This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

CFD results are subject to considerable uncertainty associated with the operating conditions. Even when the operational uncertainty is omitted under very controlled circumstances during wind tunnel experiments, substantial disagreement between experimental and CFD results persists. This discrepancy must be attributed to model uncertainty. This report discusses the various sources of uncertainty. The need for advanced uncertainty modeling is illustrated by means of a computationally inexpensive 1-D Burgers equation model. We specifically address the uncertainty due to missing variables (inexact or incomplete differential equations). To this extent a random field model is used for the viscosity and the fundamental differences between the solutions of the stochastic differential equations and a simple random variable model is highlighted. The Burgers equation theoretically needs to be integrated over an infinite domain. In a deterministic approach, the integration domain is cut off at some far field boundary. This truncation effectively ignores all variability outside this far field boundary. We present a practical treatment for the uncertainty on the boundary conditions. The results indicate that ignoring the boundary condition uncertainty dramatically underestimates the variance of the velocity u#x# in the interior of the domain.

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.

This paper describes methods for parallel stochastic finite element analysis using distributed memory multiprocessors. The conventional finite element method is deterministic and fails to account for uncertainties in loads, material properties, geometry and boundary conditions. Stochastic fem can be used to compute the variability in the structural response. It requires considerable more computational effort than the deterministic method, making it necessary to exploit parallelism in order to solve large, realistic structural problems. This paper describes parallel versions of the First Order Second Moment method as well as Monte Carlo simulation for analyzing structures with random material properties. Stochastic fem is composed of several sub-problems such as solution of a banded system of equations, the real symmetric eigenvalue problem, sensitivity analysis and structural reanalysis. Parallelism can be exploited at all stages of the solution process. These methods were implemented ...

this paper, an analysis of the stochastic distribution of structural resistance of a reinforced concrete one-span girder is presented. The stochastic characteristics of the structural resistance are determined with three different methods: second-moment analysis, Monte-Carlo simulation and stochastic finite element method. The results derived from the different methods are compared with each other and also with the structural resistance calculated according to EUROCODE 2

Abstract. We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos (ME-gPC) method for elliptic equations with random coefficients for a moderate number (O(10)) of random dimensions. We employ loworder (p ≤ 3) polynomial chaos and refine the solution using adaptivity in the parametric space. We first study the approximation error of ME-gPC and prove its hpconvergence. We subsequently generate local and global a posteriori error estimators. In order to resolve the error equations efficiently, we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC approximation. Based on the a posteriori estimators, we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random coefficients. Numerical results for convergence and efficiency are also presented. AMS subject classifications: 65C20, 65C30 Key words: Stochastic PDE, a posteriori error estimate, elliptic problems, adaptive numerical

In manufacturing processes, it is widely accepted that uncertainty plays an important role and should be taken into account during analysis and design processes. However, uncertainty quantification of its effects on an end-product is a very challenging task, especially when an expensive computational effort is already needed in deterministic models such as sheet metal forming simulations. In this paper, we focus our work on the variance estimation of the system response. A weighted three-point-based strategy is proposed to efficiently and effectively estimate the variance of the system response. Three first-order derivatives for each variable are used to estimate the nonlinear behavior and variance of the system. The details of the derivation of the approach are presented in the paper. The optimal locations of the three points along each axis in the standard normal space and weights for input variables following normal distributions are proposed as (-1.8257,0.0,+1.8257) and (0.075,0.850,0.075), respectively. For input variables following uniform distributions U(-1,1), the optimal locations and weights are proposed as (-0.84517, 0.0,+0.84517) and (0.04667,0.90666,0.04667), respectively. The proposed approach is applicable to nonlinear and multivariable systems as well as problems having no explicit function such as those design simulations based on finite element methods. The significant accuracy improvement over the traditional first-order approximation is demonstrated with a number of test problems. The proposed method requires significantly less computational effort compared with the Monte Carlo simulations. Discussions and conclusions of this work are given at the end of the paper. Key words: weighted three-point-based strategy; first order approximation; uncertainty propagation; variance estimation; design under uncertainty. 1.

Equations of motion for flexible multibody systems with model uncertainty are developed using the Lagrangian formulation. The dynamic model developed serves to exemplify systems which possess undesirable flexibility by representing it as a stochastic process. Further, a novel methodology for treating uncertainties in flexible systems is proposed. The method uses concepts from stochastic finite elements to address the problem of uncertain parameters in flexible mechanical systems. In particular, the problem of random bending rigidity of a flexible beam is studied. The spatial random process is represented using the Karhunen-Loeve expansion. The response process of the beam, comprising of large rotation and elastic vibration is expressed as a projection on the Homogeneous Chaos. Expressions for the response statistics, including the coupling between the elastic vibration and the large displacements are derived. Comparison of the results obtained with those from Monte-Carlo simulation sho...

In this paper, we first review the impact of the powerful finite element method (FEM) in structural engineering, and then address the shortcomings of FEM as a tool for riskbased decision making and incomplete-data-based failure analysis. To illustrate the main shortcoming of FEM, i.e., the computational results are point estimates based on “deterministic” models with equations containing mean values of material properties and prescribed loadings, we present the FEM solutions of two classical problems as reference benchmarks: (RB-101) The bending of a thin elastic cantilever beam due to a point load at its free end and (RB-301) the bending of a uniformly loaded square, thin, and elastic plate resting on a grillage consisting of 44 columns of ultimate strengths estimated from 5 tests. Using known solutions of those two classical problems in the literature, we first estimate the absolute errors of the results of four commercially available FEM codes (ABAQUS, ANSYS, LSDYNA, and MPAVE) by comparing the known with the FEM results of two specific parameters, namely, (a) the maximum displacement and (b) the peak stress in a coarse-meshed geometry. We then vary the mesh size and element type for each code to obtain grid convergence and to answer two questions on FEM and failure analysis in

Abstract. CFD results are subject to considerable uncertainty associated with the operating conditions. Even when the operational uncertainty is omitted under very controlled circumstances during wind tunnel experiments, substantial disagreement between experimental and CFD results persists. This discrepancy must be attributed to model uncertainty. This report discusses the various sources of uncertainty. The need for advanced uncertainty modeling is illustrated by means of a computationally inexpensive 1-D Burgers equation model. We specifically address the uncertainty due to missing variables (inexact or incomplete differential equations). To this extent a random field model is used for the viscosity and the fundamental differences between the solutions of the stochastic differential equations and a simple random variable model is highlighted. The Burgers equation theoretically needs to be integrated over an infinite domain. In a deterministic approach, the integration domain is cut off at some far field boundary. This truncation effectively ignores all variability outside this far field boundary. We present a practical treatment for the uncertainty on the boundary conditions. The results indicate that ignoring the boundary condition uncertainty dramatically underestimates the variance of the velocity u(x) in the interior of the domain.