Shape derivatives for the compressible Navier–Stokes equations in variational form

Shape optimization based on surface gradients and the Hadamard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the “function composition” approach involving local shape derivatives and an alternate methodology based on the weak form of the state equation. The resulting gradient expressions are found to be equal only if the existence of a strong form solution is assumed. Surface shape derivatives based on both formulations are implemented within a Discontinuous Galerkin flow solver of variable order. The gradient expression stemming from the variational approach is found to give superior accuracy when compared to finite differences.