This work addresses a topology optimization problem associated with an aircraft fuel tank’s active cooling system. We developed a hierarchical physical model to accurately capture the conjugate heat transfer process while maintaining computational efficiency. As illustrated in Fig. 1, the model cross-section features a typical sandwich-type composite layout.

The outer layer is composed of an advanced thermal insulation material to withstand high external temperatures from the hot outside airflow encountered during flight. Beneath this lies a cast aluminum skin, which provides structural integrity and high thermal conductivity. The attenuated heat flux then conducts through the aluminum skin to reach the inner cooling channel surface. The innermost layer is the fuel distribution layer, which is the design domain for topology optimization in this study (denoted as Ωd). In this layer, the coolant (the fuel) and the solid channel walls form interpenetrating phases. A pressurized flow drives the fuel through the embedded channel network, absorbing the heat conducted from the outer skin via convection at the fluid-solid interfaces, thereby regulating the temperature and maintaining the overall thermal balance of the structure.

Solving 3D CHT problems with topology optimization presents a fundamental trade-off between accuracy and computational cost. To significantly reduce the computational cost while preserving solution accuracy, a “pseudo-3D” modeling strategy is employed. This approach comprises two key aspects. First, the flow field is simplified to two dimensions (2D) by assuming the coolant flow is predominantly in-plane, neglecting the through-thickness velocity component. This allows the flow equations to be solved in a 2D domain. Second, the thermal field is treated in a pseudo-3D manner. Although the geometry is reduced to 2D, heat transfer in the thickness direction is retained. The external heat flux passes through the insulation and aluminum skin before entering the channel layer. By incorporating equivalent source and conduction terms into the energy equation, the model captures this through-thickness heat flow. This maintains critical 3D thermal effects, such as wall-to-fluid conduction, rather than simplifying the thermal load to a purely in-plane 2D source.

Following these assumptions, the design domain Ωd is defined as a 2D region comprising two design-variable-dependent subdomains: the fluid domain Ωf and the solid domain Ωs. Its boundary consists of the inlet Γin, the outlet Γout, and the solid walls Γwall, as shown schematically in Fig. 2.

Conventional fluid topology optimization typically employs the Navier-Stokes (N-S) equations. For steady, incompressible, Newtonian flow, the momentum conservation equation is given by:(1)ρ(u⋅∇)u=−∇p+μ∇2u+fwhere ρ is the fluid density, u the velocity field, p the pressure field, μ the dynamic viscosity, and f an external body force. The convective term (u⋅∇)u introduces nonlinearity. However, in topology optimization, the flow channel layout undergoes significant topological changes, including channel disconnection, merging, and formation. Concurrently, the velocity approaches zero in solid or low-permeability regions, often leading to numerical singularities and spurious oscillations.

To address these numerical challenges, the Darcy flow model from porous media theory is adopted to approximate the macroscopic flow behavior. This model provides a numerically stable and simplified framework for representing fluid flow in topology optimization. It establishes a linear relationship where the fluid velocity within the porous medium is proportional to the pressure gradient and inversely proportional to the dynamic viscosity:(2)uf=−κμ∇pwhere uf is the fluid velocity and κ is the permeability of the medium. While permeability is generally a second-order tensor, it reduces to a scalar under the assumption of isotropy.

It is important to emphasize that the Darcy model is not intended to directly resolve the underlying turbulent flow physics in high-Reynolds-number regimes. Instead, it is employed here as an effective macroscopic resistance model that establishes a simplified relationship between pressure gradient and flow rate. In this context, the permeability parameter does not necessarily correspond to a physical porous medium, but rather serves as an equivalent parameter to represent the overall flow resistance. The nonlinear inertial effects inherent in turbulent flow are implicitly incorporated into this parameter through calibration against high-fidelity simulations, as detailed in Section 3. In this way, the proposed approach avoids directly solving the turbulent governing equations while still capturing their macroscopic influence. It bridges the gap between the linear Darcy model and high-Reynolds-number turbulent flows.

For topology optimization, the permeability κ is the key parameter to implicitly distinguish between fluid and solid regions. In regions defined as flow channels, κ is assigned a very large value (κ→∞), which greatly reduces flow resistance and approximates unobstructed fluid motion. Conversely, in solid regions, κ is set to approach zero (κ→0), rendering the flow resistance effectively infinite and thereby forcing the fluid velocity uf to near zero. This method uses a spatially varying permeability field κ(x) to implicitly define the fluid-solid interface, where x denotes the position vector. It avoids the computationally expensive and algorithmically complex explicit interface tracking required in traditional moving boundary formulations.

For an incompressible fluid, mass conservation is expressed by the continuity equation:(3)∇⋅uf=0

Substituting the Darcy model from Eq. (2) into Eq. (3) yields a governing equation for the pressure field:(4)∇⋅(−κμ∇p)=0

Eq. (4) is a second-order linear elliptic partial differential equation. Compared to the N-S equations, which include nonlinear convective terms, this formulation possesses significantly more favorable numerical properties. First, for any κ>0, the equation stays positive definite, guaranteeing stable convergence even with distorted meshes or poor initial guesses. Moreover, upon discretization, it yields a symmetric positive-definite linear system that can be solved efficiently using standard iterative or direct solvers.

Well-posed boundary conditions are imposed based on the physical conditions. At the inlet, a normal velocity (Neumann) condition is specified:(5)n⋅uf=uin,x∈Γinwhere uin is the prescribed average inlet velocity.

A zero-pressure (Dirichlet) condition is set at the outlet:(6)p(x)=0,x∈Γout

This defines the pressure field relative to the outlet, simplifying the subsequent calculation of pressure drop.

All remaining boundaries are treated as impermeable walls. The no-penetration condition is enforced by setting the normal velocity to zero, corresponding to a homogeneous Neumann condition:(7)n⋅uf=0⟹n⋅(−κμ∇p)=0,x∈Γwall

The heat transfer within the design domain is a CHT process. The steady-state energy conservation equation for the fluid is given by:(8)ρcpuf⋅∇Tf−∇⋅(k∇Tf)=Qwhere cp is the specific heat capacity at constant pressure, k is the thermal conductivity, Tf is the fluid temperature, and Q is the volumetric heat source representing heat flux from the channel walls.

Substituting the Darcy velocity field from Eq. (2) into Eq. (8) yields the explicit governing equation for the coupled fluid-thermal problem:(9)ρcp(−κμ∇p)⋅∇Tf−∇⋅(k∇Tf)=Q

In the pseudo-3D model, the volumetric heat source term Q maps the actual 3D thermal load onto the 2D domain. Specifically, it provides an equivalent representation of the surface heat flux applied to the external boundaries of the structure. In practical aerodynamic heating scenarios, thermal loads are imposed as surface heat fluxes on the windward and leeward surfaces. To incorporate this effect into the reduced-order formulation, this boundary heat flux is mathematically transformed into an equivalent internal volumetric heat source. By integrating the heat transfer process along the thickness direction, the surface heat flux is converted into an equivalent volumetric heat source according to Q=Qf/t or Q=Qb/t, where Qf and Qb are the prescribed surface heat fluxes on the windward and leeward surfaces, respectively, and t is the effective thickness of the layer. This approximation implies a uniform distribution of heat across the thickness, which is consistent with the pseudo-3D modeling framework. While this approach neglects detailed boundary-layer effects near the wall, it preserves the global thermal behavior.

Moreover, aerodynamic heating on the fuel tank surface is highly non-uniform in actual flight conditions. For instance, the windward side experiences significantly higher heat flux than the leeward side. To capture this spatial non-uniformity in the optimization, Q is defined as a spatially varying function Q(x). This spatially varying heat load drives the optimization toward generating channel structures with asymmetric or locally intensified features, thereby concentrating thermal transport where it is most needed.

The following thermal boundary conditions are applied. At the fluid inlet, the coolant enters at a prescribed constant temperature Tin, giving the Dirichlet condition:(10)Tf(x)=Tin,x∈Γin

The lateral walls and the outlet are assumed to be adiabatic, resulting in a homogeneous Neumann condition:(11)n⋅∇Tf=0,x∈Γwall∪Γout

In this work, the material layout within the design domain Ωd is optimized via topology optimization. A continuous design variable field γ(x)∈ [0, 1] represents the material distribution, where γ= 0 corresponds to pure fluid, γ= 1 to pure solid, and intermediate values 0 <γ<1 represent intermediate states.

A mapping must be defined between the material properties and the design variable γ. Scalar properties such as thermal conductivity, density, and specific heat capacity are interpolated linearly:(12)k(γ)=kf+γ(ks−kf) (13)ρ(γ)=ρf+γ(ρs−ρf) (14)cp(γ)=cpf+γ(cps−cpf)where the subscripts f and s denote the fluid and solid properties, respectively.

However, a strongly penalized interpolation scheme is required for the permeability κ to drive the design toward a clear solid/fluid distribution. This penalization encourages binary designs, i.e., clear distinctions between flow channels (κ→∞) and solid regions (κ→0). A convex penalization function is adopted:(15)κ(γ)=κf+γpκ(κs−κf)with a penalization factor pκ=3, which significantly penalizes the permeability of intermediate material. Theoretically, κs should be zero for solid regions. However, this would make the system matrix singular, causing numerical instability. In practice, a small positive value is used for κs to maintain numerical robustness while effectively blocking flow.

The sensitivity filtering technique based on partial differential equation is employed. The filtered design variable γ~ is obtained by solving the following Helmholtz-type partial differential equation:(16)−Rmin2∇2γ~+γ~=γ, in Ωdwhere Rmin represents the filter radius, which effectively defines the minimum feature size of the design. This parameter implicitly enforces a minimum length scale in the design, effectively suppressing checkerboard patterns and imposing a lower bound on channel width. However, filtering typically produces gray, blurred interfaces. To recover a crisp binary distribution, a hyperbolic tangent projection is applied to γ~:(17)γ¯=tanh⁡(β(γ~−η))+tanh⁡(βη)tanh⁡(β(1−η))+tanh⁡(βη)where β controls the projection slope and η is the threshold. In the present study, β= 2 and η= 0.5 are employed. This projection sharpens the intermediate gray regions, yielding a manufacturable fluid-solid interface.

The filtering and projection steps not only improve numerical stability but also play a crucial role in controlling the geometric characteristics of the optimized topology. Specifically, the filter introduces a minimum length scale, which suppresses variations in the design field and prevents the formation of checkerboard patterns and disconnected microstructures. It can be interpreted as an approximate lower bound for the channel width, which is critical for both numerical stability and manufacturability. The projection operation further promotes the discreteness of the material distribution by enforcing a clear distinction between fluid and solid regions. These combined effects are clearly reflected in the optimized results, which are characterized by smooth boundaries, clear connectivity, and physically consistent flow pathways.

The objective is to design a flow channel layout that maximizes heat dissipation efficiency. The problem is mathematically formulated as minimizing the average temperature subject to multiple constraints:(18)F⁡i⁢n⁢d:γi∈[0,1],i=1,2,…,NM⁢i⁢n⁢i⁢m⁢i⁢z⁢e:Φ=∫ΩdT⁡(γ)dΩΩdS⁢u⁢b⁢j⁢e⁢c⁡t⁢t⁢o:{Governing Eqns:{∇⋅uf=0uf=−κ⁡(γ)μ∇pρ⁡(γ)⁢cp⁡(γ)⁢uf⋅∇T−∇⋅(k⁡(γ)∇T)=QVolume Constraint:∫Ωd(1−γ)⁢d⁢Ω≤Vf⋅ΩdPressure Drop:h1=1|Γi⁢n|⁢∫Γi⁢np⁢d⁢Γ−h1∗≤0Temp. Limits:h2=Tmax−h2∗≤0Temp. Uniformity:h3=(Tmax−Tmin)−h3∗≤0

Here, the objective function Φ represents the spatially averaged temperature across the entire design domain Ωd. The optimization is subject to several constraints. First, a volume constraint limits the material usage, where Vf denotes the maximum allowable volume fraction. Second, the pressure drop constraint h1 requires that the average pressure at the inlet boundary Γin does not exceed the specified limit h1∗. Regarding thermal performance, Tmax and Tmin represent the maximum and minimum temperatures within the domain, respectively. The maximum temperature constraint h2 ensures that Tmax does not exceed the critical threshold h2∗ to avoid boiling or thermal decomposition of the coolant. Finally, the temperature uniformity constraint h3 restricts the global temperature difference (Tmax−Tmin) to a prescribed bound h3∗, thereby mitigating thermal stress and maintaining structural safety.

The topology optimization problem defined in Eq. (18) is solved using the Method of Moving Asymptotes. The optimization process is terminated when the relative change in the objective function between successive iterations falls below 10⁻³ or when the maximum number of iterations (e.g., 300) is reached. The sensitivities of the objective function and constraints with respect to the design variables are computed via the adjoint method. The filtering and projection techniques are incorporated into the sensitivity analysis through the chain rule to ensure consistency between design variables and physical properties.

All optimization cases in this section employ the material properties listed in Table 1. The design domain is a 2D rectangular region of size 1.0 m × 1.25 m. Inlet velocity is fixed at uin= 0.35 m/s, and outlet pressure is set to Pout= 0 Pa. All other boundaries (excluding inlets and outlets) are modeled as adiabatic no-slip walls, consistent with the physical conditions.

The non-uniform heat sources are applied to simulate the concentrated heating encountered in aircraft thermal protection systems. The domain is divided vertically into high- and low-heat-flux regions. The central region, y∈ [0.3 L, 0.7 L] is given a high heat flux of Qhigh= 4700 W/m². The side regions, y∈ [0, 0.3 L) and y∈ (0.7 L, L], receive a lower heat flux of Qlow= 3300 W/m². This step-wise heat source provides a simple representation of the concentrated heating often seen in aerospace thermal protection systems. Moreover, in the present study, the optimization formulation uses the following parameters: volume fraction Vf= 0.6, maximum temperature h2∗= 373.15 K, temperature uniformity h3∗= 40 K, and pressure drop limits h1∗= 0.01 MPa (double-inlet) or 0.02 MPa (single-inlet).

First, we consider a SISO optimization case. Fluid enters through a centered inlet on the left boundary and exits through a symmetrically placed outlet on the right boundary. The design domain and boundary conditions are shown in Fig. 7.

Fig. 8 shows the optimized flow channel topology for a SISO symmetric structure under a non-uniform heat source, along with the corresponding pressure, temperature, and velocity fields. The optimized design exhibits a typical hierarchical branching structure (Fig. 8a). Near the inlet, a wide channel maintains a large fluid-filled region to promote bulk heat removal. The flow subsequently splits into multiple finer branches, ensuring efficient heat transport. Within the high-heat-flux region (y∈ [0.3L, 0.7L]), the channels are more concentrated than in the adjacent low-flux zones, creating an increased solid-fluid contact area for heat transfer. This configuration shortens conduction paths and fully utilizes the coolant’s cooling capacity. Toward the outlet, the branches gradually merge into wider collecting channels. The pressure, temperature, and velocity fields in Fig. 8 demonstrate that this channel design effectively enhances heat transfer.

To quantitatively characterize the optimized channel design, several geometric metrics are introduced. The fluid volume fraction represents the proportion of the flow region within the design domain. The wetted perimeter is defined as the total length of the fluid–solid interface. The number of effective channels is estimated based on the connectivity of the flow paths in the optimized topology. These metrics provide a quantitative basis for comparing different designs beyond qualitative descriptions.

The optimized design exhibits a fluid volume fraction of 58%. The number of effective flow channels increases to 14, and the wetted perimeter is increased by 28% compared to the straight channel. This increase in channel number and wetted perimeter indicates that the optimization process promotes the formation of distributed branching structures, particularly in regions with high thermal loads. Moreover, similar quantitative trends are observed in other configurations. As a result, the coolant can more effectively access critical regions, thereby improving temperature uniformity and mitigating thermal hotspots.

This example examines a configuration with a single centered inlet on the left side and dual outlets on the upper and lower sides, as shown in Fig. 9. The two outlets are placed asymmetrically about the domain centerline. The upper outlet is located nearer the outer boundary, whereas the lower outlet is positioned closer to the centerline. This configuration presents a more complex scenario where the flow must distribute into multiple downstream branches under inherently unbalanced conditions.

Fig. 10 shows the topology optimization results for the SIDO asymmetric flow channel under non-uniform heat source conditions, along with the pressure, temperature, and velocity field distributions. Near the inlet, the channel structure covers a wide region, enhancing heat dissipation. Here, the fluid temperature stays low and can absorb heat efficiently. Further inside, the flow splits into multiple fine channels to ensure uniform flow distribution and effective heat dissipation. During this process, the fluid is heated by the heat source and its temperature gradually rises. Near the outlet, the channels merge again and the fluid temperature reaches its peak. Moreover, channels are more densely distributed in the middle region with higher heat flux. This design effectively enhances heat dissipation. This design promotes rapid heat transfer to the fluid, effectively enhancing dissipation and preventing local overheating. This demonstrates that the optimization algorithm prioritizes local cooling efficiency by maximizing the solid-fluid contact area in critical regions, while simultaneously adapting the global flow distribution to the imposed outlet constraints.

The DISO configuration is widely adopted in engineering to improve flow and temperature uniformity, particularly for wider cooling structures. Here, fluid enters from inlets on both sides and converges at a centered outlet on the right boundary. Fig. 11 shows the topology optimization results for the symmetric DISO structure under non-uniform heat source conditions, together with the distributions of pressure, temperature, and velocity fields. Compared to the single-inlet design, the dual-inlet layout allows coolant to enter from both sides simultaneously, ensuring coolant coverage across the full width of the structure and promoting more uniform and rapid convergence toward the central high-heat region. Within this region, a dense channel network maximizes the solid-fluid contact area, accelerating heat transfer. Near the outlet, the branching channels merge into thicker collectors.

To examine the influence of inlet placement, we optimize the structures by varying the inlet location relative to the top boundary. Three cases are examined, with the distance l1 from the inlet to the top boundary set to 0.1, 0.25, and 0.4 m, respectively. The corresponding model configurations and heat source distributions are shown in Fig. 12.

The optimized topologies and field distributions for different inlet positions are shown in Fig. 13. Table 5 compares the pressure drop, maximum temperature, and maximum flow velocity across the four inlet positions examined. When the inlet is close to the side boundary, the main channel forms near that boundary, forcing the fluid to travel along the edge before turning toward the central hot spot. This increases the flow path length and, consequently, the pressure drop. As the inlet is moved inward, the main channel aligns more directly toward the central high-heat-flux region, yielding a more direct flow path that reduces both flow length and bending losses. Despite these variations in inlet position, all optimized designs maintain a high channel concentration in the high-heat-flux region. This demonstrates that the optimization algorithm is more sensitive to the heat source distribution than to the inlet location, with the primary objective remaining the effective mitigation of hot spots.

The DIDO configuration presents the most challenging fluid-thermal topology optimization problem in this study. It involves both multiple inputs and multiple outputs. This offers greater design flexibility but also demands that the optimization algorithm balance flow distribution among multiple branches. The design domain is shown in Fig. 14, where the upper outlet is positioned nearer the outer boundary, while the lower outlet lies closer to the domain centerline.

Fig. 15 presents the topology optimization results for the DIDO asymmetric structure under non-uniform heat source conditions, together with the corresponding distributions of pressure, temperature, and velocity fields. The optimized design exhibits a complex network that is highly interconnected and hierarchical branching. As the fluid enters from the two inlets, it begins to split immediately. Some of the flow goes straight to the outlets, while the rest converges in the high-heat-flux region. After completing the main heat exchange there, it then splits and exits through the outlets. The channel layout consists of multiple parallel flow paths. This arrangement significantly improves the system’s cooling capacity by balancing flow distribution and maximizing heat transfer area.

Although the pseudo-3D Darcy model with parameter calibration captures macroscopic behavior, it has inherent limitations. To validate the reliability and performance benefits of the optimized designs for real engineering applications, we perform a full 3D reconstruction and simulation.