## 4 Model Validation  
### Verification of Steady-State Heat Conduction for the FGM Model  
As shown in Figure 7, a 1 m \(\times\) 1 m model is established and discretized into 914 triangular meshes. The initial temperature of the model is 0 \({}^{\circ}\)C, the top boundary is \(T_{1}\) = 100 \({}^{\circ}\)C, the bottom boundary is \(T_{1}\) = 0 \({}^{\circ}\)C, and the left and right boundaries are adiabatic boundaries. The thermal conductivity is assumed to obey the square distribution in the \(y\) direction, expressed as follows:  
\[k(y)=k_{0}(1+ay)^{2} \tag{17}\]  
where \(k_{0}\) is the heat conduction coefficient at \(y\) = 0 and \(a\) is the gradient coefficient [56].  
The analytical solution to this problem is [56]:  
\[T=\frac{(1+aL)T_{1}y}{(1+ay)L} \tag{18}\]  
To verify the correctness of the model for solving the steady-state heat conduction problem of FGMs, the temperature distribution obtained by the FDEM model is compared with the analytical solution under different gradient coefficients, as shown in Figure 8. The numerical solution is in good agreement with the analytical solution. In addition, it can be seen that a larger gradient coefficient leads to a larger temperature gradient from bottom to top along the \(y\) direction. The reason is that the larger the gradient coefficient, the greater the thermal conductivity from the top, and the smaller the temperature gradient. When \(a\) = 0, the thermal conductivity is constant, so the steady-state temperature gradient in the plate is the same. Figure 9 shows the steady-state heat conduction temperature distribution of the FGM plate under the square distribution with different \(a\). As the gradient coefficient increases, the temperature gradient near the bottom becomes larger.  
Next, we consider the temperature distribution inside the plate when the thermal conductivity is an exponential function. The model size is 0.4 m \(\times\) 0.4 m and the initial temperature and boundary conditions are the same as in the square distribution example. The thermal conductivity is exponentially distributed in the \(y\) direction, and its gradient function is as follows:  
\[k(y)=k_{0}e^{2 by} \tag{19}\]  
where \(b\) is the gradient coefficient.  
Figure 7: Steady-state heat conduction model of the FGM plate.  
The analytical solution (\(y>0\)) to this problem is [56]:  
\[T=T_{1}\frac{e^{-2 by}-1}{e^{-2 bL}-1} \tag{20}\]  
Figure 10 illustrates the numerical and analytical solutions of the temperature distribution in the plate under the exponential distribution. As the gradient coefficient increases, the slope of the curve also increases. Since the thermal conductivity is exponentially distributed from top to bottom and positively correlated with the gradient coefficient, the temperature gradient at the bottom is greater than that at the top. Figure 11 presents the steady-state conduction temperature distribution of the plate under exponential distribution.  
Figure 8: Numerical and analytical solutions along the \(y\) direction for the steady-state heat conduction with different gradient coefficients under the square distribution.  
Figure 9: Temperature distribution in the FGM plate with different gradient coefficients.  
The above analysis compares the numerical solution to the analytical solution while considering the effects of two function distribution methods and gradient coefficients on the steady-state heat conduction of the plate. It can be concluded that the model has a high accuracy for simulating heterogeneous steady-state heat conduction.