Test-Based Uncertainty Quantification and Propagation Using Hurty/Craig-Bampton Substructure Representations

In recent years, the demand to provide best-estimate predictions with confident bounds is increasing for the nuclear reactor performance and safety analysis. The Organisation for Economic Co-operation and Development Nuclear Energy Agency has been developing an international benchmark of the light water reactor (LWR) uncertainty analysis in modeling (UAM) for the examination of uncertainty quantification and propagation methodologies with various modeling and simulation code systems. The objective of the present work is to develop an uncertainty propagation mechanism based on the stochastic sampling method by taking into account the uncertainties of both basic nuclear data and fuel modeling parameters in the simulation of pressurized water reactors (PWRs) that can be incorporated in the conventional LWR simulation approach. More specifically, the Three Mile Island Unit 1–related exercises from the LWR-UAM benchmark have been modeled using the coupled TRACE/PARCS code system in the three-dimensional core representation. The input uncertainties of the neutronics simulation include few-group cross sections and kinetics parameters generated using the Sampler/Polaris sequence of SCALE 6.2.1. Several heat transfer–related variables for the fuel modeling were considered as sources of input uncertainty of the thermal-hydraulics simulations, including the thermal conductivity of fuel and cladding, fuel heat capacity, and the gap conductance. Dakota was used to sample input parameters of the coupled code system and to perform the uncertainty analysis. Two types of simulations were conducted: steady-state calculation at hot full-power condition and transient scenario initiated by the spatially asymmetric rod ejection accident. Quantities of interest for the steady-state calculation, including core multiplication factor and power peaking factors, were calculated with associated uncertainties. For transient calculations, best-estimate plus uncertainty results of the time evolution of core reactivity, core power, and peak fuel temperature were generated and analyzed. The Wilks’ formula was used to determine the necessary sample size to achieve a 95% confidence of the 95% limit of output quantities of interest. Although the uncertainty propagation and quantification method presented in this paper was developed for PWRs, it could be in general applicable to the multiphysics uncertainty quantification of other types of LWR cores.

In this paper, a novel random matrix approach is utilized to characterize the uncertainty propagation in the articulated robotic systems. This method permits the probabilistic formulation to take into account the complexity and structural inter-dependencies of the kinematic manipulators. However, conventional random variable/vector based approaches are unable to address this issue. The kinematic Jacobian matrix of the manipulator is probabilistically formulated so that it adopts a random symmetric positive definite perturbation matrix. The density function of the perturbation matrix is constructed using maximum entropy principle that yields a Wishart distribution with specific parameters. The only information used in the construction of the Jacobian matrix probability model are mean and a scalar value dispersion parameter. We propose to use this framework to study the propagation of the uncertainties in both serial and parallel manipulators. Parallel architecture manipulators offer superior manipulation accuracy and structural rigidity compared to serial chain manipulators but feature more complex physical layout and limited workspace. Therefore, for the design purposes, a trade off between two different architectures is required. This necessitates the mathematical characterization of their capability with respect to the uncertainty propagation. As a case study, the developed random matrix method is implemented on a planar RRR serial and a planar 3-RRR parallel manipulator, that are performing an identical task. A Monte Carlo analysis is conducted and the statistics of the response of both systems provides appropriate measures to compare their dexterity dealing with the uncertainties. The final results verify superiority of the parallel architecture.

Uncertainty quantification and propagation with probability boxes

Robust uncertainty quantification in structural dynamics under scarse experimental modal data: A Bayesian-interval approach

The idea underlying the SICODYN (SImulations credibility via test-analysis COrrelation and uncertainty quantification in structural DYNamics) funded research project is to give easy tools, based on tested methodologies, to a priori estimate the confidence associated to a dynamical simulation-based prediction. The project is based on a complex built-up pump in industrial environment. Gathering 13 French academic and industrial partners, it is organized in 6 parts. In Part 1, an inventory of the benchmarks in structure dynamics and a review of methods leading to credible models are performed. Experimental benchmarks based on in-situ measurements permit the quantification of experimental variability, related to nominally-identical structures or due to operator and modal identification methodology (Part 2). Numerical benchmarks (Part 3) lead to observe the corresponding total numerical variability. Part 4 is devoted to test-analysis correlation methods. In Part 5, both parametric and non parametric methods are confronted in order to quantify the parametrical and model-form uncertainties, either in a deterministic (method of intervals…) or probabilistic context. The objective of Part 6 is to estimate the capacity of uncertainty quantification methods to represent the observed numerical variability, and to select, or adapt, some of them, to propose simple tools usable in industrial context.

Surrogate models are statistical or conceptual approximations for more complex simulation models. In this context, it is crucial to propagate the uncertainty induced by limited simulation budget and surrogate approximation error to predictions, inference, and subsequent decision-relevant quantities. However, quantifying and then propagating the uncertainty of surrogates is usually limited to special analytic cases or is otherwise computationally very expensive. In this paper, we propose a framework enabling a scalable, Bayesian approach to surrogate modeling with thorough uncertainty quantification, propagation, and validation. Specifically, we present three methods for Bayesian inference with surrogate models given measurement data. This is a task where the propagation of surrogate uncertainty is especially relevant, because failing to account for it may lead to biased and/or overconfident estimates of the parameters of interest. We showcase our approach in three detailed case studies for linear and nonlinear real-world modeling scenarios. Uncertainty propagation in surrogate models enables more reliable and safe approximation of expensive simulators and will therefore be useful in various fields of applications.

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of dierent observation or measurement operators on the update.

Quantification and propagation of neutronics uncertainties of the Kozloduy-6 VVER-1000 fuel assembly using SCALE 6.2.1 within the NEA/OECD benchmark for uncertainty analysis in modelling of LWRs

The Bayesian framework for hierarchical modeling is applied to quantify uncertainties, arising mainly due to manufacturing variability, for a group of identical structural components. Parameterized Gaussian models of uncertainties in structural model parameters are introduced that depend on a set of hyperparameters. Laplace methods of asymptotic approximation and, more accurate, stochastic simulation algorithms are used for estimating posterior joint and marginal distributions of structural model parameters and hyper-parameters. The posterior distribution of the hyperparameters is formulated as the product of as many multi-dimensional integrals as the number of structural components tested, with the dimension in each integral equal to the number of physical uncertain model parameters. Novel analytical approximations of these integrals are proposed that are shown to be computationally efficient, accurate and highly parallelized. A parallelized blocked Gibbs sampler is also proposed to efficiently generate samples simultaneously along the different uncertain parameter subspaces associated with the different number of structural component tested. Theoretical and computational developments are illustrated using simulated data for modal frequencies obtained from a number of identical bar structures.

ICME approaches provide decision support for materials design by establishing quantitative process-structure-property relations. Confidence in the decision support, however, must be achieved by establishing uncertainty bounds in ICME model chains. The quantification and propagation of uncertainty in computational materials science, however, remains a rather unexplored aspect of computational materials science approaches. Moreover, traditional uncertainty propagation frameworks tend to be limited in cases with computationally expensive simulations. A rather common and important model chain is that of CALPHAD-based thermodynamic models of phase stability coupled to phase-field models for microstructure evolution. Propagation of uncertainty in these cases is challenging not only due to the sheer computational cost of the simulations but also because of the high dimensionality of the input space. In this work, we present a framework for the quantification and propagation of uncertainty in a CALPHAD-based elastochemical phase-field model. We motivate our work by investigating the microstructure evolution in Mg2SixSn1−x thermoelectric materials. We first carry out a Markov Chain Monte Carlo-based inference of the CALPHAD model parameters for this pseudobinary system and then use advanced sampling schemes to propagate uncertainties across a high-dimensional simulation input space. Through high-throughput phase-field simulations we generate 200,000 time series of synthetic microstructures and use machine learning approaches to understand the effects of propagated uncertainties on the microstructure landscape of the system under study. The microstructure dataset has been curated in the Open Phase-field Microstructure Database (OPMD), available at http://microstructures.net.

Uncertainty Propagation in a Multiscale CALPHAD-Reinforced Elastochemical Phase-Field Model

One of the main issues addressed in any engineering design problem is to predict the performance of the component or system as accurately and realistically as possible, taking into account the variability of operating conditions or the uncertainty on input data (boundary conditions or geometry tolerance). In this paper, the propagation of uncertainty on boundary conditions through a numerical model of supersonic nozzle is investigated. The evaluation of the statistics of the problem response functions is performed following ‘Surrogate-Based Uncertainty Quantification’. The approach involves: (a) the generation of a response surface starting from a DoE in order to approximate the convergent–divergent ‘physical’ model (expensive to simulate), (b) the application of the UQ technique based on the LHS to the meta-model. Probability Density Functions are introduced for the inlet boundary conditions in order to quantify their effects on the output nozzle performance. The physical problem considered is very relevant for the experimental tests on the UQ approach because of its high non-linearity. A small perturbation to the input data can drive the solution to a completely different output condition. The CFD simulations and the Uncertainty Quantification were performed by coupling the open source Dakota platform with the ANSYS Fluent® CFD commercial software: the process is automated through scripting. The procedure adopted in this work demonstrate the applicability of advanced simulation techniques (such as UQ analysis) to industrial technical problems. Moreover, the analysis highlights the practical use of the uncertainty quantification techniques in predicting the performance of a nozzle design affected by off-design conditions with fluid-dynamic complexity due to strong nonlinearity.

Particle Image Velocimetry (PIV) is a non-invasive measurement technique which resolves the flow velocity by taking instantaneous snapshots of tracer particle motion in the flow and uses digital image cross-correlation to estimate the particle shift up to subpixel accuracy. The measurement chain incorporates numerous sets of parameters, such as the particle displacements, the particle image size, the flow shear rate, the out-of-plane motion for planar PIV and image noise to name a few, and these parameters are interrelated and influence the final velocity estimate in a complicated way. In the last few decades, PIV has become widely popular by virtue of developments in both the hardware capabilities and correlation algorithms, especially with the scope of 3-component (3C) and 3-dimensional (3D) velocity measurements using stereo-PIV and tomographic-PIV techniques, respectively. The velocity field measurement not only leads to other quantities of interest such as Pressure, Reynold stresses, vorticity or even diffusion coefficient, but also provides a reference field for validating numerical simulations of complex flows. However, such a comparison with CFD or applicability of the measurement to industrial design requires one to quantify the uncertainty in the PIV estimated velocity field. Even though the PIV community had a strong impetus in minimizing the measurement error over the years, the problem of uncertainty estimation in local instantaneous PIV velocity vectors have been rather unnoticed. A typical norm had been to assign an uncertainty of 0.1 pixels for the whole field irrespective of local flow features and any variation in measurement noise. The first article on this subject was published in 2012 and since then there has been a concentrated effort to address this gap. The current dissertation is motivated by such a requirement and aims to compare the existing 2D PIV uncertainty methods, propose a new method to directly estimate the planar PIV uncertainty from the correlation plane and subsequently propose the first comprehensive methods to quantify the measurement uncertainty in stereo-PIV and 3D Particle Tracking Velocimetry (PTV) measurements.The uncertainty quantification in a PIV measurement is, however, non-trivial due to the presence of multitude of error sources and their non-linear coupling through the measurement chain transfer function. In addition, the advanced algorithms apply iterative correction process to minimize the residual which increases the complexity of the process and hence, a simple data-reduction equation for uncertainty propagation does not exist. Furthermore, the calibration or a reconstruction process in a stereo or volumetric measurement makes the uncertainty estimation more challenging. Thus, current uncertainty quantification methods develop a-posterior models utilizing the evaluated displacement information and combine it with either image information, correlation plane information or even calibration “disparity map” information to find the desired uncertainties in the velocity estimates.

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

An adaptive data-driven subspace polynomial dimensional decomposition for high-dimensional uncertainty quantification based on maximum entropy method and sparse Bayesian learning