The thermal management of an EV battery pack is a critical determinant of its operational stability, energy efficiency, and driving range, representing a significant performance bottleneck in new energy vehicles. Excessive maximum temperature within the pack can lead to accelerated aging, reduced capacity, and in severe cases, thermal runaway. This necessitates advanced design methodologies to enhance heat dissipation. While topology optimization has been widely applied to steady-state heat conduction problems, practical operating conditions for EV battery packs often involve transient thermal loads due to varying discharge rates and ambient conditions. This transient effect is frequently neglected in conventional design. Furthermore, common performance metrics like thermal compliance do not directly control the peak temperature, which is often the primary failure criterion. This work addresses these gaps by formulating a topology optimization framework for minimizing the maximum temperature in a specified region of a structure under transient thermal loading, a problem highly relevant to EV battery pack cooling system design.
The core challenge in optimizing for maximum temperature is its non-differentiable nature, as the location of the hot spot shifts with material distribution. To overcome this, we introduce a Regional Temperature Control Function based on the P-norm aggregation scheme. For a design domain $\Omega$ with a control region $\Omega_c$, the approximate maximum temperature over the total operational time $t_f$ is defined as:
$$ g(\mathbf{X}) = cp_2 \left[ \frac{1}{t_f} \int_{0}^{t_f} \left( cp_1 \int_{\Omega_c} T^P(\mathbf{x}, t) d\Omega \right)^{1/P} dt \right] $$
where $\mathbf{X}$ is the vector of design variables (element densities), $T(\mathbf{x}, t)$ is the temperature field, and $P$ is the aggregation exponent. The coefficients $cp_1$ and $cp_2$ are correction factors applied spatially and temporally, respectively, to ensure $g(\mathbf{X})$ closely approximates the actual maximum temperature in $\Omega_c$ over $[0, t_f]$:
$$ cp_1 = \frac{\max(T(\mathbf{x}, t))}{\left( \int_{\Omega_c} T^P d\Omega \right)^{1/P}}, \quad cp_2(t) = \frac{\max(T(\mathbf{x}, t))}{ \left( cp_1 \int_{\Omega_c} T^P d\Omega \right)^{1/P} }, \quad t \in [0, t_f]. $$
This function provides a smooth, differentiable surrogate for the non-smooth max operator, enabling gradient-based optimization.
The physical process is governed by the transient heat conduction equation. For a two-dimensional domain with convection boundaries, the strong form is:
$$ \rho c \frac{\partial T}{\partial t} – \nabla \cdot (k \nabla T) = Q \quad \text{in } \Omega $$
with boundary conditions:
$$ \begin{aligned}
T &= \bar{T} \quad &\text{on } \Gamma_1, \\
-k \frac{\partial T}{\partial n} &= \bar{q} \quad &\text{on } \Gamma_2, \\
-k \frac{\partial T}{\partial n} &= h(T – T_{\infty}) \quad &\text{on } \Gamma_3,
\end{aligned} $$
and initial condition $T|_{t=0} = T_0$. Here, $\rho$, $c$, and $k$ are density, specific heat, and thermal conductivity, respectively. $Q$ is the internal heat generation, highly relevant for modeling the heat source from an operating EV battery pack.
Using the Finite Element Method (FEM) for spatial discretization and the Crank-Nicolson scheme ($\theta=0.5$) for temporal discretization, the semi-discrete system is:
$$ \mathbf{C}_h \dot{\mathbf{T}} + \mathbf{K}_h \mathbf{T} = \mathbf{P}(t), $$
which leads to the time-stepping equation:
$$ \hat{\mathbf{K}}_h \mathbf{T}^{t+1} = \hat{\mathbf{Q}}^{t+1}, $$
where
$$ \begin{aligned}
\hat{\mathbf{K}}_h &= \mathbf{C}_h + \theta \Delta t \mathbf{K}_h, \\
\hat{\mathbf{Q}}^{t+1} &= [\mathbf{C}_h – (1-\theta)\Delta t \mathbf{K}_h] \mathbf{T}^{t} + \Delta t [\theta \mathbf{P}^{t+1} + (1-\theta)\mathbf{P}^{t}].
\end{aligned} $$
This allows for the efficient computation of the transient temperature field $\mathbf{T}(t)$.
The material properties are interpolated using the Solid Isotropic Material with Penalization (SIMP) model:
$$ k(\rho_e) = k_{\min} + \rho_e^p (k_0 – k_{\min}), \quad c(\rho_e) = c_{\min} + \rho_e^p (c_0 – c_{\min}), $$
where $\rho_e$ is the element density (design variable), $k_0$ and $c_0$ are the properties of the solid conductive material, $k_{\min}$ and $c_{\min}$ are small values to prevent singularity, and $p$ is the penalty factor (typically $p=3$).
The topology optimization problem is formally stated as:
$$ \begin{aligned}
& \underset{\mathbf{X}}{\text{minimize:}} & & g(\mathbf{X}) = cp_2 \left[ \frac{1}{t_f} \int_{0}^{t_f} \left( cp_1 \int_{\Omega_c} T^P d\Omega \right)^{1/P} dt \right] \\
& \text{subject to:} & & \hat{\mathbf{K}}_h \mathbf{T}^{t+1} = \hat{\mathbf{Q}}^{t+1}, \quad t=0,1,\dots,N_t-1 \\
& & & V(\mathbf{X}) = \frac{\sum \rho_e v_e}{V_0} \leq f, \\
& & & 0 < \rho_{\min} \leq \rho_e \leq 1, \quad e=1,\dots,N_e.
\end{aligned} $$
Here, $V(\mathbf{X})$ is the material volume, $V_0$ is the domain volume, $f$ is the allowed volume fraction, and $\rho_{\min}$ is a lower bound to ensure numerical stability.
The sensitivity of the objective $g(\mathbf{X})$ with respect to the design variables $\rho_e$ is derived using the adjoint method. The augmented Lagrangian $\mathcal{L}$ is constructed:
$$ \mathcal{L} = g(\mathbf{X}) + \sum_{t=0}^{N_t-1} (\boldsymbol{\lambda}^{t+1})^T \left( \hat{\mathbf{K}}_h \mathbf{T}^{t+1} – \hat{\mathbf{Q}}^{t+1} \right), $$
where $\boldsymbol{\lambda}^{t}$ are adjoint variables. The derivative $\partial \mathcal{L} / \partial \rho_e = \partial g / \partial \rho_e$ if the state equations are satisfied. Setting the variation of $\mathcal{L}$ with respect to the state variables $\mathbf{T}^{t}$ to zero yields the adjoint equations, which are solved backward in time:
$$ \hat{\mathbf{K}}_h^T \boldsymbol{\lambda}^{t} = -\frac{\partial g}{\partial \mathbf{T}^{t}} + \left[ \mathbf{C}_h^T – (1-\theta)\Delta t \mathbf{K}_h^T \right] \boldsymbol{\lambda}^{t+1}, \quad \text{with } \boldsymbol{\lambda}^{N_t} = \mathbf{0}. $$
The final sensitivity is then:
$$ \frac{\partial g}{\partial \rho_e} = \sum_{t=0}^{N_t-1} (\boldsymbol{\lambda}^{t+1})^T \left( \frac{\partial \hat{\mathbf{K}}_h}{\partial \rho_e} \mathbf{T}^{t+1} – \frac{\partial \hat{\mathbf{Q}}^{t+1}}{\partial \rho_e} \right). $$
This gradient information is supplied to the Method of Moving Asymptotes (MMA) optimizer to update the design variables iteratively.
A numerical case study inspired by EV battery pack cooling is presented. The design domain is a square representing a simplified cold plate attached to a battery module. The central region $\Omega_c$ is subjected to a transient heat flux $P(t)$ simulating battery heat generation during a drive cycle:
$$ P(t) = C_{rate} \cdot \left(6 + 6 \sin\left(\pi \frac{t}{t_f}\right)\right), \quad 0 \leq t \leq t_f, $$
where $C_{rate}$ models the discharge rate. All four edges are maintained at a constant coolant temperature ($T=0^\circ$C). The goal is to distribute a limited volume of high-conductivity material (volume fraction $f=0.5$) to minimize the maximum temperature in $\Omega_c$. The performance is compared against the traditional objective of minimizing thermal compliance (weighted average temperature).
The results demonstrate the critical importance of considering transient effects. Table 1 shows the optimal topologies for both objectives at different operational times $t_f$.
| Time (s) | Target I: Min Max Temp ($g(\mathbf{X})$) | Target II: Min Thermal Compliance |
|---|---|---|
| 50 | Topology A1 | Topology B1 |
| 100 | Topology A2 | Topology B2 |
| 250 | Topology A3 | Topology B3 |
| 500 | Topology A4 | Topology B4 |
| 1000 | Topology A5 | Topology B5 |
| 5000 | Topology A6 | Topology B6 |
| 10000 | Topology A7 | Topology B7 |
| 20000 | Topology A8 | Topology B8 |
For short times (e.g., 50s, 100s), the heat has not diffused fully, leading to distinct, non-equilibrium topologies for both targets. As $t_f$ increases, the structures evolve, with material accumulating near heat sinks and developing more branched pathways. Notably, after approximately $t_f \geq 1000$s, the topologies converge to a steady-state form, indicating that the transient effect diminishes as the system approaches thermal equilibrium. This convergence validates the numerical stability of the method. Crucially, the optimal topology for a short $t_f$ is not optimal for a long $t_f$, and vice-versa. Since real-world EV battery pack loads are often transient and may not reach steady state, optimizing for the specific transient profile is essential.
A key finding is the structural difference between the two objectives. Target I (min max temp) consistently produces topologies with more uniform material distribution in the heated central region and thicker, more direct paths to the boundaries. Target II (min compliance) tends to create slender, branching structures that optimize for overall heat flux but can leave localized hot spots. This is reflected in the peak temperatures. For $t_f=1000$s, the maximum temperature for Target I’s design is significantly lower than for Target II’s design, with a reduction exceeding $10^\circ$C. The proposed regional temperature control function is thus effective in directly mitigating peak temperatures, a critical requirement for EV battery pack safety.
Table 2 illustrates the temperature distribution at $t_f=1000$s for designs obtained under different discharge rates ($C_{rate}$).
| Discharge Rate ($C_{rate}$) | Target I: Temperature Contour & Max Temp | Target II: Temperature Contour & Max Temp |
|---|---|---|
| 0.5C | Contour I1, Max: ~6.0°C | Contour II1, Max: ~7.5°C |
| 1C | Contour I2, Max: ~14.9°C | Contour II2, Max: ~29.7°C |
| 2C | Contour I3, Max: ~61.8°C | Contour II3, Max: ~117.5°C |
Under higher discharge rates (1C, 2C), which generate more heat, the advantage of Target I becomes even more pronounced. Its design maintains the maximum temperature within a safer operating range compared to Target II, which exhibits severe local overheating. This demonstrates the robustness of the proposed method across various operating conditions of an EV battery pack.
Further analyses on volume fraction and ambient temperature variations confirm the generality of the approach. For a given allowable temperature, Target I leads to more material-efficient (lighter) designs. Under higher ambient temperatures, which reduce the cooling gradient, Target I’s designs continue to maintain lower and more uniform temperatures compared to Target II, which struggles with thermal management.
In conclusion, this work presents a robust topology optimization framework for the design of heat dissipation structures under transient thermal loading, with direct application to EV battery pack thermal management systems. By introducing a differentiable Regional Temperature Control Function, the method successfully minimizes the maximum temperature in a critical region over a specified operational period. The numerical examples conclusively show that: (1) Transient effects are crucial and yield different optimal designs compared to steady-state assumptions; (2) Minimizing the maximum temperature directly leads to topologies that are superior in preventing hot spots compared to minimizing thermal compliance; (3) The method performs effectively across various discharge rates, volume constraints, and environmental conditions. This approach provides a powerful design tool for developing next-generation, high-performance cooling solutions for electric vehicle battery packs, enhancing their safety, longevity, and efficiency.