Low-order models. Optimal sampling and linearized control strategies

Lombardi, Edoardo; Bergmann, Michel; Camarri, Simone; Iollo, Angelo

doi:10.3166/jesa.45.575-593

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…The POD is an efficient technique of dimension reduction based on spectral decomposition for high dimensional, multivariate and nonlinear data set. A wide range of applications can be found in the literature such as human face characterization [34], data compression [35] or optimal control [36]. The POD term was first introduced in 1967 [37] to study dominant turbulent eddies, also called Coherent Structures.…”

Section: B Proper Orthogonal Decompositionmentioning

confidence: 99%

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

2018

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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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Section: B Proper Orthogonal Decompositionmentioning

confidence: 99%

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

2018

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“…In the present work, we adopt the approach first proposed by Lombardi et al (2011), which couples Constrained Centroidal Voronoi Tessellations (CCVT) and Greedy methods. In this strategy, new well-spaced points are added iteratively, enriching the database in those regions where a certain error indicator exceeds a fixed tolerance.…”

Section: Sampling Strategymentioning

confidence: 99%

Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation

Salmoiraghi

Scardigli

Telib

et al. 2018

International Journal of Computational Fluid Dynamics

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In this work we provide an integrated pipeline for the model order reduction of turbulent flows around parametrised geometries in aerodynamics. In particular, Free-Form Deformation is applied for geometry parametrisation, whereas two different reduced-order models based on Proper Orthogonal Decomposition (POD) are employed in order to speed-up the full-order simulations: the first method exploits POD with interpolation, while the second one is based on domain decomposition. For the sampling of the parameter space, we adopt a Greedy strategy coupled with Constrained Centroidal Voronoi Tessellations, in order to guarantee a good compromise between space exploration and exploitation.The proposed framework is tested on an industrially relevant application, i.e. the front-bumper morphing of the DrivAer car model, using the finite-volume method for the full-order resolution of the Reynolds-Averaged Navier-Stokes equations.

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“…It may be possible to improve the accuracy of the proposed nonlinear ROMs for compressible cavity problems by applying some recently proposed ideas, e.g., through the incorporation of fine-scales into the ROM basis [5,76,22,15,92], through the addition of LES turbulence closure terms to the ROM equations [94], through the incorporation of boundary condition terms in the ROM equations [39,57] (Appendix A.5), and/or through an adaptive h-refinement of the ROM basis [26]. It may also be worthwhile to see if the situation can be improved by devising specialized snapshot collection/sampling methods (e.g., methods based on "optimal" sampling strategies [70]; methods in which low-energy modes are included in the POD basis [82,15]). It is conjectured that using a set of snapshots spaced closer together in time (i.e., with a smaller ∆t snap ) to construct the POD basis may yield a more accurate and stable ROM for the viscous laminar cavity problem [14].…”

Section: Prospects For Future Workmentioning

confidence: 99%

“…It is noted that a ROM with much better stability properties was obtained when the snapshot set employed to compute the POD basis includes the initial transient present in the high-fidelity solution between time t = 0 and time t = 5.0 × 10 −2 seconds (results not shown here). Given that the quality of the ROM seems to be highly dependent on which snapshots are employed to construct the POD basis, it would be worthwhile to examine the effect of various snapshot collection strategies (e.g., [70,82,15]) on ROM stability and accuracy in future work. e At BB T e A T t dt, (A It will be assumed herein that the matrix A defining the full order system (4.18) is stable, i.e., it has no eigenvalues with a positive real part.…”

Section: A5 Boundary Conditions For Compressible Fluid Roms Construcmentioning

confidence: 99%

Reduced Order Modeling for Prediction and Control of Large-Scale Systems.

Kalashnikova

Arunajatesan²,

Barone³

et al. 2014

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This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe-spot analysis, control and decision-making in the presence of uncertainty. Of particular interest to Sandia is the use of ROMs for the quantification of the compressible captive-carry environment, simulated for the design and qualification of nuclear weapons systems. It is an unfortunate reality that many ROM techniques are computationally intractable or lack an a priori stability guarantee for compressible flows. For this reason, this LDRD project focuses on the development of techniques for building provably stable projection-based ROMs. Model reduction approaches based on continuous as well as discrete projection are considered.

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Low-order models. Optimal sampling and linearized control strategies

Cited by 5 publications

References 17 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

Free-form deformation, mesh morphing and reduced-order methods: enablers for efficient aerodynamic shape optimisation

Reduced Order Modeling for Prediction and Control of Large-Scale Systems.

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