Surrogate Modeling of Aerodynamic Simulations for Multiple Operating Conditions Using Machine Learning

Abstract

This article presents an original methodology for the prediction of steady turbulent aerodynamic fields. Due to the important computational cost of high-fidelity aerodynamic simulations, a surrogate model is employed to cope with the significant variations of several inflow conditions. Specifically, the Local Decomposition Method presented in this paper has been derived to capture nonlinear behaviors resulting from the presence of continuous and discontinuous signals. A combination of unsupervised and supervised learning algorithms is coupled with a physical criterion. It decomposes automatically the input parameter space, from a limited number of high-fidelity simulations, into subspaces. These latter correspond to different flow regimes. A measure of entropy identifies the subspace with the expected strongest non-linear behavior allowing to perform an active resampling on this low-dimensional structure. Local reduced-order models are built on each subspace using Proper Orthogonal Decomposition coupled with a multivariate interpolation tool. The methodology is assessed on the turbulent two-dimensional flow around the RAE2822 transonic airfoil. It exhibits a significant improvement in term of prediction accuracy for the Local Decomposition Method compared with the classical method of surrogate modeling for cases with different flow regimes. Nomenclature A= matrix of the reduced coordinates a k = k-th reduced coordinate B = matrix of the reduced coordinates of the sensor b k = k-th reduced coordinate of the sensorthe quantity of interest E = averaged normalized error f = high fidelity model g = acceleration due to the gravity or normal distribution H = global entropy h = altitude L = temperature lapse rate l = latent function matrix l = latent function M = Mach number m = number of predictions N = Gaussian probability distribution * PhD. Student, Embedded Systems Department, 118 route de Narbonne, Toulouse. n = number of training samples p = dimension of an input parameter or static pressure Q 2 = predictivity coefficient q = number of clusters r = specific gaz constant or correlation function S = matrix of the snapshots s i = quantity of interet at node i T = temperature U = velocity w = weight of the Gaussian Mixture Model X = horizontal coordinate along the chord Y = vertical coordinate y = target value α = angle of attack Γ = spatial domain δ = Kronecker symbol = energy ratio θ = hyperparameters λ = eigenvalues matrix λ = eigenvalues µ = mean of the Gaussian Process ρ = density Σ = covariance matrix σ 2 0 = prior covariance σ = sigmoid function τ w = wall shear stress Φ = mixture coefficient φ = proper orthogonal decomposition matrix χ = input parameter 1 C = hard splitting function = Subscripts = t = training p = prediction 0 = sea level ∞ = freestream = Superscripts = (k) = k-th component or element = fluctuating part = Operators= · = surrogate model · = mean | · | = absolute value · 2 = Euclidian norm (· , · ) = canonical inner product

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