Proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization

Shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations. Reduced-order models based on proper orthogonal decomposition (POD) constitute an economical alternative. However the truncation of the POD basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results. In our former contribution (Xiao and Breitkopf, 2013), we have introduced a constrained POD projector allowing for exact linear constraint verification for a reduced order model. Nevertheless, this approach was limited a to relatively low numbers constraints. Therefore, in the present paper, we propose an approach for a high number of constraints. The main idea is to extend the snapshot POD by introducing a new constrained projector in order to reduce both the physical field and the constraint space. This allows us to search for the Pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine-tuning of the reduced model for a particular purpose. We illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables.