Implementation of Intrusive Polynomial Chaos in CFD Codes and Application to 3D Navier-Stokes

Lacor, Christian; Dinescu, Cristian; Hirsch, Charles; Smirnov, Sergey

doi:10.1007/978-3-319-00885-1_5

Cited by 11 publications

(6 citation statements)

References 31 publications

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“…As a first numerical example we consider the lid driven cavity from [18]. We choose the dynamic viscosity of the fluid to be random, i.e.…”

Section: Lid Driven Cavitymentioning

confidence: 99%

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Flows

Beck,

Dürrwächter,

Kuhn

et al. 2018

Preprint

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We propose a novel hp-Multilevel Monte Carlo method for the quantification of uncertainties in compressible flows using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes jointly with an increasing polynomial degree for the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the hp-Multilevel Monte Carlo method to compute the mean field with prescribed accuracy is of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overshooting the optimal number of samples. The method is in particular adapted to the needs of high-performance computing. Our theoretical results are verified by numerical experiments for the two dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

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“…As a first numerical example we consider the lid driven cavity from [18]. We choose the dynamic viscosity of the fluid to be random, i.e.…”

Section: Lid Driven Cavitymentioning

confidence: 99%

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Flows

Beck,

Dürrwächter,

Kuhn

et al. 2018

Preprint

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“…Intrusive methods, such as stochastic FE [61][62][63], modify the FE solver to directly solve the parametric physical equations. Stochastic FE use traditional finite elements to discretize space, but make the FE coefficients polynomial functions of the parameters.…”

Section: Introductionmentioning

confidence: 99%

Physics-informed neural networks for solving parametric magnetostatic problems

Beltrán-Pulido¹,

Bilionis²,

Aliprantis³

2022

Preprint

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The optimal design of magnetic devices becomes intractable using current computational methods when the number of design parameters is high. The emerging physics-informed deep learning framework has the potential to alleviate this curse of dimensionality. The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. We derive the variational principle for 2-D parametric magnetostatic problems, and prove the existence and uniqueness of the solution that satisfies the equations of the governing physics, i.e., Maxwell's equations. We use a deep neural network (DNN) to represent the magnetic field as a function of space and a total of ten parameters that describe geometric features and operating point conditions. We train the DNN by minimizing the physics-informed loss function

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“…In general, most UQ methods have a cost that scales with a power of the number M of the uncertain variables c i , i ∈ [1, M], making UQ computationally feasible for problems with only a moderate M . This paper focuses on some variants of the Method of Moments (MoM), [2,3,4,5], and non-Intrusive Polynomial Chaos Expansion (niPCE) [6,7,8,9] with a potential for a relatively low UQ cost and compares them in terms of cost and accuracy of computation of the statistical moments of the QoI; Monte Carlo (MC) acts as the reference method for computing the latter.…”

Section: Introductionmentioning

confidence: 99%

Assessment of Variants of the Method of Moments and Polynomial Chaos Approaches to Aerodynamic Uncertainty Quantification

Papoutsis‐Kiachagias¹,

Asouti²,

Giannakoglou³

2021

4th International Conference on Uncertainty Quantification in Computational Sciences and Engineering

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In real life problems uncertainties, for instance through uncertain boundary conditions or manufacturing imperfections, may affect the performance of an aerodynamic shape significantly. In order to measure the impact of uncertainties on the performance of an aerodynamic shape, statistical moments (usually the mean value and variance) of the Quantity of Interest (QoI, e.g. the lift or drag forces) have to be quantified through Uncertainty Quantification (UQ) techniques. This paper compares a number of variants of the Method of Moments (MoM) and the non-intrusive polynomial chaos (niPCE) approaches to UQ. Regarding the MoM, firstand second-order derivatives of the QoI with respect to the uncertain variables are necessary for formulating its first-(FOSM) and second-order (SOSM) variants, respectively. These are computed using a combination of continuous adjoint and direct differentiation of the governing (flow) equations. The statistical moments of the QoI can, then, easily be computed in terms of the QoI value and derivatives computed at the mean values of the uncertain variables. The results of the above-mentioned MoM variants are additionally compared to a number of niPCE variants developed and utilized by the authors in the past. The UQ variants of MoM and niPCE are then compared in terms of cost and accuracy of the computed statistical moments of the QoI. Comparisons also include results of the Monte Carlo method which acts as the reference method. The two benchmark cases used for the comparison include an airfoil under uncertain flow conditions and fluid properties as well as the DrivAer car model, under uncertain flow conditions.

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Implementation of Intrusive Polynomial Chaos in CFD Codes and Application to 3D Navier-Stokes

Cited by 11 publications

References 31 publications

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Flows

$hp$-Multilevel Monte Carlo Methods for Uncertainty Quantification of Compressible Flows

Physics-informed neural networks for solving parametric magnetostatic problems

Assessment of Variants of the Method of Moments and Polynomial Chaos Approaches to Aerodynamic Uncertainty Quantification

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