Scaling strategies for Brinkman penalization in fluid topology optimization

Topology optimization provides a rigorous framework for determining optimal flow-path designs in fluid mechanics. A popular technique is the density-based approach, which treats fluid–solid interfaces by considering the solid phase as a porous medium with low permeability. Mathematically, such a problem is governed by the Navier–Stokes equations combined with the Brinkman penalization. To consistently solve the optimization problem, this study examines the theoretical foundations of the method, focusing on dimensionless parameters such as the Reynolds number and a specific Darcy number, Da*. Through dimensional analysis, we derive scaling guidelines for the Brinkman penalization in relation to fluid properties and domain geometry. Numerical simulations show that incorrect scaling can yield nonphysical results, including excessive fluid penetration into solid regions, undermining the optimization. Our findings demonstrate that the invariance of the solution can be maintained by appropriately scaling Da* with the relative velocity in the porous region, ensuring accurate and reliable results in different scenarios. This work provides a systematic framework for parameter selection in fluid topology optimization, addressing key modeling and computational challenges. By emphasizing the importance of dimensional analysis, it contributes to a broader understanding of topology optimization, paving the way for its more robust and consistent application in fluid mechanics.