A Virtual Element Method for Topology Optimization Problem in Fluid Dynamics

This paper introduces a topology optimization framework for steady incompressible Stokes flow based on the non-conforming Virtual Element Method, VEM. The proposed framework combines the geometric flexibility of VEM with an optimality criteria update scheme to minimize viscous and Darcy dissipation under a prescribed volume constraint. The method is applied to the Stokes-flow pipe bend benchmark with parabolic inlet velocity, no-slip wall, and prescribed outlet velocity boundary conditions. By allowing general polygonal elements, including concave and semi-structured polygonal meshes, the method alleviates mesh-related restrictions commonly encountered in conventional finite element discretizations. The methodology is demonstrated through Stokes-flow benchmark problems on different polygonal meshes. The numerical results show that the proposed VEM-based formulation can obtain stable and mesh-insensitive optimized flow channels for Stokes-flow topology optimization. This work offers a systematic approach to obtaining accurate, efficient, and mesh-independent optimal designs for complex fluid systems, providing a stable numerical tool for low-energy-consumption flow channel design in microfluidics, heat exchangers, and biomedical engineering. Extensions to Navier–Stokes and non-Newtonian flow models are left for future work. It should be clarified that the proposed method is only validated for steady Stokes flow and has not been validated for complex fluid models including unsteady Navier–Stokes and non-Newtonian flow models; extensions to these complex models are left for future work.