Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

Abstract

In this article we propose a scalable shape optimization algorithm which is tailored for large scale problems and geometries represented by hierarchically refined meshes. Weak scalability and grid independent convergence is achieved via a combination of multigrid schemes for the simulation of the PDEs and quasi Newton methods on the optimization side. For this purpose a self-adapting, nonlinear extension operator is proposed within the framework of the method of mappings. This operator is demonstrated to identify critical regions in the reference configuration where geometric singularities have to arise or vanish. Thereby the set of admissible transformations is adapted to the underlying shape optimization situation. The performance of the proposed method is demonstrated for the example of drag minimization of an obstacle within a stationary, incompressible Navier-Stokes flow.

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• Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing
• Shape optimization in $W^{1,\infty }$ with geometric constraints: a study in distributed-memory systems