Multiphysics Topology Optimization — Simultaneous Optimization of Structure, Heat, and Fluid Using the SIMP Method

Of course it can. Optimization that simultaneously considers structure + heat + fluid is possible. Roughly speaking, it's a mathematical method to find out "where to place material so that the structure doesn't break and heat is properly dissipated."

Good question. A representative example is the shape optimization of a heat sink. It automatically generates fin layouts that maximize cooling performance while maintaining structural strength using the SIMP method. When humans design intuitively, they tend to create "straight fins with equal spacing," but optimization yields complex branching shapes that are not intuitively obvious.

Like the shape of a tree branch. Thick branches come out from the trunk and branch out finer towards the tips. This makes sense when considering heat flow and matches the optimal structure of "constructal theory," which efficiently collects heat from the source and transports it to the heat dissipation surface. In an actual industrial example, GE Aviation used this for a jet engine fuel nozzle, consolidating 20 parts into 1 part and reducing weight by 25%.

That's precisely where the compatibility with metal 3D printing (Additive Manufacturing, AM) comes in. Complex internal flow paths that cannot be made by conventional machining can be integrally formed with AM. That's why topology optimization and AM are considered the "ultimate combination," and their practical application has advanced rapidly over the last 10 years.

First, as a review, in the usual SIMP method, the density of each element $\rho_e \in [0, 1]$ is the design variable, and Young's modulus is interpolated as $E_e = \rho_e^p E_0$ ($p$ is the penalty exponent, usually 3). When extending to multiphysics, the material properties for each physical field are penalized simultaneously:

Here, $E_0$ is Young's modulus, $k_0$ is thermal conductivity, $\kappa_0$ is permeability. The penalty exponents $p_s, p_t, p_f$ may take different values for each physical field. Then it is formulated as a multi-objective optimization problem:

$C_s = \mathbf{f}_s^T \mathbf{u}$ is the structural compliance (softness of the structure); minimizing this maximizes stiffness. $C_t = \mathbf{q}_t^T \mathbf{T}$ is the thermal compliance; minimizing this makes the temperature distribution more uniform. $\Phi_f$ is the fluid pressure loss. $w_s, w_t, w_f$ are weighting coefficients; changing these determines which point on the Pareto optimal front is selected.

Roughly speaking, it means "let's lower the overall temperature by cleverly placing material." For example, imagine placing a heat sink on top of a CPU. It optimizes "where and how much" to place material with high thermal conductivity from the heat generation surface to the heat dissipation surface.

This equation can be read as "the larger the temperature gradient, the worse it is." Minimizing thermal compliance eliminates hot spots and homogenizes the temperature distribution. In practice, this is a very commonly used method in cooling design for power electronics.

Yes, that's the difficult part. To dissipate heat efficiently, you want to place material thinly and widely, but to increase structural strength, you want material thick and concentrated. This conflicting trade-off is adjusted with the weights $w_s, w_t$. Running optimization multiple times with different weights yields the Pareto optimal front.

Solve the simultaneous equations using direct methods (LU decomposition, Cholesky decomposition) or iterative methods (CG method, GMRES method). For large-scale problems, preconditioned iterative methods are effective.

Yes, it can. Using a method proposed by Borrvall et al. in 2003, a Darcy term is used to make low-density regions ($\rho \to 0$) "impermeable." A virtual resistance term is added to the Navier-Stokes equations:

The last term $\alpha(\rho)\mathbf{v}$ is the Darcy resistance term. In regions with density $\rho=1$, $\alpha \to 0$ (easy to flow); in regions with $\rho=0$, $\alpha \to \infty$ (no flow = solid wall). This automatically designs "where to place walls and where to make flow channels." It is very effective for designing cooling channels in liquid-cooled heat sinks, automatically generating shapes like serpentine flow paths or porous-like structures.

Serpentine flow paths designed by humans are often "equally spaced, equal width," but optimization yields denser flow paths in areas with higher heat generation and sparser ones in areas with lower heat generation. Furthermore, branching and merging are included, resulting in irregular yet rational shapes that minimize pressure loss while maximizing cooling efficiency. Tesla is said to use a method similar to this for cooling power units in electric vehicles.

Founded in Sweden in 1986. Started as FEMLAB with MATLAB integration, later renamed COMSOL. Strong in multiphysics.

Developed in 1970 by Swanson Analysis Systems Inc. (SASI). Based on APDL (Ansys Parametric Design Language).