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Proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization. (English) Zbl 1338.76042
Summary: 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 [M. Xiao et al., ibid. 223, 254–263 (2013; Zbl 1329.76147)], 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.

76G25 General aerodynamics and subsonic flows
76N25 Flow control and optimization for compressible fluids and gas dynamics
74P15 Topological methods for optimization problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
49Q12 Sensitivity analysis for optimization problems on manifolds
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