A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible inviscid aerodynamics

Pisaroni, Michele; Nobile, Fabio; Leyland, Pénélope

doi:10.1016/j.cma.2017.07.030

Cited by 55 publications

(60 citation statements)

References 28 publications

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“…We achieve this using the multilevel Monte Carlo (MLMC) method [17,18]. In previous works, the potential of the MLMC method has already been demonstrated in context of the inviscid compressible flow in [19] for propagating lower-dimensional geometric and operational uncertainties. In the current work, we use two local stochastic models based on a random eddy viscosity and a random Reynolds stress tensor.…”

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confidence: 99%

Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo

2020

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A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method.The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al.(2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.The Reynolds-Averaged Navier-Stokes (RANS) equations combined with turbulence closure models are widely utilized in engineering to predict flows with high Reynolds number. These turbulence closure models are used to obtain an approximate Reynolds stress tensor that is responsible for coupling the mean flow with turbulence. Although many turbulence models exist in the literature, there is no single model that generalizes well to all classes of turbulent flows [1,2]. Specifically, the performance depends on the modeling assumptions and the type of flow used to calibrate the so-called closure coefficients that are needed as inputs to a turbulence model.Since the dominant source of error in the flow prediction comes from the turbulence modeling, a number of approaches have already been developed for the model-form uncertainty quantification (UQ) of RANS simulations, see e.g. [3,4] for recent reviews. The majority of these approaches are based on the perturbation of baseline RANS models. One way to achieve this is by injecting uncertainties in the closure coefficients [5,6,7,8] of turbulence models. Another more general physics-based approaches exists, which typically introduces randomness directly into the modeled Reynolds Stress Tensor (RST), either by perturbing its eigenvalues [9,10,11], tensor invariants [12,13] or the entire RST field [14]. One can also classify these stochastic models in terms of global and local perturbation (in space). For global approaches, such as in [5,6,7,10], the magnitude of perturbations in closure coefficients, eigenvalues of RST, etc. is the same throughout the flow domain. This translates to a low-dimensional UQ problem which can be efficiently handled by deterministic sampling techniques like stochastic collocation or just by simulating flows for Email addresses: pkumar@cwi.nl (Prashant Kumar), m.schmelzer@tudelft.nl (Martin Schmelzer), r.p.dwight@tudelft.nl (Richard P. Dwight) arXiv:1811.00872v1 [physics.comp-ph] 1 Nov 2018

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confidence: 99%

Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo

2020

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“…For many engineering problems such parameters are generally estimated through a computational expensive screeening procedure performed before the actual uncertainty analysis. In a previous work we presented a robust and efficient Continuation Multi Level Monte Carlo (C-MLMC) approach, 9 following the idea of Collier et al, 10 which is capable of propagating the operational and geometrical uncertainties in compressible inviscid flow problems. The key parameters that control the number of levels and the number of realizations per level are computed on the fly using an online least squares fitting.…”

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confidence: 99%

“…9,10 We assume that the sequence of discretizations (with parameters M 0 < M 1 < ... < M L = M ) provide errors that decrease algebraically with M l , with cost increasing algebraically in M l (see Figure 1). More precisely:…”

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confidence: 99%

“…Such PQ-covariances can be used to update the terms that appear in the bias and statistical error contributions, the MC estimators E MC and V MC that appear in the MLMC estimators in (9) and (11) at each iteration of the C-MLMC algorithm and when the simulations are performed in parallel without any storage of informations in the cache memory as they are updated whenever the results of simulations are available.…”

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confidence: 99%

“…Additionally if the number of screening levels and samples per level are chosen too small, the extrapolation of the convergence rates might be quite unstable, leading to an unreliable estimation or an over conservative (more computational demanding) simulation. We assessed the efficiency and dimension independent (number of uncertain parameters) convergence rates of the C-MLMC approach on compressible inviscid aerodynamics problems 9 and recently on viscous problems modeled by RANS equations. Thanks to the rigorous mean square error estimators that exist for the MLMC estimators for mean and variance we are able to approximate, with different levels of accuracy, in just one run of our C-MLMC algorithm the following quantities:…”

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See 2 more Smart Citations

A Multilevel Monte Carlo Algorithm for Robust Multi Objective Aerodynamic Shape Optimization

Pisaroni

Nobile

Leyland

2017

18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference

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The majority of problems in aircraft production and operation require decisions made in the presence of uncertainty. For this reason aerodynamic designs obtained with traditional deterministic optimization techniques seeking only optimality in a specific set of conditions may have very poor off-design performances or may even be unreliable. In this work we present a novel approach for robust optimization of aerodynamic shapes based on the combination of single and multi-objective Evolutionary Algorithms and a Continuation Multi Level Monte Carlo methodology to estimate robust designs, without relying on derivatives and meta-models. Detailed numerical studies are presented for a transonic airfoil design affected by geometrical and operational uncertainties.

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