Cooperation and competition are natural laws that regulate the interactions between agents in numerous physical, or social phenomena. By analogy, we transpose these laws to devise efficient multi-objective algorithms applied to shape optimization problems involving two or more disciplines. Two efficient strategies are presented in this paper: a multiple gradient descent algorithm (MGDA) and a Nash game strategy based on an original split of territories between disciplines. MGDA is a multi-objective extension of the steepest descent method. The use of a gradient-based algorithm that exploits the cooperation principle aims at reducing the number of iterations required for classical multi-objective evolutionary algorithms to converge to a Pareto optimal design. On the other hand side, the Nash game strategy is well adapted to typical aeronautical optimization problems, when, after having optimized a preponderant or fragile discipline (e.g. aerodynamics), by the minimization of a primary objective-function, one then wishes to reduce a secondary objective-function, representative of another discipline, in a process that avoids degrading excessively the original optimum. Presently, the combination of the two approaches is exploited, in a method that explores the entire Pareto front. Both algorithms are first analyzed on analytical test cases to demonstrate their main features and then applied to the optimum-shape design of a low-boom/low-drag supersonic business jet design problem. Indeed, sonic boom is one of the main limiting factors to the development of civil supersonic transportation. As the driving design for low-boom is not compliant with the low-drag one, our goal is to provide a trade-off between aerodynamics and acoustics. Thus Nash games are adopted to define a low-boom configuration close to aerodynamic optimality w.r.t. wave drag.

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The shaped sonic boom theory is a valuable, efficient, computationally economical and robust tool in preliminary design of low-boom aircraft configurations. Instead of introducing a new F-function parameterization, as it has been investigated already in the past, the paper adopts a more general formulation proposed in the literature and focuses on reducing the limitations of the inverse method in the design process. Three main contributions are proposed: 1) a revisited procedure based on optimization to solve the coefficients of the F-function that enables to switch between different parameterizations; 2) a definition of the geometry corresponding to the equivalent area distribution combined with a fuselage tailoring process based on direct shape optimization; 3) a strategy to introduce a generic acoustic metrics in the definition of an optimum F-function. The proposed strategy enables the designer to evaluate the geometry of a low-boom configuration that corresponds to a desired F-function in a complete inverse design approach. In this way, the usual limits of the inverse method are significantly alleviated by the present method.

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International audienceA novel numerical method for multi-objective differentiable optimization, the Multiple-Gradient Descent Algorithmm (MGDA), has been proposed in [8] [11] to identify Pareto fronts. In MGDA, a direction of search for which the directional gradients of the objective functions are all negative, and often equal by construction [12], is identified and used in a steepest-descent-type iteration. The method converges to Pareto-optimal points. MGDA is here briefly reviewed to outline its principal theoretical properties and applied first to a classical mathematical test-case for illustration. The method is then ex-tended encompass cases where the functional gradients are approximated via meta-models, as it is often the case in complex situations, and demonstrated on three optimum-shape design problems in compressible aerodynamics.The first problem is purely related to aerodynamic performance. It is a wing shape optimization exercise w.r.t. lift and drag in typical transonic cruis