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We present a level set based adaptive mesh method for solving the drag minimization problem of incompressible flow governed by Stokes equations. Shape sensitivity analysis of the cost functional is derived. During the optimization process, two levels of meshes are adopted. A uniform coarse mesh for evolving the level set function is defined over the whole computational domain. The level set function also serves as the refinement indicator with an easy strategy. The coarse mesh that contains the interfaces then further divided into a uniform fine mesh and the computation is mainly near the interfaces. Therefore, the computational cost is significantly reduced compared with the uniform mesh over the whole domain that achieves the same resolution. Above all, the value of the shape derivative on the boundary can be obtained implicitly, which is a very difficult task in classical optimal shape design problems. The overall optimization procedure is presented and verified with benchmark examples.

Adaptive level set method applied to drag minimization problems constrained by Stokes equations