Predicting shock dynamics in the presence of uncertainties

Lin, Guang; Su, Chi‐Cheung; Karniadakis, George

doi:10.1016/j.jcp.2006.02.009

Cited by 52 publications

(32 citation statements)

References 15 publications

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“…(1) with Monte Carlo simulations (MCS) [11,12] and stochastic finite elements [13][14][15][16][17][18]. For transient nonlinear systems such as (1) these direct approaches are computationally expensive, and often prohibitively so, especially when the parameter fields have small correlation lengths and high variances.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification

2014

Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures

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a b s t r a c tWe develop a probabilistic approach to quantify parametric uncertainty in first-order hyperbolic conservation laws (kinematic wave equations). The approach relies on the derivation of a deterministic equation for the cumulative density function (CDF) of a system state, in which probabilistic descriptions (probability density functions or PDFs) of system parameters and/or initial and boundary conditions serve as inputs. In contrast to PDF equations, which are often used in other contexts, CDF equations allow for straightforward and unambiguous determination of boundary conditions with respect to sample variables. The accuracy and robustness of solutions of the CDF equation for one such system, the SaintVenant equations of river flows, are investigated via comparison with Monte Carlo simulations.

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Section: Introductionmentioning

confidence: 99%

Uncertainty quantification

2014

Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures

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“…In this more general case, the solution of linear hyperbolic problems may have lower regularity than those of elliptic, parabolic and hyperbolic problems with constant random coefficients. There are also recent works on stochastic nonlinear conservation laws, see for instance [25,26,34,39,40].…”

Section: Introductionmentioning

confidence: 99%

A stochastic collocation method for the second order wave equation with a discontinuous random speed

2012

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In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general This work was supported by the King Abdullah University of Science and Technology (AEA project "Bayesian earthquake source validation for ground motion simulation"), the VR project "Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar", and the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). The second author was partially supported by the Italian grant FIRB-IDEAS (Project no. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".

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“…For this problem, they have also derived a collocation technique that reduces the computational burden associated with high-order non-linearities. Some research work in the supersonic regime have also been performed by Lin et al [12] dealing with 2D Euler equations for a stochastic wedge flow (random inflow velocity and random oscillations of the wedge around its apex). Loeven et al [13] make use of a deterministic compressible RANS code which is coupled to a probabilistic collocation solver to propagate freestream aerodynamic (Mach number) uncertainty through a subsonic steady flow around a NACA0012 airfoil.…”

Section: Introductionmentioning

confidence: 99%

A gPC-based approach to uncertain transonic aerodynamics

Simon

Guillen

Sagaut

et al. 2010

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThe present paper focus on the stochastic response of a two-dimensional transonic airfoil to parametric uncertainties. Both the freestream Mach number and the angle of attack are considered as random parameters and the generalized Polynomial Chaos (gPC) theory is coupled with standard deterministic numerical simulations through a spectral collocation projection methodology. The results allow for a better understanding of the flow sensitivity to such uncertainties and underline the coupling process between the stochastic parameters. Two kinds of non-linearities are critical with respect to the skin-friction uncertainties: on one hand, the leeward shock movement characteristic of the supercritical profile and on the other hand, the boundary-layer separation on the aft part of the airfoil downstream the shock. The sensitivity analysis, thanks to the Sobol' decomposition, shows that a strong non-linear coupling exists between the uncertain parameters. Comparisons with the one-dimensional cases demonstrate that the multi-dimensional parametric study is required to get the correct shape and magnitude of the standard deviation distributions of the flow quantities such as pressure and skin-friction.

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Predicting shock dynamics in the presence of uncertainties

Cited by 52 publications

References 15 publications

Uncertainty quantification

Uncertainty quantification

A stochastic collocation method for the second order wave equation with a discontinuous random speed

A gPC-based approach to uncertain transonic aerodynamics

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