new high-order mathematical model (the time step is20Δt)
124031
253
The hardware environment: microcomputer.


## 3.1. 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. First, the number of casting meshes is 124031 and this number remains constant. The different time steps (Δt=0.000026s, 20Δt=0.00052s, and 200Δt=0.0052s) are adopted to compute the temperature simulation of point A. Second, the time step is 20Δt and this remains constant. The different mesh numbers (6825, 124031, and 672963) are adopted to compute the temperature simulation of point A. The hamming distance can be used to evaluate the degrees of similarity between the simulation results and the experimental results. The figure analysis is shown in Figure 5.Comparison results.
(a)
The mesh number remains constant
(b)
The time step remains constantLetU be the universe of discourse, U={t1=0.0s,t2=0.2s,t3=0.4s,t4=0.6s,…,t10=1.8s}, with a generic element of U denoted by ti; T∞=671°C denotes “typical” temperature; A(ti)=(TAti/T∞) and B(ti)=(TBti/T∞) are two membership functions; TAti represents a temperature measured from the experimental method, and TBti represents the temperature  derived from the simulation method. These include the different time steps and the different mesh numbers.The fuzzy setA can be described as follows:
(8)A={(t1,(669.2671)),(t2,(585.4671)),(t3,(576.2671)),(t4,(564.8671)),(t5,(551.9671)),(t6,(538.4671)),(t7,(525.0671)),(t8,(511.6671)),(t9,(498.7671)),(t10,(483.0671))}.The mesh number remains constant.(1)
The time step isΔt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(9)B1={(t1,(669.5671)),(t2,(585.6671)),(t3,(577.0671)),(t4,(566.7671)),(t5,(555.3671)),(t6,(539.8671)),(t7,(526.9671)),(t8,(512.9671)),(t9,(499.9671)),(t10,(485.1671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(10)N(A,B1)=1-(110)∑i=110|A(ti)-B(ti)|=0.997839.(2)
The time step is20Δt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(11)B2={(t1,(669.8671)),(t2,(590.7671)),(t3,(583.3671)),(t4,(573.2671)),(t5,(560.8671)),(t6,(548.9671)),(t7,(534.7671)),(t8,(520.5671)),(t9,(508.1671)),(t10,(490.3671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(12)N(A,B2)=1-(110)∑i=110|A(ti)-B(ti)|=0.988659.(3)
The time step is200Δt and the number of casting meshes is 124031. The fuzzy set B can be described as follows:
(13)B3={(t1,(669.9671)),(t2,(607.8671)),(t3,(599.1671)),(t4,(590.5671)),(t5,(580.3671)),(t6,(570.2671)),(t7,(556.9671)),(t8,(545.3671)),(t9,(533.2671)),(t10,(523.0671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(14)N(A,B3)=1-(110)∑i=110|A(ti)-B(ti)|=0.959463.The time step remains constant.(1)
The time step is20Δt and the number of casting meshes is 6825. The fuzzy set B can be described as follows:
(15)B4={(t1,(669.9671)),(t2,(592.7671)),(t3,(584.3671)),(t4,(575.5671)),(t5,(562.8671)),(t6,(549.9671)),(t7,(537.7671)),(t8,(521.5671)),(t9,(510.1671)),(t10,(492.3671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(16)N(A,B4)=1-(110)∑i=110|A(ti)-B(ti)|=0.986140.(2)
The time step is20Δt and the number of casting meshes is 672963. The fuzzy set B can be described as follows:
(17)B5={(t1,(669.1671)),(t2,(586.9671)),(t3,(578.9671)),(t4,(567.7671)),(t5,(555.7671)),(t6,(542.5671)),(t7,(527.9671)),(t8,(514.9671)),(t9,(502.1671)),(t10,(485.7671))}.According to (7), the degree of similarity between sets A and B can be evaluated:
(18)N(A,B5)=1-(110)∑i=110|A(ti)-B(ti)|=0.995946.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:(19)(N(A,B1)=0.997839,N(A,B2)=0.988659,N(A,B3)=0.959463).(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:(20)(N(A,B4)=0.986140,N(A,B2)=0.988659,N(A,B5)=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 Figure5, 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.


## 3.2. 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 (Δt=0.000026s, 20Δt=0.00052s, and 200Δt=0.0052s) are adopted to compute the temperature simulation of point B. Second, the time step is 20Δt 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.Comparison results.
(a)
The mesh number remains constant
(b)
The time step remains constantThe 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 Table3.Table 3
The comparison results of calculation time.
Method
The number of casting meshes
Calculation time/second
The EFD method
124031
2579
This new high-order mathematical model (the time step is5Δt)
124031
1742
This new high-order mathematical model (the time step is20Δt)
124031
253
The hardware environment: microcomputer.


## 4. 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 (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.



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*Source: 894384-2013-11-24.xml*