Uncertainty Quantification Analysis Research Articles

Recurrent neural network-induced Gaussian process

In this study, we develop a recurrent neural network-induced Gaussian process (RNNGP) to model sequence data. We derive the equivalence between infinitely wide neural networks and Gaussian processes (GPs) for a relaxed recurrent neural network (RNN) with untied weights. We compute the covariance function of the RNNGP using an analytical iteration formula derived through the RNN procedure with an error-function-based activation function. To simplify our discussion, we use the RNNGP to perform Bayesian inference on vanilla RNNs for various problems, such as Modified National Institute of Standards and Technology digit identification, Mackey–Glass time-series forecasting, and lithium-ion battery state-of-health estimation. The results demonstrate the flexibility of the RNNGP in modeling sequence data. Furthermore, the RNNGP predictions typically outperform those of the original RNNs and GPs, demonstrating the efficiency of the RNNGP as a data-driven model. Moreover, the RNNGP can quantify the uncertainty in the predictions, which implies the significant potential of the RNNGP in uncertainty quantification analyses.

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.

Physics-informed machine learning assisted uncertainty quantification for the corrosion of dissimilar material joints

Jointing techniques like the Self-Piercing Riveting (SPR), Resistance Spot Welding (RSW) and Rivet-Weld (RW) joints are used for mass production of dissimilar material joints due to their high performance, short cycle time, and adaptability. However, the service life and safety usage of these joints can be largely impacted by the galvanic corrosion due to the difference in equilibrium potentials between the metals with the presence of electrolyte. In this paper, we focus on Al-Fe galvanic corrosion and develop physics-informed machine learning based surrogate model for statistical corrosion analysis, which enables the reliability analysis of dissimilar material joints under corrosion environment. In this study, a physics-based finite element (FE) corrosion model has been developed to simulate the galvanic corrosion between a Fe cathode and an Al anode. Geometric and environmental factors including crevice gap, roughness of anode, conductivity, and the temperature of the electrolyte are investigated. Further, a thorough Uncertainty Quantification (UQ) analysis is conducted for the overall corrosion behavior of the Fe-Al joints. It is found that the electrolyte conductivity has the largest effects on the material loss and needs to be managed closely for better corrosion control. This will help in designing and manufacturing joints with improved corrosion performance.

Uncertainty analysis and visualization for a coupled thermohydraulics-neutronics simulation by deterministic sampling

Uncertainty quantification and sensitivity analysis by deterministic sampling is performed on a thermohydraulics-neutronics coupling simulation of a 5×5 rod bundle in pressurized water reactor. Thermohydraulics calculation is conducted by a CFD simulation and neutronics calculation is conducted by method of characteristic (MOC). The coupling simulation of CFD and MOC, together with large number of calculations needed by random sampling in uncertainty quantification, leads to high computational cost. Therefore, deterministic sampling is introduced in this study and proves its efficiency. Moreover, the calculated results usually have some distribution pattern on the selected section, using mass average value as output is unable to reflect the distribution feature. Therefore, a method realized by Python program is introduced to perform uncertainty quantification and sensitivity analysis of the outputs on the outlet section in fine-mesh level. It is found that the uncertainty of outputs and the inputs’ impact on outputs vary in different areas at the outlet section.

Input parameter and thermal load representation uncertainties effects on the buckling and dynamic characteristics of cylindrical tubes

This research considers different methods in sensitivity analysis and uncertainty quantification applied to cylindrical tubes subject to three types of thermal load representation, namely, uniform, linear, and nonlinear in the radial direction. Sensitivity analysis techniques are used to quantify and rank the parameters that are most influential in altering the critical buckling temperatures, the post-buckled configuration amplitudes, and the natural frequencies in the pre- and post-buckling regimes. The uncertainty quantification and sensitivity analysis strategies show a great potential and usefulness in terms of determining the most influential input parameters for cylindrical tubes subject to thermal loads. Based on the set of nominal parameters in this study, it is shown that the length is the most sensitive parameter in altering the critical thermal buckling load. Additionally, it is demonstrated how uncertainty in the thermal load representation can lead to over or underestimation in both sensitivity analysis and uncertainty quantification findings. The outcome of this analysis can be utilized by other researchers in the design and optimization of cylindrical tubes in thermal environment.

Uncertainty analysis of turbulence model in capturing flows involving laminarization and retransition

Flows experiencing laminarization and retransition are universal and crucial in many engineering applications. The objective of this study is to conduct an uncertainty quantification and sensitivity analysis of turbulence model closure coefficients in capturing laminarization and retransition for a rapidly contracting channel flow. Specifically, two commonly used turbulence models are considered: the Spalart-Allmaras (SA) one-equation model and the Menter Shear Stress Transport (SST) two-equation model. Thereby, a series of steady Reynolds Averaged Navier-Stokes (RANS) predictions of aero-engine intake acceleration scenarios are carried out with the purposely designed turbulence model closure coefficients. As a result, both SA and SST models fail to capture the retransition phenomenon though they achieve pretty good performance in laminarization. Using the non-intrusive polynomial chaos method, solution uncertainties in velocity, pressure, and surface friction are quantified and analyzed, which reveals that the SST model possesses much great uncertainty in the non-laminar regime, especially for the logarithmic law prediction. Besides, a sensitivity analysis is performed to identify the critical contributors to the solution uncertainty, and then the correlations between the closure coefficients and the deviations of the outputs of interest are obtained via the linear regression method. The results indicate that the diffusion-related constants are the dominant uncertainty contributors for both SA and SST models. Furthermore, the remarkably strong correlation between the critical closure coefficients and the outputs might be a good guide to recalibrate and even optimize the commonly used turbulence models.

An Efficient Sequential Strategy for Non-probabilistic Reliability-based Topology Optimization (NRBTO) of Continuum Structures with Stress Constraints

In the study, the sequential strategy is proposed for non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with stress constraints. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. Then the deterministic topology optimization formulation by Stress-Influence-Function (SIF) approach is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the elemental stresses under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization formulation. The sequential strategy is proposed to handle the NRBTO problem and thus the original nested process of topology optimization and reliability analysis was decoupled, in which the update of design variables and the shift of the strength feature in the stress influence function are conducted alternately in the optimization process. The Performance Measure Approach (PMA) is applied to calculate the shift value of the strength feature and a shift value function is presented to avoid the numerical oscillations caused by the large variation of strength feature. Three numerical examples are eventually given to illustrate the applicability and validity of the developed sequential strategy for NRBTO with stress constraints.

McCARD/MIG stochastic sampling calculations for nuclear cross section sensitivity and uncertainty analysis

In this study, a cross section stochastic sampling (S.S.) capability is implemented into both the McCARD continuous energy Monte Carlo code and MIG multiple-correlated data sampling code. The ENDF/B-VII.1 covariance data based 30 group cross section sets and the SCALE6 covariance data based 44 group cross section sets are sampled by the MIG code. Through various uncertainty quantification (UQ) benchmark calculations, the McCARD/MIG results are verified to be consistent with the McCARD stand-alone sensitivity/uncertainty (S/U) results and the XSUSA S.S. results. UQ analyses for Three Mile Island Unit 1, Peach Bottom Unit 2, and Kozloduy-6 fuel pin problems are conducted to provide the uncertainties of keff and microscopic and macroscopic cross sections by the McCARD/MIG code system. Moreover, the SNU S/U formulations for uncertainty propagation in a MC depletion analysis are validated through a comparison with the McCARD/MIG S.S. results for the UAM Exercise I-1b burnup benchmark. It is therefore concluded that the SNU formulation based on the S/U method has the capability to accurately estimate the uncertainty propagation in a MC depletion analysis.

Full-scale validation of CFD simulations of buoyancy-driven ventilation in a three-story office building

Computational fluid dynamics (CFD) is frequently used to support the design of naturally ventilated buildings; however, the model accuracy should be thoroughly assessed, ideally through validation with full-scale measurements. The present study aims to (1) validate transient CFD simulations with uncertainty quantification (UQ) for buoyancy-driven natural ventilation against full-scale experiments in an operational atrium building, and (2) quantify the sensitivity of the CFD results to the thermal boundary conditions. The UQ and sensitivity analysis consider uncertainties in the initial and boundary conditions for the temperatures. Considering the volume-averaged air temperature on each floor, the predictions and measurements agree well with discrepancies less than 0.3 °C. When considering the temperature averaged over smaller zones on each floor, two trends can be observed. First, in zones not adjacent to windows, the discrepancies between the CFD and measurements can be explained by uncertainty in the boundary conditions and the measurements. Second, in zones adjacent to windows, higher discrepancies are observed due to oscillations in the inflow jets just downstream of the windows, and due to geometrical simplifications in the CFD model. The sensitivity analysis demonstrates that the boundary conditions for the thermal mass surface temperature and the outdoor temperature have a dominant effect on the indoor air temperature predictions, with their relative importance varying as a function of proximity to the windows.

Bayesian Imaging with Data-Driven Priors Encoded by Neural Networks

This paper proposes a new methodology for performing Bayesian inference in imaging inverse problems where the prior knowledge is available in the form of training data. Following the manifold hypothesis, we adopt a data-driven prior that is supported on a submanifold of the ambient space, which we can learn from the training data using a generative model, such as a variational autoencoder or generative adversarial network. We establish the existence and well-posedness of the associated posterior distribution and posterior moments under easily verifiable conditions, providing a rigorous underpinning for Bayesian estimators and uncertainty quantification analyses. Bayesian computation is performed using a parallel tempered version of the pCN algorithm on the manifold, which is shown to be ergodic and robust to the nonconvex nature of these data-driven models. In addition to point estimators and uncertainty quantification analyses, we derive a model misspecification test to automatically detect situations where the data-driven prior is unreliable, and we explain how to identify the dimension of the latent space directly from the training data. The proposed approach is illustrated with a range of experiments with the MNIST dataset and is compared with some variational and message passing image reconstruction approaches from the state of the art that also use data-driven regularization. A model accuracy analysis suggests that the Bayesian probabilities reported by the proposed data-driven models are also accurate under a frequentist definition of probability, suggesting that the learnt prior is close to the true marginal distribution of the unknown image.

Predictive Design and Analysis of Drug Transport by Multiscale Computational Models Under Uncertainty.

Computational modeling of drug delivery is becoming an indispensable tool for advancing drug development pipeline, particularly in nanomedicine where a rational design strategy is ultimately sought. While numerous in silico models have been developed that can accurately describe nanoparticle interactions with the bioenvironment within prescribed length and time scales, predictive design of these drug carriers, dosages and treatment schemes will require advanced models that can simulate transport processes across multiple length and time scales from genomic to population levels. In order to address this problem, multiscale modeling efforts that integrate existing discrete and continuum modeling strategies have recently emerged. These multiscale approaches provide a promising direction for bottom-up in silico pipelines of drug design for delivery. However, there are remaining challenges in terms of model parametrization and validation in the presence of variability, introduced by multiple levels of heterogeneities in disease state. Parametrization based on physiologically relevant in vitro data from microphysiological systems as well as widespread adoption of uncertainty quantification and sensitivity analysis will help address these challenges.

Bayesian, frequentist, and information geometric approaches to parametric uncertainty quantification of classical empirical interatomic potentials

In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs) using both Bayesian (Markov Chain Monte Carlo) and frequentist (profile likelihood) methods. We interface these tools with the Open Knowledgebase of Interatomic Models and study three models based on the Lennard-Jones, Morse, and Stillinger-Weber potentials. We confirm that IPs are typically sloppy, i.e., insensitive to coordinated changes in some parameter combinations. Because the inverse problem in such models is ill-conditioned, parameters are unidentifiable. This presents challenges for traditional statistical methods, as we demonstrate and interpret within both Bayesian and frequentist frameworks. We use information geometry to illuminate the underlying cause of this phenomenon and show that IPs have global properties similar to those of sloppy models from fields, such as systems biology, power systems, and critical phenomena. IPs correspond to bounded manifolds with a hierarchy of widths, leading to low effective dimensionality in the model. We show how information geometry can motivate new, natural parameterizations that improve the stability and interpretation of uncertainty quantification analysis and further suggest simplified, less-sloppy models.

Adaptive surrogates of crashworthiness models for multi-purpose engineering analyses accounting for uncertainty

Uncertainty Quantification (UQ) is a booming discipline for complex computational models based on the analysis of robustness, reliability and credibility. UQ analysis for nonlinear crash models with high dimensional outputs presents important challenges. In crashworthiness, nonlinear structural behaviours with multiple hidden modes require expensive models (18 h for a single run). Surrogate models (metamodels) allow substituting the full order model, introducing a response surface for a reduced training set of numerical experiments. Moreover, uncertain input and large number of degrees of freedom result in high dimensional problems, which derives to a bottle neck that blocks the computational efficiency of the metamodels. Kernel Principal Component Analysis (kPCA) is a multidimensionality reduction technique for non-linear problems, with the advantage of capturing the most relevant information from the response and improving the efficiency of the metamodel. Aiming to compute the minimum number of samples with the full order model. The proposed methodology is tested with a practical industrial problem that arises from the automotive industry.

Investigations on Turbulence Model Uncertainty for Hypersonic Shock-Wave/Boundary-Layer Interaction Flows

Shock-wave/boundary-layer interaction (SWBLI) is a fundamental scientific problem that restricts the breakthrough of high-speed flight technology. Owing to the complex shock structure and boundary-layer separation in the interaction region, the widely used eddy-viscosity turbulence models have large uncertainties and model errors in the simulation of such flows. To make more reasonable aerodynamic/aerothermal predictions and identify the critical parameters affecting the prediction results, Bayesian uncertainty quantification and sensitivity analysis are conducted for the three turbulence models [Spalart–Allmaras (SA), SA with quadratic constitutive relation (SA-QCR), and shear-stress transport (SST) models] and the hypersonic SWBLI cases. First, the prior variances and Sobol indices are calculated to obtain a preliminary understanding of the parameter variability for turbulence models. Then, the posterior distributions of the model closure coefficients and posterior uncertainties of the wall pressure and thermal flux are systematically analyzed and compared. The results indicate that the prediction performances of the three models can be effectively enhanced by Bayesian estimation. The peak of the posterior uncertainty of the SST model is the largest among the three models, whereas that of the SA-QCR model is the smallest. The results of the Bayesian model evaluation demonstrate that the SA-QCR model has the highest reliability.

Robustness Analysis on the Aerothermal Performance of Turbine Blade Squealer Tip

Abstract An improved efficient uncertainty quantification (UQ) analysis framework is proposed by the combination of sparse polynomial chaos expansion (PCE) and universal Kriging (UK) metamodel to obtain the surrogate model (UK-PCE). Moreover, a challenging analytical test function and an engineering test are considered to investigate the response performance of UK-PCE method. The results show that the UK-PCE method reduces the computational cost by more than 70% in comparison to the typical PCE method. Then this method was applied to the UQ of the aerodynamic and heat transfer performance of GE-E3 rotor blade squealer tip. Additionally, a series of uncertainty quantities visualization methods based on the data mining method, parallel computing method, and Delaunay triangulation method is proposed to reveal more enlightening uncertainty phenomena in the actual operation. The results of UQ show that under the influence of uncertain inputs, the leakage flow rate and downstream entropy increase will be significantly increased. The statistical average of tip heat flux has increased by 8.56% relative to the design value, and the probability of it deviating from the design value by 10% is as high as 43.27%. In addition, the three-dimensional tip heat flux deviation distributions calculated by the proposed uncertainty quantities visualization method reveal a coupling of the hot corrosion and thermal fatigue of the squealer tip. It is also indicated that under the influence of the uncertain inputs, there is a marked increase in blade tip flux, and the blade tip flux deviation has been maintained at a high value, about 13.0%. The results of sensitivity analysis show that the largest contributor to the uncertainty of the blade tip aerodynamic performance is the tip clearance deviation and its variance index to the uncertainty of leakage flow rate and downstream entropy increase is as high as 88.21% and 62.63%. Therefore, the geometric accuracy of the tip clearance should be strictly ensured in the turbine blade assembly and marching process. The influence of the inlet total temperature deviation on the uncertainty of the heat transfer performance of the squealer tip must also be taken into account. So a satisfactory control system should be designed in the actual operation of the gas turbine to make sure that the fluctuation of inlet total temperature can be attenuated rapidly.

Uncertainty quantification (UQ) for CFD simulation of OECD-NEA cold leg mixing benchmark

An uncertainty quantification (UQ) analysis was conducted for the unsteady Reynolds Average Navier Stokes Simulation (URANS) of the OECD-NEA cold leg mixing benchmark. The methodology is based on the American Society of Mechanical Engineers Verification and Validation (ASME V&V) standard for UQ in Computational Fluid Dynamics (CFD) applications. The numerical, input and experimental uncertainties were calculated to determine the validation uncertainty for the open test (T1). Furthermore, based on validation uncertainty, model uncertainty was calculated. Finally, on the basis of open test (T1) the symmetric and asymmetric uncertainty band was achieved for the blind test (T2). The validation uncertainty closely covers the PIV data in the cold leg. Besides, the simulation results of the blind test, together with the uncertainty band, closely bound the PIV data of the mean velocity in the cold leg. However, a large simulation uncertainty band was obtained in the downcomer due to a large validation error.

Uncertainty analysis of NOx and CO emissions in industrial ethylene cracking furnace using high-precision sparse polynomial chaos expansion

ABSTRACT Traditional designing of ethylene-cracking furnaces by computational fluid dynamics (CFD) does not consider the uncertainties in actual engineering. This paper introduces one uncertainty analysis framework based on non-intrusive polynomial chaos expansion (NIPCE) and multi-order Sobol indices to perform uncertainty quantification (UQ) and sensitivity analysis for NOx and CO emissions in the combustion process. In this framework, several operating parameters of the bottom burners that easily cause uncontrollable fluctuations in the flame and emission characteristics are selected as uncertainty variables and characterized as a probability distribution. Computationally expensive CFD simulation of cracking furnace is used to generate a small number of samples, which are carried out to develop high-precision sparse PCE models by the degree-adaptive scheme and least angle regression (LAR) algorithm. And multi-order Sobol sensitivity analysis based on PCE models is performed efficiently to research the influence of uncertain parameters on pollution generation. Under 3% of burner’s uncertainty operating parameters, the results find that the excess air coefficient of 1.20 can obviously reduce the generation of NOx and CO emissions and control the fluctuation caused by uncertainties. Moreover, sensitivity analysis determines the critical variables that affect pollutant emissions to help the actual process avoid the unknown effects of uncertainties.

Uncertainty quantification of wind turbine airfoil aerodynamics with geometric uncertainty

An artificial neural network based reduced order model (ROM) is developed to predict the load coefficients and performance of wind turbine airfoils. The model is trained using a representative database of 972 wind turbine airfoil shapes generated by perturbing the design parameters in each of 12 baseline airfoils defining commercially relevant modern wind turbines. The predictions from our ROM show excellent agreement with the CFD data, with a 99th percentile maximum errors of 0.03 in lift-coefficient, 2 in lift-to-drag ratio and 0.002 in pitching moment coefficient. A Monte-Carlo based uncertainty quantification (UQ) and global sensitivity analysis (GSA) framework is developed using this computationally economical ROM. Using UQ, we observed the stall behavior to be very sensitive to geometric uncertainty, with more than 10% deviation in lift coefficient associated to 5% deviation in geometric features. Sobol’s analysis is used to identify the most influencing geometric feature for the stall behavior to be concentrated at the maximum thickness location on the airfoil suction surface.

Integrated uncertainty quantification and sensitivity analysis of single-component dynamic column breakthrough experiments

We have carried out the traditional analysis of a set of dynamic breakthrough experiments on the hbox {CO}_2/He system adsorbing onto activated carbon by fitting a 1D dynamic column breakthrough model to the transient experimental profiles. We have quantified the uncertainties in the fitted model parameters using the techniques of Bayesian inference, and have propagated these parametric uncertainties through the dynamic model to assess the robustness of the modelling. We have found significant uncertainties in the outlet mole fraction profile, internal temperature profile and internal adsorption profiles of approximately pm 15%. To assess routes to reduce these uncertainties we have applied a global variance-based sensitivity analysis to the dynamic model using the Sobol method. We have found that approximately 70% of the observed variability in the modelling outputs can be attributed to uncertainties in the adsorption isotherm parameters that describe its temperature dependence. We also make various recommendations for practitioners, using the developed Bayesian statistical tools, regarding the choice of the isotherm model, the choice of the fitting data for the extraction of system specific parameters and the simplification of the wall energy balance.

Assessing uncertainties and identifiability of foam displacement models employing different objective functions for parameter estimation

Foam injection in porous media is often used to control the gas fingering in multi-phase flow. Mathematical models of foam dynamics involve non-Newtonian formulations. To numerically simulate these complex phenomena, experimental data is gathered and used to estimate the parameter values of models via optimization techniques. The present work improves this procedure by introducing a new objective function based on the mobility reduction factor and does not require further experimental observations other than those usually obtained in core-flooding experiments. A series of numerical experiments were carried out to show the features and robustness of the proposed approach. In particular, the identifiability analysis results show that key parameters of the models are practically non-identifiable when using traditional objective functions that rely only on apparent viscosity. When the proposed objective function is used for parameter estimation, the identifiability issues are solved. In addition, the uncertainty quantification analysis revealed that lower uncertainty is achieved when using the new objective function when compared to the one that uses only apparent viscosity. In summary, we show that the new objective function generates better-calibrated models with high fidelity and low uncertainties and alleviates parameter non-identifiability issues. Therefore, the results suggest the new objective function is better suited for calibrating foam displacement models for enhanced oil recovery.