Forward Uncertainty Propagation Research Articles

Embedded-Error Bayesian Calibration of Thermal Decomposition of Organic Materials

Abstract Organic materials are an attractive choice for structural components due to their light weight and versatility. However, because they decompose at low temperatures relative to traditional materials, they pose a safety risk due to fire and loss of structural integrity. To quantify this risk, analysts use chemical kinetics models to describe the material pyrolysis and oxidation using thermogravimetric analysis (TGA). This process requires the calibration of many model parameters to closely match experimental data. Previous efforts in this field have largely been limited to finding a single best-fit set of parameters even though the experimental data may be very noisy. Furthermore, the chemical kinetics models are often simplified representations of the true decomposition process. The simplification induces model-form errors that the fitting process cannot capture. In this work, we propose a methodology for calibrating decomposition models to TGA data that accounts for uncertainty in the model-form and experimental data simultaneously. The methodology is applied to the decomposition of a carbon fiber epoxy composite with a three-stage reaction network and Arrhenius kinetics. The results show a good overlap between the model predictions and TGA data. Uncertainty bounds capture deviations of the model from the data. The calibrated parameter distributions are also presented. The distributions may be used in forward propagation of uncertainty in models that leverage this material.

Optimal sensor placement for uncertain inverse problem of structural parameter estimation

This paper presents an optimal sensor placement approach for the uncertain inverse problem of structural parameter estimation, aiming to mitigate the ill-posedness problem that often exists in inverse procedures. The key idea is to select the sensor positions with the most sensitive measured responses to the structural parameters and the least correlated measured responses at the selected positions. Our optimization strategy is to convert the traditional minimum variance criterion of structural parameters to be identified into a new maximum independent mean-variance criterion of structural responses, so that the complex optimal sensor placement problem is transformed into a forward uncertainty propagation problem. Then, two orthogonal matching pursuit (OMP) methods based on Monte Carlo simulation (MCS) and dimension reduction integration (DRI) method are presented in this work, in which the MCS and the much more efficient DRI method are employed to solve the forward uncertainty propagation problem, and the OMP methods in sample form and moment form are developed to determine the optimal sensor placement for the identification procedure by considering the uncertainties in the measured responses. A Markov Chain Monte Carlo algorithm is adopted to identify the distributions of structural parameters eventually. Numerical and experimental examples are presented to verify the practicability and effectiveness of the proposed methods.

Assessing uncertainties from physical parameters and modelling choices in an atmospheric large eddy simulation model.

In this study, we investigate uncertainties in a large eddy simulation of the atmosphere, employing modern uncertainty quantification methods that have hardly been used yet in this context. When analysing the uncertainty of model results, one can distinguish between uncertainty related to physical parameters whose values are not exactly known, and uncertainty related to modelling choices such as the selection of numerical discretization methods, of the spatial domain size and resolution, and the use of different model formulations. While the former kind is commonly studied e.g. with forward uncertainty propagation, we explore the use of such techniques to also assess the latter kind. From a climate modelling perspective, uncertainties in the convective response and cloud formation are of particular interest, since these affect the cloud-climate feedback, one of the dominant sources of uncertainty in current climate models. Therefore we analyse the DALES model in the RICO case, a well-studied convection benchmark. We use the VECMA toolkit for uncertainty propagation, assessing uncertainties stemming from physical parameters as well as from modelling choices. We find substantial uncertainties due to small random initial state perturbations, and that the choice of advection scheme is the most influential of the modelling choices we assessed.This article is part of the theme issue ‘Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico’.

Bayesian method and polynomial chaos expansion based inverse uncertainty quantification of spent fuel using decay heat measurements

This work presents a new application of decay heat measurement based on the calibration and inverse uncertainty quantification (IUQ) of modeling parameters of pressurized water reactor (PWR) fuel assemblies. This work (i) solves the problem encountered in forward UQ i.e., the lack of fuel vendor proprietary information (manufacturing tolerances of fuel assembly design) and operating condition uncertainties which are based on ad-hoc expert judgement or personal opinion (ii) calibrates the parameters of a fuel assembly model for improved code calculation-to-experiment agreement. The IUQ is conducted under the Bayesian framework and finds the fuel assembly model parameters that are consistent with decay heat measurements. The forward model implements a polynomial chaos expansion (PCE) based surrogate model for a computationally efficient inverse analysis. The approach introduced is then tested with the decay heat calculations of the deterministic code STREAM. Eight input parameters: fuel density, enrichment, pellet radius, clad outer radius, fuel temperature, power density, moderator temperature and boron concentration, are considered. The outcomes of this work include quantification of uncertainties and determination of probability distribution function (PDF) of these parameters, in addition to reducing code calculation-to-experiment discrepancies. These outcomes are important for future studies on the forward propagation of model parameters uncertainties.

Multilevel Decomposition Framework for Reliability Assessment of Assembled Stochastic Linear Structural Systems

AbstractTo reduce the computational cost of assembled stochastic linear structural dynamic systems, a three-staged reduced order model-based framework for forward uncertainty propagation was develo...

An integrated sensitivity‐uncertainty quantification framework for stochastic phase‐field modeling of material damage

Materials accumulate energy around voids and defects under external loading, causing the formation of microcracks. With increasing or repeated loads, those microcracks eventually coalesce to form macrocracks, which in a brittle material can cause catastrophic failure without apparent permanent deformation. At the continuum level, a stochastic phase-field model is employed to simulate failure through introducing damage and fatigue variables. The damage phase-field is introduced as a continuous dynamical variable representing the volumetric portion of fractured material and fatigue is treated as a continuous internal field variable to model the effects of microcracks arising from energy accumulation. We formulate a computational-mathematical framework for quantifying the corresponding model uncertainties and sensitivities in order to unfold and mitigate the salient sources of unpredictability in the model, hence, leading to new possible modeling paradigms. Considering an isothermal isotropic linear elastic material with viscous dissipation under the hypothesis of small deformations, we employed Monte Carlo and probabilistic collocation methods to perform the forward uncertainty propagation, in addition to local-to-global sensitivity analysis. We demonstrate that the model parameters associated with free-energy potentials contribute significantly more to the total model output uncertainties, motivating further investigations for obtaining more predictable model forms, representing the damage diffusion.

Selecting Technologies in Aircraft Conceptual Design Using Probabilistic Inversion

Technology forecasting is an essential starting point for conceptual design of any complex engineering system. In fact, many research projects are focused on developing a small set of promising technologies to a suitable readiness level. However, selecting a set of technologies from a larger pool is a nontrivial task, opposed by uncertainty and subjective tradeoffs. This paper proposes a probabilistic method to represent technologies and quantify their effects, while accounting for uncertainty. Using probabilistic inversion, technologies can be selected from a larger set to meet a certain combination of requirements. Several test cases illustrate the method and how it may be used in conceptual design projects. It is concluded that probabilistic inversion enables answering technology development and selection queries, which would be challenging to answer with traditional deterministic approaches, or purely forward uncertainty propagation approaches.

Sensitivity-based adaptive sequential sampling for metamodel uncertainty reduction in multilevel systems

Decomposition-based technique is often used in the analysis and design of complex engineering systems for reducing the computational complexity by studying the subsystems decomposed from multilevel systems. Metamodels, as a replacement of original simulation models, can further alleviate the computational burden. However, discrepancy between the simulation models and metamodels, which is defined as metamodel uncertainty, may be introduced in the analysis process of multilevel systems owing to the lack of data. The metamodel uncertainties of sub-models will be further amplified because of the hierarchical uncertainty propagation and interaction between uncertainties, which will have a great impact on the system results. An adaptive sequential sampling strategy based on sensitivity is proposed in this paper so as to improve the prediction accuracy of system response. In this strategy, polynomial-chaos expansion is used to realize the forward propagation of metamodel uncertainty quantified by the Kriging model. The forward propagation is combined with optimization based on maximum variance criterion for searching the input locations that results in the largest variance of system response. Then, the indices of subsystems are obtained to make decisions about which subsystem needs extra samples by combining Karhunen-Loeve expansion and sensitivity analysis. The effectiveness of the proposed sequential sampling strategy method is verified by two mathematical examples and a multiscale composite material.

Conformally mapped polynomial chaos expansions for Maxwell's source problem with random input data

AbstractGeneralized Polynomial Chaos (gPC) expansions are well established for forward uncertainty propagation in many application areas. Although the associated computational effort may be reduced in comparison to Monte Carlo techniques, for instance, further convergence acceleration may be important to tackle problems with high parametric sensitivities. In this work, we propose the use of conformal maps to construct a transformed gPC basis, in order to enhance the convergence order. The proposed basis still features orthogonality properties and hence, facilitates the computation of many statistical quantities such as sensitivities and moments. The corresponding surrogate models are computed by pseudo‐spectral projection using mapped quadrature rules, which leads to an improved cost accuracy ratio. We apply the methodology to Maxwell's source problem with random input data. In particular, numerical results for a parametric finite element model of an optical grating coupler are given.

Learning on dynamic statistical manifolds.

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback-Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

SAPIUM: A Generic Framework for a Practical and Transparent Quantification of Thermal-Hydraulic Code Model Input Uncertainty

Uncertainty analysis is a key element in nuclear power plant deterministic safety analysis using best-estimate thermal-hydraulic codes and best-estimate-plus-uncertainty methodologies. If forward uncertainty propagation methods have now become mature for industrial applications, the input uncertainty quantification (IUQ) on the physical models still requires further investigations. The Organisation for Economic Co-operation and Development/Nuclear Energy Agency PREMIUM project attempted to benchmark the available IUQ methods, but observed a strong user effect due to the lack of best practices guidance. The SAPIUM project has been proposed toward the construction of a clear and shared systematic approach for IUQ. The main outcome of the project is a first “good-practices” document that can be exploited for safety study in order to reach consensus among experts on recommended practices as well as to identify remaining open issues for further developments. This paper describes the systematic approach that consists of five elements in a step-by-step approach to perform a meaningful model IUQ and validation as well as some good-practice guideline recommendations for each step.

A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations

Image-based computational fluid dynamics (CFD) modeling enables derivation of hemodynamic information (e.g., flow field, wall shear stress, and pressure distribution), which has become a paradigm in cardiovascular research and healthcare. Nonetheless, the predictive accuracy largely depends on precisely specified boundary conditions and model parameters, which, however, are usually uncertain (or unknown) in most patient-specific cases. Quantifying the uncertainties in model predictions due to input randomness can provide predictive confidence and is critical to promote the transition of CFD modeling in clinical applications. In the meantime, forward propagation of input uncertainties often involves numerous expensive CFD simulations, which is computationally prohibitive in most practical scenarios. This paper presents an efficient bi-fidelity surrogate modeling framework for uncertainty quantification (UQ) in cardiovascular simulations, by leveraging the accuracy of high-fidelity models and efficiency of low-fidelity models. Contrary to most data-fit surrogate models with several scalar quantities of interest, this work aims to provide high-resolution, full-field predictions (e.g., velocity and pressure fields). Moreover, a novel empirical error bound estimation approach is introduced to evaluate the performance of the surrogate a priori. The proposed framework is tested on a number of vascular flows with both standardized and patient-specific vessel geometries, and different combinations of high- and low-fidelity models are investigated. The results show that the bi-fidelity approach can achieve high predictive accuracy with a significant reduction of computational cost, exhibiting its merit and effectiveness. Particularly, the uncertainties from a high-dimensional input space can be accurately propagated to clinically relevant quantities of interest (e.g., wall shear stress) in the patient-specific case using only a limited number of high-fidelity simulations, suggesting a good potential in practical clinical applications.

Inverse parametric uncertainty identification using polynomial chaos and high-order moment matching benchmarked on a wet friction clutch

Inconsistent behavior of mechatronic applications is often related to uncertainties ingrained in the system. Stochastic models spawn an output distribution for any given deterministic input. A stochastic model can be obtained by associating a probability distribution to several uncertain model parameters and by propagating the parametric input uncertainty through the deterministic model. Typically a maximum likelihood identification procedure is used to estimate the parametric input uncertainty from observations. Such inverse identification procedures require to iterate the expensive forward input uncertainty propagation. In this paper we establish an efficient forward uncertainty propagation method by combining Polynomial Chaos and moment matching. The high-order statistical moments of the output distribution are estimated using the generalized Polynomial Chaos framework and by using the maximum entropy strategy a distribution can be associated to them. This strategy is numerically very attractive due to reduced forward sampling and deterministic nature of the propagation strategy. The strategy is integrated in the inverse uncertainty identification of a wet clutch system for which certain model parameters exhibit inconsistent behaviour and are therefore considered as stochastic. The number of required model simulations to achieve the same accuracy as the brute force methodologies is decreased by one order of magnitude. The probability model identified with the high order estimates resulted into a true log-likelihood increase of about 4% since the accuracy of the estimated output probability density function could be improved up to 47%.

GOAL-ORIENTED MODEL ADAPTIVITY IN STOCHASTIC ELASTODYNAMICS: SIMULTANEOUS CONTROL OF DISCRETIZATION, SURROGATE MODEL AND SAMPLING ERRORS

The presented adaptive modeling approach aims to jointly control the level of refinement for each of the building blocks employed in a typical chain of finite element approximations for stochastically parametrized systems, namely: (i) finite error approximation of the spatial fields, (ii) surrogate modeling to interpolate quantities of interest(s) in the parameter domain, and (iii) Monte Carlo sampling of associated probability distribution(s). The control strategy seeks accurate calculation of any statistical measure of the distributions at minimum cost, given an acceptable margin of error as the only tunable parameter. At each stage of the greedy-based algorithm for spatial discretization, the mesh is selectively refined in the subdomains with highest contribution to the error in the desired measure. The strictly incremental complexity of the surrogate model is controlled by enforcing preponderant discretization error integrated across the parameter domain. Finally, the number of Monte Carlo samples is chosen such that either (a) the overall precision of the chain of approximations can be ascertained with sufficient confidence or (b) the fact that the computational model requires further mesh refinement is statistically established. The efficiency of the proposed approach is discussed for a frequency-domain vibration structural dynamics problem with forward uncertainty propagation. Results show that locally adapted finite element solutions converge faster than those obtained using uniformly refined grids.

Overlapping Clustering Based Technique for Scalable Uncertainty Quantification in Physical Systems

Forward propagation of uncertainty in physics-based model is nontrivial and a necessary undertaking. This paper provides a methodology for decomposing the state space of scalable dynamical systems ...

Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such process can be computational prohibitive for most real-time applications (e.g., clinical diagnosis and surgery planning) and many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures, success of which heavily relies on the large amount of training data and interpolatory nature of the problem, fail to operate when the data becomes sparse. Unfortunately, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Navier–Stokes equations. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations (i.e., Navier–Stokes equations) are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations.

Forward and backward uncertainty quantification with active subspaces: Application to hypersonic flows around a cylinder

We perform a Bayesian calibration of the freestream velocity and density starting from measurements of the pressure and heat flux at the stagnation point of a hypersonic high-enthalpy flow around a cylinder. The objective is to explore the possibility of using stagnation heat flux measurements, together with pressure measurements, to rebuild freestream conditions since such measurements are available for recent space missions but not exploited for freestream characterization. First, we formulate an algorithm of mesh adaptation, enabling accurate numerical solutions in an automatic way for a given set of inputs. Secondly, active subspaces are used to find a low-dimensional dependence structures in the input-to-output map of the forward numerical solver. Then, surrogate models on the active variables are used to accelerate the forward uncertainty propagation by Monte Carlo sampling and the Markov Chain Monte Carlo sampling of the posterior distribution for Bayesian inversion. A preliminary sensitivity analysis with sparse Polynomial Dimensional Decomposition is performed on the chemical model of the air mixture, to determine the most influential uncertain chemical parameters in the forward problem. Then, the forward and backward methodologies are applied to the simulation of a hypersonic flow around a cylinder, in conditions for which experimental data are available, revealing new insights towards the potential exploitation of heat flux data for freestream rebuilding.

Stochastic modeling techniques for textile yarn distortion and waviness with 1D random fields

Within the concept of simulation approaches for manufacturing-induced imperfections of composite structures, this work proposes modeling frameworks for the consideration of stochastic deviations concerning the yarns of textile composite materials. The random distortion of a yarn’s cross-section, is addressed by flexible 1D Fourier-based random fields, with the potential to be calibated from measurements of the deviations from the nominal yarn shape and their statistical characteristics. Furthermore, a Kriging-based modeling approach is presented, able to randomize any nominal yarn path in short or long range problems, considering data for the correlation and variance in a straightforward manner. The effects of defects due to stochastic yarn distortion and waviness, are investigated by simulating a forward uncertainty propagation problem of a triaxially braided composite material. The response variability concerning stiffness and strength for different uncertainty levels is highlighted, while several comments are offered regarding numerical issues and potential surrogate modeling techniques.

Semi-intrusive uncertainty propagation for multiscale models

A family of semi-intrusive forward uncertainty propagation (UP) methods for multiscale models is introduced. The methods are semi-intrusive in the sense that inspection of the model is limited up to the level of the single scale systems, and viewing these single scale components as black-boxes. The goal is to estimate uncertainty in the result of multiscale models at a reduced amount of time as compared to black-box Monte Carlo (MC).In the resulting semi-intrusive MC method, the required number of samples of an expensive single scale model is minimized in order to reduce the execution time for the overall uncertainty estimation. In the metamodeling approach the expensive model component is replaced completely by a computationally much cheaper surrogate model.These semi-intrusive algorithms have been tested on two case studies based on reaction-diffusion dynamics. The results demonstrate that the proposed semi-intrusive methods can reduce significantly the computational time for multiscale UP, while still computing accurately the estimates of uncertainties.The semi-intrusive methods can therefore be a valid alternative, when uncertainties of a multiscale model cannot be estimated by the black-box MC methods with a high precision in a feasible amount of time.

Causality and Bayesian Network PDEs for multiscale representations of porous media

Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (continuum- or Darcy-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media, and the design of novel materials, e.g., for energy storage. Microscopic properties exhibit complex statistical correlations and geometric constraints that present challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian Networks. The proposed framework allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that are informed by domain knowledge and emulate engineering processes, e.g., the design of hierarchical nanoporous materials. These PDFs also serve as input for the forward propagation of parametric uncertainty thereby yielding a Bayesian Network PDE. To assess the impact of causal relationships and microscale correlations on macroscopic material properties, we propose a moment-independent GSA and corresponding effect rankings for Bayesian Network PDEs, based on the differential Mutual Information, that leverage the structure of Bayesian Networks and account for both correlated inputs and complex non-Gaussian (skewed, multimodal) QoIs. Our findings from numerical experiments, which feature a non-intrusive uncertainty quantification workflow, indicate two practical outcomes. First, the inclusion of correlations through structured priors based on causal relationships informed by domain knowledge impacts predictions of QoIs and has important implications for engineering design. Second, structured priors with non-trivial correlations yield different effect rankings than independent priors; these rankings are more consistent with the anticipated physics.