# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 6 Conclusions  
In the present work, we first assessed various gradient-based methods for uncertainty quantifications, in view of their use for the robust design optimization (RDO) of CFD problems. Specifically, the capability of a Bayesian gradient-enhanced kriging surrogate model and a first-order method of moments to accurately and efficiently compute the lower order statistical moments of a quantity of interest was evaluated for an inexpensive test problem, representative of a supersonic divergent nozzle, for which the results can be compared with well-converged Monte Carlo sampling. The results show that GEK allows computational gains of a factor of two or more with respect to a Bayesian kriging surrogate not using gradient information, when the gradients are efficiently computed using an adjoint method. In the present work, both a discrete and a continuous adjoint method were used for building GEK surrogates. The first one provides more accurate results, but it is somewhat more costly and requires intrusive automatic derivation of the CFD code, which is not possible if, for instance, a commercially available CFD solver is to be employed. The continuous adjoint method allows developing a non-intrusive adjoint solver, but it is less accurate due to inconsistencies in the numerical treatment of the direct and adjoint equations. In the present implementation, the continuous adjoint solver benefits from a direct solution of the linear system of adjoint equations, and it is therefore quicker than the discrete adjoint solver, which converges by means of a pseudo-transient technique. The first-order moment method, based on either discrete or continuous adjoint gradients, is less accurate than GEK, but it still provides satisfactory estimates of the QoI for the present shocked flow problem, due to the relatively small variation ranges of the uncertain parameters. The UQ methods are then combined with a genetic algorithm for solving the RDO problem. The computations are sped up by constructing a Bayesian kriging surrogate model of the design space. The surrogate is enriched during GA iterations by means of a multi-objective expected improvement (MOEI) infill criterion. For the test problem at hand, the RDO results are found to be similar for the BK, GEK, and MoM methods, the latter being much less expensive in terms of CPU time, but slightly less accurate than the former ones. In order to benefit from the computational efficiency of the MoM and the accuracy of the GEK UQ solvers at the same time, a multi-fidelity surrogate model is built by fusing together the low-fidelity MoM and the high-fidelity GEK. An MOEI infill is used again to enrich the surrogate during convergence, with preference for low-fidelity infills. The multi-fidelity approach successfully identifies the RDO optimum, while dividing by a factor of \(3\div 4\) the computational cost  
Figure 8: Baseline geometry and optimized geometries (with \(3\sigma\) uncertainty intervals) for various RDO strategies. with respect to GEK. Such an approach is then identified as a promising candidate for more complex RDO problems using CFD models. Further work is however required to assess its effectiveness for more realistic and complex CFD problems. For this aim, its application to the RDO of the stator row of an organic Rankine cycle turbine is underway and will be reported in the near future. In such a context, the introduction of multi-fidelity UQ solvers combining BK and/or GEK based on different numbers of samples could be an interesting future development for further speed up of the RDO procedure. Conceptualization, A.S. and P.C. ; methodology, A.S.; software, A.S.; validation, P.C. and B.O. ; formal analysis, A.S.; investigation, P.C. ; resources, P.C. ; data curation, A.S.; writing, original draft preparation, A.S.; writing, review and editing, P.C. ; visualization, A.S.; supervision, P.C. and B.O. ; project administration, P.C. ; funding acquisition, B.O. All authors read and agreed to the published version of the manuscript.