Uncertainty Quantification Methodology Research Articles

Democratizing uncertainty quantification

Uncertainty Quantification (UQ) is vital to safety-critical model-based analyses, but the widespread adoption of sophisticated UQ methods is limited by technical complexity. In this paper, we introduce UM-Bridge (the UQ and Modeling Bridge), a high-level abstraction and software protocol that facilitates universal interoperability of UQ software with simulation codes. It breaks down the technical complexity of advanced UQ applications and enables separation of concerns between experts. UM-Bridge democratizes UQ by allowing effective interdisciplinary collaboration, accelerating the development of advanced UQ methods, and making it easy to perform UQ analyses from prototype to High Performance Computing (HPC) scale.In addition, we present a library of ready-to-run UQ benchmark problems, all easily accessible through UM-Bridge. These benchmarks support UQ methodology research, enabling reproducible performance comparisons. We demonstrate UM-Bridge with several scientific applications, harnessing HPC resources even using UQ codes not designed with HPC support.

Uncertainty quantification of structural blade parameters for the aeroelastic damping of wind turbines: a code-to-code comparison

Abstract. Uncertainty quantification (UQ) is a well-established category of methods to estimate the effect of parameter variations on a quantity of interest based on a solid mathematical foundation. In the wind energy field most UQ studies focus on the sensitivity of turbine loads. This article presents a framework, wrapped around a modern Python UQ library, to analyze the impact of uncertain turbine properties on aeroelastic stability. The UQ methodology applies a polynomial chaos expansion surrogate model. A comparison is made between different wind turbine simulation tools on the engineering model level (alaska/Wind, Bladed, HAWC2/HAWCStab2, and Simpack). Two case studies are used to demonstrate the effectiveness of the method to analyze the sensitivity of the aeroelastic damping of an unstable turbine mode to variations of structural blade cross-section parameters. The code-to-code comparison shows good agreement between the simulation tools for the reference model, but also significant differences in the sensitivities.

Global Sensitivity Analysis for Segmented Inverse Uncertainty Quantification in the Safety Analysis of Nuclear Power Plants

Within the Best Estimate Plus Uncertainty framework for the safety analysis of Nuclear Power Plants, the quantification of the uncertainties affecting the Thermal-Hydraulics (T-H) codes used is crucial. For this, Inverse Uncertainty Quantification (IUQ) methodologies are being developed for determining the probability density functions of relevant T-H codes input parameters, based on experimental data from Separate Effect Tests (SETs) experimental facilities. In practice, IUQ is challenged by the large range of variability of the experimental data in terms of Initial and Boundary Conditions (ICs & BCs), because the experimental campaigns are designed to cover the widest possible domain of conditions with the smallest number of experiments, so that same or similar ICs and BCs are seldomly repeated. To address this issue, we propose to use global sensitivity analysis, to tailor the IUQ on specific sub-regions described by segmented ICs & BCs domains. The methodology proposed is exemplified on two SETs, namely Sozzi-Sutherland and Super Moby Dick, whose experimental databases have been made available in the ATRIUM (Application Tests for Realization of Inverse Uncertainty quantification and validation Methodologies in thermal hydraulics) project promoted by the OECD/NEA/CSNI. The results obtained are superior to those of traditional IUQ methodologies for models highly sensitive to ICs & BCs.

Kriging Methodology for Uncertainty Quantification in Computational Electromagnetics

Kriging Methodology for Uncertainty Quantification in Computational Electromagnetics

A fast uncertainty quantification methodology and sampling technique for joint probability distribution of the Arrhenius rate expression: a case study applied to H2/CO kinetic mechanism

This work proposes a fast, novel, mathematically robust, and elegant unconstrained Method of Uncertainty Quantification (MUQ) for the temperature-dependent Arrhenius rate constant using Cholesky Decomposition (CD) of the covariance matrix of the Arrhenius parameters. The Arrhenius parameters of a reaction are treated as normally distributed correlated random variables. The MUQ method lends itself to an approach for Sampling of Arrhenius Curves (SAC), which automatically ensures that the generated samples are consistent with the distribution of Arrhenius parameters. The Method of Uncertainty Quantification and Sampling of Arrhenius Curves (MUQ-SAC) is used to train a quadratic polynomial response surface (PRS). Three important classes of Arrhenius curves possible within the SAC method are identified. From the total set of targets, a small subset of targets is first used to study the combinations of different classes of Arrhenius curves and their proportions within the design matrix, which is used to train PRS. Based on this understanding, response surfaces are generated for 64 ignition delay targets (Tig), which comprises of a wide range of pressure (P: 1--32.7 atm), temperature (T: 916-DIFadd-2869 K), and equivalence ratio (ϕ: 0.5--6.11), as well as for 74 laminar flame speed targets (Fls), spanning P: 1--25 atm, T: 285--600 K, and ϕ: 0.6--5. The H 2 /CO sub-mechanism of FFCM1.0 is chosen for the case study in which 22 reactions (66 Arrhenius parameters) are considered active parameters. The quality of the generated PRS is measured using the relative Maximum Residual Error (MRE). The quality of PRS is also checked against the search iterations encountered during mechanism optimisation. Accurate PRS could be generated over the search iterations, validating that the SAC method is able to span the entire uncertainty domain while training the PRS. The implementation of the MUQ-SAC method is made available at https://github.com/KrunalPanchal1995/MUQ-SAC.git.

Probabilistic aeroelastic analysis of high-fidelity composite aircraft wing with manufacturing variability

Safety margins of aerospace structures can be improved through altering the laminate parameters of composite materials to increase flutter and divergence velocities. Existing work demonstrates the impact of material uncertainties on low-fidelity structural models that are not sufficient to represent realistic aircraft designs. A gap exists in quantifying laminate parameter uncertainties on aeroelasticity for high-fidelity three-dimensional composite structures in realistic tailored designs. This paper puts forward an efficient methodology for uncertainty quantification on the aeroelastic characteristics of three-dimensional composite structures using FE-based parametric composite models and advanced Kriging surrogate models. The methodology is tested on both low and high fidelity case studies to represent the composite wing structure. Similarities between the case studies are observed in the coefficient of variance of all hard flutter modes being within 0.15-1.4% of each other. The difference was found for divergence and soft flutter velocities where the coefficient of variance could be over ten times higher in the high fidelity case. Global sensitivity results revealed similar physical behavior cases can be produced from both studies at early design stages.

Intrusive and non-intrusive uncertainty quantification methodologies for pyrolysis modeling

In this work we discuss and compare multiple uncertainty analysis methodologies for pyrolysis modeling. Using Arrhenius equation as the pyrolysis model and kinetic parameters for PMMA material as uncertain variables, different approaches for stochastic uncertainty quantification and their computational effort are analyzed. First non-intrusive (NI) methods are compared through convergence analysis where the deterministic model can be used as is to ascertain the statistical moments, followed by an intrusive polynomial chaos (IPC) approach which requires reformulation of the deterministic model, but solution of the model only once. The applied NI methods include Monte Carlo based simulations, regression and projection based non-intrusive polynomial chaos (NIPC) methods. The uncertainty analysis and convergence comparison is performed for three materials including the effects of overlapping reactions and dependent parameters. In general for different uncertain parameters and materials, IPC gave accurate statistical information in a single model run compared to NIPC projection requiring runs ranging from tens to hundreds and NIPC regression requiring runs in range of thousands to generate converged statistics. However, in case of only a few uncertain parameters, presence of non-polynomial functions in the model and small-scale simulations, projection based NIPC outshines IPC in terms of simplicity and applicability.

Main outcomes of the Phebus FPT1 uncertainty and sensitivity analysis in the EU-MUSA project

The Management and Uncertainties of Severe Accidents (MUSA) project was funded in HORIZON 2020 and is coordinated by CIEMAT (Spain). The project aims at consolidating a harmonized approach for the analysis of uncertainties and sensitivities associated with Severe Accidents (SAs) analysis, focusing on source term figures of merit. The Application of Uncertainty Quantification (UQ) Methods against Integral Experiments (AUQMIE – Work Package 4 (WP4)), led by ENEA (Italy), was devoted to apply and test UQ methodologies adopting the internationally recognized PHEBUS FPT1 test. FPT1 was chosen to test UQ methodologies because, even though it is a simplified SA scenario, it was representative of the in-vessel phase of a severe accident initiated by a break in the cold leg of a PWR primary circuit.WP4 served as a platform to identify and discuss the issues encountered in the application of UQ methodologies to SA analyses (e.g. discuss the UQ methodology, perform the coupling between the SA codes and the UQ tools, define the results post-processing methods, etc.). The purpose of this paper is to describe the MUSA PHEBUS FPT1 uncertainty application exercise with the related specifications and the methodologies used by the partners to perform the UQ exercise. The main outcomes and lessons learned of the analysis are: scripting was in general needed for the SA code and uncertainty tool coupling and to have more flexibility; particular attention should be devoted to the proper choice of the input uncertain parameters; outlier values of figures of merit should be carefully analyzed; the computational time is a key element to perform UQ in SA; the large number of uncertain input parameters may complicate the interpretation of correlation or sensitivity analysis; there is the need for a statistically solid handling of failed calculations.

Exploring global uncertainty quantification and sensitivity analysis methodologies: CO2 capture absorber model with MEA solvent as a test case

A set of global metamodeling uncertainty quantification (UQ) techniques belonging to non-intrusive and forward propagation categories; Polynomial Chaos Expansion (PCE), Kriging, Canonical Low Rank Approximation (LRA), and Polynomial Chaos Kriging (PC-Kriging) and global sensitivity analysis (SA) techniques, Sobol’ indices, Borgonovo, and Morris were compared to a benchmark methodology of direct Monte Carlo Simulation (MCS). The comparative analysis was demonstrated on a CO2 capture absorber model with MEA solvent. Our analysis concluded as follows; (1) although significant variation in the CO2 capture ratio in the prediction profile is shown, there is no apparent advantage of using a larger sample size via direct MCS except achieving a higher clarity in the output distribution, (2) while a very little effect on the convergence was observed which confirms the number of sample size, the fraction of computational time is enhanced by using Sobol sampling method, (3) the benefit of using metamodeling techniques compared to direct MCS was proven as they provide equal predictions in the output statistical measures with fewer evaluations of the original model and are therefore computationally cheaper, and (4) the global SA results showed Sobol indices is computationally more efficient while sharing similar ranking for the most influential parameter with Borgonovo and Morris.

Uncertainty in Modeling Ablation Heat Transfer in Rocket Nozzles

The ablation of carbon cloth phenolic insulators used in solid rocket motor (SRM) nozzles involves highly complex phenomena that are difficult to accurately predict. Historical and even more modern ablation predictions rely heavily on anchoring to SRM test data to improve predictability and SRM reliability. Accelerated schedules, reductions in static SRM testing prior to flight, and a highly competitive global market are placing a substantial onus on computational capability and predictive uncertainty. Quantifying uncertainty in SRM nozzle ablation predictions is essential for motor reliability. This paper provides the details of a modern uncertainty quantification methodology applied to ablation predictions in carbon cloth phenolic insulators exposed to SRM nozzle environments. A particular historical test motor is used as a demonstration case. The system response quantities of interest are the erosion depth and char depth. A representative model and input uncertainty are provided, and a sensitivity analysis is performed to identify influential parameters. Uncertainties in the numerical models and inputs are propagated through a two-dimensional uncertainty quantification methodology using a Latin Hypercube Sampling approach. The results show that the primary sources of uncertainty in SRM nozzle thermal modeling are the heat transfer coefficient, incident radiation heat flux, char material thermal conductivity, virgin material density, char material density, char material specific heat, and pyrolysis gas enthalpy. Uncertainties in the predictions of nozzle insulation erosion and char for the test case are provided relative to the nozzle location at the 99 th percentile and 95th confidence interval. Uncertainty in the char depth is roughly ±28% along the entire axial length of the nozzle. Uncertainty in the erosion depth ranges from about ±45% for the entrance and throat regions to ±80% at the nozzle exit.

Uncertainty Assessment in the Prediction of Selected Fission Products in the Case of a Postulated Severe Accident Scenario at VVER-1000

The aim of the paper is to present the results and insights gained from the analysis of the postulated severe accident scenario for the Nuclear Power Plant (NPP) equipped with a VVER-1000, focusing on the uncertainty assessment in the prediction of selected Fission Products (FPs). To this end, the simulation of the Large Break Loss of Coolant Accident (LB LOCA) along with a Station Blackout (SBO) scenario was carried out. The capabilities of the ASTECv2.2.0.1 [1] severe accident (SA) code along with the uncertainty quantification tool – SUNSET [2] in the coupled mode was used. The analysis was carried out for the eight input variables and ten output variables. The uncertainty quantification method incorporated within the SUNSET statistical tool was used in order to conduct the present analysis. Based on the Wilks’ formula, the number of ASTEC code runs was determined. The main statistical data are obtained for the selected fission products as figures of merit. The main steps of the analysis are outlined. Furthermore, a brief overview of the SUNSET/ASTECv2.2.0.1 coupling procedures [3] is also presented and discussed. This work demonstrates the application of the uncertainty quantification methodology in the analyses of the severe accidents in the Nuclear Power Plants and provides the specific recommendations of its use. The main outcomes from the assessment provided will be useful and may be a base for the future analyses related to the evaluation of the safety of the VVER-1000.

Uncertainty quantification in the Permian Basin using conventional and modified bootstrap methodology

Various uncertainty quantification methodologies are presented using a combination of several deterministic decline curve analysis models and two bootstrapping algorithms. These probabilistic models are applied to 126 sample wells from the Permian basin. Results are presented for 12–72 months of production hindcast given an average well production history of 103 months. Based on the coverage rate and the forecast error (with the coverage rate being more significant in our choice of the best probabilistic models) and using up to one-half of the available production history for a group of sample wells from the Permian Basin, we find that the CBM-SEPD combination is the best probabilistic model for the Central Basin Platform, the MBM-Arps combination is the best probabilistic model for the Delaware Basin, the CBM-Arps is the best probabilistic model for the Midland Basin, and the best probabilistic model for the overall Permian Basin is the CBM-Arps when early time data is used as hindcast and CBM-SEPD for when one-quarter to one-half of the data is used as hindcast. When three-quarters or more of the available production history is used for analysis, the MBM-SEPD probabilistic model is the best combination in terms of both coverage rate and forecast error for all the sub-basins in the Permian. The novelty of this work lies in its extension of bootstrapping methods to other decline curve analysis models. This work also offers the engineer guidance on the best choice of probabilistic model whilst attempting to forecast production from the Permian Basin.

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacement data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

Validation of Suction Velocity Analysis for Active Laminar Flow Control with Uncertainties

This study provides a validation of a recently developed uncertainty quantification methodology for suction velocities with the example of a novel laminar flow control design referred to as a tailored skin single duct (TSSD) panel. The TSSD panel is located on the leading edge of a vertical fin model of an Airbus A320 that is designed to maintain and extend laminar flow over the fin by means of boundary-layer suction. The suction velocities are highly dependent on the local porosity, which is only partly known in practice; therefore, the associated uncertainties need to be quantified. This is achieved by combining experimental and numerical data. The experimental data are based on the wind-tunnel experiments on a full-scale Airbus A320 vertical fin equipped with the TSSD system performed in the large low-speed facility of the German–Dutch Wind Tunnels. The measurements also feature highly accurate mass flow measurements of the integral mass flow through the TSSD system that serve as a validation reference for the uncertainty analysis. Uncertainties are provided for the predicted and the measured total mass flows to validate the proposed method, and a good agreement is obtained. A sensitivity study reveals the most influential contributions to the variance of the computed suction velocities. Depending on the difference between plenum and outer pressure, either uncertainties in the pressure distribution or uncertainties in the porosity parameters dominate the variance of the suction velocity.

Probabilistic Modeling and Uncertainty Quantification of Detailed Combustion Simulation for a Swirl Stabilized Spray Burner

To enable risk informed decisions in the simulation-based design and development of novel combustors, uncertainties in the simulation results must be considered. However, due to the high computational costs for their quantification, these uncertainties are commonly not taken into account. Therefore, this work aims at applying an efficient methodology for uncertainty quantification based on Polynomial Chaos Expansion to a semi-technical spray burner reflecting characteristics typically found in modern aeroengine combustors. This requires accurate treatment of the multicomponent liquid fuel, a combustion model relying on finite rate chemistry and a scale resolving hybrid turbulence model to account for highly unsteady flow features and combustion. To overcome the need for costly experimental data for the spray boundary conditions, an algebraic primary breakup model is utilized. The resulting reduction in a priori information is compensated through probabilistic modeling and uncertainty quantification. Due to their importance in the design process, temperature distribution in the gas phase as well as the flame position are considered as the primary quantities of interest. For these quantities of interest, moderate uncertainties are found in the probabilistic simulation results. Further, the predictive capability of the simulation model under uncertainties is quantitively assessed by defining accurary metrics for the gas phase temperature prediction. The study further reveals that the imposed input uncertainties affect quantities of interest in both the dispersed and the gas phase through phase coupling effects.

Uncertainty quantification for a severe accident sequence in a SFP in the frame of the H-2020 project MUSA: First outcomes

The Management and Uncertainties of Severe Accidents (MUSA) project, funded in HORIZON 2020 and coordinated by CIEMAT (Spain), aims at consolidating a harmonized approach for the analysis of uncertainties and sensitivities associated with Severe Accidents (SAs) focusing on Source Term (ST). In this framework, the objectives of the Innovative Management of Spent Fuel Pool Accidents (IMSFP – WP6), led by IRSN (France), are to quantify and rank the uncertainties affecting accident analyses in a Spent Fuel Pool (SFP), to review existing and contemplated SA management measures and systems and to assess their possible benefits in terms of reduction of radiological consequences.To quantify the propagation of the uncertainties of the input parameters to the output uncertainties of severe accident codes (ASTEC, MELCOR, RELAP/SCDAP), a diverse set of uncertainty quantification (UQ) tools (DAKOTA, RAVEN, SUNSET, SUSA) are used. The statistical framework used by the different UQ-tools is similar e.g. pure random (Monte Carlo) and Latin hypercube sampling (LHS).Fourteen partners from three different world regions are involved in the WP6 activities. The target of this paper is to describe the achievements during the first three years of the project. In a first part, a description is given of the SFP accidental scenario, of the key target variables and radionuclides chosen as ST Figures of Merit (FoM) and of the identified uncertainty sources in models and input parameters. A key element when defining the SFP scenario has been the consideration (or not) of the reactor building, as it is expected to significantly affect analyses. In a second part, the first insights coming out from the calculation phase of the project are presented. The review of existing SA management measures is also exposed, as well as systems whose benefits will be assessed in the second phase of the project. Finally, challenges that arise from such an exercise are discussed, as well as major difficulties found when applying UQ methodologies to SFP scenarios and solutions adopted.

Statistical Inference of Equivalent Initial Flaw Size Distribution for Fatigue Analysis of an Anisotropic Material

A novel methodology for the fatigue life uncertainty quantification of anisotropic structures is presented in this work. The concept of the equivalent initial flaw size distribution (EIFSD) is employed to overcome the difficulties in small cracks detection and fatigue prediction. This EIFSD concept is combined with the dual boundary element method (DBEM) to provide an efficient methodology for modelling the fatigue crack growth. Bayesian inference is used to infer the EIFSD based on inspection data from the routine maintenance of the structure, simulated with the DBEM. A large amount of DBEM simulations were required for the Bayesian inference. Therefore, surrogate models are used as part of the inference to further improve computational efficiency. A numerical example featuring an anisotropic plate is investigated for demonstrating the proposed methodology. When considering a low level of uncertainty in the crack propagation parameters, an error of 0.12% was found between the estimated fatigue life obtained using the proposed method compared to actual fatigue life, and only 0.35% error when considering high level of uncertainty. The application of the estimated fatigue life can be used to determine an appropriate inspection interval for aircraft maintenance.

Overview of Arbitrarily High-Order Adjoint Sensitivity and Uncertainty Quantification Methodology for Large-Scale Systems

This work reviews from a unified viewpoint the concepts underlying the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) methodologies. The practical application of the nth-CASAM-L methodology is illustrated for an OECD/NEA reactor physics benchmark, while the practical application of the nth-CASAM-N methodology is illustrated for a nonlinear model of reactor dynamics that exhibits periodic and chaotic oscillations. As illustrated both by the general theory and by the examples reviewed in this work, both the nth-CASAM-L and nth-CASAM-N methodologies overcome the curse of dimensionality in sensitivity analysis. The availability of efficiently and exactly computed sensitivities of arbitrarily high order can lead to major advances in all areas that need such high-order sensitivities, including data assimilation, model calibration, uncertainty reduction, and predictive modeling.

Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design

Uncertainty quantification has proven to be an indispensable study for enhancing reliability and robustness of engineering systems in the early design phase. Single and multi-fidelity surrogate modelling methods have been used to replace the expensive high fidelity analyses which must be repeated many times for uncertainty quantification. However, since the number of analyses required to build an accurate surrogate model increases exponentially with the number of random input variables, most surrogate modelling methods suffer from the curse of dimensionality. As an alternative approach, the Low-Rank Approximation method can be applied to high-dimensional uncertainty quantification studies with a low computational cost, where the number of coefficients for building the surrogate model increases only linearly with the number of random input variables. In this study, the Low-Rank Approximation method is implemented for multi-fidelity applications with additive and multiplicative correction approaches to make the high-dimensional uncertainty quantification analysis more efficient and accurate. The developed uncertainty quantification methodology is tested on supersonic aircraft design problems and its predictions are compared with the results of single- and multi-fidelity Polynomial Chaos Expansion and Monte Carlo methods. For the same computational cost, the Low-Rank Approximation method outperformed both in surrogate modeling and uncertainty quantification cases for all the benchmarks and real-world engineering problems addressed in the present study.

Probabilistic deep learning methodology for uncertainty quantification of remaining useful lifetime of multi-component systems

For dealing with uncertainty in Remaining Useful Life (RUL) predictions, numerous studies in literature use stochastic models to characterize the degradation process and predict the RUL distribution. However, in practice, it is difficult to derive stochastic models to capture degradation mechanisms of complex physical systems. Besides, the outstanding achievements in sensing technologies have facilitated the development of data-driven methods. Among them, deep learning methods become one of the most popular trends in recent studies; but they usually provide point predictions without quantifying the output uncertainties. In this paper, we present a new probabilistic deep leaning methodology for uncertainty quantification of multi-component systems’ RUL. It is a combination of a probabilistic model and a deep recurrent neural network to predict the components’ RUL distributions. Then, using the information about the system’s architecture, the formulas to quantify system reliability or system-level-RUL uncertainty are derived. The performance of the proposed methodology is investigated through the benchmark data provided by NASA. The obtained results highlight the point prediction accuracy and the uncertainty management capacity of the proposed methodology. In addition, thanks to the explicit RUL distributions of components, the system reliability for different structures is obtained with high accuracy, especially for series structures.