Uncertainty Quantification Methodology Research Articles

Nuclear data uncertainty propagation and modeling uncertainty impact evaluation in neutronics core simulation

Uncertainty analysis is a critical requirement in reactor simulation as it is used to quantify the reliability of best-estimate calculation. A comprehensive uncertainty analysis should characterize all sources of uncertainties in a computationally-feasible and scientifically-defendable manner. This manuscript employs a well-established reduced order modeling (ROM) based uncertainty quantification methodology to propagate uncertainties throughout neutronic calculations. ROM relies on recent advances in randomized data mining techniques applied to large data streams. In our proposed implementation, the nuclear data uncertainties are first propagated from multi-group level through lattice physics calculation to generate few-group parameter uncertainties, described using a vector of mean values and a covariance matrix. Employing an ROM-based compression of the covariance matrix, the few-group uncertainties are then propagated through downstream core simulation in a computationally efficient manner. This straightforward approach, albeit efficient as compared to brute force forward and/or adjoint-based methods, often employs a number of assumptions that have been unquestioned in the literature of neutronic uncertainty analysis. This manuscript argues that these assumptions could introduce another source of uncertainty referred to as modeling uncertainties, whose magnitude needs to be quantified in tandem with nuclear data uncertainties. Thus, our primary goal is to explore the interactions between these two uncertainty sources in order to assess whether modeling uncertainties have an impact on parameter uncertainties. To explore this endeavor, the impact of a number of modeling assumptions on core attributes uncertainties is quantified. The study employs a CANDU reactor model, with Serpent and NEWT as lattice physics solvers and NESTLE-C as core simulator. The modeling assumptions investigated include those related with the uncertainty propagation method employed, e.g., deterministic vs. stochastic, the few-group energy structure employed to represent the cross-sections, the resonance treatment in lattice physics calculation, the reference values for the cross-section, and the number of samples employed to render ROM compression. Results indicate that some of the modeling assumptions could have a non-negligible impact on the core responses propagated uncertainties, highlighting the need for a more comprehensive approach to combine parameter and modeling uncertainties.

Contribution to Online System-level Prognostics based on Adaptive Degradation Models

Considering traditional model-based prognostics approaches, a previously defined model is required to estimate the system’s health state and then propagate it to predict the system remaining useful life (SRUL). Following a Bayesian framework, the result of this prior estimation is updated by in-field measurements without changing the model parameters. Nevertheless, in the case of prognostics at system-level, solely updating prior health state, based on the pre-determined model, is no longer sufficient because numerous mutual interactions between components cause multiple uncertainties in system degradation modeling, and then can lead to inaccurate SRUL prediction. Therefore, this paper proposes a new methodology for online joint uncertainty quantification and model estimation based on particle filtering (PF) and gradient descent (GD). In detail, the inoperability input-output model (IIM) is used to characterize system degradations considering interactions between components and effects of the mission profile; and then the inoperability of system components is estimated in a probabilistic manner using PF. In the case of consecutive discrepancy between the prior and posterior estimates of the system health state, GD is used to correct and to adapt the IIM parameters. To illustrate the effectiveness of the proposed methodology and its suitability for an online implementation, the Tennessee Eastman Process is investigated as a case study.

Surrogate-assisted Bayesian inversion for landscape and basin evolution models

Abstract. The complex and computationally expensive nature of landscape evolution models poses significant challenges to the inference and optimization of unknown model parameters. Bayesian inference provides a methodology for estimation and uncertainty quantification of unknown model parameters. In our previous work, we developed parallel tempering Bayeslands as a framework for parameter estimation and uncertainty quantification for the Badlands landscape evolution model. Parallel tempering Bayeslands features high-performance computing that can feature dozens of processing cores running in parallel to enhance computational efficiency. Nevertheless, the procedure remains computationally challenging since thousands of samples need to be drawn and evaluated. In large-scale landscape evolution problems, a single model evaluation can take from several minutes to hours and in some instances, even days or weeks. Surrogate-assisted optimization has been used for several computationally expensive engineering problems which motivate its use in optimization and inference of complex geoscientific models. The use of surrogate models can speed up parallel tempering Bayeslands by developing computationally inexpensive models to mimic expensive ones. In this paper, we apply surrogate-assisted parallel tempering where the surrogate mimics a landscape evolution model by estimating the likelihood function from the model. We employ a neural-network-based surrogate model that learns from the history of samples generated. The entire framework is developed in a parallel computing infrastructure to take advantage of parallelism. The results show that the proposed methodology is effective in lowering the computational cost significantly while retaining the quality of model predictions.

Uncertainty Quantification in Small-Timescale Model-Based Fatigue Crack Growth Analysis Using a Stochastic Collocation Method

Due to the uncertainties originating from the underlying physical model, material properties and the measurement data in fatigue crack growth (FCG) processing, the prediction of fatigue crack growth lifetime is still challenging. The objective of this paper was to investigate a methodology for uncertainty quantification in FCG analysis and probabilistic remaining useful life prediction. A small-timescale growth model for the fracture mechanics-based analysis and predicting crack-growth lifetime is studied. A stochastic collocation method is used to alleviate the computational difficulties in the uncertainty quantification in the small-timescale model-based FCG analysis, which is derived from tensor products based on the solution of deterministic FCG problems on sparse grids of collocation point sets in random space. The proposed method is applied to the prediction of fatigue crack growth lifetime of Al7075-T6 alloy plates and verified by fatigue crack-growth experiments. The results show that the proposed method has the advantage of computational efficiency in uncertainty quantification of remaining life prediction of FCG.

Uncertainty quantification in density estimation from background-oriented Schlieren measurements

We present an uncertainty quantification methodology for density estimation from background-oriented Schlieren (BOS) measurements, in order to provide local, instantaneous, a posteriori uncertainty bounds on each density measurement in the field of view. Displacement uncertainty quantification algorithms from cross-correlation-based particle image velocimetry are used to estimate the uncertainty in the dot pattern displacements obtained from cross-correlation for BOSs and assess their feasibility. In order to propagate the displacement uncertainty through the density integration procedure, we also develop a novel methodology via the Poisson solver using sparse linear operators. Testing the method using synthetic images of a Gaussian density field showed agreement between the propagated density uncertainties and the true uncertainty. Subsequently, the methodology is experimentally demonstrated for supersonic flow over a wedge, showing that regions with sharp changes in density lead to an increase in density uncertainty throughout the field of view, even in regions without these sharp changes. The uncertainty propagation is influenced by the density integration scheme, and for the Poisson solver the density uncertainty on average increases on moving away from the regions where the Dirichlet boundary conditions are specified.

Probabilistic Risk-Based Operational Safety Bound for Rotary-Wing Unmanned Aircraft Systems Traffic Management

A novel method to determine probabilistic operational safety bound for rotary-wing unmanned aircraft systems (UAS) traffic management is proposed in this paper. The key idea is to combine a deterministic model for rotary-wing UAS flying distance estimation to avoid conflict and a probabilistic uncertainty quantification methodology to evaluate the risk level (defined as the probability of failure) of separation loss between UAS. The proposed methodology results in a dynamic and probabilistic airspace reservation to ensure the safety and efficiency for future UAS operations. The model includes UAS performance, system updating frequency and accuracy, and weather conditions. Also, the parameterized probabilistic model includes various uncertainties from different sources and develops an anisotropic operational safety bound. Monte Carlo simulations are used to illustrate the operational safety bound determination with a specified risk level (i.e., probability of failure). It is known that uncertainty plays an important role in determining the operational safety bound size, and the proposed methodology provides a simple and efficient quantification of uncertainty impact on the safety bound with a prescribed risk level. It is also providing a useful tool to quantify uncertainty reduction with additional information and measurements in future UAS operations.

Uncertainty study on thermal and energy performances of a deterministic parameters based optimal aerogel glazing system using machine-learning method

Uncertainty and sensitivity analyses of deterministic parameters based optimal aerogel glazing system are necessary due to multi-dimensional uncertainties in the real working condition, whereas thermal and energy performances of aerogel glazing system, in the academia, are normally characterized by deterministic parameters. In this study, a generic uncertainty quantification methodology was proposed using the two-dimensional Markov Chain Monte Carlo to quantify both aleatory and epistemic uncertainties of scenario parameters in the aerogel glazing system. A surrogate model, trained by mathematical heat and optical models using the machine-learning based data-driven method, was developed to predict the thermal and energy performances under multi-level scenario uncertainties. Results showed that, the developed surrogate model is efficient to deal with computational complexity of sophisticated light and heat transfer processes. When considering scenario uncertainties, the annual value of heat flux is reduced from 237.2 to 185.3 kWh/(m2.a) by 21.9%, and the annual value of total heat gain is reduced from 267.2 to 209.5 kWh/m2.a by 21.6%. This study proposes a generic methodology for multi-dimensional uncertainties’ quantification and a surrogate model for thousands of cases-based uncertainty analysis. Approaches for the stochastic uncertainty analysis on aerogel glazing system were presented, which can promote the optimal design in buildings.

Uncertainty Quantification Analysis of a Pressurized Fuel Cell Hybrid System

Abstract Pressurized solid oxide fuel cell (SOFC) systems are a sustainable opportunity for improvement over conventional systems, featuring high electric efficiency, potential for cogeneration applications, and low carbon emissions. Such systems are usually analyzed in deterministic conditions. However, it is widely demonstrated that such systems are affected significantly by uncertainties, both in component performance and operating parameters. This paper aims to study the propagation of uncertainties related both to the fuel cell (ohmic losses, anode ejector diameter, and fuel gas composition) and the gas turbine cycle characteristics (compressor and turbine efficiencies, recuperator pressure losses). The analysis is carried out on an innovative hybrid system layout, where a turbocharger is used to pressurize the fuel cell, promising better cost effectiveness then a microturbine-based hybrid system, at small scales. Due to plant complexity and high computational effort required by uncertainty quantification methodologies, a response surface (RS) is created. To evaluate the impact of the aforementioned uncertainties on the relevant system outputs, such as overall efficiency and net electrical power, the Monte Carlo method is applied to the RS. Particular attention is focused on the impact of uncertainties on the opening of the turbocharger wastegate valve, which is aimed at satisfying the fuel cell constraints at each operating condition.

Machine-learning based study on the on-site renewable electrical performance of an optimal hybrid PCMs integrated renewable system with high-level parameters’ uncertainties

The uncertainty and sensitivity analyses for multivariables of the optimal system based on deterministic parameters are necessary, as multivariables are full of uncertainties in the real operation. However, the generic methodology for multi-dimensional uncertainties quantification is rare, and the energy performance simulation is normally at high computational cost, especially considering a huge amount of parameters’ uncertainties. In this study, the on-site renewable electricity generation of an optimal hybrid renewable system based on deterministic parameters, was investigated, under high-level parameters’ uncertainties. A generic uncertainty quantification methodology was proposed using the two-dimensional Markov Chain Monte Carlo to quantify multi-dimensional uncertainties. A machine-learning based data-driven model, using the supervised machine learning with high computational efficiency, was developed to predict the on-site renewable electricity generation, and thereafter used for the uncertainty and sensitivity analyses. Compared with the deterministic scenario parameters, the cases with the scenario uncertainties can increase the peak power and the total amount of the on-site renewable electricity generation. This study proposes a novel generic uncertainty quantification methodology, together with a machine-learning based data-driven model for conducting the uncertainty analysis of an optimal renewable system based on deterministic parameters, which are important for the promotion of renewable and sustainable buildings.

Surrogate modeling of advanced computer simulations using deep Gaussian processes

The continuous advancements in computer power and computational modeling through high-fidelity and multiphysics simulations add more challenges on assessing their predictive capability. In this work, metamodeling or surrogate modeling through deep Gaussian processes (DeepGP) is performed to construct surrogates of advanced computer simulations drawn from the nuclear engineering area. This work is centered around three major ideas: (1) surrogate modeling through deep Gaussian processes (DeepGP), (2) simulation assessment through surrogate-based uncertainty quantification (UQ) methodologies, and (3) drawing conclusions regarding the underlying uncertainty of the four simulations considered in this paper. First, DeepGP models are trained, optimized, and validated to yield variety of features: (1) achieving high accuracy (small error metrics) on the validation set, (2) automatically capturing the surrogate model uncertainty (i.e. interpolation errors), (3) fitting multiple outputs with different scales simultaneously, (4) handling high dimensional input spaces, and (5) learning from small data amounts. Second, the validated DeepGP surrogates are utilized to efficiently perform UQ tasks such as uncertainty propagation (through Monte Carlo sampling), parameter screening (through Morris screening), and variance decomposition (through Sobol Indices) to investigate the selected simulations. Third, the thermal-hydraulics (fluid flow) results demonstrate the importance of inlet temperature uncertainty in void fraction predictions. For the reactor physics application (fuel depletion/consumption), DeepGP accurately captures the uncertainty in criticality calculations, which is about 0.6% (i.e. a considerable value for this application). For the application of kinetic parameters (nuclear data), DeepGP successfully explains 95% or more of the variance in all 12 outputs. Finally, DeepGP-based UQ analysis of the fuel performance application (materials science) shows the importance of the clad surface temperature, fuel porosity, and linear heat rate in explaining the variance of the maximum fuel centerline and surface temperatures.

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.

Prediction and reduction of runtime in non-intrusive forward UQ simulations

To foster predictive simulations, a variety of methods have recently been developed to efficiently tackle uncertainty quantification (UQ) in complex, computational intensive problems. Many of these methods are non-intrusive and, thus, result in a (large) number of embarrassingly parallel black-box evaluations of the underlying simulation codes. While the focus of development is typically on the number of black-box evaluations, which represents the bulk of the computational workload, an additional level of potential performance gains exists. In many scenarios, uncertain input leads not only to uncertain outputs, but also to a varying and thus stochastic runtime of the simulation codes. For scheduling the individual black-box runs, this information is typically not taken into account, resulting in non-negligible idling times on parallel systems. In this contribution, we compare a variety of different scheduling strategies for non-intrusive UQ scenarios using the non-intrusive polynomial chaos approach. In particular, we propose to construct a surrogate model for the runtime of the application using the identical UQ methodology as for the original problem. Using this model to predict the runtimes for subsequent black-box runs allows for (heuristical) optimization of the scheduling. The method has been tested for the forward quantification of uncertainty on academic models and on a pedestrian simulation in the context of evacuation scenarios. This approach allows speed-up factors of about two for the total runtime and can be generalised to a large variety of applications that incorporate parameter-dependent runtime.

Uncertainty Quantification of Non-Dimensional Parameters for a Film Cooling Configuration in Supersonic Conditions

In transonic high-pressure turbine stages, oblique shocks originating from vane trailing edges impact the suction side of each adjacent vane. High-pressure vanes are cooled to tolerate the combustor exit-temperature levels, then it is highly probable that shock impingement will occur in proximity to a row of cooling holes. The presence of such a shock, together with the inevitable manufacturing deviations, alters the location of the shock impingement and of the performance parameters of each cooling hole. The present work provides a general description of the aero-thermal field that occurs on the rear suction side of a cooled vane. Computational Fluid Dynamics (CFD) is used to evaluate the deterministic response of the selected configurations in terms of adiabatic effectiveness, discharge coefficient, blowing ratio, density ratio, and momentum ratio. Turbulence is modelled by using both the Shear Stress Transport method (SST) and the Reynolds Stress Model (RSM) implemented in ANSYS® FLUENT®. The obtained results are compared with the experimental data obtained by the Institut für Thermische Strömungsmaschinen in Karlsruhe. Two uncertainty quantification methodologies based on Hermite polynomials and Padè–Legendre approximants are used to consider the probability distribution of the geometrical parameters and to evaluate the response surfaces for the system response quantities. Trailing-edge and cooling-hole diameters have been considered to be aleatory unknowns. Uncertainty quantification analysis allows for the assessment of the mutual effects on global and local parameters of the cooling device. Obtained results demonstrate that most of the parameters are independent by the variation of the aleatory unknowns while the standard deviation of the blowing ratio associated with the hole diameter uncertainty is around 12%, with no impact by the trailing-edge thickness. No relevant advantages are found using either SST model or RSM in combination with Hermite polynomials and Padè–Legendre approximants.

Uncertainty quantification for acoustic wave propagation in a shallow water environment

Sound wave propagation in a shallow water environment is complex due to e.g. the uncertainties of sound speed profile being inhomogeneous and imprecisely measured, the bottom reflections, etc. The propagation and influence of several uncertainty parameters are quantified in this paper. A four-layer model, which can approximately represent a wide range of shallow water environments, is considered; six parameters representing sound speed profile and water depth are considered as random variables. We investigate how the wave field (pressure) in this model is influenced by these uncertainties. For this purpose, the sound field is computed for different realizations of the random variables, when the medium is excited with sources whose frequencies are appropriate, for example, for marine seismic exploration applications. Since classical Monte Carlo methods require a huge sample size to converge, we use three surrogate modeling techniques (Kriging, Polynomial Chaos, and Polynomial Chaos-based Kriging). The proposed methods require much smaller sample sizes, which makes the uncertainty quantification (UQ) possible. Wavelength-to-depth ratio (λ∕d) is introduced as the key parameter that defines the degree of interaction (reflection and transmission) of the sound waves with the boundaries of the shallow water waveguide. The results show that for small and large values of λ∕d, the wave field is more sensitive to the variations of the water depth and the velocity of the bottom layer, respectively. The robustness (precision) of the surrogate models is shown to decrease for lower values of λ∕d. The proposed UQ methodology can be used for more complicated underwater environments; it is even more advantageous because it can efficiently deal with a large number of model uncertainty parameters and identify the most influential ones.

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

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

Modeling Average Pressure and Volume Fraction of a Fluidized Bed Using Data-Driven Smart Proxy

Simulations can reduce the time and cost to develop and deploy advanced technologies and enable their rapid scale-up for fossil fuel-based energy systems. However, to ensure their usefulness in practice, the credibility of the simulations needs to be established with uncertainty quantification (UQ) methods. The National Energy Technology Laboratory (NETL) has been applying non-intrusive UQ methodologies to categorize and quantify uncertainties in computational fluid dynamics (CFD) simulations of gas-solid multiphase flows. To reduce the computational cost associated with gas-solid flow simulations required for UQ analysis, techniques commonly used in the area of artificial intelligence (AI) and data mining are used to construct smart proxy models, which can reduce the computational cost of conducting large numbers of multiphase CFD simulations. The feasibility of using AI and machine learning to construct a smart proxy for a gas-solid multiphase flow has been investigated by looking at the flow and particle behavior in a non-reacting rectangular fluidized bed. The NETL’s in house multiphase solver, Multiphase Flow with Interphase eXchanges (MFiX), was used to generate simulation data for the rectangular fluidized bed. The artificial neural network (ANN) was used to construct a CFD smart proxy, which is able to reproduce the CFD results with reasonable error (about 10%). Several blind cases were used to validate this technology. The results show a good agreement with CFD runs while the approach is less computationally expensive. The developed model can be used to generate the time averaged results of any given fluidized bed with the same geometry with different inlet velocity in couple of minutes.

Active subspace analysis and uncertainty quantification for a polydomain ferroelectric phase-field model

We perform parameter subset selection and uncertainty analysis for phase-field models that are applied to the ferroelectric material lead titanate. A motivating objective is to determine which parameters are influential in the sense that their uncertainties directly affect the uncertainty in the model response, and fix noninfluential parameters at nominal values for subsequent uncertainty propagation. We employ Bayesian inference to quantify the uncertainties of gradient exchange parameters governing 180° and 90° tetragonal phase domain wall energies. The uncertainties of influential parameters determined by parameter subset selection are then propagated through the models to obtain credible intervals when estimating energy densities quantifying polarization and strain across domain walls. The results illustrate various properties of Landau and electromechanical coupling parameters and their influence on domain wall interactions. We employ energy statistics, which quantify distances between statistical observations, to compare credible intervals constructed using a complete set of parameters against an influential subset of parameters. These intervals are obtained from the uncertainty propagation of the model input parameters on the domain wall energy densities. The investigation provides critical insight into the development of parameter subset selection, uncertainty quantification, and propagation methodologies for material modeling domain wall structure evolution, informed by density functional theory simulations.

Bayeslands: A Bayesian inference approach for parameter uncertainty quantification in Badlands

Bayesian inference provides a rigorous methodology for estimation and uncertainty quantification of unknown parameters in geophysical forward models. Badlands is a landscape evolution model that simulates topography development at various space and time scales. Badlands consists of a number of geophysical parameters that needs estimation with appropriate uncertainty quantification; given the observed present-day ground truth such as surface topography and the stratigraphy of sediment deposition through time. The inference of the unknown parameters is challenging due to the scarcity of data, sensitivity of the parameter setting, and complexity of the model. In this paper, we take a Bayesian approach to provide inference using Markov chain Monte Carlo sampling (MCMC). We present Bayeslands; a Bayesian framework for Badlands that fuses information obtained from complex forward models with observational data and prior knowledge. As a proof-of-concept, we consider a synthetic and real-world topography with two parameters for Bayeslands; namely, precipitation and erodibility. We demonstrate the challenge in sampling irregular and multi-modal posterior distributions using a likelihood surface that has a range of sub-optimal modes. The results of the experiments show that Bayeslands yields a promising distribution of the selected Badlands parameters.

Uncertainty Quantification method for CFD applied to the turbulent mixing of two water layers – II: Deterministic Sampling for input uncertainty

Computer simulations are frequently used for design and safety analyses of nuclear installations. In such computer simulations, uncertainties in the outcome are present, for instance, due to uncertainties in the input parameters of the analyses. It is important to assess and quantify these uncertainties, especially when the simulations form the basis for nuclear reactor safety or licensing.Quantification of the uncertainties in Computational Fluid Dynamics (CFD) analyses is currently seldom done, since the existing and robust way of Uncertainty Quantification (UQ) involves a large number of computations, and CFD computations are computationally very demanding. The long run-times and large computational efforts do simply not allow for thousands of CFD simulations. Therefore, smart UQ approaches for CFD are being tested, developed and validated at NRG, which allow a full quantification of the uncertainties, while the required number of computations is still acceptable.The UQ methodology used by NRG is based on the ASME Verification and Validation (V&V) standard for UQ in CFD applications, combined with a propagation technique to evaluate the input uncertainty, and Richardson extrapolation to evaluate the spatial discretization uncertainty. This CFD-UQ methodology proved to be efficient thanks to the Latin Hypercube Sampling (LHS) technique, which samples the uncertain input parameters prior to CFD propagation, in contrast to the computationally expensive and fully random Monte Carlo sampling (MCS) technique. Nevertheless, MCS and LHS are both Random Sampling (RS) techniques, which will always be affected to some degree by a random sampling error. Therefore, Deterministic Sampling (DS) techniques are tested in this work to evaluate the input uncertainty, instead of LHS. The DS techniques proved to be more efficient, producing similar results with less computations.

Eigenvector perturbation methodology for uncertainty quantification of turbulence models

Reynolds-averaged Navier-Stokes (RANS) models are the primary numerical recourse to investigate complex engineering turbulent flows in industrial applications. However, to establish RANS models as reliable design tools, it is essential to provide estimates for the uncertainty in their predictions. In the recent past, an uncertainty estimation framework relying on eigenvalue and eigenvector perturbations to the modeled Reynolds stress tensor has been widely applied with satisfactory results. However, the methodology for the eigenvector perturbations is not well established. Evaluations using only eigenvalue perturbations do not provide comprehensive estimates of model form uncertainty, especially in flows with streamline curvature, recirculation, or flow separation. In this article, we outline a methodology for the eigenvector perturbations using a predictor-corrector approach, which uses the incipient eigenvalue perturbations along with the Reynolds stress transport equations to determine the eigenvector perturbations. This approach was applied to benchmark cases of complex turbulent flows. The uncertainty intervals estimated using the proposed framework exhibited substantial improvement over eigenvalue-only perturbations and are able to account for a significant proportion of the discrepancy between RANS predictions and high-fidelity data.