Global sensitivity analysis using polynomial chaos expansions

Global sensitivity analysis (SA) aims at quantifying the respective effects of input random variables (or combinations thereof) onto the variance of the response of a physical or mathematical model. Among the abundant literature on sensitivity measures, the Sobol’ indices have received much attention since they provide accurate information for most models. The paper introduces generalized polynomial chaos expansions (PCE) to build surrogate models that allow one to compute the Sobol’ indices analytically as a post-processing of the PCE coefficients. Thus the computational cost of the sensitivity indices practically reduces to that of estimating the PCE coefficients. An original non intrusive regression-based approach is proposed, together with an experimental design of minimal size. Various application examples illustrate the approach, both from the field of global SA (i.e. well-known benchmark problems) and from the field of stochastic mechanics. The proposed method gives accurate results for various examples that involve up to eight input random variables, at a computational cost which is 2–3 orders of magnitude smaller than the traditional Monte Carlo-based evaluation of the Sobol’ indices.     

Time response of structures with uncertain parameters under stochastic inputs based on coupled polynomial chaos expansion and component mode synthesis methods

This article presents a numerical procedure to compute the stochastic dynamic response of large finite element models with uncertain parameters based on polynomial chaos and component mode synthesis methods. Polynomial chaos expansions at higher orders are used to derive the statistical solution of the dynamic response as well as the Monte Carlo simulation procedure. Based on various component mode synthesis methods, the size of the model is reduced. These methods are coupled with polynomial chaos expansion and the explicit mathematical formulations are given. Numerical results illustrating the accuracy and efficiency of the proposed coupled methodological procedures are presented.     

An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e.of Galerkin type) or non-intrusive) unaffordable when the deterministic finite element model is expensive to evaluate.     

On the accuracy of the polynomial chaos approximation

Polynomial chaos representations for non-Gaussian random variables and stochastic processes are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. Finite truncations of these series are referred to as polynomial chaos (PC) approximations. This paper explores features and limitations of PC approximations. Metrics are developed to assess the accuracy of the PC approximation. A collection of simple, but relevant examples is examined in this paper. The number of terms in the PC approximations used in the examples exceeds the number of terms retained in most current applications. For the examples considered, it is demonstrated that (1) the accuracy of the PC approximation improves in some metrics as additional terms are retained, but does not exhibit this behavior in all metrics considered in the paper, (2) PC approximations for strictly stationary, non-Gaussian stochastic processes are initially nonstationary and gradually may approach weak stationarity as the number of terms retained increases, and (3) the development of PC approximations for certain processes may become computationally demanding, or even prohibitive, because of the large number of coefficients that need to be calculated. However, there have been many applications in which PC approximations have been successful.     

Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation

Uncertainty quantification (UQ) is the process of determining the effect of input uncertainties on response metrics of interest. These input uncertainties may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties resulting from a lack of knowledge. When both aleatory and epistemic uncertainties are mixed, it is desirable to maintain a segregation between aleatory and epistemic sources such that it is easy to separate and identify their contributions to the total uncertainty. Current production analyses for mixed UQ employ the use of nested sampling, where each sample taken from epistemic distributions at the outer loop results in an inner loop sampling over the aleatory probability distributions. This paper demonstrates new algorithmic capabilities for mixed UQ in which the analysis procedures are more closely tailored to the requirements of aleatory and epistemic propagation. Through the combination of stochastic expansions for computing statistics and interval optimization for computing bounds, interval-valued probability, second-order probability, and Dempster-Shafer evidence theory approaches to mixed UQ are shown to be more accurate and efficient than previously achievable.     

Towards error bounds of the failure probability of elastic structures using reduced basis models

Structural reliability methods aim at computing the probability of failure of systems with respect to prescribed limit state functions. A common practice to evaluate these limit state functions is using Monte Carlo simulations. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. Surrogate models, which are built from a limited number of runs of the original model, have been developed, as substitute of the original model, to reduce the computational cost. However, these surrogate models, while decreasing drastically the computational cost, may fail in computing an accurate failure probability. In this paper, we focus on the control of the error introduced by a reduced basis surrogate model on the computation of the failure probability obtained by a Monte Carlo simulation. We propose a technique to determine bounds of this failure probability, as well as a strategy of enrichment of the reduced basis, based on limiting the bounds of the error of the failure probability for a multi‐material elastic structure. Copyright © 2017 John Wiley & Sons, Ltd.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random field case

This paper is the second of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. The concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in the first paper for the special case of material property variations modeled by random variables to more general problems involving random fields. Specifically, a hierarchy of spectral- and probability-distribution-free upper bounds on the mean, variance, and exceedance values of the response of stochastic systems is established when only the coefficient of variation and lower limit of the stochastic material properties are known. Furthermore, a hierarchy of probability-distribution-free upper bounds on these quantities is established when the spectral density function describing the stochastic material properties is known in addition to the coefficient of variation and the lower limit.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

Fatigue life prediction of carbon/epoxy laminates by stochastic numerical simulation

A three dimensional (3D) finite element model is developed to predict the progressive fatigue damage and the life of a plain carbon/epoxy laminate (AS4/3501-6) based on the longitudinal, transverse and in-plane shear fatigue characteristic. The model takes into account stress analysis, fatigue failure analysis, random distribution and material property degradation. Different cross- and angle-ply laminates including [0₈], [90₈], [0/90₂]_s, [0/90₄]_s, [0₂/90₂]_s, [30₁₆], [45/−45]_2s with the available experimental data are considered for the fatigue life simulation. In order to consider the random distribution of the laminate’s properties from element to element in the model, the laminate’s stiffness, and strength are randomly generated using a Gaussian distribution function. Sudden and gradual material properties degradation are considered during the fatigue simulation. The progressive fatigue damage and failure analysis is implemented in ABAQUS through user subroutines UMAT (user-defined material) and USDFLD (user-defined field variables). The predicted fatigue life of the simulation for different laminates is in good agreement with the experimental results.     

A computationally efficient method for the buckling analysis of shells with stochastic imperfections

A computationally efficient method is presented for the buckling analysis of shells with random imperfections, based on a linearized buckling approximation of the limit load of the shell. A Stochastic Finite Element Method approach is used for the analysis of the “imperfect” shell structure involving random geometric deviations from its perfect geometry, as well as spatial variability of the modulus of elasticity and thickness of the shell, modeled as random fields. A corresponding eigenproblem for the prediction of the buckling load is solved at each MCS using a Rayleigh quotient-based formulation of the Preconditioned Conjugate Gradient method. It is shown that the use of the proposed method reduces drastically the computational effort involved in each MCS, making the implementation of such stochastic analyses in real-world structures affordable.     

Efficient wave propagation simulation on quadtree meshes using SBFEM with reduced modal basis

We apply a combination of the transient scaled boundary finite element method (SBFEM) and quadtree‐based discretization to model dynamic problems at high frequencies. We demonstrate that the current formulation of the SBFEM for dynamics tends to require more degrees of freedom than a corresponding spectral element discretization when dealing with smooth problems on regular domains. Thus, we improve the efficiency of the SBFEM by proposing a novel approach to reduce the number of auxiliary variables for transient analyses. Based on this improved SBFEM, we present a modified meshing procedure, which creates a quadtree mesh purely based on the geometry and allows arbitrary sizes and orders of elements, as well as an arbitrary number of different materials. The discretization of each subdomain is created automatically based on material parameters and the highest frequency of interest. The transition between regions of different properties is straightforward when using the SBFEM. The proposed approach is applied to image‐based analysis with a particular focus on geological models. Copyright © 2016 John Wiley & Sons, Ltd.     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

A stochastic collocation method for large classes of mechanical problems with uncertain parameters

The paper presents a self-contained and didactic approach to the stochastic collocation method. The method relies on the Lagrange polynomials and the Gauss quadrature rule. It is presented for large classes of mechanical problems, i.e. static problems, dynamic problems and spectral problems. After a general presentation of each of them, examples and results are provided. Numerical results show the high rate of convergence of the proposed method.     

Spectral stochastic analysis of structures with uncertain parameters

We present a probabilistic analysis of a structure with uncertain parameters subject to arbitrary stochastic excitations in a frequency domain. The problem of stochastic dynamic analysis of a linear system in a frequency domain is formulated by taking into consideration the uncertainty of structural parameters. The solution is based on the idea of a random frequency response vector for stationary input excitation and a transient random frequency response vector for nonstationary one which are used in the context of spectral analysis in order to determine the influence of structural uncertainty on the random response of structure. The numerical spectral analysis of the building structure under wind and earthquake excitation is provided to demonstrate the described algorithms in the context of computer implementation.     

A reduced SAND method for optimal design of non-linear structures

An SQP-based reduced Hessian method for simultaneous analysis and design (SAND) of non-linearly behaving structures is presented and compared with conventional nested analysis and design (NAND) methods. It is shown that it is possible to decompose the SAND formulation to take advantage of the particular structure of the problem at hand. The resulting reduced SAND method is of the same size as the conventional NAND method but it is computationally more efficient. The presentation here builds on previous research on SAND methods generalizing the solution approach to cases with both equality and inequality constraints. The new version of the reduced SAND method is tested in the context of weight minimization of 3-D truss structures with geometrically non-linear behaviour. © 1997 John Wiley & Sons, Ltd.     

Comparative study of finite element formulations for the semiconductor drift-diffusion equations

A number of transient and steady-state finite element formulations of the semiconductor drift-diffusion equations are studied and compared with respect to their accuracy and efficiency on a simple test structure (the Mock diode). A new formulation, with a consistent interpolation function used to represent the electron and hole carrier densities throughout the set of semiconductor drift-diffusion and Poisson's equations, is introduced. Results highlight the advantages in using consistent interpolation functions showing an increased accuracy in the calculated values and a saving in data storage and execution time. The results also illustrate how the use of different time integration methods affect the number of time steps required during transient simulations. The combination of the fully consistent DFUS with appropriate time integration methods is found to yield a saving of up to 80 per cent of the execution time required for standard spatial/temporal discretization techniques. © 1997 by John Wiley & Sons, Ltd.     

Radial basis functional model for multi-objective sheet metal forming optimization

Fracture and wrinkling are two major defects in sheet metal forming and can be eliminated via an appropriate drawbead design. This article proposes to adopt a multi-objective particle swarm optimization (MOPSO) approach, which differs from traditional multi-objective optimization with construction of a single cost function. MOPSO shows a certain advantage over other single cost function or population-based algorithms. While radial basis function (RBF) has shown considerable promise in highly non-linear problems, there has been no report in sheet metal forming design. Here RBF is attempted to establish the metamodels for fracture and wrinkling criteria in sheet metal forming design. In this article, a sophisticated automobile inner stamping case is exemplified, which demonstrated that RBF provides a better surrogate accuracy and MOPSO is more effective than the other methods studied. The use of RBF driven MOPSO procedure significantly improved the formability and can be recommended for sheet metal process design.