Generalized Polynomial Chaos Expansion Research Articles

Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain

Perfect structural periodicity is disturbed in presence of imperfections. The present paper is based on a realistic modeling of imperfections, using uncertainties, to investigate the robustness of the collective nonlinear dynamics of a periodic coupled pendulums chain. A generic discrete analytical model combining multiple scales method and standing-wave decomposition is proposed. To propagate uncertainties through the established model, the generalized Polynomial Chaos Expansion is used and compared to the Latin Hypercube Sampling method. Effects of uncertainties are investigated on the stability and nonlinearity of two and three coupled pendulums chains. Results prove the satisfying approximation given by the generalized Polynomial Chaos Expansion for a significantly reduced computational time, with respect to the Latin Hypercube Sampling method. Dispersion analysis of the frequency responses show that the nonlinear aspect of the structure is strengthened, the multistability domain is wider, more stable branches are obtained and thus multimode solutions are enhanced. More fine analysis is allowed by the quantification of the variability of the attractors’ contributions in the basins of attraction. Results demonstrate benefits of presence of imperfections in such periodic structure. In practice, imperfections can be functionalized to generate energy localization suitable for several engineering applications such as vibration energy harvesting.

Uncertainty quantification/propagation in nonlinear models

PurposeThe purpose of this paper is to develop robust metamodels, which allow propagating parametric uncertainties, in the presence of localized nonlinearities, with reduced cost and without significant loss of accuracy.Design/methodology/approachThe proposed metamodels combine the generalized polynomial chaos expansion (gPCE) for the uncertainty propagation and reduced order models (ROMs). Based on the computation of deterministic responses, the gPCE requires prohibitive computational time for large-size finite element models, large number of uncertain parameters and presence of nonlinearities. To overcome this issue, a first metamodel is created by combining the gPCE and a ROM based on the enrichment of the truncated Ritz basis using static residuals taking into account the stochastic and nonlinear effects. The extension to the Craig–Bampton approach leads to a second metamodel.FindingsImplementing the metamodels to approximate the time responses of a frame and a coupled micro-beams structure containing localized nonlinearities and stochastic parameters permits to significantly reduce computation cost with acceptable loss of accuracy, with respect to the reference Latin Hypercube Sampling method.Originality/valueThe proposed combination of the gPCE and the ROMs leads to a computationally efficient and accurate tool for robust design in the presence of parametric uncertainties and localized nonlinearities.

Comparison of stochastic fault detection and classification algorithms for nonlinear chemical processes

This paper presents a comparative study of two methods to identify and classify intermittent stochastic faults occurring in a dynamic nonlinear chemical process. The methods are based on two popular stochastic modelling techniques, i.e., generalized polynomial chaos expansion (gPC) and Gaussian Process (GP). The goal is to assess which method is more efficient for fault detection and diagnosis (FDD) when using models with parametric uncertainty, and to show the capabilities and drawbacks of each method. The first method is based on a first-principle model combined with a gPC expansion to represent the uncertainty. Resulting statistics such as probability density functions (PDFs) of the measured variables is further used to infer the intermittent faults. For the second method, a GP model is used to project multiple inputs into a univariate model response from which the fault can be identified based on a minimum distance criterion. The performance of the proposed FDD algorithms is illustrated through two examples: (i) a chemical process involving two continuous, stirred tank reactors (CSTRs) and a flash tank separator, and (ii) the Tennessee Eastman benchmark problem.

Wiener–Hermite polynomial expansion for multivariate Gaussian probability measures

This paper introduces a new, transformation-free, generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

Modal response uncertainty in structural acoustic systems using generalized polynomial chaos expansion

When knowledge of material design parameters is lacking, it is important to understand how this uncertainty effects the system response. Generalized polynomial chaos (gPC) expansions provide a means for quantifying this uncertainty and typically show good agreement with Monte Carlo techniques at a much reduced cost [K. Sepahvand et al. / Applied Acoustics 87 (2015) 23-29]. In this work, we apply gPC expansions to study the effects of design parameter uncertainty on a structure’s modal characteristics—both eigenvalues and eigenfunctions. The gPC expansions permit the propagation of uncertainty from the design parameters to the modes and natural frequencies, which then characterize the vibration response of the system as a function of the random parameters. The response uncertainty can then be described using the solutions of the deterministic system sampled carefully over the parameter space. Uncertainty in the forcing function can also be included in this formulation. Numerical calculations demonstrate these techniques to predict the structural acoustic uncertainty of a vibrating plate with the results validated against Monte Carlo simulations. The effects of uncertain forcing functions will also be discussed.

Implicit sampling combined with reduced order modeling for the inversion of vadose zone hydrological data

Bayesian inverse modeling techniques are computationally expensive because many forward simulations are needed when sampling the posterior distribution of the parameters. In this paper, we combine the implicit sampling method and generalized polynomial chaos expansion (gPCE) to significantly reduce the computational cost of performing Bayesian inverse modeling. There are three steps in this approach: (1) find the maximizer of the likelihood function using deterministic approaches; (2) construct a gPCE-based surrogate model using the results from a limited number of forward simulations; and (3) efficiently sample the posterior distribution of the parameters using implicit sampling method. The cost of constructing the gPCE-based surrogate model is further decreased by using sparse Bayesian learning to reduce the number of gPCE coefficients that have to be determined. We demonstrate the approach for a synthetic ponded infiltration experiment simulated with TOUGH2. The surrogate model is highly accurate with mean relative error that is <0.035% in predicting saturation and <0.25% in predicting the likelihood function. The posterior distribution of the parameters obtained using our proposed technique is nearly indistinguishable from the results obtained from either an implicit sampling method or a Markov chain Monte Carlo method utilizing the full model.

Stochastic analysis of moderately thick plates using the generalized polynomial chaos and element free Galerkin method

In this paper, the element free Galerkin method is combined with the generalized polynomial chaos to quantify the uncertainties in the bending analysis of shear deformable plates with elastically restrained edges resting on a Pasternak elastic foundation with random system properties. The plate modules of elasticity, stiffnesses of the elastically restrained edges and the foundation stiffnesses are considered as random processes and are represented by using the Karhunen–Loève expansion. It is shown that the results obtained by the presented method are in a very good agreement with the results of Monte Carlo simulations in spite of using low order polynomials in the generalized polynomial chaos expansion. Also the applicability and versatility of the presented method are demonstrated by solving numerical examples for various values of coefficient of variations, aspect ratio, thickness and several combinations of boundary conditions, different types of lateral loading and various values of stiffnesses of restrained edges and elastic foundation.

Computation of higher‐order moments of generalized polynomial chaos expansions

SummaryBecause of the complexity of fluid flow solvers, non‐intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher‐order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin‐type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third‐order, indeed fourth‐order moments of the polynomials are needed in this analysis. Besides, both intrusive and non‐intrusive approaches call for their computation provided that higher‐order moments of the quantities of interest need to be post‐processed. In most applications, they are evaluated by Gauss quadratures and eventually stored for use throughout the computations. In this paper, analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature‐free procedures instead. Matlab© codes have been developed for this purpose and tested by comparisons with Gauss quadratures. Copyright © 2017 John Wiley &amp; Sons, Ltd.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models subject to uncertain input parameters. The standard sampling approach for recovering sparse polynomials is to use Monte Carlo (MC) sampling of the density of orthogonality. However MC methods result in poor function recovery when the polynomial degree is high. Here we propose a general algorithm that can be applied to any admissible weight function on a bounded domain and a wide class of exponential weight functions defined on unbounded domains. Our proposed algorithm samples with respect to the weighted equilibrium measure of the parametric domain, and subsequently solves a preconditioned $\ell^1$-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided that demonstrate that our proposed Christoffel Sparse Approximation algorithm leads to comparable or improved accuracy even when compared with Legendre and Hermite specific algorithms.

Performance Comparison of the SGM and the SCM in EMC Simulation

Uncertainty analysis methods are widely used in today's electromagnetic compatibility simulations in order to take account of the nonideality and unpredictability in reality and improve the reliability of simulation results. The Stochastic Galerkin Method (SGM) and the Stochastic Collocation Method (SCM), both based on the generalized polynomial chaos expansion theory, have become two prevailing types of uncertainty analysis methods thanks to their high accuracy and high computational efficiency. This paper, by using the feature selective validation method, presents the quantitative accuracy comparison between the foregoing two methods, with the commonly used Monte–Carlo method (MCM) used as the comparison reference. This paper also introduces SCM into the CST software simulation as an example of performing uncertainty analysis. The advantages and limitations of SGM and SCM are discussed in detail in this paper. Finally, the strategy of how to choose among SGM, SCM, and MCM under different situations is proposed in Section VII

Uncertainty quantification of overpressure buildup through inverse modeling of compaction processes in sedimentary basins

This study illustrates a procedure conducive to a preliminary risk analysis of overpressure development in sedimentary basins characterized by alternating depositional events of sandstone and shale layers. The approach rests on two key elements: (1) forward modeling of fluid flow and compaction, and (2) application of a model-complexity reduction technique based on a generalized polynomial chaos expansion (gPCE). The forward model considers a one-dimensional vertical compaction processes. The gPCE model is then used in an inverse modeling context to obtain efficient model parameter estimation and uncertainty quantification. The methodology is applied to two field settings considered in previous literature works, i.e. the Venture Field (Scotian Shelf, Canada) and the Navarin Basin (Bering Sea, Alaska, USA), relying on available porosity and pressure information for model calibration. It is found that the best result is obtained when porosity and pressure data are considered jointly in the model calibration procedure. Uncertainty propagation from unknown input parameters to model outputs, such as pore pressure vertical distribution, is investigated and quantified. This modeling strategy enables one to quantify the relative importance of key phenomena governing the feedback between sediment compaction and fluid flow processes and driving the buildup of fluid overpressure in stratified sedimentary basins characterized by the presence of low-permeability layers. The results here illustrated (1) allow for diagnosis of the critical role played by the parameters of quantitative formulations linking porosity and permeability in compacted shales and (2) provide an explicit and detailed quantification of the effects of their uncertainty in field settings.

Shape and topology optimization of a permanent-magnet machine under uncertainties

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

Application of an Adaptive Polynomial Chaos Expansion on Computationally Expensive Three-Dimensional Cardiovascular Models for Uncertainty Quantification and Sensitivity Analysis.

When applying models to patient-specific situations, the impact of model input uncertainty on the model output uncertainty has to be assessed. Proper uncertainty quantification (UQ) and sensitivity analysis (SA) techniques are indispensable for this purpose. An efficient approach for UQ and SA is the generalized polynomial chaos expansion (gPCE) method, where model response is expanded into a finite series of polynomials that depend on the model input (i.e., a meta-model). However, because of the intrinsic high computational cost of three-dimensional (3D) cardiovascular models, performing the number of model evaluations required for the gPCE is often computationally prohibitively expensive. Recently, Blatman and Sudret (2010, "An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis," Probab. Eng. Mech., 25(2), pp. 183-197) introduced the adaptive sparse gPCE (agPCE) in the field of structural engineering. This approach reduces the computational cost with respect to the gPCE, by only including polynomials that significantly increase the meta-model's quality. In this study, we demonstrate the agPCE by applying it to a 3D abdominal aortic aneurysm (AAA) wall mechanics model and a 3D model of flow through an arteriovenous fistula (AVF). The agPCE method was indeed able to perform UQ and SA at a significantly lower computational cost than the gPCE, while still retaining accurate results. Cost reductions ranged between 70-80% and 50-90% for the AAA and AVF model, respectively.

Modelling uncertainty in incompressible flow simulation using Galerkin based generalized ANOVA

This paper presents a new algorithm, referred to here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA) for modelling input uncertainties and its propagation in incompressible fluid flow. The proposed approach utilizes ANOVA to represent the unknown stochastic response. Further, the unknown component functions of ANOVA are represented using the generalized polynomial chaos expansion (PCE). The resulting functional form obtained by coupling the ANOVA and PCE is substituted into the stochastic Navier–Stokes equation (NSE) and Galerkin projection is employed to decompose it into a set of coupled deterministic ‘Navier–Stokes alike’ equations. Temporal discretization of the set of coupled deterministic equations is performed by employing Adams–Bashforth scheme for convective term and Crank–Nicolson scheme for diffusion term. Spatial discretization is performed by employing finite difference scheme. Implementation of the proposed approach has been illustrated by two examples. In the first example, a stochastic ordinary differential equation has been considered. This example illustrates the performance of proposed approach with change in nature of random variable. Furthermore, convergence characteristics of GG-ANOVA has also been demonstrated. The second example investigates flow through a micro channel. Two case studies, namely the stochastic Kelvin–Helmholtz instability and stochastic vortex dipole, have been investigated. For all the problems results obtained using GG-ANOVA are in excellent agreement with benchmark solutions.

Global sensitivity analysis of a model for venous valve dynamics

Chronic venous disease is defined as dysfunction of the venous system caused by incompetent venous valves with or without a proximal venous obstruction. Assessing the severity of the disease is challenging, since venous function is determined by various interacting hemodynamic factors. Mathematical models can relate these factors using physical laws and can thereby aid understanding of venous (patho-)physiology. To eventually use a mathematical model to support clinical decision making, first the model sensitivity needs to be determined. Therefore, the aim of this study is to assess the sensitivity of the venous valve model outputs to the relevant input parameters. Using a 1D pulse wave propagation model of the tibial vein including a venous valve, valve dynamics under head up tilt are simulated. A variance-based sensitivity analysis is performed based on generalized polynomial chaos expansion. Taking a global approach, individual parameter importance on the valve dynamics as well as importance of their interactions is determined. For the output related to opening state of the valve, the opening/closing pressure drop (dpvalve,0) is found to be the most important parameter. The venous radius (rvein,0) is related to venous filling volume and is consequently most important for the output describing venous filling time. Finally, it is concluded that improved assessment of rvein,0 and dpvalve,0 is most rewarding when simulating valve dynamics, as this results in the largest reduction in output uncertainty. In practice, this could be achieved using ultrasound imaging of the veins and fluid structure interaction simulations to characterize detailed valve dynamics, respectively.

Classification of Normal and Apoptotic Cells from Fluorescence Microscopy Images Using Generalized Polynomial Chaos and Level Set Function.

Accurate automated quantitative analysis of living cells based on fluorescence microscopy images can be very useful for fast evaluation of experimental outcomes and cell culture protocols. In this work, an algorithm is developed for fast differentiation of normal and apoptotic viable Chinese hamster ovary (CHO) cells. For effective segmentation of cell images, a stochastic segmentation algorithm is developed by combining a generalized polynomial chaos expansion with a level set function-based segmentation algorithm. This approach provides a probabilistic description of the segmented cellular regions along the boundary, from which it is possible to calculate morphological changes related to apoptosis, i.e., the curvature and length of a cell's boundary. These features are then used as inputs to a support vector machine (SVM) classifier that is trained to distinguish between normal and apoptotic viable states of CHO cell images. The use of morphological features obtained from the stochastic level set segmentation of cell images in combination with the trained SVM classifier is more efficient in terms of differentiation accuracy as compared with the original deterministic level set method.

Sequential experimental design based generalised ANOVA

Over the last decade, surrogate modelling technique has gained wide popularity in the field of uncertainty quantification, optimization, model exploration and sensitivity analysis. This approach relies on experimental design to generate training points and regression/interpolation for generating the surrogate. In this work, it is argued that conventional experimental design may render a surrogate model inefficient. In order to address this issue, this paper presents a novel distribution adaptive sequential experimental design (DA-SED). The proposed DA-SED has been coupled with a variant of generalised analysis of variance (G-ANOVA), developed by representing the component function using the generalised polynomial chaos expansion. Moreover, generalised analytical expressions for calculating the first two statistical moments of the response, which are utilized in predicting the probability of failure, have also been developed. The proposed approach has been utilized in predicting probability of failure of three structural mechanics problems. It is observed that the proposed approach yields accurate and computationally efficient estimate of the failure probability.

Sensitivity of the two-dimensional shearless mixing layer to the initial turbulent kinetic energy and integral length scale.

The shearless mixing layer is generated from the interaction of two homogeneous isotropic turbulence (HIT) fields with different integral scales ℓ_{1} and ℓ_{2} and different turbulent kinetic energies E_{1} and E_{2}. In this study, the sensitivity of temporal evolutions of two-dimensional, incompressible shearless mixing layers to the parametric variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} is investigated. The sensitivity methodology is based on the nonintrusive approach; using direct numerical simulation and generalized polynomial chaos expansion. The analysis is carried out at Re_{ℓ_{1}}=90 for the high-energy HIT region and different integral length scale ratios 1/4≤ℓ_{1}/ℓ_{2}≤4 and turbulent kinetic energy ratios 1≤E_{1}/E_{2}≤30. It is found that the most influential parameter on the variability of the mixing layer evolution is the turbulent kinetic energy while variations of the integral length scale show a negligible influence on the flow field variability. A significant level of anisotropy and intermittency is observed in both large and small scales. In particular, it is found that large scales have higher levels of intermittency and sensitivity to the variations of ℓ_{1}/ℓ_{2} and E_{1}/E_{2} compared to the small scales. Reconstructed response surfaces of the flow field intermittency and the turbulent penetration depth show monotonic dependence on ℓ_{1}/ℓ_{2} and E_{1}/E_{2}. The mixing layer growth rate and the mixing efficiency both show sensitive dependence on the initial condition parameters. However, the probability density function of these quantities shows relatively small solution variations in response to the variations of the initial condition parameters.

Fragility Analysis of Base-Isolated Liquid Storage Tanks under Random Sinusoidal Base Excitation Using Generalized Polynomial Chaos Expansion–Based Simulation

AbstractGeneralized polynomial chaos (gPC) expansion–based simulation technique is used to investigate the influence of input parameter uncertainty, on peak response quantities and fragility curves of base-isolated liquid storage tanks. Unidirectional horizontal sinusoidal base excitation is considered to develop the fragility curves for the base-isolated liquid storage tanks. Extensively used laminated rubber bearing (LRB), with linear force-deformation behavior, is considered as the isolation system. The liquid storage tank is modeled using a widely accepted lumped mass model. The failure of the liquid storage tank is defined corresponding to the elastic buckling of the tank wall. The uncertainties are considered in the isolator parameters and in the base excitation. Considerable difference in the peak response estimation is observed when the input parameters are represented using different probability distributions, especially when the uncertainties are higher. It is also observed that when the uncerta...

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.