Surrogate models and mixtures of experts in aerodynamic performance prediction for aircraft mission analysis

Liem, Rhea Patricia; Mader, Charles A.; Martins, Joaquim R. R. A.

doi:10.1016/j.ast.2015.02.019

Cited by 109 publications

(37 citation statements)

References 75 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Indeed, small errors in the multivariate interpolation step can amplify POD modes associated with a physical regime which does not exist for the considered prediction. For this reason, approaches based on local reduced-order models have emerged in the literature by considering only restrictions to the total amount of snapshots [22][23][24][25][26][27][28]. This paper describes an original active local method, called "Local Decomposition Method" (LDM), extending the classical reduced-order modeling method using POD and data fit method to particular steady problems with different physical regimes.…”

Section: Introductionmentioning

confidence: 99%

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

2018

View full text Add to dashboard Cite

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

show abstract

Section: Introductionmentioning

confidence: 99%

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

2018

View full text Add to dashboard Cite

show abstract

“…In addition, comparisons of the hypervolumes show that the solution of additional samples by the single-fidelity/multi-objective EGO continues to stall earlier. To investigate the reason for the superiority of the proposed multi-fidelity/multi-objective EGO, the cross-validation [29,30] of f 1 and f 2 was compared, as shown in Figure 6. It can be seen that the linear regression line nearly coincides with the predicted line in the case of the proposed multi-fidelity/multi-objective EGO.…”

Section: Two-objective Test Function Resultsmentioning

confidence: 99%

Multi-Fidelity Multi-Objective Efficient Global Optimization Applied to Airfoil Design Problems

Ariyarit

Kanazaki

2017

Applied Sciences

View full text Add to dashboard Cite

Abstract:In this study, efficient global optimization (EGO) with a multi-fidelity hybrid surrogate model for multi-objective optimization is proposed to solve multi-objective real-world design problems. In the proposed approach, a design exploration is carried out assisted by surrogate models, which are constructed by adding a local deviation estimated by the kriging method and a global model approximated by a radial basis function. An expected hypervolume improvement is then computed on the basis of the model uncertainty to determine additional samples that could improve the model accuracy. In the investigation, the proposed approach is applied to two-objective and three-objective optimization test functions. Then, it is applied to aerodynamic airfoil design optimization with two objective functions, namely minimization of aerodynamic drag and maximization of airfoil thickness at the trailing edge. Finally, the proposed method is applied to aerodynamic airfoil design optimization with three objective functions, namely minimization of aerodynamic drag at cruising speed, maximization of airfoil thickness at the trialing edge and maximization of lift at low speed assuming a landing attitude. XFOILis used to investigate the low-fidelity aerodynamic force, and a Reynolds-averaged Navier-Stokes simulation is applied for high-fidelity aerodynamics in conjunction with a high-cost approach. For comparison, multi-objective optimization is carried out using a kriging model only with a high-fidelity solver (single fidelity). The design results indicate that the non-dominated solutions of the proposed method achieve greater data diversity than the optimal solutions of the kriging method. Moreover, the proposed method gives a smaller error than the kriging method.

show abstract

“…Kriging models can be extended to utilize gradient information when available, which improves the accuracy of the model. Such methods are known in the literature as gradient-enhanced kriging (GEK) [Liem et al, 2015b], cokriging [Chung and Alonso, 2002;Laurenceau and Sagaut, 2008], or first-order kriging [Lewis, 1998]. GEK has been shown to be effective in various studies [Chung and Alonso, 2002;Laurenceau and Sagaut, 2008;Lewis, 1998;Liu, 2003], and are especially advantageous when the gradient is computed with an adjoint method, where the cost of computing the gradient is independent of the number of independent variables [Martins and Hwang, 2013].…”

Section: Symbol Meaning Dmentioning

confidence: 99%

“…Despite this performance, the number of input variables was still low (2 to 6) because of the exorbitant computational cost required to build GEK for larger inputs. Liem et al [2015b] used a mixture of experts method using GEK to approximate the drag coefficients on a surrogate-based aircraft mission analysis. This method is compared to conventional surrogate models showing the superiority of GEK models, especially in terms of accuracy.…”

Section: Symbol Meaning Dmentioning

confidence: 99%

Gradient-enhanced kriging for high-dimensional problems

Bouhlel

Martins

2018

Engineering with Computers

104

View full text Add to dashboard Cite

Surrogate models provide a low computational cost alternative to evaluating expensive functions. The construction of accurate surrogate models with large numbers of independent variables is currently prohibitive because it requires a large number of function evaluations. Gradient-enhanced kriging has the potential to reduce the number of function evaluations for the desired accuracy when efficient gradient computation, such as an adjoint method, is available. However, current gradient-enhanced kriging methods do not scale well with the number of sampling points due to the rapid growth in the size of the correlation matrix where new information are added for each sampling point in each direction of the design space. They do not scale well with the number of independent variables either due to the increase in the number of hyperparameters that needs to be estimated. To address this issue, we develop a new gradient-enhanced surrogate model approach that drastically reduced the number of hyperparameters through the use of the partial-least squares method that maintains accuracy. In addition, this method is able to control the size of the correlation matrix by adding only relevant points defined through the information provided by the partial-least squares method. To validate our method, we compare the global accuracy of the proposed method with conventional kriging surrogate models on two analytic functions with up to 100 dimensions, as well as engineering problems of varied complexity with up to 15 dimensions. We show that the proposed method requires fewer sampling points than conventional methods to obtain a desired accuracy, or provides more accuracy for a fixed budget of sampling points. In some cases, we get over 3 times more accurate models than a bench of surrogate models from the literature, and also over 3200 times faster than standard gradient-enhanced kriging models.

show abstract

Surrogate models and mixtures of experts in aerodynamic performance prediction for aircraft mission analysis

Cited by 109 publications

References 75 publications

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

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

Multi-Fidelity Multi-Objective Efficient Global Optimization Applied to Airfoil Design Problems

Gradient-enhanced kriging for high-dimensional problems

Contact Info

Product

Resources

About