Framework For Uncertainty Propagation Research Articles

Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and Legendre interval inclusion function

This paper investigates a non-intrusive hybrid uncertainty propagation framework in mechanical dynamic systems, utilizing polynomial chaos expansion (PCE) and Legendre inclusion function. Uncertainties with substantial knowledge and information are conceptualized as stochastic parameters and described using PCE, while Legendre polynomials are employed to represent uncertain, bounded parameters, specifically interval uncertainties. During the computation, the PCE model for each time step is developed through Galerkin projection and sparse grid numerical integration. Consequently, the statistical moments of the dynamic responses relative to the stochastic parameters are derived from the orthogonality of the PCE and transformed into functions reflecting the interval parameters. The interval bounds for these statistical moments are further determined using the Legendre inclusion function, which is obtained through optimal Latin hypercube sampling and collocation methods. Three dynamics examples, respectively, modeled by differential–algebraic equations, finite element method, and commercial software proves its applicability and superiority. Detailed assessment demonstrates that this method offers high computational efficiency and accuracy. As a non-intrusive approach, it poses no particular limitations on numerical methods for solving differential equations, making it universally applicable.

Composite Pressure Vessel Failure Simulation Considering Spatial Variability

Carbon fiber-reinforced polymers offer lightweight solutions for demanding applications, but material imperfections affect structural reliability. In this study, an efficient uncertainty propagation framework is applied to predict composite behavior. The framework accounts for spatial variability of fiber misalignment, uneven fiber distribution, and single-fiber strength. Spatial variability is represented at both the micro- and mesoscale. Macroscale simulations incorporate this spatial variability indirectly using homogenized material properties. The framework was applied to composite pressure vessels, whose stochastic burst pressure was predicted. The predictions were validated by experimental measurements. These measurements show that the actual burst pressure was underpredicted by an average of 5.8%. Several hypotheses were investigated to explain this discrepancy.

Reliability of inelastic wind excited structures by dynamic shakedown and adaptive fast nonlinear analysis (AFNA)

The ever-growing interest in performance-based wind engineering has created a need for assessment frameworks that can efficiently deal with inelasticity. The computationally efficient strain-driven dynamic shakedown approach has provided a solution that is not only capable of identifying failure mechanisms that are potentially critical during extreme winds, e.g., low cycle fatigue and ratcheting, but also allows direct estimation of inelastic deformations. This approach, however, can only solve problems at dynamic shakedown, i.e., with limited nonlinearity, and is not capable of providing response time histories. To address these limitations, this paper presents an efficient framework for reliability assessment of inelastic structures at dynamic shakedown and beyond. To this end, a novel step-by-step integration algorithm is developed for rapid response time history analysis within the setting of dynamic shakedown. The method is based on advancing fast nonlinear analysis through introducing schemes for enabling at each time step the adaptive selection of the step size, number of modes to be included, and number of potentially nonlinear elements. Inelasticity is modeled as distributed at the level of the stress resultants through a return mapping scheme based on the Haar–Kàrmàn principle, therefore enabling the integrated estimation of the state of dynamic shakedown. The scheme is seen to preserve the efficiency of recently developed strain-driven dynamic shakedown algorithms while providing a full range of response time histories at and beyond the state of dynamic shakedown. To enable reliability analysis, the scheme is embedded in a general uncertainty propagation framework.

Hierarchical uncertainty quantification of hybrid (riveted/bonded) single lap aluminum-CFRP joints with structural multiscale characteristic

Due to multiple inevitable uncertainty sources from materials and geometry, the mechanical properties of the hybrid joints (HJs) are stochastic. This paper proposes an innovative method to predict the uncertainty of the mechanical properties of the HJs considering the structural multiscale characteristic. The proposed approach is based on multiscale numerical models ranging from the mesoscopic scale of the plain-woven CFRP to the macrostructure scale of the joint. In the mechanical performance simulation, multiple experimentally quantified uncertainty sources arising from different aspects (materials, geometry, and assembly) and different scales (mesoscopic, macroscopic) are introduced. The probabilistic uncertainty of high scale mechanical properties could be quantified scale by scale from low scale uncertainty variables with the adoption of the multiscale uncertainty propagation method. The quantitative results are in high agreement with the experiment, which proves the effectiveness of this method. Furthermore, the PCE-based uncertainty propagation framework makes it straightforward to implement the global sensitivity analysis of each variable, which helps identify crucial variables influencing the mechanical properties of the HJs. This work helps to deepen the understanding of the mechanism of fluctuations in the mechanical properties of the HJs.

SUP: a probabilistic framework to propagate genome sequence uncertainty, with applications.

Genetic sequencing is subject to many different types of errors, but most analyses treat the resultant sequences as if they are known without error. Next generation sequencing methods rely on significantly larger numbers of reads than previous sequencing methods in exchange for a loss of accuracy in each individual read. Still, the coverage of such machines is imperfect and leaves uncertainty in many of the base calls. In this work, we demonstrate that the uncertainty in sequencing techniques will affect downstream analysis and propose a straightforward method to propagate the uncertainty. Our method (which we have dubbed Sequence Uncertainty Propagation, or SUP) uses a probabilistic matrix representation of individual sequences which incorporates base quality scores as a measure of uncertainty that naturally lead to resampling and replication as a framework for uncertainty propagation. With the matrix representation, resampling possible base calls according to quality scores provides a bootstrap- or prior distribution-like first step towards genetic analysis. Analyses based on these re-sampled sequences will include a more complete evaluation of the error involved in such analyses. We demonstrate our resampling method on SARS-CoV-2 data. The resampling procedures add a linear computational cost to the analyses, but the large impact on the variance in downstream estimates makes it clear that ignoring this uncertainty may lead to overly confident conclusions. We show that SARS-CoV-2 lineage designations via Pangolin are much less certain than the bootstrap support reported by Pangolin would imply and the clock rate estimates for SARS-CoV-2 are much more variable than reported.

Quantifying the uncertainty of lake-groundwater interaction using the forward uncertainty propagation framework: The case of Lake Urmia

The interaction between a lake and groundwater has important implications to the quantity and quality of water in both environments. Quantification of lake-groundwater interaction (LGI) has been challenging in regions with limited in-situ data. LGI can be quantified by physically-based models, direct measurement of seepage, measurements of conservative chemical or isotopic tracers, and lake water balance. Despite the accuracy of the methods based on hydrochemical or isotopic measurements and analysis, they require extensive field data that are costly to collect in large lakes. Instead, the data required to quantify LGI by the lake water budget method can be obtained via typical ground measurements and satellite remote sensing. However, the influence on the estimated LGI imposed by the uncertainty associated with the other components of lake water budget (e.g., precipitation, evaporation, and inflow/outflow) is still poorly understood. In this study, we coupled a forward uncertainty propagation framework with the lake water budget to estimate the uncertainty of LGI using Latin-Hypercube sampling. We implemented the above framework in the hypersaline Lake Urmia (LU), located in the northwestern Iran, over five consecutive study periods between September 2017 and May 2020. The results revealed the exchange of flow from groundwater to the LU in all studied periods, ranging between 0.2 million cubic meters (MCM)/day and 7.6 MCM/day. The relative contribution of LGI in the water budget varied between 3% and 37%. The results of the uncertainty analysis further demonstrated that the cumulative probability of LGI < 0, which indicates flow direction from the LU to groundwater, was insignificant in all the study periods ranging between 0.001 and 0.35.

Reliability analysis of SHS compression members in shipbuilding: derivation of probabilistic buckling curves

ABSTRACT The compressive resistance of pillars used in ship structures is studied under a reliability perspective. A particular focus is placed on the study of Square Hollow Section (SHS) welded profiles by examining sample cases and postulating over the practical scantling range of interest. Uncertainty in the material and production imperfections have been probabilistically modelled by considering representative statistical structures. A pre-defined number of non-linear finite element simulations has been performed based on DOE sampling schemes in accordance with the response surface methodology. The generated surrogates were used within a Monte Carlo Simulation (MCS) framework for uncertainty propagation. The obtained statistical structures allowed for the generalised reliability assessment of SHS pillars which further allows for the subsequent proposal of deterministic utilisation factors for practical design.

Uncertainty propagation in coupled hydrological models using winding stairs and null-space Monte Carlo methods

By integrating disciplinary sub-models, coupled hydrological models allow the exchange of output and input fluxes among their sub-models. While such coupling of models can mirror the conceptual representation of the water-cycle, potential uncertainty propagation and aggregation across the sub-models may limit their overall performance. There are limited studies dealing with uncertainty in coupled hydrological models due to the high computational needs, the absence of detailed data, and the lack of efficient uncertainty propagation frameworks. This study presents an effective uncertainty propagation framework using a combination of statistical techniques, multi-variable calibration, and parallel computation. The framework was tested using a synthetic mathematical coupled model and a real‐world, coupled surface water (PRMS) and subsurface (MODFLOW) model. For the synthetic coupled model, the framework has shown its effectiveness to reveal the interplay of input variables, quantify the uncertainties within each sub-model and track their propagation through the coupled modeling system. For the PRMS-MODFLOW model, the framework has demonstrated how uncertainty in input precipitation, surface water, and subsurface water sub-models influences the different segments of a hydrograph. The results also indicate that improved predictions of high flow require a better quantification of input uncertainty. In contrast, baseflow and recession flow uncertainties largely depend on the subsurface and surface water sub-models, respectively. The presented framework, in addition to providing relatively comprehensive uncertainty information on the integrated model outputs, it helps to identify the potential sources of uncertainty that can be used for further model improvement and guide new data collection campaigns.

Application of Cross Sections Uncertainty Propagation Framework to Light and Heavy Water Reactor Systems

Abstract Uncertainty quantification has been recognized by the community as an essential component of best-estimate reactor analysis simulation because it provides a measure by which the credibility of the simulation can be assessed. In a companion paper, a framework for the propagation of nuclear data uncertainties from the multigroup level through lattice physics and core calculations and ultimately to core responses of interest has been developed. The overarching goal of this framework is to automate the propagation, prioritization, mapping, and reduction of uncertainties for reactor analysis core simulation. This paper employs both heavy and light water reactor systems to exemplify the application of this framework. Specifically, the paper is limited to the propagation of the nuclear data starting with the multigroup cross section covariance matrix and down to core responses, e.g., eigenvalue and power distribution, in steady-state core wide calculations. The goal is to demonstrate how the framework employs reduction techniques to compress the uncertainty space into a very small number of active degrees-of-freedom (DOFs), which renders the overall process computationally feasible for day-to-day engineering evaluations.

Spatially varying fuzzy multi-scale uncertainty propagation in unidirectional fibre reinforced composites

This article presents a non-probabilistic fuzzy based multi-scale uncertainty propagation framework for studying the dynamic and stability characteristics of composite laminates with spatially varying system properties. Most of the studies concerning the uncertainty quantification of composites rely on probabilistic analyses, where the prerequisite is to have the statistical distribution of stochastic input parameters. In many engineering problems, these statistical distributions remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based approach could be appropriate to characterize the effect of uncertainty. A novel concept of fuzzy representative volume element (FRVE) is developed here for accounting the spatially varying non-probabilistic source-uncertainties at the input level. Such approach of uncertainty modelling is physically more relevant than the prevalent way of modelling non-probabilistic uncertainty without considering the ply-level spatial variability. An efficient radial basis function based stochastic algorithm coupled with the fuzzy finite element model of composites is developed for the multi-scale uncertainty propagation involving multi-synchronous triggering parameters. The concept of a fuzzy factor of safety (FFoS) is discussed in this paper for evaluation of safety factor in the non-probabilistic regime. The results reveal that the present physically relevant approach of modelling fuzzy uncertainty considering ply-level spatial variability obtains significantly lower fuzzy bounds of the global responses compared to the conventional approach of non-probabilistic modelling neglecting the spatially varying attributes.

A Flexible Uncertainty Propagation Framework for General Multiphysics Systems

We present an efficient “module-based” uncertainty propagation framework for stochastic multiphysics systems. Our proposed framework facilitates modeling flexibility and independence within each module of a stochastic multiphysics solver, and extends the framework, previously introduced by the authors, to general (nonlinear) multiphysics systems via generic restriction and prolongation operators, which transform between the global and local polynomial chaos representations of the input/output data. Moreover, these operators preserve the convergence rate of monolithic generalized-polynomial-chaos--based methods, and lower the rate of growth of the overall computational costs w.r.t. the global dimension. Following a discussion of these mathematical and algorithmic details, we implement our framework on a test problem, and demonstrate its superior computational performance over an implementation of the standard monolithic framework. (An erratum is attached.)

Dealing with uncertainty in risk assessments in early stages of a CO2 geological storage project: comparison of pure-probabilistic and fuzzy-probabilistic frameworks

CO2 capture and storage is recognized as a promising solution among others to tackle greenhouse gas emissions. This technology requires robust risk assessment and management from the early stages of the project (i.e. during the site selection phase, prior to injection), which is a challenging task due to the high level of aleatory and epistemic uncertainties. This paper aims at implementing and comparing two frameworks for dealing with uncertainties: a classical probabilistic framework and a probabilistic-fuzzy-based (i.e. jointly combining fuzzy sets and probabilities) one. The comparison of both frameworks is illustrated for assessing the risk related to leakage of brine through an abandoned well on a realistic site in the Paris basin (France). For brine leakage flow computation, a semi-analytical model, requiring 25 input parameters, is used. Depending on the framework, available data is represented in a different manner (either using classical probability laws or interval-valued tools). Though the fuzzy-probabilistic framework for uncertainty propagation is computationally more expensive, it presents the major advantage to highlight situations of high degree of epistemic uncertainty: this enables nuancing a too-optimistic decision-making only supported by a single probabilistic curve (i.e. using the Monte-Carlo results). On this basis, we demonstrate how fuzzy-based sensitivity analysis can help identifying how to reduce the imprecision in an effective way, which has useful applications for additional studies. This study highlights the importance of choices in the mathematical tools for representing the lack of knowledge especially in the early phases of the project, where data is scarce, incomplete and imprecise.

A General Framework for Uncertainty Propagation Based on Point Estimate Methods

A general framework to approach the challenge of uncertainty propagation in model based prognostics is presented in this work. It is shown how the so-called Point Estimate Methods (PEMs) are ideally suited for this purpose because of the following reasons: 1) A credible propagation and representation of Gaussian (normally distributed) uncertainty can be done with a minimum of computational effort for non-linear applications. 2) Also non-Gaussian uncertainties can be propagated by evaluating suitable transfer functions inherently. 3) Confidence intervals of simulation results can be derived which do not have to be symmetrically distributed around the mean value by applying PEM in conjunction with the Cornish-Fisher expansion. 4) Moreover, the entire probability function of simulation results can be reconstructed efficiently by the proposed framework. The joint evaluation of PEM with the Polynomial Chaos expansion methodology is likely to provide good approximation results. Thus, non-Gaussian probability density functions can be derived as well. 5) The presented framework of uncertainty propagation is derivativefree, i.e. even non-smooth (non-differentiable) propagation problems can be tackled in principle. 6) Although the PEM is sample-based the overall method is deterministic. Computational results are reproducible which might be important to safety critical applications. - Consequently, the proposed approach may play an essential part in contributing to render the prognostics and health management into a more credible process. A given study of a generic uncertainty propagation problem supports this issue illustratively.

Hypersonic Aeroelastic and Aerothermoelastic Studies Using Computational Fluid Dynamics

A framework for aerothermoelastic-stability-boundary calculation for hypersonic configurations using computational fluid dynamics combined with radial basis functions for mesh deformation is developed. Application of computational fluid dynamics enables one to consider different turbulence conditions, laminar or turbulent, and different models of the air mixture, in particular real-gas model, which accounts for dissociation of molecules at high temperature. The effect of transition on the flutter margin of the heated structure is also considered using an uncertainty-propagation framework. The aerothermoelastic-stability margin of a three-dimensional low-aspect-ratio wing, representative of a control surface of a hypersonic vehicle, is investigated for various flight conditions. The system is found to be sensitive to turbulence modeling, as well as the location of the transition from laminar to turbulent flow. Real-gas effects play a minor role for the flight conditions considered. This study demonstrates the advantages of accounting for uncertainty at an early stage of the analysis, and emphasizes the important relation between transition from laminar to turbulent, thermal stresses, and stability margins of hypersonic vehicles.

A framework for propagation of uncertainty contributed by parameterization, input data, model structure, and calibration/validation data in watershed modeling

Failure to consider major sources of uncertainty may bias model predictions in simulating watershed behavior. A framework entitled the Integrated Parameter Estimation and Uncertainty Analysis Tool (IPEAT), was developed utilizing Bayesian inferences, an input error model and modified goodness-of-fit statistics to incorporate uncertainty in parameter, model structure, input data, and calibration/validation data in watershed modeling. Applications of the framework at the Eagle Creek Watershed in Indiana shows that watershed behavior was more realistically represented when the four uncertainty sources were considered jointly without having to embed watershed behavior constraints in auto-calibration. Accounting for the major sources of uncertainty associated with watershed modeling produces more realistic predictions, improves the quality of calibrated solutions, and consequently reduces predictive uncertainty. IPEAT is an innovative tool to investigate and explore the significance of uncertainty sources, which enhances watershed modeling by improved characterization and assessment of predictive uncertainty.

Uncertainty Propagation in Hypersonic Aerothermoelastic Analysis

A framework for uncertainty propagation in hypersonic aeroelastic and aerothermoelastic analyses is presented. First, the aeroelastic stability of a typical section representative of a control surface on a hypersonic vehicle is examined. Variability in the uncoupled natural frequencies of the system is modeled using beta probability distributions. Uncertainty in the flutter Mach number is computed using stochastic collocation. Next, the stability of an aerodynamically heated panel representing a component of the skin of a hypersonic vehicle is considered. In this case, uncertainty is due to the location of transition from laminar to turbulent flow and the heat flux prediction. The effect of propagating these uncertainties on vehicle behavior is determined. For both cases, uncertainty is treated using stochastic collocation, which is a new and effective approach for incorporating uncertainty in this class of problems.

Polynomial-Chaos-Based Bayesian Approach for State and Parameter Estimations

Two new recursive approaches have been developed to provide accurate estimates for posterior moments of both parameters and system states while making use of the generalized polynomial-chaos framework for uncertainty propagation. The main idea of the generalized polynomial-chaos method is to expand random state and input parameter variables involved in a stochastic differential/difference equation in a polynomial expansion. These polynomials are associated with the prior probability density function for the input parameters. Later, Galerkin projection is used to obtain a deterministic system of equations for the expansion coefficients. The first proposed approach provides means to update prior expansion coefficients by constraining the polynomial-chaos expansion to satisfy a specified number of posterior moment constraints derived from Bayes’s rule. The second proposed approach makes use of the minimum variance formulation to update generalized polynomial-chaos coefficients. The main advantage of the proposed methods is that they not only provide a point estimate for the states and parameters, but they also provide the associated uncertainty estimates along these point estimates. Numerical experiments involving four benchmark problems are considered to illustrate the properties of the proposed methods.

Estimating statistical uncertainty of Monte Carlo efficiency-gain in the context of a correlated sampling Monte Carlo code for brachytherapy treatment planning with non-normal dose distribution

Correlated sampling Monte Carlo methods can shorten computing times in brachytherapy treatment planning. Monte Carlo efficiency is typically estimated via efficiency gain, defined as the reduction in computing time by correlated sampling relative to conventional Monte Carlo methods when equal statistical uncertainties have been achieved. The determination of the efficiency gain uncertainty arising from random effects, however, is not a straightforward task specially when the error distribution is non-normal. The purpose of this study is to evaluate the applicability of the F distribution and standardized uncertainty propagation methods (widely used in metrology to estimate uncertainty of physical measurements) for predicting confidence intervals about efficiency gain estimates derived from single Monte Carlo runs using fixed-collision correlated sampling in a simplified brachytherapy geometry. A bootstrap based algorithm was used to simulate the probability distribution of the efficiency gain estimates and the shortest 95% confidence interval was estimated from this distribution. It was found that the corresponding relative uncertainty was as large as 37% for this particular problem. The uncertainty propagation framework predicted confidence intervals reasonably well; however its main disadvantage was that uncertainties of input quantities had to be calculated in a separate run via a Monte Carlo method. The F distribution noticeably underestimated the confidence interval. These discrepancies were influenced by several photons with large statistical weights which made extremely large contributions to the scored absorbed dose difference. The mechanism of acquiring high statistical weights in the fixed-collision correlated sampling method was explained and a mitigation strategy was proposed.