# Topology Optimization of Functionally Graded Structure for Thermal Management of Cooling Plate
## 3 Evaluation and Analysis
### Numerical Example 1  
The first case is to evaluate the proposed method for uniform thermal load management, the design domain and boundary condition are given in Figure 3, where the length and height of the cooling plate are 120 and 60 mm, respectively, and the inlet and outlet of the rectangular area define the width of the inlet and outlet at 10 mm. The local filter radius \(R\) was set to 2 mm. The velocity of the flow for inlet was set to 0.1 m/s.  
To demonstrate the advancement of the optimized structure, the classic straight micro-channel cooling plate is introduced for a comparison, which can be found in Figure 4. We can observe that the right part of the structure suffered from high temperature, and the temperature of the middle part of the design was comparatively lower due to the different velocities of flow. The maximum temperature approached 365.38 K. Figure 4b shows the layout of the traditional SIMP-based topology optimization. It was constructed with several macro-runners and numerous micro-channels, and the channel distribution  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\rho\) **(kg/m\({}^{3}\))** & \(k\) **(W/(m\cdot K))** & \(C_{p}\) **(J/(kg\cdot K))** & \(\
u\) **(Pa\(\cdot s\))** \\ \hline solid & 2700 & 217 & 880 & - \\ fluid & 1000 & 0.6 & 4200 & \(10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material property for numerical simulation.  
Figure 3: Design domain for cooling plate optimization.  
Figure 2: Layout of battery and cooling plate.  
ended up being more complicated and novel when compared with the classical straight channel one. As the topology optimization technology introduced more freedom for the structure design, and the material distribution of the design was comparatively random, the maximum temperature of the optimal structure was lower than the classical straight channel cooling plate, which was 357.85 K. Figure 4c shows the optimal cooling plate with its lattice structure constructed using the proposed method. It can be found that the optimal design has been constructed by channels with similar widths, the distribution of the channel is even, and fewer slender runners exist when compared with the result of the traditional SIMP-based topology optimization. Thus, the velocity of flow in the structure constructed using the proposed method is consistent, and the maximum temperature of the optimal design came out to 324.12 K.  
Figure 4: (**a**) Traditional cooling plate, (**b**) design using SIMP method, and (**c**) design using proposed lattice structure. The left column shows the optimal design of the cooling plate, and the right shows the temperature response of the cooling plate design.  
Table 2 records the average temperature, the maximum temperature of the cooling plate designs, and shows the standard error of the temperature distribution in the cooling plate. It can be observed that the proposed lattice structure performed best among the cooling plate designs, which not only significantly decreased the maximum temperature of the cooling plate but also ensured the thermal homogeneity of the design. The standard error of the temperature distribution decreased from 21.80 to 7.88, which demonstrates that the proposed method is of good performance in the control of the temperature rise and homogeneity of the cooling plate.