Shape Optimization for Incompressible Fluid Flows With Diffuse Domain Method

Shape optimization plays a central role in scientific and engineering design, where one seeks to improve the performance of a system by altering its geometries subject to the governing partial differential equations (PDEs). This setting poses challenges for traditional numerical methods, as the optimization process may involve grid regeneration on evolving and irregular domains. The diffuse domain method (DDM) provides a flexible alternative by embedding complex geometries into a rectangular background domain using a smooth phase‐field function and then extending the governing PDEs to the new domain. This eliminates the need for re‐meshing and facilitates the implementation of standard finite difference or finite element discretization. In this work, we develop a DDM‐based method for shape optimization for fluid flows governed by the incompressible Navier–Stokes equations. The method incorporates variational formulations of the PDEs, the associated adjoint problem, sensitivity analysis, and gradient‐based optimization strategies within the diffuse domain setting. Numerical experiments show that the DDM approach achieves robust convergence in space. Furthermore, efficient handling of topological changes by the proposed DDM‐based shape optimization method is demonstrated through several benchmark problems. The results highlight its potential as a versatile tool for solving complex shape optimization problems in fluid flows and related applications.