International Journal for Numerical Methods in Engineering

Ghanem, Roger G.

doi: 10.1002/nme.2751pmid: N/A

Soize, Christian

doi: 10.1002/nme.2563pmid: N/A

The paper is devoted to elliptic boundary value problems with uncertainties. Such a problem has already been analyzed in the context of the parametric probabilistic approach of system parameters uncertainties or for random media. Model uncertainties are induced by the mathematical–physical process, which allows the boundary value problem to be constructed from the design system. If experiments are not available, the Bayesian approach cannot be used to take into account model uncertainties. Recently, a nonparametric probabilistic approach of both the model uncertainties and system parameters uncertainties has been proposed by the author to analyze uncertain linear and non‐linear dynamical systems. Nevertheless, the use of this concept that has to be developed for dynamical systems cannot directly be applied for elliptic boundary value problem, for instance, for a linear elastostatic problem relative to an elastic bounded domain. We then propose an extension of the nonparametric probabilistic approach in order to take into account model uncertainties for strictly elliptic boundary value problems. The theory and its validation are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Red‐Horse, John R.; Ghanem, Roger G.

doi: 10.1002/nme.2643pmid: N/A

Three hundred‐plus years of successful theoretical development and application of probability theory provide sufficient justification for it as the mathematical context in which to analyze the uncertainty in the performance of engineering and scientific systems. In this document, we propose a joint probabilistic and deterministic function analytic approach as the means for the development of advanced techniques that feature a strong connection between classical deterministic and probabilistic methods. We know of no other means to achieve simultaneous, balanced approximations across these two constituents. We present foundational materials on the general approach to particular aspects of functional analysis, which are relevant to probability, and emphasize the common elements it shares, and the close connections it provides, to various classical deterministic mathematical analysis elements. Finally, we describe how to use the joint approach as a means to augment deterministic analysis methods in a particular Hilbert space context, and thus enable a rigorous framework for commingling deterministic and probabilistic analysis tools in an application setting. Copyright © 2009 John Wiley & Sons, Ltd.

Jiang, Xiaomo; Mahadevan, Sankaran

doi: 10.1002/nme.2550pmid: N/A

Uncertainty quantification is playing an increasingly important role in assessing the performance, safety, and reliability of complex physical systems in the absence of adequate amount of experimental data. Simulation of a complex system involves multiple levels of modeling, such as material (lowest level) to component to subsystem to system (highest level). This paper presents a Bayesian structural equation modeling approach to quantify both epistemic and aleatoric uncertainties in hierarchical model development. A generalized structural equation modeling with latent variables is presented to model three sets of relationships in hierarchical model development, namely, model predictions vs experimental observations at each individual level, model predictions at lower vs higher levels, and experimental data at lower vs higher levels. The three sets of relationships are represented by a hierarchical Bayes network, and the influencing factors between them are estimated by a Bayesian regression approach. Both measurement and prediction errors at various levels are quantified through the Bayesian method. The variability of input variables in the computational model is updated and quantified using various levels of measurement data via Bayesian inference and the structural equation modeling parameters. The proposed methodology is illustrated with a transient heat conduction example problem. Copyright © 2009 John Wiley & Sons, Ltd.

Lin, G.; Karniadakis, G. E.

doi: 10.1002/nme.2582pmid: N/A

We study the parametric uncertainties involved in plasma flows and apply stochastic sensitivity analysis to rank the importance of all inputs to guide large‐scale stochastic simulations. Specifically, we employ different gradient‐based sensitivity methods, namely Morris, multi‐element probabilistic collocation method on sparse grids, Quasi‐Monte Carlo and Monte Carlo methods. These approaches go beyond the standard ‘One‐At‐a‐Time’ sensitivity analysis and provide a measure of the non‐linear interaction effects for the uncertain inputs. The objective is to perform systematic stochastic simulations of plasma flows treating only as stochastic processes the inputs with the highest sensitivity index, hence reducing substantially the computational cost. Two plasma flow examples are presented to demonstrate the capability and efficiency of the stochastic sensitivity analysis. The first one is a two‐fluid model in a shock tube whereas the second one is a one‐fluid/two‐temperature model in flow past a cylinder. Copyright © 2009 John Wiley & Sons, Ltd.

Kenny, Sean; Crespo, Luis; Giesy, Dan

doi: 10.1002/nme.2591pmid: N/A

Dimensionality reduction is a beneficial step to alleviate some of the computation burden as well as to improve the accuracy associated with complex system analyses. This paper investigates dimensionality reduction techniques for linear, time‐invariant systems subject to general non‐linear parameter dependencies. In the context of this paper, dimensionality reduction refers to simultaneous reductions in both model state order and parameter order, i.e. number of uncertain parameters. Two complementary approaches will be presented, one based on the worst‐case H‐infinity norm error associated with both model state and parameter‐order reductions, and another, which is essentially the inverse problem, that considers the largest allowable parameter bounds for a given total H‐infinity norm error for the dimensionally reduced problem. Although applicable to larger‐order systems, a simple low‐order spring–mass example is used to demonstrate the usefulness of the techniques developed herein. Published in 2009 by John Wiley & Sons, Ltd.

Najm, H. N.; Debusschere, B. J.; Marzouk, Y. M.; Widmer, S.; Le Maître, O. P.

doi: 10.1002/nme.2551pmid: N/A

We demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non‐isothermal ignition of a methane–air system. We employ Bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process. We analyze the time evolution of moments and probability density functions of the solution. We also examine the role and significance of dependence among the uncertain parameters. We finish with a discussion of the role of non‐linearity and the performance of the algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

Arbelaez, D.; Zohdi, T. I.; Dornfeld, D. A.

doi: 10.1002/nme.2573pmid: N/A

In this work a multibody collision model, amenable to large‐scale computation, is developed to simulate a jet of near‐field grains impinging on a surface. This model is developed by computing momentum exchange for grain–grain and grain–surface interactions. The grain–grain interactions consist of collisions as well as near‐field interactions. The analysis of these flows is separated into three components: (1) volume averaged quantities; (2) average surface tractions; and (3) average outflow conditions. For the surface stress calculations, parametric studies are performed on the properties of the surface and the grains through their coefficients of restitution, the strength of the near‐field interactions, and the angle of attack of the jet. For the outflow calculations the flux of momentum through the simulation space is performed for varying near‐field forces between the grains and varying degrees of surface roughness. Copyright © 2009 John Wiley & Sons, Ltd.

Estep, D.; Målqvist, A.; Tavener, S.

doi: 10.1002/nme.2547pmid: N/A

In this paper, we develop and apply an efficient adaptive algorithm for computing the propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation with a randomly perturbed diffusion coefficient. The algorithm is well‐suited for problems for which limited information about the random perturbations is available and when an approximation of the probability distribution of the output is desired. We employ a nonparametric density estimation approach based on a very efficient method for computing random samples of elliptic problems described and analyzed in (SIAM J. Sci. Comput. 2008. DOI: JCOMP‐D‐08‐00261). We fully develop the adaptive algorithm suggested by the analysis in that paper, discuss details of its implementation, and illustrate its behavior using a realistic data set. Finally, we extend the analysis to include a ‘modeling error’ term that accounts for the effects of the resolution of the statistical description of the random variation. We modify the adaptive algorithm to adapt the resolution of the statistical description and illustrate the behavior of the adaptive algorithm in several examples. Copyright © 2009 John Wiley & Sons, Ltd.

Constantine, Paul G.; Doostan, Alireza; Iaccarino, Gianluca

doi: 10.1002/nme.2564pmid: N/A

We present a numerical method to study convective heat transfer in a high Reynolds number incompressible flow around a cylinder subject to uncertain boundary conditions. We exploit the one‐way coupling of the energy and momentum transport to derive a semi‐intrusive uncertainty propagation scheme, which combines Galerkin and collocation approaches for computing approximate statistics of the stochastic temperature field. The hybrid scheme converges rapidly and dramatically reduces the overall computational cost compared with the conventional uncertainty propagation schemes. Copyright © 2009 John Wiley & Sons, Ltd.