Aerodynamic shape optimization under uncertainties using embedded methods and adjoint techniques

(English) This thesis develops a framework to perform shape optimization under uncertainties for a body under the action of aerodynamic forces. The solution of the flow is performed with finite elements using the full potential equation with an embedded approach, where the object of study is defined implicitly with a level set function. The optimization problem is solved by combining different software packages to perform the solution of the flow, advance in the optimization loop and perform uncertainty quantification. The first contribution of the thesis is the development of a full embedded approach for the solution of the full potential equation. Due to the inviscid hypothesis of potential solvers, these require the definition of a gap in the computational mesh in order to generate lift, known as the wake. Based on previous works where the wake is defined implicitly with an embedded approach, this work also considers the geometry as an embedded body. Mesh refinement and numerical terms are employed to improve the definition of the geometry in the mesh and ensure the definition of the Kutta condition. The solver is validated for two and three dimensions for subsonic and transonic flows with different reference data. Another contribution of the thesis is the development of the adjoint analysis for the subsonic full potential equation with embedded geometries in two dimensions. Each coordinate of the object of study is considered a design parameter in the adjoint analysis, where the effect of the level set function is considered. The sensitivities of the objective function with respect to the design parameters are validated by comparing them to the sensitivities obtained by using a finite differences approach. A shape optimization problem where the lift coefficient is maximized with geometrical constraints is solved as an example of application of the adjoint sensitivities. The embedded shape optimization problem is extended to consider uncertainties in the inlet condition. The optimization problem is reformulated by choosing a risk measure, the Conditional Value-at-risk, which is minimized. The adjoint sensitivities are adapted for the stochastic case, considering the selected risk measure. The estimation of the risk measure is performed thanks to an external uncertainty quantification library, by applying a novel approach which uses Monte Carlo methods to estimate the Conditional Value-at-risk. The stochastic case is solved in a distributed environment, where each optimization step deploys a Monte Carlo hierarchy to estimate the objective function and its gradients.