Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics

Round 1

Reviewer 1 Report

The paper proposes and compares several strategies to reduce the computational time for robust design optimization of problems described by costly computer models, such as those encountered in Computational Fluid Dynamics. The proposed methodologies are applied to the inverse robust design of an expansion nozzle, which is a relatively simple problem but suitable to probe the potentialities of the proposed approaches.
The paper is clear and well presented and the subject is of interest to the scientific community. Some minor revisions are recommended.

row 106: "Exemples" should be "Examples"
row 140, row 142, Eq.1: J should be bold
row 175: "outlet static pressure PT,in= 0.6 bar" should be "outlet static pressure Pout= 0.6 bar"
Eq.4 gives the steady-state Euler equations for quasi-1D flows, but at row 185 a four-stage explicit Runge-Kutta time-stepping is mentioned: is the flow steady or unsteady? Can you explain?
row 233: J should be bold
row 264: \csi should be bold
Fig.2: Why S1(\mu_s^(i))? As it is written, it seems the response surface is computed using only i-th observation. Similarly for S2. When MOEI infill is yes, how many new points are computed? From the diagram it seems n_init, but from the text one. Is that correct? Change the figure according to the answer.
Algorithm 1: same considerations on S1 and S2.
Algorithm 2: similar considerations on S_LF,1 , S_LF,2 , S_MF,1 , S_MF,2is . The superscript (i) for s within the parentheses of response surface is misleading.
Table 3: in the column time in the first row you use a different notation, use the same notation (3.0E+06).
Figure 6: explain the colors even if they are intuitable.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The present paper presents an extension/modification of a nested approach for robust optimization of nozzles. The modification consists in using a multifidelity surrogate that combines a gradient enhanced surrogate and the method of moments for the uncertainty quantification part.

General comments

• The article is well written. The English is correct but can be improved.
• The introduction is too dense. The authors should state clearly the novelty of the paper and position it versus the existing literature.
• They should be more specific about the proposed methodology : they should state that it is an extension or a modification of a previous methodology [8,9,33] and stress out its novelty.
• The authors should consider adding to their bibliography the following references
• Nassim Razaaly, Giulio Gori, Giacomo Persico, Pietro Congedo. Quantile-based robust optimization of a supersonic nozzle for Organic Rankine Cycle turbines. [Research Report] RR-9293, Inria Saclay. 2020.
• Schulz V., Schillings C. (2013) Optimal Aerodynamic Design under Uncertainty. In: Eisfeld B., Barnewitz H., Fritz W., Thiele F. (eds) Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36185-2_13
• Dimitrios I. Papadimitriou, Costas Papadimitriou, Aerodynamic shape optimization for minimum robust drag and lift reliability constraint, Aerospace Science and Technology,Volume 55,2016,Pages 24-33, ISSN 1270-9638, https://doi.org/10.1016/j.ast.2016.05.005.(http://www.sciencedirect.com/science/article/pii/S1270963816301754)

• Sebastian Rojas-Gonzalez, Inneke Van Nieuwenhuyse, A survey on kriging-based infill algorithms for multiobjective simulation optimization, Computers & Operations Research, Volume 116, 2020,104869, ISSN 0305-0548, https://doi.org/10.1016/j.cor.2019.104869.
• Korondi, P.Z., Marchi, M., Parussini, L. et al.Multi-fidelity design optimisation strategy under uncertainty with limited computational budget. Optim Eng (2020). https://doi.org/10.1007/s11081-020-09510-1

Specific comments

• Line 48 « Nevertheless, its accuracy is limited to Gaussian or weakly non-Gaussian processes with small uncertainties ». The authors should add a reference clarifying why the method of moments or Taylor Expansion Method is only valide for Gaussian or weakly non-Gaussian random variables.
• Lines 96-98 The authors should clarify why a GEK based MOGA is not straightforward in the context of RDO.
• In their comparison of UQ methods the authors should consider adding the multifidelity solver to show how it performs compared to BK and GEK.
• Total Sobol indexes should be computed to assess the impact of each variable on the output.
• For the infill criterion, the authors should comment on the choice of the MOEI as infill criterion as many others exist.
• The authors should compare their approach with at least one multifidelity method for robust optimization to show the efficiency of their method compared to the state of the art.
• In their presentation of their approach, the authors do not specify if the UQ solver is improved too during the optimization process and if not what are the main reasons for this choice.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The English of the article has been improved and many previously raised issues have been answered. However, there are still two remaining points I am not convinced about.

• The authors claim that they are using the adjoint formulation just for the UQ part, meaning that they compute the gradient of the objective function only versus the vector of random variables treating them as parameters. In Appendix A.1 they present the discrete adjoint approach that has been implemented using an AD technique. What prevents the authors to compute the gradient versus the design variables too? In such a way a GEK-based MOGA can be possible. The authors should justify this point more clearly.
• The authors build a multifidelity solver for the UQ step as explained in §4.2. This multifidelity solver fuses a GEK surrogate and the Method of Moments. So, it gives the statistical moments of the objective function. What prevents the authors to compare this multifidelity UQ solver with BK and GEK in Table 3 for example as they did in Table 4 for the optimal solution.
•  

Author Response

Please see the attachment.

Author Response File: Author Response.pdf