PhDIn the context of gradient-based numerical optimisation, the adjoint method is an e cient
way of computing the gradient of the cost function at a computational cost independent
of the number of design parameters, which makes it a captivating option for industrial
CFD applications involving costly primal solves. The method is however a ected by
instabilities, some of which are inherited from the primal solver, notably if the latter does
not fully converge. The present work is an attempt at curbing primal solver limitations
with the goal of indirectly alleviating adjoint robustness issues.
To that end, a novel discretisation scheme for the steady-state incompressible Navier-
Stokes problem is proposed: Mixed Hybrid Finite Volumes (MHFV). The scheme draws
inspiration from the family of Mimetic Finite Di erences and Mixed Virtual Elements
strategies, rid of some limitations and numerical artefacts typical of classical Finite Volumes
which may hinder convergence properties. Derivation of MHFV operators is illustrated
and each scheme is validated via manufactured solutions: rst for pure anisotropic
di usion problems, then convection-di usion-reaction and nally Navier-Stokes. Traditional
and novel Navier-Stokes solution algorithms are also investigated, adapted to
MHFV and compared in terms of performance.
The attention is then turned to the discrete adjoint Navier-Stokes system, which is assembled
in an automated way following the principles of Equational Di erentiation, i.e. the
di erentiation of the primal discrete equations themselves rather than the algorithm
used to solve them. Practical/computational aspects of the assembly are discussed, then
the adjoint gradient is validated and a few solution algorithms for the MHFV adjoint
Navier-Stokes are proposed and tested. Finally, two examples of full shape optimisation
procedures on internal ow test cases (S-bend and U-bend) are reported.European Union's Seventh Framework Programme
grant agreement number 317006
