Polynomial Chaos Coefficients Research Articles

Rosenbrock-W methods for stochastic Galerkin systems

The stochastic Galerkin approach is a widely used method for quantifying uncertainty in dynamical systems. A polynomial chaos expansion of physical variables is used to represent uncertainty. Thus the evolution equations for the dynamics of physical quantities are replaced by evolution equations for the polynomial chaos coefficients. The dimension of the stochastic Galerkin system is m times larger than the dimension of the original physical system, where m is the number of polynomial chaos coefficients. The cost of implicit time integration methods, required for stiff dynamical systems, increases considerably, for example, by a factor m3 when direct linear solvers are used.This work studies efficient implicit methods for the time integration of stiff stochastic Galerkin systems consisting of ordinary differential equations. Linearly-implicit Rosenbrock–Wanner schemes are considered. A block-diagonal approximation of the Jacobian of the stochastic Galerkin system is employed, where the diagonal blocks are the Jacobian of the physical system, evaluated at the mean solution value. We perform a theoretical study of numerical stability for this approximation, using a stochastic extension of the Dahlquist test equation, and propose a concept of linear stochastic Galerkin stability. A rigorous analysis is done for methods of order one and two. A third-order scheme is designed to investigate the stability properties by numerical experiments. Numerical results confirm the theoretical findings.

Reliability analysis of bridges under different loads using polynomial chaos and subset simulation

AbstractReliability analysis of bridges is essential for the design of civil engineering structures. The classical methods, such as Monte Carlo Simulation (MCS) and subset simulation techniques, may provide accurate results. However, since the finite element model of the large‐scale civil engineering structures usually consists of a large number of degrees of freedom, structural reliability analysis of such structures is time‐consuming and computational intensive, which may restrict the use of these methods. This paper proposes a novel method for the reliability analysis of bridge under different types of loads by combining Polynomial Chaos (PC) and subset simulation techniques. The surrogate model of the limit state function is approximated by using the PC expansion, in which the PC coefficients are obtained from the least‐squares method. The subset simulation with a PC‐based surrogate model is then used to estimate the rare failure probability. Reliability analyses of bridge structures under different loading types, that is, static load, dynamic moving load, and seismic load are performed. Studies by using the MCS method and the classical subset simulation are also conducted, and the results from the proposed approach are compared with those from MCS and subset simulation to demonstrate the accuracy and efficiency of the proposed method.

Gegenbauer reconstruction method with edge detection for multi-dimensional uncertainty propagation

This paper proposes an edge-detection-based method for discontinuous functions in multi-dimensional uncertainty propagation problems. We develop the Gegenbauer reconstruction method for multivariate functions to resolve the Gibbs phenomenon. To this end, we extend the concentration edge detector to approximate discontinuity hypersurfaces and use the Rosenblatt transformation to treat irregular space decomposition. Numerical experiments for an algebraic test function and an aerodynamic design problem of the supersonic biplane airfoil flow show that the proposed method can reconstruct spectral expansions that are consistently accurate and free from the Gibbs phenomenon from given polynomial chaos coefficients and without additional model computation.

A non-Gaussian stochastic model from limited observations using polynomial chaos and fractional moments

The reasonable representation of input random fields is the key element in the reliability analysis of practical engineering systems. In most engineering applications, the characterization of a random field often relies on limited measurements. Although the simulation of random fields with complete probabilistic information has been quite well-established, reconstructing a random field from limited observations is still a challenging task. In this paper, we develop a methodology for constructing non-Gaussian random model from limited observations based on polynomial chaos (PC) and fractional moments for real-life problems. Our method begins with the reduce-order representation of measurements by Karhunen-Loève (KL) expansion, followed by the PC representation of KL coefficients. The PC coefficients are further modeled as random variables, whose distributions are determined by a modified maximum entropy principle with fractional moments (ME-FM) procedure and a ME-FM-based bootstrapping. In this way, the developed non-Gaussian model enables to quantify the inherent randomness and the statistical uncertainty of the observed non-Gaussian field simultaneously. Since the developed non-Gaussian model is embedded into the well-established PC framework, our method facilitates the implementation of PC-based stochastic analysis in practical engineering applications, in which only limited probabilistic measures are available. Two numerical examples demonstrate the application of the developed method.

Recent advances in polynomial chaos method for uncertainty propagation

Uncertainty exists widely in engineering design. As one of the key components of engineering design, uncertainty propagation and quantification has always been an important research topic. Polynomial chaos (PC) is a highly efficient uncertainty propagation method which has been widely studied and applied. Therefore, this paper reviews recent advances in the PC method. First, the fundamentals of PC are introduced, including the construction of an orthogonal polynomial basis and the calculation of PC coefficients. Second, strategies such as basis truncation, sparse reconstruction, sparse grid and multi-fidelity modeling are described to address the curse of dimensionality issue of PC. Local and global sensitivity analyses based on PC are then introduced. Finally, the research prospects of PC are given.

An Adaptive Sparse Anisotropic Polynomial-Chaos Expansion Algorithm Applied to EMC Problems

Various popular methods for Uncertainty Quantification (UQ) rely on the computation of Polynomial-Chaos (PC) expansions, which represent stochastic processes via convergent polynomial series. The computational cost for determining the expansion coefficients in such methods is related to the number of input variables, and is significantly enlarged in high-dimensional problems. On the other hand, the PC coefficients of the basis functions can be considered, in many cases, to form a sparse vector. This property is exploited in this paper to build PC models that are accurate and computationally efficient, by implementing the Orthogonal Matching Pursuit (OMP) as the solver, which can deal with underdetermined linear systems, assuming that their solution is sparse. In order to enhance the computational efficiency even further, the presented PC method is applied adaptively for various PC model orders, and the best solution is chosen according to a leave-one-out criterion. Finally, an optimally-designed multilayer electromagnetic filter for EMC applications is examined, considering ten or twenty stochastic design variables.

Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations

The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

Convergence rates of high dimensional Smolyak quadrature

We analyse convergence rates of Smolyak integration for parametric maps u: U → X taking values in a Banach space X, defined on the parameter domain U = [−1,1]N. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to N-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to a priori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands’ Taylor gpc coefficients. For so-called “(b,ε)-holomorphic” integrands u with b∈lp(∕) for some p ∈ (0, 1), we prove the dimension-independent convergence rate 2/p − 1 in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate 2/p − 1 is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate 2/p − 1 for practically relevant numbers of quadrature points.

Stochastic constitutive modeling of elastic-plastic materials with uncertain properties

A new deterministic constitutive model is developed that has a form that is efficient for use in a stochastic constitutive model in which the material properties and the stress and stiffness fields are considered uncertain and modeled as random variables. Simplifications are made to the deterministic model to allow for efficient propagation of uncertainty of model inputs using the Polynomial Chaos (PC) method to quantify the effect of uncertainty in the model parameters on the system response. With large uncertainties in the model parameters that are typical in soil models, the use of a simplified model is justified because the effect of the simplification will be masked by the uncertainties in the model parameters. Each component of the stress and stiffness tensor is expanded along the PC basis, and a small number of PC coefficients are updated along loading/unloading increments. The simplified deterministic model gives similar results to other traditional models, but the equations it involves are computationally much more efficient when extended to the stochastic model and solved with PC. The results are PC coefficients of each stress and stiffness component along the loading/unloading parts of the curve, from which probability distributions of the stress and stiffness can be reconstructed. Compared with traditional Monte-Carlo simulation, the PC method gives similar results while being several orders of magnitude faster computationally. The PC expansion method can also be extended to the non-linear stochastic finite element method.

Uncertainty Quantification of an Explicitly Coupled Multiphysics Simulation of In-Situ Pyrolysis by Radio Frequency Heating in Oil Shale

SummaryIn-situ pyrolysis provides an enhanced oil recovery (EOR) technique for exploiting oil and gas from oil shale by converting in-place solid kerogen into liquid oil and gas. Radio-frequency (RF) heating of the in-place oil shale has previously been proposed as a method by which the electromagnetic energy gets converted to thermal energy, thereby heating in-situ kerogen so that it converts to oil and gas. In order to numerically model the RF heating of the in-situ oil shale, a novel explicitly coupled thermal, phase field, mechanical, and electromagnetic (TPME) framework is devised using the finite element method in a 2D domain. Contemporaneous efforts in the commercial development of oil shale by in-situ pyrolysis have largely focused on pilot methodologies intended to validate specific corporate or esoteric EOR strategies. This work focuses on addressing efficient epistemic uncertainty quantification (UQ) of select thermal, oil shale distribution, electromagnetic, and mechanical characteristics of oil shale in the RF heating process, comparing a spectral methodology to a Monte Carlo (MC) simulation for validation. Attempts were made to parameterize the stochastic simulation models using the characteristic properties of Green River oil shale. The geologic environment being investigated is devised as a kerogen-poor under- and overburden separated by a layer of heterogeneous yet kerogen-rich oil shale in a target formation. The objective of this work is the quantification of plausible oil shale conversion using TPME simulation under parametric uncertainty; this, while considering a referenced conversion timeline of 1.0 × 107 seconds. Nonintrusive polynomial chaos (NIPC) and MC simulation were used to evaluate complex stochastically driven TPME simulations of RF heating. The least angle regression (LAR) method was specifically used to determine a sparse set of polynomial chaos coefficients leading to the determination of summary statistics that describe the TPME results. Given the existing broad use of MC simulation methods for UQ in the oil and gas industry, the combined LAR and NIPC is suggested to provide a distinguishable performance improvement to UQ compared to MC methods.

A fuzzy uncertainty model for analytical and numerical homogenization of transversely fiber reinforced plastics

AbstractThis work is directed to aleatoric and epistemic uncertainties in the framework of mean‐field and numerical homogenization, in order to determine the overall effective properties of transversely linear elastic fiber reinforced composite (FRC) which are taken into account by stochastic and fuzzy analysis. The stochastic part of material parameters is expanded with the multivariate polynomial chaos expansion, where for epistemic uncertainties the polynomial chaos coefficients are defined as design variables modeled as fuzzy sets. In this work, the combination of both types of uncertainty are treated with fuzzy‐random variables. Then, the fuzzy response for each selected α‐level is obtained from the minimum and maximum values of the quantity of interest (QoI). However, the QoI becomes a fuzzy‐random variable and depends on the design variables as well as the random variables. In order to avoid stochastic optimizations, so‐called surrogate QoIs are used, which can be described by ordinary and central moments. The representative example deals with a fuzzy‐random analysis of a unidirectional boron/aluminum FRC to demonstrate the versatility of the proposed formulation.

Multi-scale stochastic dynamic response analysis of offshore risers with lognormal uncertainties

This paper presents a multi-scale stochastic dynamic analysis method for offshore structures. The uncertainties in the structural material parameters, such as mass density and Young's modulus, are considered. They are assumed to be lognormal distributions and represented by using the Karhunen–Loeve (KL) and Polynomial Chaos (PC) expansions. Since the variance of the output responses is unknown, the output vibration response is represented by using PC expansion. The multi-scale stochastic analysis is conducted with PC expansions of different orders representing responses at different DOFs defined as three categories, namely, important, less important and the least important ones. Iterated Order Reduced (IOR) model reduction technique is employed to remove the PC coefficients of slave DOFs. Two numerical examples are taken to verify the accuracy and efficiency of the proposed method for the multi-scale stochastic dynamic response analysis of offshore risers. The response statistics such as mean value and variance can be obtained from the proposed method. The results are compared with those from Monte Carlo Simulation (MCS) and Stochastic Finite Element Method (SFEM). Results demonstrate that the computational demand for uncertainty evaluation is greatly reduced, and the accuracy of the results is maintained.

Stochastic dynamic analysis of marine risers considering fluid-structure interaction and system uncertainties

Stochastic dynamic analysis of offshore structures considering both fluid-structure interaction and system uncertainties is a challenging research task. This paper proposes a novel method to evaluate the statistical characteristics of offshore structural responses with consideration of uncertainties in the fluid and structure. The fluid behavior is simulated by a finite number of particles with particle finite element method (PFEM), and the dynamic behavior of the offshore structure is modelled with finite element method (FEM). A PFEM-FEM scheme is used to model the fluid-structure interaction. To evaluate the statistical characteristics, spectral representations method is used for uncertainty propagation. The output of fluid particles position/pressure, structural vibration, etc. are represented by using polynomial chaos (PC) expansion, and their coefficients are obtained from the least squares method. Statistical characteristics of the responses, such as mean value and variance, can be evaluated with the obtained PC coefficients. Three numerical examples are studied in this paper. The first example is a simple structural model, which is used to demonstrate the convergence and accuracy of the uncertainty analysis method. The second example is a benchmark dam break problem. Statistical characteristics of the fluid particles position due to uncertainties in the mass density are evaluated. Numerical results are verified with experimental data and observations. In the third example, PFEM-FEM scheme is used to conduct fluid–structure interaction analysis. The marine riser structure is modelled with beam elements. In the fluid domain, PFEM is used. Uncertainties in both the fluid domain and structural domain are considered. Results demonstrate that the proposed approach can be used to evaluate the statistical characteristics of responses in the fluid-structure interaction analysis accurately and efficiently.

Efficient Propagation of Epistemic Uncertainty in the Median Ground‐Motion Model in Probabilistic Hazard Calculations

Abstract A computationally efficient methodology for propagating the epistemic uncertainty in the median ground motion in probabilistic seismic hazard analysis is developed using the polynomial chaos (PC) approach. For this application, the epistemic uncertainty in the median ground motion for a specific scenario is assumed to be lognormally distributed and fully correlated across earthquake scenarios. In the hazard calculation, a single central ground‐motion model (GMM) is used for the median along with the epistemic standard error of the median for each scenario. A set of PC coefficients is computed for each scenario and each test ground‐motion level. The additional computation burden in computing these PC coefficients depends on the order of the approximation but is less than computing the median ground motion from one additional GMM. With the PC method, the mean and fractiles of the hazard due to the epistemic uncertainty distribution of the median ground motion are computed as a postprocess that is very fast computationally. For typical values of the standard deviation of epistemic uncertainty in the median ground motion (<0.2 natural log units), the methodology accurately estimates the epistemic uncertainty distribution of the hazard over the 1%–99% range. This full epistemic range is not well modeled with just a small number of GMM branches uses in the traditional logic‐tree approach. The PC method provides more accuracy, faster computation, and reduced memory requirements than the traditional approach. For large values of the epistemic uncertainty in the median ground motion, a higher order of the PC expansion may be needed to be included to capture the full range of the epistemic uncertainty.

An efficient adaptive forward–backward selection method for sparse polynomial chaos expansion

As an efficient uncertainty quantification (UQ) methodology for moment propagation and probability analysis of quantities of interest, polynomial chaos (PC) expansions have received broad and sustained attentions. However, the exponentially increasing cost of building PC representations with increasing dimension of uncertainty, i.e., the curse of dimensionality, seriously restricts the practical application of PC at the industrial level. Some efficient strategies applying adaptive basis selection algorithm for sparse optimization (or l1-minimization) of PC show great potential compared to the classical full PC. However, these strategies mainly focus on forward selection algorithms, which are incapable of correcting any error made by these algorithms. Hence, this paper develops a novel adaptive forward–backward selection (AFBS) algorithm for reconstructing sparse PC. The proposed algorithm by a reasonable combination of forward selection and adaptively backward elimination technique can efficiently correct mistakes made by earlier forward selection steps, which retains the most significant PC terms and discards redundant or insignificant ones. The accuracy of built PC metamodel is checked by a cross-validation procedure. As a consequence, the most significant PC terms are detected sequentially and corresponding PC coefficients are accurately recovered. It largely enhances the sparsity of PC and improves the prediction accuracy compared to the popular forward selection algorithms, e.g., least angle regressions (LARs). To validate the efficiency of the proposed algorithm, a complex analytical function with Gaussian distribution inputs and two challenging aerodynamic applications including a sonic boom propagation analysis considering atmospheric uncertainty and a natural-laminar-flow (NLF) airfoil computation under both geometrical and operational uncertainties are elaborated. With an in-depth comparison with some popular PC reconstruction methodologies, the performance of the devised AFBS method is assessed comprehensively. The results demonstrate that the proposed AFBS method can efficiently identify the significant PC contributions describing the problems, and reproduce sparser PC metamodel and more accurate UQ results than those classical full PC and LARs methods.

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

Abstract. In this paper, we develop computationally efficient techniques to calculate statistics used in wind farm optimization with the goal of enabling the use of higher-fidelity models and larger wind farm optimization problems. We apply these techniques to maximize the annual energy production (AEP) of a wind farm by optimizing the position of the individual wind turbines. The AEP (a statistic) is the expected power produced by the wind farm over a period of 1 year subject to uncertainties in the wind conditions (wind direction and wind speed) that are described with empirically determined probability distributions. To compute the AEP of the wind farm, we use a wake model to simulate the power at different input conditions composed of wind direction and wind speed pairs. We use polynomial chaos (PC), an uncertainty quantification method, to construct a polynomial approximation of the power over the entire stochastic space and to efficiently (using as few simulations as possible) compute the expected power (AEP). We explore both regression and quadrature approaches to compute the PC coefficients. PC based on regression is significantly more efficient than the rectangle rule (the method most commonly used to compute the expected power). With PC based on regression, we have reduced on average by a factor of 5 the number of simulations required to accurately compute the AEP when compared to the rectangle rule for the different wind farm layouts considered. In the wind farm layout optimization problem, each optimization step requires an AEP computation. Thus, the ability to compute the AEP accurately with fewer simulations is beneficial as it reduces the cost to perform an optimization, which enables the use of more computationally expensive higher-fidelity models or the consideration of larger or multiple wind farm optimization problems. We perform a large suite of gradient-based optimizations to compare the optimal layouts obtained when computing the AEP with polynomial chaos based on regression and the rectangle rule. We consider three different starting layouts (Grid, Amalia, Random) and find that the optimization has many local optima and is sensitive to the starting layout of the turbines. We observe that starting from a good layout (Grid, Amalia) will, in general, find better optima than starting from a bad layout (Random) independent of the method used to compute the AEP. For both PC based on regression and the rectangle rule, we consider both a coarse (∼225) and a fine (∼625) number of simulations to compute the AEP. We find that for roughly one-third of the computational cost, the optimizations with the coarse PC based on regression result in optimized layouts that produce comparable AEP to the optimized layouts found with the fine rectangle rule. Furthermore, for the same computational cost, for the different cases considered, polynomial chaos finds optimal layouts with 0.4 % higher AEP on average than those found with the rectangle rule.

Nonintrusive Global Sensitivity Analysis for Linear Systems With Process Noise

The focus of this paper is on the global sensitivity analysis (GSA) of linear systems with time-invariant model parameter uncertainties and driven by stochastic inputs. The Sobol' indices of the evolving mean and variance estimates of states are used to assess the impact of the time-invariant uncertain model parameters and the statistics of the stochastic input on the uncertainty of the output. Numerical results on two benchmark problems help illustrate that it is conceivable that parameters, which are not so significant in contributing to the uncertainty of the mean, can be extremely significant in contributing to the uncertainty of the variances. The paper uses a polynomial chaos (PC) approach to synthesize a surrogate probabilistic model of the stochastic system after using Lagrange interpolation polynomials (LIPs) as PC bases. The Sobol' indices are then directly evaluated from the PC coefficients. Although this concept is not new, a novel interpretation of stochastic collocation-based PC and intrusive PC is presented where they are shown to represent identical probabilistic models when the system under consideration is linear. This result now permits treating linear models as black boxes to develop intrusive PC surrogates.

On the use of derivatives in the polynomial chaos based global sensitivity and uncertainty analysis applied to the distributed parameter models

Sensitivity Analysis (SA) and Uncertainty Quantification (UQ) are important components of modern numerical analysis. However, solving the relevant tasks involving large-scale fluid flow models currently remains a serious challenge. The difficulties are associated with the computational cost of running such models and with large dimensions of the discretized model input. The common approach is to construct a metamodel – an inexpensive approximation to the original model suitable for the Monte Carlo simulations. The polynomial chaos (PC) expansion is the most popular approach for constructing metamodels. Some fluid flow models of interest are involved in the process of variational estimation/data assimilation. This implies that the tangent linear and adjoint counterparts of such models are available and, therefore, computing the gradient (first derivative) and the Hessian (second derivative) of a given function of the state variables is possible. New techniques for SA and UQ which benefit from using the derivatives are presented in this paper. The gradient-enhanced regression methods for computing the PC expansion have been developed recently. An intrinsic step of the regression method is a minimization process. It is often assumed that generating ‘data’ for the regression problem is significantly more expensive that solving the regression problem itself. This depends, however, on the size of the importance set, which is a subset of ‘influential’ inputs. A distinguishing feature of the distributed parameter models is that the number of the such inputs could still be large, which means that solving the regression problem becomes increasingly expensive. In this paper we propose a derivative-enhanced projection method, where no minimization is required. The method is based on the explicit relationships between the PC coefficients and the derivatives, complemented with a relatively inexpensive filtering procedure. The method is currently limited to the PC expansion of the third order. Besides, we suggest an improved derivative-based global sensitivity measure. The numerical tests have been performed for the Burgers' model. The results confirm that the low-order PC expansion obtained by our method represents a useful tool for modeling the non-gaussian behavior of the chosen quantity of interest (QoI).

A fuzzy‐stochastic model for transversely fiber reinforced plastics

AbstractThis work is directed to aleatoric and epistemic uncertainties in the framework of constitutive modeling, which are taken into account by stochastic and fuzzy analysis. The stochastic part of material parameters are expanded with the multivariate polynomial chaos expansion, where for epistemic uncertainty the polynomial chaos coefficients are defined as design variables, which are modeled as fuzzy sets. The resulting min‐max optimization problem for the fuzzy analysis is approximated by α‐level discretization. A numerical example for the hybrid fuzzy‐stochastic analysis of material parameters is concerned with two different uncertain approximations to demonstrate the versatility of the proposed formulation.

Using polynomial chaos expansion for uncertainty and sensitivity analysis of bridge structures

Abstract The quantification of the uncertainty effect of random system parameters, such as the loading conditions, material and geometric properties, on the system output response has gained significant attention in recent years. One of the well-known methods is the first-order second-moment (FOSM) method, which can be used to determine the mean value and variance of the system output. However, this method needs to derive the formulas for calculating the local sensitivity and it can only be used for systems with low-level uncertainties. Polynomial Chaos (PC) expansion is a new non-sampling-based method to evaluate the uncertainty evolution and quantification of a dynamical system. In this paper, PC expansion is used to represent the stochastic system output responses of civil bridge structures, which could be the natural frequencies, linear and nonlinear dynamic responses. The PC coefficients are obtained from the non-intrusive regression based method, and the statistical characteristic can be evaluated from these coefficients. The results from the proposed approach are compared with those calculated with commonly used methods, such as Monte Carlo Simulation (MCS) and FOSM. The accuracy and efficiency of the presented PC based method for uncertainty quantification and global sensitivity analysis are investigated. Global sensitivity analysis is performed to quantify the effect of uncertainty in each random system parameter on the variance of the stochastic system output response, which can be obtained directly from the PC coefficients. The results demonstrate that PC expansion can be a powerful and efficient tool for uncertainty quantification and sensitivity analysis in linear and nonlinear structure analysis.