Model-form Uncertainty Quantification Research Articles

Model-form uncertainty quantification of Reynolds-averaged Navier–Stokes modeling of flows over a SD7003 airfoil

Reynolds-averaged Navier–Stokes (RANS) models are known to be inaccurate in complex flows, for instance, laminar-turbulent transition, and RANS uncertainty quantification (UQ) is essential to estimate the uncertainty in their predictions. In this study, a recent physics-based UQ framework that introduces eigenvalue, eigenvector, and turbulence kinetic energy perturbations to the modeled Reynolds stress tensor has been used to estimate the uncertainty in the flow field. We introduce a regression-based marker function that focuses on the turbulence kinetic energy perturbation for the simulation of laminar-turbulent transitional flows over an Selig–Donovan 7003 airfoil. We observed a monotonic behavior of the magnitude of the predicted uncertainty bounds varying with the turbulence kinetic energy perturbation. Importantly, the predicted uncertainty bounds show a synergy behavior that dramatically increases the size of uncertainty bounds and can successfully encompass the reference data when the eigenvalue perturbations are augmented with the marker function.

Data-driven quantification of model-form uncertainty in Reynolds-averaged simulations of wind farms

Computational fluid dynamics using the Reynolds-averaged Navier–Stokes (RANS) remains the most cost-effective approach to study wake flows and power losses in wind farms. The underlying assumptions associated with turbulence closures are the biggest sources of errors and uncertainties in the model predictions. This work aims to quantify model-form uncertainties in RANS simulations of wind farms at high Reynolds numbers under neutrally stratified conditions by perturbing the Reynolds stress tensor through a data-driven machine-learning technique. To this end, a two-step feature-selection method is applied to determine key features of the model. Then, the extreme gradient boosting algorithm is validated and employed to predict the perturbation amount and direction of the modeled Reynolds stress toward the limiting states of turbulence on the barycentric map. This procedure leads to a more accurate representation of the Reynolds stress anisotropy. The data-driven model is trained on high-fidelity data obtained from large-eddy simulation of a specific wind farm, and it is tested on two other (unseen) wind farms with distinct layouts to analyze its performance in cases with different turbine spacing and partial wake. The results indicate that, unlike the data-free approach in which a uniform and constant perturbation amount is applied to the entire computational domain, the proposed framework yields an optimal estimation of the uncertainty bounds for the RANS-predicted quantities of interest, including the wake velocity, turbulence intensity, and power losses in wind farms.

Model-form uncertainty quantification in RANS simulations of wakes and power losses in wind farms

Reynolds-averaged Navier-Stokes (RANS) is one of the most cost-efficient approaches to simulate wind-farm-atmosphere interactions. However, the applicability of RANS-based methods is always limited by the accuracy of turbulence closure models, which introduce various uncertainties into the models. In this study, we estimate model-form uncertainties in RANS simulations of wind farms. For this purpose, we compare different RANS models to a large-eddy simulation (LES). We find that the realizable k −ϵ model is a representative RANS model for predicting the mean velocity, the turbulence intensity, and the power losses within the wind farm. We then investigate the model-form uncertainty associated with this turbulence model by perturbing the Reynolds stress tensor. The focus is placed on perturbing the shape of the tensor represented by its eigenvalues. The results show that the perturbed RANS model successfully estimates the region bounding the LES results for quantities of interest (QoIs). We also discuss the effect of perturbation magnitude on various QoIs.

Multi-objective Design and Optimization of Power Electronics Converters With Uncertainty Quantification—Part II: Model-Form Uncertainty

The second part of the multi-objective design and optimization with parametric and model-form uncertainty quantification (MDO with P&MF-UQ) is dedicated to incorporating MF-UQ into the MDO framework. MF-UQ is used to estimate the error due to modeling inaccuracies and to validate the mathematical models used in the MDO framework. In this approach, the sensitivity index introduced in the first part of the article is modified too so that it is a quantitative measure of system design robustness with regards to modeling inaccuracies as well as the manufacturing variability in the design of systems with multiple performance functions. The optimum design solution is finally realized by exploring the Pareto Front of the enhanced performance space, where the model-form error associated with each design is used to modify the estimated performance measures and the parametric sensitivity of each design point is considered to discern between cases and help identify the most parametrically robust of the Pareto-optimal design solutions. To demonstrate the benefits of incorporating uncertainty quantification analysis into the design optimization from a more practical standpoint, design of a robust high-efficiency high-power-density 1.25 kW Vienna-type rectifier is used as a case study to compare the theoretical analysis with a comprehensive experimental validation. It is shown that the final design selected using the MDO with P&MF-UQ reduces the design sensitivity and system loss by 33% and 5%, respectively, compared to the optimal design selected using the conventional MDO.

On the Modeling and Design of Modular Multilevel Converters With Parametric and Model-Form Uncertainty Quantification

Modeling and design with parametric and model-form uncertainty quantification is an alternative to conventional model-based design approaches as it improves the existing modeling practice and validates the model used in the design of power converters. However, in the case of modular multilevel converters (MMCs), uncertainty quantification, as the main step in this design methodology, becomes challenging due to the inherent complexity and sheer size of such units. In this article, these limitations are discussed and a systematic study for developing a simplified testbed for uncertainty quantification of an MMC is presented. To this end, first sensitivity analysis is conducted to identify the key parameters whose tolerances contribute the most to the parametric uncertainty of the selected design variables. Second, the effect of increasing the number of power cells in each arm on the estimated total uncertainty, and thus the predictive capability of the MMC simulation models for medium- and high-voltage applications is studied. A simplified testbed for model validation of the power cell in the design of an MMC is developed accordingly. The development of this simplified testbed allows validating the models used and estimating uncertainties in the design with less computational cost and hardware prototyping.

Hierarchical model form uncertainty quantification for turbulent combustion modeling

All models invoke assumptions that result in model errors. The objective of model form uncertainty quantification is to translate these assumptions into mathematical statements of uncertainty. If each assumption in a model can be isolated, then the uncertainties associated with each assumption can be independently assessed. In situations where a series of assumptions leads to a hierarchy of models with nested assumptions, physical principles from a higher-fidelity model can be used to directly estimate a physics-based uncertainty in a lower-fidelity model. Turbulent nonpremixed combustion models fall into a natural hierarchy from the full governing equations to Conditional Moment Closure to flamelet-like models to thermodynamic equilibrium, and, at each stage of the hierarchy, a single assumption can be isolated. In this work, estimates are developed for the uncertainties associated with each assumption in the hierarchy. The general method identifies a trigger parameter that can be obtained with information only from the lower-fidelity model that is used to estimate the error in the lower-fidelity model using only physical principles from the higher-fidelity model. Starting from the lowest fidelity model, the trigger parameters identified in this work are the product of a chemical time scale and the scalar dissipation rate for the equilibrium chemistry model, which characterizes the errors associated with neglecting finite-rate chemistry and transport; the reciprocal of the product of a generalized Lagrangian flow time and the scalar dissipation rate for the steady flamelet model, which characterizes the errors associated with neglecting flow history effects; and the relative magnitude of the conditional fluctuations for Conditional Moment Closure. The approach is applied in LES to a turbulent nonpremixed simple jet flame to quantify the errors associated with the equilibrium chemistry model and the steady flamelet model. The results indicate that errors associated with neglecting finite-rate chemistry and transport in the equilibrium chemistry model are dominant upstream while errors associated with neglecting flow history in the steady flamelet model become more important downstream.

Multi-objective Design and Optimization of Power Electronics Converters With Uncertainty Quantification—Part I: Parametric Uncertainty

This article presents a robust multi-objective design and optimization approach with parametric and model-form uncertainty quantification (MDO with P&MF-UQ). The first part of the article focuses on incorporating parametric uncertainty quantification (P-UQ) into the MDO framework, where a sensitivity index is defined as a quantitative measure of system design robustness with regards to manufacturing variability in the design of systems with multiple performance functions. To demonstrate the benefits of incorporating P-UQ analysis into the design optimization framework, this article presents the design and optimization of a robust high-efficiency high-power-density 1.25 kW Vienna-type rectifier. The optimum design solution is realized by exploring the Pareto Front of the enhanced performance space where the parametric sensitivity of each design point is considered to discern between cases and help identify the most parametrically-robust of the Pareto-optimal design solutions. It is shown that the design sensitivity is reduced by 39% when the optimum design is selected using MDO with P-UQ rather than the conventional MDO.

Feasible Probabilistic Learning Method for Model-Form Uncertainty Quantification in Vibration Analysis

Recently, a novel probabilistic method for modeling and quantifying model-form uncertainties in a deterministic high-dimensional computational model was proposed and demonstrated for numerous applications in linear vibrations and nonlinear structural dynamics. The method relies on a stochastic projection-based reduced-order model (SPROM) grounded in a randomized reduced-order basis to accelerate the stochastic computations it incurs, and it accounts for all modeling uncertainties introduced by model reduction. It builds the probability model of the SPROM using a small number of hyperparameters, and it determines them so that the mean value and statistical fluctuations of quantities of interest predicted using the SPROM match target values obtained from available data. Hence, it can be interpreted as a probabilistic model learning or updating method. Unfortunately, even though it is performed using a reduced-order model, the hyperparameter identification process can be cost prohibitive. For this reason, the method is reformulated in this paper around the concept of hyper-reduction. Specifically, a reduced version of the mesh underlying the high-dimensional model is constructed and used to hyper-reduce the SPROM. The feasibility of the resulting approach is demonstrated for the model-form uncertainty quantification of generalized eigencomputations performed during what-if scenarios associated with shape changes of a jet engine nozzle.

Physics-informed covariance kernel for model-form uncertainty quantification with application to turbulent flows

In uncertainty quantification of computational models (e.g., turbulence modeling) with Bayesian inferences, Gaussian processes are commonly used as the prior for model discrepancies. However, constructing the covariance kernel is a challenging task that requires significant physical knowledge of the problem. On the other hand, often the model discrepancies are described by partial differential equations (PDEs) of known structures (e.g. Reynolds stress transport equations for turbulent flows), albeit with unclosed terms (e.g, velocity triple correlation and press–strain-rate correlation). In this work, we utilize such PDEs to construct physics-informed covariance kernels by exploiting the fundamental connection between PDEs and covariance functions. We demonstrate the merits of the physics-informed covariance kernel with the application to turbulent flows over periodic hills. The covariance kernel and the modes (eigenfunctions) obtained from the physics-informed approach are physically more realistic than those obtained with commonly used squared exponential kernels. This method also has the potential of improving the performance of Bayesian model-form uncertainty quantification in applications beyond turbulent flows.

Advances in Parametric and Model-Form Uncertainty Quantification in Canonical Aeroelastic Systems

Uncertainty quantification is going to play a crucial role in the aeroelastic design and optimization of aircraft. Stochastic aeroelastic models are currently being considered to account for manufacturing tolerance in material properties, variability in flight conditions or uncertainty in the aeroelastic model itself. In this paper, some challenging issues in the development of efficient and robust stochastic solvers are reported within the framework of canonical aeroelastic systems. First, independent or correlated parametric uncertainties are propagated to compute the probability density function of the critical flutter velocity or the limit cycle oscillations in the presence of discontinuous responses. Secondly, inverse stochastic aeroelastic problems are addressed, in which experimental data are used to calibrate several stochastic aerodynamic models within a Bayesian framework. Studied configurations concern linear and non-linear pitching and plunging airfoils, and the stochastic flutter of a cantilevered straight composite wing subject to ply angle and thickness uncertainties.

Quantifying Model Form Uncertainty of Epidemic Forecasting Models from Incidence Data

ObjectiveWe present a mathematical framework for non-parametric estimation of the force of infection, together with statistical upper and lower confidence bands. The resulting estimates allow to assess how well simpler models, such as SEIR, fit the observed time series of incidence data.IntroductionUncertainty Quantification (UQ), the ability to quantify the impact of sample-to-sample variations and model misspecification on predictions and forecasts, is a critical aspect of disease surveillance. While quantifying the impact of stochastic uncertainty in the data is well understood, quantifying the impact of model misspecification is significantly harder. For the latter, one needs a "universal model" to which more restrictive parametric models are compared too.MethodsThis talk presents a useful modeling framework for time series of incidence data from contagious diseases that enables one to identify and quantify the impact of model form uncertainty. Specifically, we propose to focus on estimating the timedependent force of infection. The latter is a universal parameters for all contagious disease model. Using a machine learning technique for estimating monotone functions, i.e., isotonic regression and its variants, one can estimate the force ofinfection without addtional assumptions. We note that most contagious disease model do satisfy this monotonicity assumption, due to a combination of factors: depletion of susceptibles, implementation of mitigation strategies, behavior change, etc. Comparing the resulting "non-parametric" estimate with parametric estimates, obtained by fitting an SEIR for example, can reveal model deficiencies and help quantify model form uncertainties.Finally, we discuss how ideas from "strict bound theory" can be used to develop upper and lower uncertainty bands for force of infection that acknowledge the intrinsic stochasticity in the data.ResultsWe demonstrate the application of the methodology to weekly Influenza Like Illness (ILI) incidence data from France andcompare the results to fitted SIR and SEIR models. This comparison can be seen as a nonparametric goodness of fit test, providing one with tools to do simple model selection.ConclusionsWe present a novel and flexible model to statistically describe the force of infection as a function of time. Comparing the fit to incidence data of that model with the fit of simpler parametric models enables the quantification of model form uncertainty and associated model selection.

Power Electronics Modeling and Design: Using Parametric and Model-Form Uncertainty Quantification to Assess Predictive Accuracy of Power Converter Models

This article presents an enhanced modeling framework for the design of power electronics converters and systems. This methodology provides the means to assess and quantify the predictive accuracy of power converter models, a capability that is attained by the proper identification, characterization, and quantification of the different, and often numerous, sources of uncertainty in the modeling and simulation process. For illustrative purposes, an in-depth example is presented using a power electronics building block (PEBB) as the modeling test bed, demonstrating how, through this formal modeling procedure-and taking into consideration the uncertainties in the system-the model in question can predict the variable of interest within a known range with a given level of confidence, without requiring heuristic safety factors. This predicted range can be used as the required design margin when the selected output is a design variable, evincing the inherent potential of having a less conservative and more reliable power converter using the design methodology with uncertainty quantification.

Model form uncertainty quantification in turbulent combustion simulations: Peer models

Turbulent combustion simulations invoke a number of component models for chemical kinetics, turbulence, flame structure, etc., each of which has an error associated with its structural form and contributes to overall uncertainties in simulation results. These model form errors arise from the necessity of making assumptions in deriving a model. Conventional approaches to estimating model form errors rely on an ad hoc additive error that is then calibrated against experimental or computational data. These approaches inherently neglect any a priori knowledge of physics in developing the model error estimate. Instead, in this work, an inherently physics-based approach to estimating model form error is developed based on the notion of “peer” models. In the generic approach, the error in a candidate model is determined by taking the difference between it and an equally plausible alternative “peer” model with a different set of assumptions. The generic approach is applied in this work to the modeling of the subfilter mixture fraction dissipation rate, which is typically modeled as the ratio of the subfilter mixture fraction variance and a time scale. The typical time scale approximation invokes a turbulent time scale, and a “peer” model is proposed in which a chemical time scale is invoked to estimate the model form error. Using stochastic collocation, the subfilter mixture fraction dissipation rate model form error as well as the uncertainty in a model parameter are propagated through LES calculations of the Sandia D Flame utilizing the steady flamelet model. The results indicate that the mixture fraction, temperature, and carbon monoxide uncertainties increase with downstream distance due to an increase in the relative subfilter mixture fraction variance and increased sensitivity to the time scale approximations, which diverge in magnitude with downstream distance. Uncertainties in these quantities arising from the model form error are shown to be more significant than uncertainties arising from the model constant uncertainty. For the temperature, uncertainties due to chemical kinetic rate uncertainty are shown to be slightly smaller than uncertainties due to the subfilter mixture fraction dissipation rate model error; for the carbon monoxide mass fraction, uncertainties due to chemical kinetic rate uncertainty are twice as large as uncertainties due to the subfilter mixture fraction dissipation rate error since carbon monoxide is more kinetically-controlled than the temperature.

A random matrix approach for quantifying model-form uncertainties in turbulence modeling

With the ever-increasing use of Reynolds-Averaged Navier–Stokes (RANS) simulations in mission-critical applications, the quantification of model-form uncertainty in RANS models has attracted attention in the turbulence modeling community. Recently, a physics-based nonparametric approach for quantifying model-form uncertainty in RANS simulations has been proposed, where Reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits (Xiao et al., 2016). However, a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints. In this work we propose a random matrix approach for quantifying model-form uncertainties in RANS simulations with the realizability of the Reynolds stress guaranteed, which is achieved by construction from the Cholesky factorization of the normalized Reynolds stress tensor. Furthermore, the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints. We demonstrate that the proposed approach is able to ensure the realizability of the Reynolds stress, albeit in a different manner from the physics-based approach. Monte Carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent Gaussian random fields to the Reynolds stress random field with the marginal distributions and correlation structures as specified. Numerical simulations on a typical flow with separation have shown physically reasonable results, which verify the proposed approach. Therefore, the proposed method is a promising alternative to the physics-based approach for model-form uncertainty quantification of RANS simulations. The method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering.

Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach

Despite their well-known limitations, Reynolds-Averaged Navier–Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in today's engineering analysis, design and optimization. While the predictive capability of RANS models depends on many factors, for many practical flows the turbulence models are by far the largest source of uncertainty. As RANS models are used in the design and safety evaluation of many mission-critical systems such as airplanes and nuclear power plants, quantifying their model-form uncertainties has significant implications in enabling risk-informed decision-making. In this work we develop a data-driven, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., realizability, smoothness, and symmetry). An iterative ensemble Kalman method is used to assimilate the prior knowledge and observation data in a Bayesian framework, and to propagate them to posterior distributions of velocities and other Quantities of Interest (QoIs). We use two representative cases, the flow over periodic hills and the flow in a square duct, to evaluate the performance of the proposed framework. Both cases are challenging for standard RANS turbulence models. Simulation results suggest that, even with very sparse observations, the obtained posterior mean velocities and other QoIs have significantly better agreement with the benchmark data compared to the baseline results. At most locations the posterior distribution adequately captures the true model error within the developed model form uncertainty bounds. The framework is a major improvement over existing black-box, physics-neutral methods for model-form uncertainty quantification, where prior knowledge and details of the models are not exploited. This approach has potential implications in many fields in which the governing equations are well understood but the model uncertainty comes from unresolved physical processes.

Quantification of uncertainties in turbulence modeling: A comparison of physics-based and random matrix theoretic approaches

Numerical models based on Reynolds-Averaged Navier-Stokes (RANS) equations are widely used in engineering turbulence modeling. However, the RANS predictions have large model-form uncertainties for many complex flows, e.g., those with non-parallel shear layers or strong mean flow curvature. Quantification of these large uncertainties originating from the modeled Reynolds stresses has attracted attention in the turbulence modeling community. Recently, a physics-based Bayesian framework for quantifying model-form uncertainties has been proposed with successful applications to several flows. Nonetheless, how to specify proper priors without introducing unwarranted, artificial information remains challenging to the current form of the physics-based approach. Another recently proposed method based on random matrix theory provides the prior distributions with maximum entropy, which is an alternative for model-form uncertainty quantification in RANS simulations. This method has better mathematical rigorousness and provides the most non-committal prior distributions without introducing artificial constraints. On the other hand, the physics-based approach has the advantages of being more flexible to incorporate available physical insights. In this work, we compare and discuss the advantages and disadvantages of the two approaches on model-form uncertainty quantification. In addition, we utilize the random matrix theoretic approach to assess and possibly improve the specification of priors used in the physics-based approach. The comparison is conducted through a test case using a canonical flow, the flow past periodic hills. The numerical results show that, to achieve maximum entropy in the prior of Reynolds stresses, the perturbations of shape parameters in Barycentric coordinates are normally distributed. Moreover, the perturbations of the turbulence kinetic energy should conform to log-normal distributions. Finally, the result sheds light on how large the variance of each physical variable should be compared with each other to achieve the approximate maximum entropy prior. The conclusion can be used as a guidance for specifying proper priors in the physics-based, Bayesian uncertainty quantification framework.

Quantification of model form uncertainty in the calculation of solar diffuse irradiation on inclined surfaces for building energy simulation

Traditional uncertainty quantification (UQ) in the prediction of building energy consumption has been limited to the propagation of uncertainties in model input parameters. Models by definition ignore, at least to some degree, and, in almost all cases, simplify the physical processes that govern the reality of interest, thereby introducing additional uncertainty in model predictions that cannot be captured as input parameter uncertainty. Quantification of this type of uncertainty (which we will refer to as model form uncertainty) is a necessary step towards the complete UQ of model predictions. This paper introduces a general framework for model form UQ and shows its application to the widely used sky irradiation model developed by Perez et al. [1990. “Modeling Daylight Availability and Irradiance Components from Direct and Global Irradiance.” Solar Energy 44 (5): 271–289], which computes solar diffuse irradiation on inclined surfaces. We collected a data set of one-year measurements of solar irradiation at one location in the USA. The measurements were done at surfaces with different tilt angles and orientations, for a wide spectrum of sky conditions. A statistical analysis using both this data set and published studies worldwide suggests that the Perez model performs non-uniformly across different locations and produces a certain bias in its predictions. Based on the same data, we then use a two-phase regression model to express model form uncertainty in the use of the Perez model at this particular location. Using a holdout validation test, we demonstrate that the two-phase regression model considerably reduces the model bias errors and root mean square errors for every tilted surface. Lastly, we discuss the significance of including model form uncertainty in the energy consumption predictions obtained with whole building simulation.

The deviation from parallel shear flow as an indicator of linear eddy-viscosity model inaccuracy

A marker function designed to indicate in which regions of a generic flow field the results from linear eddy-viscosity turbulence models are plausibly inaccurate is introduced. The marker is defined to identify regions that deviate from parallel shear flow. For two different flow fields it is shown that these regions largely coincide with regions where the prediction of the Reynolds stress divergence is inaccurate. The marker therefore offers a guideline for interpreting results obtained from Reynolds-averaged Navier-Stokes simulations and provides a basis for the further development of turbulence model-form uncertainty quantification methods.

Quantification of model-form uncertainty in the correlated-k distribution method for radiation heat transfer

The full-spectrum correlated-k distribution method has been shown to be exact for thermal radiation in molecular gas-particulate mixtures which obey the correlation assumption. Unfortunately, this assumption breaks down for molecular gases with strong temperature or concentration gradients. There has previously been no way to effectively quantify the model-form uncertainty introduced by the failure of the correlation assumption. This paper examines the effect of temperature gradients in CO2 and proposes a technique to generate probability density functions for the gray-gas absorption coefficients used in the full-spectrum correlated-k distribution method. The uncertainty in the absorption coefficients may then be propagated forward to ascertain confidence intervals for important quantities such as heat flux or intensity distributions.

Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures

Estimation of the uncertainty in numerical predictions by Reynolds-averaged Navier-Stokes closures is a vital step in building confidence in such predictions. An approach to model-form uncertainty quantification that does not assume the eddy-viscosity hypothesis to be exact is proposed. The methodology for estimation of uncertainty is demonstrated for plane channel flow, for a duct with secondary flows, and for the shock/boundary-layer interaction over a transonic bump.