In this paper we present a shape optimization scheme which utilizes the alternating direction method of multipliers (ADMM) to approximate a direction of steepest descent in \(W^{1,\infty }\) . The followed strategy is a combination of the approaches presented in Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022) and Müller et al. SIAM SISC 45 (2023). Here the optimization problem is expanded to include geometric constraints, which are systematically fulfilled. Simulations of a fluid dynamics case study are carried out to benchmark the novel method. Results are given to show that, compared to other methods, the proposed methodology allows for larger deformations without affecting mesh quality and convergence of the used numerical methods. The parallel scalability is tested on a distributed-memory system to illustrate the potential of the proposed techniques in a more complex, industrial setting. The main result is that both approaches are comparable in mesh quality. However, it is demonstrated that an ADMM implementation is possible without careful and time-consuming adjustment of problem and mesh-dependent parameters as in the p-Laplace case.
