Uncertainty Propagation in Dynamic Systems using Polynomial Chaos based Multiresolution Approach

High-dimensional uncertainty quantification for Mars atmospheric entry using adaptive generalized polynomial chaos

Stochastic constitutive modeling of elastic-plastic materials with uncertain properties

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.

Multiple sources of uncertainties often exist in the evaluation of slope stability. When assessing stability of slopes in the face of uncertainties, it is desirable, and sometimes necessary, to adopt reliability-based approaches that consider these uncertainties explicitly. This paper focuses on the practical procedures developed recently for the reliability-based assessment of slope stability. The statistical characterization of model uncertainty and parameter uncertainty are first described, followed by an evaluation of the failure probability of a slope corresponding to a single slip surface, and the system failure probability. The availability of site-specific information then makes it possible to update the reliability of the slope through the Bayes’ theorem. Furthermore, how to perform reliability-based design when the statistics of random variables cannot be determined accurately is also discussed. Finally, case studies are presented to illustrate the benefit of performing reliability-based design and the procedure for conducting reliability-based robust design when the statistics of the random variables are incomplete.

Aerodynamic response of structures with parametric uncertainties

This paper describes a new approach to the statistical characterization of high-speed interconnect circuits. The proposed approach is based on the idea of polynomial chaos and works by decoupling the matrices that arise from the Galerkin projection. The new approach is general in the sense that it can handle any polynomial system used in the generalized polynomial chaos(gPC) framework.

Process reliability and safety under uncertainties

Uncertainties in dynamic response of buildings with non-linear base-isolators

The random microstructure of heterogeneous materials causes spatial variations in mechanical properties, significantly influencing material behaviour, which cannot be fully captured by deterministic models with fixed input parameters. Contemporary research has increasingly shifted toward stochastic modelling to address two distinct sources of non-determinism: epistemic uncertainty (reducible due to limited knowledge) and aleatory uncertainty (irreducible, reflecting inherent randomness). Stochastic models require a specialised approach to estimate their underlying deterministic material parameters, whereas deterministic models are calibrated by treating their parameters as random variables. Within a probabilistic framework, the identification of random parameters is formulated as a stochastic inverse problem. Its objective is to reproduce the variability observed in experimental data from an ensemble of specimens using a numerical model with random material parameters. To mitigate the computational cost of the numerous model simulations required, two widely used surrogate models polynomial chaos expansions and artificial neural networks are investigated. The preferred polynomial chaos provides analytical statistics of model responses enabling efficient variable transformations and sensitivity analysis, which are critical for uncertainty quantification. This thesis explores advanced methods for identifying aleatory uncertainty, with a particular focus on their application to the calibration of heterogeneous material models. Traditional methods, often based on Bayesian hierarchical modelling, typically assume a prescribed form for the parameter probability density function. The developed stochastic inversion framework derives the parameter distribution directly from the observed data distribution, constrained only by the feasible domain of material parameters to ensure model stability and validity. Both approaches are compared through two case studies involving heterogeneous material models. Additionally, this thesis addresses the calibration of the Lattice Discrete Particle Model for concrete, which necessitates specialised treatment due to its inherent stochasticity arising from the random generation of the material microstructure.

Yucca Mountain has been proposed by the U.S. Department of Energy as the nation’s long-term, permanent geologic repository for spent nuclear fuel or high-level radioactive waste. The potential repository would be located in Yucca Mountain’s unsaturated zone (UZ), which acts as a critical natural barrier delaying arrival of radionuclides to the water table. Since radionuclide transport in groundwater can pose serious threats to human health and the environment, it is important to understand how much and how fast water and radionuclides travel through the UZ to groundwater. The UZ system consists of multiple hydrogeologic units whose hydraulic and geochemical properties exhibit systematic and random spatial variation, or heterogeneity, at multiple scales. Predictions of radionuclide transport under such complicated conditions are uncertain, and the uncertainty complicates decision making and risk analysis. This project aims at using geostatistical and stochastic methods to assess uncertainty of unsaturated flow and radionuclide transport in the UZ at Yucca Mountain. Focus of this study is parameter uncertainty of hydraulic and transport properties of the UZ. The parametric uncertainty arises since limited parameter measurements are unable to deterministically describe spatial variability of the parameters. In this project, matrix porosity, permeability and sorption coefficient of the reactive tracer (neptunium) of the UZ are treated as random variables. Corresponding propagation of parametric uncertainty is quantitatively measured using mean, variance, 5th and 95th percentiles of simulated state variables (e.g., saturation, capillary pressure, percolation flux, and travel time). These statistics are evaluated using a Monte Carlo method, in which a three-dimensional flow and transport model implemented using the TOUGH2 code is executed with multiple parameter realizations of the random model parameters. The project specifically studies uncertainty of unsaturated flow and radionuclide transport caused by multi-scale heterogeneity at the layer and local scales. Typically, in studies of Yucca Mountain, the layer scale refers to hydrogeologic layers with layer-wise average properties, and the local scale refers to the spatial variation of hydraulic properties within a layer. While most studies of radionuclide transport in the UZ have been conducted at the layer scale, the uncertainty at the local scale within a layer is also important, since it affects flow path, velocity, and travel time of radionuclide. This report first presents the uncertainty caused by layer-scale heterogeneity of matrix permeability, porosity, and sorption coefficients of reactive tracers. Homogeneous fields of the parameters are generated at each hydrogeologic layer for Monte Carlo simulations. This study is referred to as the homogeneous case. To assess the uncertainty caused by local-scale heterogeneity, the sequential Gaussian simulator (SGSIM) of GSLIB (Deutsch and Journel, 1998) is used to generate heterogeneous parameter fields within each layer, and Monte Carlo simulations are conducted. This study is referred to as the heterogeneous cases. For the homogeneous and heterogeneous cases, the mean, variance, 5th and 95th percentiles of simulated state variables are estimated for uncertainty assessment. In addition, the statistics of the two cases are compared to investigate effect of local-scale heterogeneity on the unsaturated flow and radionuclide transport. It is found that the local-scale heterogeneity increased the predictive uncertainty of percolation flux and cumulative mass arrival for computational blocks below the footprint of proposed repository, whereas mean predictions are hardly affected. The local-scale heterogeneity significantly affects travel times to the water table for both conservative and reactive tracers. In the early simulation period, tracer mean travel times are delayed, whereas the influence of local-scale heterogeneity diminishes during the late simulation period. Simulated state variables in this project are more realistic than those of using one- or two dimensional models, due to a three-dimensional numerical model used in the project to characterize hydrological conditions at the UZ. Therefore, we expect that results of this project can be used directly to facilitate DOE site performance analysis and decision making.

Discretization of a random field by the generalized polynomial chaos (GPC) begins with selecting a specific type of orthogonal polynomial (e.g. Legendre and Hermite polynomials) (e.g. Ghanem and Spanos, 1991). This selection of a type of orthogonal polynomial can be performed based on the reported experiences (e.g. Xiu and Karniadakis, 2003) or data revealing the distribution of a random field to be discretized. If such data or reported experiences are unavailable, a third way may be generating some pilot tests to study the performance of a specific type of orthogonal polynomial in discretizing this random field. This study tries to develop an evolutionary algorithm-based auxiliary tool for the implementation of such pilot tests. A similar tool (Allaix and Carbone, 2009), which is based on the singleobjective evolutionary algorithm, had been developed for constructing the Karhunen-Loeve (KL) representation of a random field. Both KL and GPC expansions are two of the popular random field discretization methods (Ghanem and Spanos, 1991). But, the KL expansion should be applied under a prerequisite of knowing the covariance matrix of a random field to be discretized (e.g. Ghanem and Spanos, 1991); while, the GPC expansions can be applied without similar prerequisites (e.g. Xiu and Karniadakis, 2003). Therefore, the development of an auxiliary tool for constructing a GPC representation of a random field would be necessary. The succeeding research considers the derivation of an GPC representation of a random field as a multi-objective (MO) problem having two goals: (a) limiting the computational efforts spent in applying the resulting GPC representation; and (b) keeping the resulting GPC representation satisfying all accuracy standards (e.g. getting the sufficiently accurate prediction of statistical parameters of a random field). The former goal will be attained by limiting the highest order of polynomial term and total number of uncorrelated random variables used to construct a GPC representation; while the latter goal will be attained by minimizing multiple error estimators. Since there are multiple goals to be attained, a multiple objective evolutionary algorithm (MOEA) is required. Among all available MOEAs, the strength Pareto evolutionary algorithm II (SPEA 2) (Zitzler, et al., 2001) is chosen. The highest order of polynomial term and total number of uncorrelated random variables used to construct an GPC representation are considered as two parameters to be identified. The remainder of this study is organized into four sections. In Sec. 2, the theoretical backgrounds of GPC expansions (e.g. Xiu and Karniadakis, 2003) are briefly reviewed. In Sec. 3, the SPEA 2 (Zitzler, et al., 2001) is used to construct a parameter identification procedure to identify the aforementioned highest order of orthogonal polynomial and total

The hazard and increased risk associated with slope failures in the natural habitat are attributed to uncertainty in soil parameters. These soil parameters, that is, cohesion, internal friction angle, and shear modulus, are estimated using the back-analysis method. The method incorporates machine learning algorithms such as particle swarm optimization and support vector machine. However, to capture the uncertainties in the soil parameters, probabilistic analysis is required. This study implements support vector machine for estimating soil parameters as the function of displacement in the soil. The Bayesian method is used to account for uncertainty in soil parameters. The Noto Hanto earthquake in 2007 and the consequent slope failure at Noto Yuryo Road, Japan, is considered as the case study for the proposed model. The finite-element method is used to simulate this slope failure. The soil parameters are treated as random variables to account for uncertainties in these parameters. The predicted displacement is comparable with the measured displacement in the field, which validates the application of the proposed model. The proposed method can efficiently predict experiment soil parameters by capturing uncertainty in soil parameters through a probabilistic approach.

The present work explores the stochastic trajectory planning problem considering the parametric and initial condition uncertainties. To address this issue, a novel stochastic trajectory planning (STP) procedure integrating the non-intrusive polynomial chaos expansion and the convex optimization technique is proposed in this paper. This is achieved by transforming the nonconvex and infinite dimensional optimization problem into convex and discrete optimization problem with high efficiency in solving optimal control. Subsequently, the stochastic parametric convex dynamics and path constraints are surrogated by deterministic versions by taking advantage of the high accuracy of the non-intrusive polynomial chaos expansion (NTPCE) without tedious modification of governing equations. The proposed STP procedure is applied to generate the optimal control profile of a three-dimensional interception with the specific impact angle. For comparative studies, the optimal control profile of original optimization problem without disturbance is also derived by the convex optimization technique based on the same scenario. The simulation results show that the optimal control profile obtained by the proposed STP approach is more robust to the parametric and initial condition uncertainties under the open-loop control policy and don’t need significant increase of computational cost. These results demonstrate the effectiveness of the proposed procedure.

Reliability based design optimization of coupled acoustic-structure system using generalized polynomial chaos

Abstract. The future of the Antarctic Ice Sheet in response to climate warming is one of the largest sources of uncertainty in estimates of future changes in global mean sea level (ΔGMSL). Mass loss is currently concentrated in regions of warm circumpolar deep water, but it is unclear how ice shelves currently surrounded by relatively cold ocean waters will respond to climatic changes in the future. Studies suggest that warm water could flush the Filchner–Ronne (FR) ice shelf cavity during the 21st century, but the inland ice sheet response to a drastic increase in ice shelf melt rates is poorly known. Here, we use an ice flow model and uncertainty quantification approach to project the GMSL contribution of the FR basin under RCP emissions scenarios, and we assess the forward propagation and proportional contribution of uncertainties in model parameters (related to ice dynamics and atmospheric/oceanic forcing) on these projections. Our probabilistic projections, derived from an extensive sample of the parameter space using a surrogate model, reveal that the FR basin is unlikely to contribute positively to sea level rise by the 23rd century. This is primarily due to the mitigating effect of increased accumulation with warming, which is capable of suppressing ice loss associated with ocean-driven increases in sub-shelf melt. Mass gain (negative ΔGMSL) from the FR basin increases with warming, but uncertainties in these projections also become larger. In the highest emission scenario RCP8.5, ΔGMSL is likely to range from −103 to 26 mm, and this large spread can be apportioned predominantly to uncertainties in parameters driving increases in precipitation (30 %) and sub-shelf melting (44 %). There is potential, within the bounds of our input parameter space, for major collapse and retreat of ice streams feeding the FR ice shelf, and a substantial positive contribution to GMSL (up to approx. 300 mm), but we consider such a scenario to be very unlikely. Adopting uncertainty quantification techniques in future studies will help to provide robust estimates of potential sea level rise and further identify target areas for constraining projections.