Generalized Polynomial Chaos Expansion Research Articles

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

Robust Feedback Stabilization of Interacting Multi-agent Systems Under Uncertainty

We consider control strategies for large-scale interacting agent systems under uncertainty. The particular focus is on the design of robust controls that allow to bound the variance of the controlled system over time. To this end we consider H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\infty $$\\end{document} control strategies on the agent and mean field description of the system. We show a bound on the H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}_\\infty $$\\end{document} norm for a stabilizing controller independent on the number of agents. Furthermore, we compare the new control with existing approaches to treat uncertainty by generalized polynomial chaos expansion. Numerical results are presented for one-dimensional and two-dimensional agent systems.

The effect of uncertainty in calibration data on burning rate predictions

Comprehensive pyrolysis models describe solid phase reaction rates with respect to the material temperature and the concentrations of components. In theory, this aspect of comprehensive pyrolysis models allows for true predictive capabilities for pyrolysis models and, more generally, fire models. To utilize these capabilities, the reaction kinetics must be defined for the material of interest and there is a lack of publicly available data from which these kinetic parameters may be determined. There is also a lack of guidance for quality and repeatability for thermal analysis experiments, which complicates the use of mean thermal analysis data found in the literature to estimate the thermal degradation kinetics. This study details a numerical investigation on the effect of repeatability of thermogravimetric analysis data on the kinetics calibrated from the data. The uncertainty in the kinetic parameters that corresponded to target repeatability values imposed on synthetic calibration data was determined through a latin hypercube sampling procedure. The generalized polynomial chaos expansions method allowed propagation of the uncertainty of the kinetic parameters through a one-dimensional model of a bench-scale experiment. The relationship between uncertainty in calibration data and uncertainty in bench-scale model predictions was linear.

Uncertainty Quantification and Sensitivity Analysis for the Self-Excited Vibration of a Spline-Shafting System

Abstract Self-excited vibrations can occur in the spline-shafting system due to internal friction of the tooth surface. However, due to manufacturing errors, design tolerances, and time-varying factors, the parameters that induce self-excited vibrations are always uncertain. This study provides new insights into the uncertainty quantification and sensitivity analysis of a spline-shaft system suffering from self-excited vibrations. The nonintrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is used to represent the propagation of uncertainties in the rotor dynamics, which allows rapid estimation of the statistics of the nonlinear responses. Furthermore, the global sensitivity analysis of the stochastic self-excited vibration response of the rotor system with probabilistic uncertain parameters is evaluated by Sobol indices. The relative influence of different random parameters on the vibration behavior and initial displacement conditions for the occurrence of self-excited vibration is investigated. The accuracy of the adopted method based on the gPCE metamodel is validated by conventional Monte Carlo simulation (MCS). Finally, the effects of parameter uncertainties considering random distribution characteristics on the stochastic vibration characteristics of the rotor system are discussed, which demonstrates the need to consider input uncertainties in analysis and design to ensure robust system performance.

Fast Uncertainty Quantification by Sparse Data Learning From Multiphysics Systems

Data learning is a burgeoning area; here, we refer to it as a gorgeous blend of scientific computing and high-dimensional statistics. In this article, we propose a generalized polynomial chaos expansion (gPCE)-based surrogate model, with a sparse dataset, for multiphysics problems with associated uncertainty. This data-driven method avoids the dilemma of traditional intrusive methods, which usually require tremendous revisions over the backend source code. To circumvent the curse of dimensionality in uncertainty quantification, we perform sparse data learning for the relationship between the system output and the gPCE bases. Numerical experiments showcase that the surrogate model exhibits superior fast performance in system-output prediction, statistical-feature extraction, and sensitivity analysis of the 3-D multiphysics problems.

Evaluation of uncertainty in direct measurement for parameterization of pyrolysis models, Part I: Thermal analysis

A comprehensive uncertainty analysis is an important part of any engineering calculation because it allows stakeholders to assess the confidence in the conclusions that come from the calculation. Without this necessary step in hypothesis testing and model validation, the true accuracy and confidence of predictions cannot be completely quantified. An experimental and analytical approach is presented in this work to determine the kinetics and energetics of pyrolysis which form the foundation for larger scale models of material burning. A detailed uncertainty quantification which utilized the generalized polynomial chaos expansion method is presented to propagate uncertainty through the pyrolysis model representation of a thermal analysis experiment. Using this methodology, the uncertainty in the predictions was found to be comparable to the uncertainty in the experimental thermal analysis data within calibration conditions. Sensitivity analyses revealed the largest contributor to uncertainty in heat flow rate predictions was the specific heat capacity of all components. Comparison of the uncertainty quantification and sensitivity analyses to well-known results validated the method of uncertainty propagation for more complex scenarios.

Impact of the Spatial Velocity Inlet Distribution on the Hemodynamics of the Thoracic Aorta.

The impact of the distribution in space of the inlet velocity in the numerical simulations of the hemodynamics in the thoracic aorta is systematically investigated. A real healthy aorta geometry, for which in-vivo measurements are available, is considered. The distribution is modeled through a truncated cone shape, which is a suitable approximation of the real one downstream of a trileaflet aortic valve during the systolic part of the cardiac cycle. The ratio between the upper and the lower base of the truncated cone and the position of the center of the upper base are selected as uncertain parameters. A stochastic approach is chosen, based on the generalized Polynomial Chaos expansion, to obtain accurate response surfaces of the quantities of interest in the parameter space. The selected parameters influence the velocity distribution in the ascending aorta. Consequently, effects on the wall shear stress are observed, confirming the need to use patient-specific inlet conditions if interested in the hemodynamics of this region. The surface base ratio is globally the most important parameter. Conversely, the impact on the velocity and wall shear stress in the aortic arch and descending aorta is almost negligible.

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.

Error propagation for quantile estimation via combining Polynomial Chaos expansions and metalog distributions

Introducing parametric uncertainties to models is a quintessential step of simulation-based risk assessment and rare event simulations. A novel quantile estimation method is proposed based on generalized Polynomial Chaos expansions and metalog distribution estimation. We propose to derive the metalog distribution based on the statistical information of an expansion of the random model. The advantage of latter lies in a reduced number of evaluations of the full model, while deriving the metalog distribution avoids sampling-related challenges of quantile estimation. Error estimates and an algorithm to choose the relevant number of statistical moments are developed and analyzed, giving a framework for assessing the method applicability and the accuracy of the quantile estimation. The proposed method and the analysis are demonstrated on numerical examples. The method is applied to a complex fluid flow problem in transpiration cooling.

A Posteriori Validation of Generalized Polynomial Chaos Expansions

Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e., for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multiparameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2 dimensional manifolds of steady states of the Swift–Hohenberg equation.

Stochastic analysis for in-plane dynamic responses of low-speed uniformity of tires due to geometric defects

Tire uniformity is defined as the dynamic mechanical characteristics of pneumatic tires due to the defects originating in tire components during manufacturing processes. These defects lead to deviations in the structural and material parameters of tires and, in turn, induce variations in the dynamic response. In this scenario, this paper proposes a probabilistic model of the low-speed uniformity analysis for predicting the impacts of the uncertainties in the geometric defects and the vertical load on the dynamic responses. The deterministic coupled rigid–flexible ring model, combined with the generalized polynomial chaos expansion theory, is developed to simulate the radial force variation due to radial run-out. In this approach, the non-uniform tire radius and the vertical load are selected as a set of random variables and are approximated by the generalized polynomial chaos expansions. The distributions of these input variables are identified from 200 measured tires by the Pearson model. A non-intrusive probabilistic collocation method is adopted to evaluate the coefficients of the generalized polynomial chaos expansions, and the selection of collocation points is also illustrated. Finally, the distributions of the radial force variation and its first harmonic due to the geometric defects and vertical loads are predicted in comparison with measurements. Moreover, the distributions of the dynamic response caused by each parameter and different variance levels are discussed and verified by the measurement data from 1130 tires.

Mixed aleatory and epistemic uncertainty propagation using Dempster–Shafer theory

In this paper, we deal with epistemic uncertainty in the framework of Dempster–Shafer theory, where basic belief assignments are used to characterize the uncertain parameters. To propagate the mixed epistemic and aleatory uncertainties, we introduce the relevant theoretical basics of DS theory, present the numerical approach based on DS theory combined with generalized polynomial chaos expansion, and conduct the error analysis for the numerical approach. Specifically, to analyze the convergence rate of the numerical solution represented with basic belief assignments, we define a measure based on the Hausdorff distance to quantify the difference between two basic belief assignments. The convergence of the numerical approximation is demonstrated with a few simple examples under different scenarios, and the presented numerical approach is applied to quantify the mixed types of uncertainty in quasi-one-dimensional flow.

Time domain characteristics of lossy multiconductor transmission lines randomly excited by electromagnetic pulse

This paper provides a time domain method for solving the induced currents (and voltages) of frequency-dependent lossy multiconductor transmission lines (MTLs) model randomly excited by electromagnetic pulse (EMP) plane wave. The model is processed based on the generalized polynomial chaos expansion (gPCE) and improved finite difference time domain (FDTD). The improved FDTD method overcomes the shortcomings of the conventional method that oversimplifies the expression of conductor impedance. In addition, when quantifying the influence of uncertainty on induced values, general formula of the global sensitivity analysis is provided by defining the gPCE basis function sequence to solve the complicated search work. The verification shows that the improved FDTD method alone or combined with gPCE to solve the lossy MTLs model excited by EMP is superior to the conventional methods in solving time and accuracy. Finally, compared with the Monte Carlo method, the effectiveness of the sensitivity method is also verified.

Multifidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos-Kriging

We propose novel methods for Conditional Value-at-Risk (CVaR) estimation for nonlinear systems under high-dimensional dependent random inputs. We develop a novel DD-GPCE-Kriging surrogate that merges dimensionally decomposed generalized polynomial chaos expansion and Kriging to accurately approximate nonlinear and nonsmooth random outputs. We use DD-GPCE-Kriging (1) for Monte Carlo simulation (MCS) and (2) within multifidelity importance sampling (MFIS). The MCS-based method samples from DD-GPCE-Kriging, which is efficient and accurate for high-dimensional dependent random inputs, yet introduces bias. Thus, we propose an MFIS-based method where DD-GPCE-Kriging determines the biasing density, from which we draw a few high-fidelity samples to provide an unbiased CVaR estimate. To accelerate the biasing density construction, we compute DD-GPCE-Kriging using a cheap-to-evaluate low-fidelity model. Numerical results for mathematical functions show that the MFIS-based method is more accurate than the MCS-based method when the output is nonsmooth. The scalability of the proposed methods and their applicability to complex engineering problems are demonstrated on a two-dimensional composite laminate with 28 (partly dependent) random inputs and a three-dimensional composite T-joint with 20 (partly dependent) random inputs. In the former, the proposed MFIS-based method achieves 104x speedup compared to standard MCS using the high-fidelity model, while accurately estimating CVaR with 1.15% error.

Bi-fidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos expansion

Digital twin models allow us to continuously assess the possible risk of damage and failure of a complex system. Yet high-fidelity digital twin models can be computationally expensive, making quick-turnaround assessment challenging. Toward this goal, this article proposes a novel bi-fidelity method for estimating the conditional value-at-risk (CVaR) for nonlinear systems subject to dependent and high-dimensional inputs. For models that can be evaluated fast, a method that integrates the dimensionally decomposed generalized polynomial chaos expansion (DD-GPCE) approximation with a standard sampling-based CVaR estimation is proposed. For expensive-to-evaluate models, a new bi-fidelity method is proposed that couples the DD-GPCE with a Fourier-polynomial expansion of the mapping between the stochastic low-fidelity and high-fidelity output data to ensure computational efficiency. The method employs measure-consistent orthonormal polynomials in the random variable of the low-fidelity output to approximate the high-fidelity output. Numerical results for a structural mechanics truss with 36-dimensional (dependent random variable) inputs indicate that the DD-GPCE method provides very accurate CVaR estimates that require much lower computational effort than standard GPCE approximations. A second example considers the realistic problem of estimating the risk of damage to a fiber-reinforced composite laminate. The high-fidelity model is a finite element simulation that is prohibitively expensive for risk analysis, such as CVaR computation. Here, the novel bi-fidelity method can accurately estimate CVaR as it includes low-fidelity models in the estimation procedure and uses only a few high-fidelity model evaluations to significantly increase accuracy.

HIGH-DIMENSIONAL STOCHASTIC DESIGN OPTIMIZATION UNDER DEPENDENT RANDOM VARIABLES BY A DIMENSIONALLY DECOMPOSED GENERALIZED POLYNOMIAL CHAOS EXPANSION

Newly restructured generalized polynomial chaos expansion (GPCE) methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions are reported. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multipoint single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus facilitating to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More importantly, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation. Numerical results affirm that the proposed methods yield accurate and computationally efficient optimal solutions of mathematical problems and design solutions for simple mechanical systems. Finally, the success in conducting stochastic shape optimization of a bogie side frame with 41 random variables demonstrates the power of the MPSS-DD-GPCE method in solving industrial-scale engineering design problems.

Improving Li-ion battery parameter estimation by global optimal experiment design

Li-ion batteries are a key enabling technology for electric vehicles and determining their properties precisely is an essential step in improving utilization and performance. Batteries are highly complex electrochemical systems, with processes occurring in parallel on many time- and length-scales. Models describing these mechanisms require extensive parametrization efforts, conventionally using a combination of ex-situ characterization and systems identification. We present a methodology that algorithmically designs current input signals to optimize parameter identifiability from voltage measurements. Our approach uses global sensitivity analysis based on the generalized polynomial chaos expansion to map the entire parameter uncertainty space, relying on minimal prior knowledge of the system. Parameter specific optimal experiments are designed to maximize sensitivity and simultaneously minimize interactions and unwanted contributions by other parameters. Experiments are defined using only three design variables making our approach computationally efficient. The methodology is demonstrated using the Doyle-Fuller-Newman battery model for eight parameters of a 2.6 Ah 18,650 cell. Validation confirms that the proposed approach significantly improves model performance and parameter accuracy, while lowering experimental burden.

Derivative-Free Observability Analysis for Sensor Placement Optimization of Bioinspired Flexible Flapping Wing System.

Observability analysis of a bioinspired flexible flapping wing system provides a measure of how well the states of flexible flapping wing micro-aerial vehicles can be estimated from real-time measurements during high-speed flight. However, the traditional observability analysis approaches have trouble in terms of lack of quantitative analysis index, high computational complexity, low accuracy, and unavailability in stochastic systems with memory, including bioinspired flexible flapping wing systems. Therefore, a novel derivative-free observability analysis method is proposed here based on the generalized polynomial chaos expansion. By formulating a surrogate model to represent the relationship between the cumulative measurement and the random initial state, the observability coefficient matrix is calculated and the observability rank condition is stated. Consequently, several observability indices are proposed to quantity the observability of the system. Altogether, the proposed method avoids the disadvantages of the traditional approaches, especially in assessing the observability degree of each state and the effect of stochastic noise on observability. The validation of the proposed method is first provided by demonstrating the equivalence between the traditional and proposed methods and subsequently by comparing the observability of the Lorenz system calculated via three different approaches. Finally, the proposed method is applied on a bioinspired flexible wing system to optimize the placement of sensors, which is consistent with the natural configuration of campaniform sensilla on the wing of the hawkmoth.

Polynomial chaos Kalman filter for target tracking applications

In this paper, an approximate Gaussian state estimator is developed based on generalised polynomial chaos expansion for target tracking applications. Motivated by the fact that calculating conditional moments in an approximate Gaussian filter involves computing integrals with respect to Gaussian density, the authors approximate the non-linear dynamics using polynomial chaos expansion. Second-order as well as third-order polynomial chaos expansions were used for approximate filtering, to derive the necessary recursive algorithm and also provide certain algebraic simplifications which reduce the computational burden without significantly affecting the filtering performance. Two comprehensive numerical experiments for multivariate systems, including one for a multi-model system, demonstrate the potential of the new algorithms.

Probabilistic constrained Bayesian inversion for transpiration cooling

AbstractTo enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant required to ensure a safe operation. To study this, we introduce an approach that determines the posterior probability distribution of the Reynolds number using an inverse problem and constraining the maximum temperature of the system under parameter uncertainties. Mathematically, this chance inequality constraint is dealt with by a generalized polynomial chaos expansion of the system. The posterior distribution will be evaluated by different Markov chain Monte Carlo based methods. A novel method for the constrained case is proposed and tested among others on two‐dimensional transpiration cooling models.