  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 M\"uller et al. SIAM SISC 45 (2023). This has appeared previously
for relatively simple elliptic PDEs with geometric constraints which were
handled using an ad-hoc projection. Here, however, the optimization problem is
expanded to include geometric constraints, which are systematically fulfilled.
Moreover, this results in a nonlinear system of equations, which is challenging
from a computational perspective. 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 the convergence of the used numerical methods.
The mesh quality is studied across the surface of the optimized obstacle, and
is further compared to previous approaches which used descents in
$W^{p,\infty}$. The parallel scalability is tested on a distributed-memory
system to illustrate the potential of the proposed techniques in a more
complex, industrial setting.
