Stochastic Approaches to Uncertainty Quantification in CFD Simulations

Mathelin, Lionel; Hussaini, M. Y.; Zang, Thomas A.

doi:10.1007/s11075-004-2866-z

Cited by 174 publications

(86 citation statements)

References 17 publications

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“…h,H,N can be given an equivalent formulation of collocation type (see [1,24] for other examples of collocation schemes for SPDEs). Indeed, thanks to (7.3), the bilinear form (.…”

Section: Discretization Of the Deterministic Formulationmentioning

confidence: 99%

“…Hence, the treatment of the stochastic variable is non-intrusive (see, e.g., [19,24]), i.e., it does not require any modification of the deterministic code used for solving (7.8). Each independent problem has the algebraic structure P discussed in Sect.…”

Section: Discretization Of the Deterministic Formulationmentioning

confidence: 99%

“…The new variables are the images of the basis random variables; they form a coordinate system in a tensor product of intervals of the real line. Discretization in the new variables can be accomplished by a Galerkin projection method ( [34]), possibly computationally softened by numerical integration, which may lead to an equivalent collocation method ( [1,24]). This paper is focussed on geometric uncertainty, i.e., the main assumption is that the PDE is posed in a domain whose boundary is described by random variables.…”

Section: Introductionmentioning

confidence: 99%

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A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Canuto

Kozubek

2007

Numer. Math.

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We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is "spectral" in the polynomial chaos order, in any subdomain which does not contain the random boundaries. Mathematics Subject Classification (1991)

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“…h,H,N can be given an equivalent formulation of collocation type (see [1,24] for other examples of collocation schemes for SPDEs). Indeed, thanks to (7.3), the bilinear form (.…”

Section: Discretization Of the Deterministic Formulationmentioning

confidence: 99%

Section: Discretization Of the Deterministic Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Canuto

Kozubek

2007

Numer. Math.

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“…More efficient nonintrusive Polynomial Chaos methods [16,27] aim at reducing the number of deterministic computations by using a global polynomial interpolation of the samples in parameter space. An effective sampling in suitable Gauss quadrature points is employed in Probabilistic Collocation (PC) approaches [2,23,28]. Other Polynomial Chaos formulations [1,13,32,39] more robust approximation is achieved by multi-element Adaptive Stochastic Finite Elements (ASFE) methods [11,19,20,31,37], which employ a piecewise polynomial interpolation of the samples.…”

Section: Introductionmentioning

confidence: 99%

Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements

Witteveen

Bijl

2009

Journal of Computational Physics

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“…The method has been successfully applied to different types of problems in various fields, such as in the context of structural analysis [3,4,34,38,54,70,79,80], representing non-Gaussian processes [67], protein labeling reactions [19], fluid flow simulation [50,53,88], identification of random shapes [74], etc.…”

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confidence: 99%

On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview

Panayirci

Schuëller

2011

Arch Computat Methods Eng

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This paper addresses the most recent developments concerning the application of the P-C expansion method within the Stochastic Finite Element (SFE) analysis, in particular considering computational solid mechanics. More specifically, the focus has been on the use of the method for the propagation of the stochastic structural responses due to the extensive amount of contributions in this context. Numerical examples presented in the literature are listed in this regard, in order to shed some light on the range of applications, especially in terms of the probabilistic dimensionality of the problem. Furthermore, the recently emerging utilization of the method within the modeling of uncertain input parameters is covered briefly. With this contribution, it is aimed to present an overview on the state-ofart of the P-C literature and the capabilities of the method.

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Stochastic Approaches to Uncertainty Quantification in CFD Simulations

Cited by 174 publications

References 17 publications

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements

On the Capabilities of the Polynomial Chaos Expansion Method within SFE Analysis—An Overview

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