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Section: New Results

Shape optimization under uncertainties

Non-Parametric Measure Approximations for Constrained Multi-Objective Optimisation under Uncertainty

M. Rivier, P.M. Congedo

In this paper, we propose non-parametric estimations of robustness and reliability measures to solve constrained multi-objective optimisation under uncertainty. These approximations with tunable fidelity permit to capture the Pareto front in a parsimonious way, and can be exploited within an adaptive refinement strategy. First, we build a non-Gaussian surrogate model of the objectives and constraints, allowing for more representativeness and detecting potential correlations. Additionally, we illustrate an efficient approach for obtaining joint representations of these robustness and reliability measures, which discriminates more sharply the Pareto-optimal designs from the others. Secondly, we propose an adaptive refinement strategy, using these tunable fidelity approximations to drive the computational effort towards the computation of the optimal area. To this extent, an adapted Pareto dominance rule and Pareto optimal probability computation are formulated. We assess the performance of the proposed strategy on several analytical test-cases against classical approaches in terms of the average distance to the Pareto front. Finally, we illustrate the performances of the method on the case of shape optimisation under uncertainty of an Organic Rankine Cycle turbine [26].

Optimization under Uncertainty of Large Dimensional Problems usingQuantile Bayesian Regression

C. Sabater, O. Le Maitre, P.M. Congedo, S. Goertz

When dealing with robust optimization problems, surrogate models are traditionally constructed to efficiently obtain the statistics of the random variable. However, when a large number of uncertainties are present, the required number of training samples to construct an accurate surrogate generally increases exponentially. The use of surrogates also requires the parametrization of the uncertainties. We present a novel approach for robust design insensitive to the number of uncertainties and that is able to deal with non-parametric uncertainties by leveraging a Bayesian formulation of quantile regression. The method does not require the use of any surrogate in the stochastic space. It is able to globally predict at every design point any given quantile of the random variable. In addition, it provides an estimation of the error in this prediction due to the limited sampling by making use of the posterior distribution of the model parameters.The framework includes an active infill that efficiently balances exploration with exploitation and accelerates the optimization process by increasing the accuracy of the statistic in the regions of interest. We validate the method in test functions and observe good convergence properties towards the determination of the location of the global optima. The framework is applied towards the aerodynamic robust design of an airfoil with a shock control bump under 382 geometrical and operational uncertainties. The framework is able to efficiently find the optimum configuration in complex, large-scale problems. An article on this topic is under preparation.