Efficient Uncertainty Propagation Research Articles

Stochastic assessment of electric powertrain whining noise under early-stage design uncertainties

Despite the advantage of being quieter than traditional internal combustion engine vehicles, electric vehicles are often distinguished by high-frequency tonal components, which can be perceived as unpleasant to the occupants and the environment. To ensure optimal acoustic comfort in electric vehicles, it is important to analyze the NVH behavior of e-powertrains during the early stages of the design process which poses inherent uncertainties, such as varying operating conditions, partial knowledge of design parameters, dispersion in measurement-based data, etc. To effectively address these uncertainties, it is necessary to use fast and comprehensive stochastic models during the design phase. In this work, a probabilistic framework is presented to estimate the electric powertrain’s interior whining noises considering the structure-borne contribution. Hence, two different stochastic metamodels are developed for efficient quantification and propagation of uncertainties from electric motor stage to powertrain mounting system. Multivariate Bayesian regression models help to incorporate prior knowledge on the uncertain parameters and generate the respective posterior distributions using Markov chains Monte Carlo (MCMC) techniques. For this particular application, the data is generated through weakly-coupled multi-physical domains estimated using semi-analytical approaches and combined with measured vehicle transfer functions. Importantly, the validation of each domain is conducted separately to ensure accurate representation. The results obtained from the developed probabilistic framework will aid in the early design stages by guiding engineers in making informed decisions to optimize NVH performance.

Probabilistic Printability Maps for Laser Powder Bed Fusion Via Functional Calibration and Uncertainty Propagation

Abstract In this work, we develop an efficient computational framework for process space exploration in laser powder bed fusion (LPBF) based additive manufacturing technology. This framework aims to find suitable processing conditions by characterizing the probability of encountering common build defects. We employ a Bayesian approach toward inferring a functional relationship between LPBF processing conditions and the unobserved parameters of laser energy absorption and powder bed porosity. The relationship between processing conditions and inferred laser energy absorption is found to have good correspondence to the literature measurements of powder bed energy absorption using calorimetric methods. The Bayesian approach naturally enables uncertainty quantification and we demonstrate its utility by performing efficient forward propagation of uncertainties through the modified Eagar–Tsai model to obtain estimates of melt pool geometries, which we validate using out-of-sample experimental data from the literature. These melt pool predictions are then used to compute the probability of occurrence of keyhole and lack-of-fusion based defects using geometry-based criteria. This information is summarized in a probabilistic printability map. We find that the probabilistic printability map can describe the keyhole and lack-of-fusion behavior in experimental data used for calibration, and is capable of generalizing to wider regions of processing space. This analysis is conducted for SS316L, IN718, IN625, and Ti6Al4V using melt pool measurement data retrieved from the literature.

A dimension reduction-based Kriging modeling method for high-dimensional time-variant uncertainty propagation and global sensitivity analysis

Surrogate models have been widely used in the uncertainty propagation and global sensitivity analysis of complex evaluation-expensive engineering problems. However, the construction of high-accuracy surrogate models for time-variant problems with high-dimensional inputs and outputs using a small number of training samples remains a challenge. To address this challenge, we propose a dimension reduction-based Kriging modeling (KMDR) method for high-dimensional time-variant uncertainty propagation and global sensitivity analysis. Singular value decomposition is performed on the original time-variant response to extract principal components from high-dimensional outputs. And the improved sufficient dimension reduction (ISDR) is performed on high-dimensional inputs to identify the latent input space with respect to each principal component of outputs. A ladle estimator with a rigorous mathematical definition is then employed to determine the number of principal components of the outputs and the dimensionalities of the latent input spaces. The ladle estimator considers variabilities in both eigenvalues and eigenvectors of a matrix and can determine the latent dimensionality more accurately and efficiently than existing approaches. Subsequently, Kriging models between high-dimensional inputs and each principal component of the outputs are constructed based on a newly devised Kriging kernel, which embeds the information of the ISDR into the kernel function and can achieve higher accuracy than directly constructing Kriging models between latent inputs and outputs. In addition, a generalized variance-based sensitivity index, which can quantify the effects of the inputs on the overall time-variant response, is defined and computed directly from Sobol’ sensitivity indices of the inputs at each time node. Finally, the surrogate model of the time-variant system constructed by the KMDR is directly adopted for efficient time-variant uncertainty propagation and global sensitivity analysis. Several examples demonstrate that the proposed approach can construct more accurate surrogate models and obtain more accurate time-variant uncertainty propagation and global sensitivity analysis results than existing methods with a small training set.

Dimensionality Reduction for Onboard Modeling of Uncertain Atmospheres

Onboard density models are a key aspect of closed-loop guidance systems for hypersonic flight. Traditional approaches model density as a deterministic function of altitude, but a recent drive toward stochastic guidance approaches motivates onboard uncertainty propagation. Existing solutions for efficient uncertainty propagation generally treat density as an exponential function of altitude, but this approach is limited in its ability to capture relevant dispersions. This work models density as a Gaussian random field that is approximated by a Karhunen–Loève expansion, enabling a relatively high-fidelity, finite-dimensional parametric representation. Alternative models are also developed using a variational autoencoder architecture, resulting in greater dimensionality reduction at the expense of analytical description. Normalization schemes are presented and compared by their efficiency in capturing density variability in a limited number of terms, and normalization by reference dynamic pressure is shown to be the most compact approach. The model alternatives are compared both by their approximations of density itself and by their predictions of peak heat flux for dispersed direct-entry and aerocapture trajectories. An extension of this approach for modeling density as a function of multiple independent variables is also presented and demonstrated. Finally, it is shown that the Karhunen–Loève density model can be sequentially updated according to noisy density observations by formulating the problem as a Kalman measurement function.

An efficient uncertainty propagation method for nonlinear dynamics with distribution-free P-box processes

The distribution-free P-box process serves as an effective quantification model for time-varying uncertainties in dynamical systems when only imprecise probabilistic information is available. However, its application to nonlinear systems remains limited due to excessive computation. This work develops an efficient method for propagating distribution-free P-box processes in nonlinear dynamics. First, using the Covariance Analysis Describing Equation Technique (CADET), the dynamic problems with P-box processes are transformed into interval Ordinary Differential Equations (ODEs). These equations provide the Mean-and-Covariance (MAC) bounds of the system responses in relation to the MAC bounds of P-box-process excitations. They also separate the previously coupled P-box analysis and nonlinear-dynamic simulations into two sequential steps, including the MAC bound analysis of excitations and the MAC bounds calculation of responses by solving the interval ODEs. Afterward, a Gaussian assumption of the CADET is extended to the P-box form, i.e., the responses are approximate parametric Gaussian P-box processes. As a result, the probability bounds of the responses are approximated by using the solutions of the interval ODEs. Moreover, the Chebyshev method is introduced and modified to efficiently solve the interval ODEs. The proposed method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering black-box problem of launch vehicle trajectory. Compared to the reference solutions based on the Monte Carlo method, with relative errors of less than 3%, the proposed method requires less than 0.2% calculation time. The proposed method also possesses the ability to handle complex black-box problems.

Hybrid uncertainty propagation based on multi-fidelity surrogate model

There always exist multiple uncertainties including random uncertainty, interval uncertainty, and fuzzy uncertainty in engineering structures. In the presence of hybrid uncertainties, the hybrid uncertainty propagation analysis can be a challenging problem, which suffers from the computational burden of double-loop procedure when numerical simulation techniques are employed. In this work, a novel method for efficient hybrid uncertainty propagation analysis with the three types of uncertainties is proposed. Generally, multi-fidelity surrogate models, such as Co-Kriging, can greatly improve the computational efficiency by leveraging information from a low-fidelity model to build a high-fidelity approximate model. However, the traditional multi-fidelity surrogate model methods always calculate the hybrid uncertainty propagation result by combining with several numerical simulation techniques. This process can introduce post-processing errors unless unlimited number of samples are used, which is impossible in engineering application. In order to address this issue, the analytical solutions of the output mean and output variance are derived based on the Co-Kriging, and the resulting mean and variance are both random variables. Moreover, a new adaptive framework is established to strengthen the estimation accuracy of the hybrid uncertainty propagation result, by combining the augmented expected improvement function and the derived mean random variable. Several applications are introduced to demonstrate the effectiveness of the proposed method for solving hybrid uncertainty propagation problems.

An efficient uncertainty propagation analysis method of non-parameterized P-boxes based on dimension-reduction integral and maximum entropy estimation

An efficient uncertainty propagation analysis method of non-parameterized P-boxes based on dimension-reduction integral and maximum entropy estimation

Stochastic optimal control for autonomous driving applications via polynomial chaos expansions

AbstractModel‐based methods in autonomous driving and advanced driving assistance gain importance in research and development due to their potential to contribute to higher road safety. Parameters of vehicle models, however, are hard to identify precisely or they can change quickly depending on the driving conditions. In this paper, we address the problem of safe trajectory planning under parametric model uncertainties motivated by automotive applications. We use the generalized polynomial chaos expansions for efficient nonlinear uncertainty propagation and distributionally robust inequalities for chance constraints approximation. Inspired by the tube‐based model predictive control, an ancillary feedback controller is used to control the deviations of stochastic modes from the nominal solution, and therefore, decrease the variance. Our approach allows reducing conservatism related to nonlinear uncertainty propagation while guaranteeing constraints satisfaction with a high probability. The performance is demonstrated on the example of a trajectory optimization problem for a simplified vehicle model with uncertain parameters.

A hybrid data-driven and model-based approach for computationally efficient stochastic unit commitment and economic dispatch under wind and solar uncertainty

Stochastic unit commitment (UC) and economic dispatch (ED) are imperative in dealing with uncertainty in renewable forecast for power system operation and planning such that the overall expected production cost is minimized over the planning horizon. However, accurate calculation of the expected production cost requires assessment of a very large number of different scenarios of uncertain renewable resources, such as solar and wind, which is practically infeasible to simulate in real time. This article proposes a hybrid data-driven and physics-based model-predictive paradigm to efficiently solve for stochastic unit commitment and economic dispatch considering uncertainty in wind and solar power forecasts. The novelty of the approach lies in decoupling the production cost estimation from the unit commitment and economic dispatch optimization problems under uncertainty without compromising on the fidelity of the solutions. A data-driven machine learning model is first developed to predict the mean optimal production cost. A physics-based inverse problem is then solved to get the stochastic UC and ED profiles from the expected cost. The presented approach considers, for the first time, solar uncertainty in UC/ED determination and enables efficient and accurate propagation of wind and solar uncertainty to estimate the statistics of the production cost. The effectiveness of the developed approach is demonstrated systematically on a stylized RTS-GMLC single-node system. The overall framework predicts the expected cost 62.5% more accurately than the existing state-of-the-art, on unforeseen days during the entire year, and yields, for the first time, the associated physically consistent UC and ED profiles. The solutions are also shown to be flexible in providing adequate daily reserves to address any statistical deviations from probabilistic power forecasts. The computational time associated with the presented method is only about 10 s compared to over 24 h needed for a conventional stochastic UC/ED determination under uncertainty on an Intel Core i9 processor with 32 GB of RAM.

Efficient Uncertainty Propagation in Model-Based Reinforcement Learning Unmanned Surface Vehicle Using Unscented Kalman Filter

This article tackles the computational burden of propagating uncertainties in the model predictive controller-based policy of the probabilistic model-based reinforcement learning (MBRL) system for an unmanned surface vehicles system (USV). We proposed filtered probabilistic model predictive control using the unscented Kalman filter (FPMPC-UKF) that introduces the unscented Kalman filter (UKF) for a more efficient uncertainty propagation in MBRL. A USV control system based on FPMPC-UKF is developed and evaluated by position-keeping and target-reaching tasks in a real USV data-driven simulation. The experimental results demonstrate a significant superiority of the proposed method in balancing the control performance and computational burdens under different levels of disturbances compared with the related works of USV, and therefore indicate its potential in more challenging USV scenarios with limited computational resources.

An efficient uncertainty quantification and propagation method through skewness and kurtosis fitting region

An efficient uncertainty quantification and propagation method through skewness and kurtosis fitting region

Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains

A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes.

A new Bayesian probabilistic integration framework for hybrid uncertainty propagation

Efficient propagation of uncertainty is one of the most critical tasks for uncertainty quantification and reliable design in the presence of multi-source uncertainties. This work presents a new methodology framework, termed as “mixed imprecise probabilistic integration”, for efficient propagation of the uncertainties characterized by non-probabilistic models, imprecise probability models and precise probability models. In this paper, the mixed-high dimensional model representation combining the cut-high dimensional model representation and random sampling-high dimensional model representation is firstly proposed, which conserves all the advantages of these two methods. Secondly, the proposed method is realized by using mixed-high dimensional model representation and Bayesian inference. The estimations and the posterior variances of the proposed method are analytically derived of high accuracy. With the proposed method, all the three kinds of uncertainty models can be efficiently propagated with one set of function evaluations. The effectiveness of the proposed methods is demonstrated by three engineering examples.

Sparse Bayesian learning for complex‐valued rational approximations

AbstractSurrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non‐linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. It has been shown that for models with discontinuities or rational dependencies, for example, frequency response functions of dynamic systems, the use of a rational (Padé) approximation can significantly improve the approximation accuracy. In order to avoid overfitting issues in previously proposed standard least squares approaches, we introduce a sparse Bayesian learning approach to estimate the coefficients of the rational approximation. Therein the linearity in the numerator polynomial coefficients is exploited and the denominator polynomial coefficients as well as the problem hyperparameters are determined through type‐II‐maximum likelihood estimation. We apply a quasi‐Newton gradient‐descent algorithm to find the optimal denominator coefficients and derive the required gradients through application of ‐calculus. The method is applied to the frequency response functions of an algebraic frame structure model as well as that of an orthotropic plate finite element model.

A novel linear uncertainty propagation method for nonlinear dynamics with interval process

Interval process is a preferable model for time-varying uncertainty propagation of dynamic systems when only the range of uncertainties can be obtained. However, for nonlinear systems, except for Monte Carlo (MC) simulation, there are still few efficient uncertainty propagation methods under the interval process model. This paper develops a non-intrusive and semi-analytical uncertainty propagation method, named the “convex model linearization method (CMLM),” by constructing a linearization formulation of a nonlinear system in a non-probabilistic sense. First, the criterion to evaluate the difference between the original system and the linearization formulation is derived, represented by the discrepancy of middle point, radius and correlations of response. By minimizing these three parameters, the coefficients of linear equations will be optimized to obtain the linearization formulation of the original system. Then, analytical equations are built to calculate uncertainty response under the interval process, without time-consuming analysis of the original system. To further improve the efficiency of the linearization process, Chebyshev polynomial is introduced to approximate the nonlinear dynamic analysis. Two numerical examples of duffing oscillators and vehicle rides are set to test the proposed CMLM. Compared to the MC method, with comparable uncertainty response precision, the CMLM just needs 1–10% times of dynamic analyses of the nonlinear system. Furthermore, a practical launch vehicle ascent trajectory problem with black-box dynamics is solved by, respectively, the CMLM and MC method. The results verify the capacity of the CMLM to deal with black-box problems and show that the CMLM performs better in terms of accuracy, efficiency and robustness.

Bayesian probabilistic propagation of hybrid uncertainties: Estimation of response expectation function, its variable importance and bounds

Uncertainties existing in physical and engineering systems can be characterized by different kinds of mathematical models according to their respective features. However, efficient propagation of hybrid uncertainties via an expensive-to-evaluate computer simulator is still a computationally challenging task. In this contribution, estimation of response expectation function (REF), its variable importance and bounds under hybrid uncertainties in the form of precise probability models, parameterized probability-box models and interval models is investigated through a Bayesian approach. Specifically, a new method, termed “Parallel Bayesian Quadrature Optimization” (PBQO), is developed. The method starts by treating the REF estimation as a Bayesian probabilistic integration (BPI) problem with a Gaussian process (GP) prior, which in turn implies a GP posterior for the REF. Then, one acquisition function originally developed in BPI and other two in Bayesian global optimization are introduced for Bayesian experimental designs. Besides, an innovative strategy is also proposed to realize multi-point selection at each iteration. Overall, a novel advantage of PBQO is that it is capable of yielding the REF, its variable importance and bounds simultaneously via a pure single-loop procedure allowing for parallel computing. Three numerical examples are studied to demonstrate the performance of the proposed method over some existing methods.

Recent advances in polynomial chaos method for uncertainty propagation

Uncertainty exists widely in engineering design. As one of the key components of engineering design, uncertainty propagation and quantification has always been an important research topic. Polynomial chaos (PC) is a highly efficient uncertainty propagation method which has been widely studied and applied. Therefore, this paper reviews recent advances in the PC method. First, the fundamentals of PC are introduced, including the construction of an orthogonal polynomial basis and the calculation of PC coefficients. Second, strategies such as basis truncation, sparse reconstruction, sparse grid and multi-fidelity modeling are described to address the curse of dimensionality issue of PC. Local and global sensitivity analyses based on PC are then introduced. Finally, the research prospects of PC are given.

Effect of turbulence model uncertainty on scramjet strut injector flow field analysis

The effect of turbulence model closure coefficient uncertainty on the Reynolds-averaged Navier–Stokes solution of a scramjet strut fuel injector flow field is investigated with an uncertainty quantification and sensitivity analysis study for the Menter-BSL and Wilcox-2006 k-ω turbulence models. Simulations were performed using the VULCAN flow solver. Nonintrusive polynomial chaos theory was used for efficient propagation of uncertainty, and Sobol indices were utilized to quantify the sensitivity of various solution metrics to the variation of each closure coefficient. The output metrics of interest were evaluated at three crossflow planes, and included the integrated quantities of mixing efficiency, circulation, total pressure recovery, and one-dimensional Mach number, as well as the pointwise vorticity and eddy viscosity distributions. Influential sets of closure coefficients were identified for each turbulence model, with the Kármán and diffusion constants being the most prominent. These results are expected to assist future efforts aimed at reducing the uncertainty in the numerical design of scramjet fuel injectors through the identification of closure coefficients and physical aspects of the flow that warrant further investigation.

Efficient Propagation of Epistemic Uncertainty for Probabilistic Seismic Hazard Analyses (PSHAs) Including Partial Correlation of Magnitude–Distance Scaling

ABSTRACT Probabilistic seismic hazard analysis (PSHA) is moving from ergodic ground-motion models (GMMs) to nonergodic GMMs that account for site-source-specific source, path, and site effects and which require a much larger number of GMM branches on the logic tree to capture the full epistemic uncertainty. An efficient method for computing PSHA with a large number of GMM branches was developed by Lacour and Abrahamson (2019) using polynomial chaos (PC) expansion with the key assumption that the epistemic uncertainty in the median ground motion is fully correlated. In the current study, we remove the assumption of full correlation using a multivariate PC expansion. The correlation structure of the available median GMMs across scenarios is computed empirically. The median ground motion is modeled as a Gaussian random process with the correlation structure of the GMMs across the range of relevant earthquake scenarios. This Gaussian random process is discretized using the Karhunen–Loeve expansion, which leads to multivariate PC expansions of uncertain hazard curves. The hazard fractiles can be reconstructed during an efficient postprocessing phase that includes the effects of partial correlation between the GMMs. Multivariate PC expansions require significantly more terms than for the fully correlated case, which increases the calculation time by about a factor of 5, but it is still much more efficient than direct sampling of the branches of the GMM logic for a large number of branches. An example hazard calculation shows that the effect of using partial correlation in place of full correlation of the GMMs is small for the Next Generation Attenuation-West2 (NGA-West2) set of GMMs, indicating that the fully correlated assumption may be adequate for many applications. The multivariate PC method can be used to evaluate the effects of the partial correlation of the GMMs for sets of GMMs that are different from the NGA-West2 GMMs.

Atomistic-to-meso multi-scale data-driven graph surrogate modeling of dislocation glide

From their birth in the manufacturing process, materials inherently contain defects that affect the mechanical behavior across multiple length and time-scales, including vacancies, dislocations, voids and cracks. Understanding, modeling, and real-time simulation of the underlying stochastic micro-structure defect evolution is therefore vital towards multi-scale coupling and propagating numerous sources of uncertainty from atomistic to eventually aging continuum mechanics. We develop a graph-based surrogate model of dislocation glide for computation of stochastic dislocation mobility. We model an edge dislocation as a random walker, jumping between neighboring nodes of a graph following a Poisson stochastic process. The network representation functions as a coarse-graining of a molecular dynamics simulation that provides dislocation trajectories for an empirical computation of jump rates. With this construction, we recover the original atomistic mobility estimates, with remarkable computational speed-up and accuracy. Furthermore, the underlying stochastic process provides the statistics of dislocation mobility associated to the original molecular dynamics simulation, allowing an efficient propagation of material parameters and uncertainties across the scales.