# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 4 Robust Design Optimization Strategy
### BK-Based Robust Design Optimization  
In [9; 10; 54], an external BK surrogate was constructed to describe variations of the mean and variance of the QoIs in the design space, by using \(n_{init}\) samples chosen according to a preliminary design of experiments (DOE). In those references, a first-level UQ BK using \(N\) samples was used to evaluate the cost functions. In the present work, a similar approach is used, whereby a GEK surrogate or the MoM may replace first-level BK for the UQ step; a flowchart of the RDO process is provided in Figure 2.  
The construction of the second-level surrogate requires \(N\times n_{init}\) deterministic runs of the direct CFD solver in the case of BK and of the direct and adjoint solvers for GEK, which can be parallelized in a straightforward manner. The moderate extra-cost of adjoint solves for GEK is in general counter-balanced by a smaller \(N\) required for achieving a given accuracy. If the MoM is used as the UQ solver, the cost of the initial surrogate is reduced to \(n_{init}\) deterministic runs of the direct (nonlinear) and adjoint CFD solvers. Numerical tests show that a sufficiently accurate BK surrogate can be obtained by setting \(n_{init}=10n_{des}\). Once \(n_{init}\) estimates of the QoI mean and variance have been obtained, a second-level BK is constructed and coupled with the MOGA. In order to control the accuracy of the approximated cost functions, an adaptive infill strategy is adopted to enrich the external BK surrogate during the evolution. For this purpose, we add to the initial new DOE samples selected during the MOGA iteration and retrain the BK model. Several infill criteria are available. For an overview about all these techniques the reader is addressed to [66; 67]. Among them, the expected improvement (EI) criterion [68] has been shown to provide a good trade-off between exploitation and exploration, and it was therefore selected for the RDO carried out in the present work. The new samples are drawn in such a way that the EI of the global minimum is maximized [69; 70]. The EI quantifies the probability to improve the surrogate on the design space parameters. A global optimization of the EI function is then carried out to find the most suitable location for the new sample (see [70] for details). Since RDO is intrinsically a multi-objective optimization process, the EI function is a surface of a hyper-dimensional parameter space, and a more complex formulation is developed, referred-to as the multi objective expected improvement (MOEI) approach (see [70]). The accuracy of the surrogate is rapidly improved with a few MOEI infills, so that \(n_{MOEI}<<n_{gen}\). An MOEI update implies running the UQ algorithm for the additional sample to be integrated to the second-level BK. The \(N\) deterministic CFD and adjoint runs required for the BK and GEK UQ solvers can be carried out in parallel. Finally, the cost of the RDO based on the adaptive infill strategy in terms of cost-function evaluation is given by \(N\times(n_{init}+n_{MOEI})\) direct CFD calculations for BK, \(N\times(n_{init}+n_{MOEI})\) direct and adjoint CFD calculations for GEK, and \((n_{init}+n_{MOEI})\) direct and adjoint calculations for MoM. The turn-around time, in the case of a perfectly-scaling parallel implementation, is approximately equal to \(n_{MOEI}+1\) runs. Based on numerical experiments, an MOEI adaptation every three to five  
Figure 2: Flowchart of the BK-based SMOGA RDO process. MOGA generations is generally sufficient to achieve an accurate approximation of the optimum, as shown in the Results Section. The full BK-based RDO loop is described in the pseudo-code provided hereafter in Algorithm 1. ```
1. Initialization: LHS DOE with \(n_{init}\) samples \(\mathcal{S}_{DOE}=\left\{s^{(1)},s^{(2)},...,s^{(n_{init})}\right\}\)  While \(i<n_{init}\)  Select design \(s^{(i)}\) in \(\mathcal{S}_{DOE}\)  Run UQ solver to compute the statistics \(\mu_{s^{(i)}}\sigma_{s^{(i)}}^{2}\)  Train surrogates \(S_{\mu}(\mathcal{S}_{DOE})\), \(S_{\sigma^{2}}(\mathcal{S}_{DOE})\) 2.