Density-based hole seeding in XFEM level-set topology optimization of fluid problems

The optimization results of level-set methods typically suffer from a strong dependence on the initial design. To mitigate this dependence, this work presents a density-based hole nucleation method for level-set topology optimization of flow problems. This is achieved by defining both the level-set and density values as functions of the design variables. To preserve the crispness of the geometry definition afforded by the level-set method, the fluid flow is modeled using the Heaviside enriched eXtended Finite Element Method (XFEM) for the laminar incompressible Navier-Stokes equation, which is augmented by a Brinkman model. The boundary conditions are enforced weakly using Nitsche's method. A face-oriented ghost stabilization scheme is applied to stabilize the XFEM formulation. Additional terms ensuring stability are added to Nitsche's method and the ghost stabilization to account for the Brinkman term in the Navier-Stokes equation. The necessity of adding these terms is highlighted by numerical studies. Two- and three-dimensional fluid manifolds are optimized to minimize fluid power dissipation while achieving a predefined mass flow distribution among the outlets. The optimization results show that the proposed method bypasses the need for an initial hole seeding and speeds up the convergence of the optimization process.