AIAA 2000–2545 Airfoil Design Optimization Using Reduced Order Models Based on Proper Orthogonal Decomposition

This paper presents a method for inviscid airfoil analysis and design optimization that uses reduced order models to reduce the cost of computation. Strong emphasis is placed on obtaining reasonably accurate solutions to the Euler equations with computational costs which are far lower than those required by traditional Computational Fluid Dynamics (CFD) techniques. The design procedure presented here begins by computing a series of flow solutions (snapshots) in which the design variables of interest are perturbed using a Design of Experiments approach. Proper Orthogonal Decomposition (POD) is then used to produce the optimal linear representation of these snapshots using a finite series of basis functions or modes. These basis modes are then used to construct arbitrary solutions to the Euler equations about modified airfoil geometries with very small computational expense. The flow solution problem is reduced in this way to a non-linear least squares fit problem with a small number of variables that can be solved efficiently. For design purposes, a gradient-based optimization procedure is used with the information supplied by the reduced order model. Results for both direct airfoil analysis and for an inverse design optimization problem are presented. Observations regarding the useability of this technique in a design environment are also discussed. Nomenclature aj generic coefficient of the j-th POD mode E total energy (internal plus kinetic) f , g Euler flux vectors H total enthalpy M number of modes used in approximation p static pressure R(x, x) autocorrelation function R autocorrelation tensor, finite-volume residual R autocorrelation matrix for method of snapshots u x-component of velocity v y-component of velocity u arbitrary function to be generated x vector of independent variables λ Lagrange multiplier, an eigenvalue ηi coefficient of the ith mode in a function expansion Ω domain of interest ρ density ϕ j (x) j-th POD basis mode