Generalized Polynomial Chaos Research Articles

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

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.

RSM-Based Approximate Dynamic Programming for Stochastic Energy Management of Power Systems

The stochastic energy management (SEM) of power systems is computationally intractable due to its randomness, nonconvexity, and nonlinearity. To solve this problem, a response surface method (RSM)-based approximate dynamic programming (ADP) algorithm is proposed in this paper. Since the value function can be directly obtained by RSM, the proposed algorithm does not need to iteratively approach them as existing ADP algorithms, which facilitates reducing the computing time. In addition, an improved generalized polynomial chaos (IgPC) method (i.e., an extension of RSM) is proposed to calculate the expectation of the value function of ADP in the stochastic environment. Compared with the Monte Carlo method, which is commonly used in existing ADP algorithms, IgPC requires fewer sampling scenarios while providing similar results. Simulation results with two modified IEEE test systems and a real 2778-bus system demonstrate the effectiveness of the proposed algorithm in terms of accuracy and computation efficiency.

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.

Numerical simulation of aortic coarctations of different grades of severity: Flow features and importance of outlet boundary conditions

Numerical simulations of the blood flow inside a patient-specific thoracic aorta in presence of coarctation are considered. Different grades of severity of the coarctation are obtained by constructing parametric geometries in which the coarctation section is circular with varying diameter values. The impact of a fine-tuning of the Windkessel model resistances, at each outlet, is also investigated. A stochastic approach based on the generalized Polynomial Chaos (gPC) is used to carry out a systematic analysis. It allows obtaining continuous response surfaces of the quantities of interest in the parameter space from a limited number of simulations. Two parameters are selected: the vessel diameter, D, at the coarctation plane and a non-dimensional parameter, α, through which it is possible to calibrate the resistance offered by organs and vessels downstream the thoracic aorta. The value of the coarctation diameter has the strongest impact on all the flow features, i.e., flow rate, pressure, velocity, and wall shear stresses. It is also shown that, as the value of D increases, the dependence on α decreases. This means that the more the geometry of the thoracic aorta approaches a healthy shape, the less significant it is to perform a fine-tuning of the Windkessel model resistances to match the patient-specific pressure waveform, whereas it should be done in cases of severe coarctations.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Stochastic dynamic analysis of composite plates in thermal environments using nonlinear autoregressive model with exogenous input in polynomial chaos expansion surrogate

The application of composite materials has increased manifolds in aerospace and high-speed vehicle industries, where structures experience dynamic loads along with temperature variation. To ensure safety of these structures, the stochastic dynamic analysis in varying thermal environments is essential. The generalized polynomial chaos (gPC) expansion method is a well-known metamodel used to quantify uncertainties in engineering systems instead of using the direct Monte Carlo simulation. Though the gPC expansion method is used extensively for engineering systems, its classical version has some limitations while implementing in dynamical systems. The accuracy of the gPC expansion method reduces in delayed time domain and can be improved to some extent by increasing the order of polynomial, which though is computationally intensive. Furthermore, the classical gPC expansion method is suitable for low-dimensional problems having independent random parameters. To overcome these limitations in the classical gPC expansion method, a system identifier, i.e., nonlinear autoregressive model with exogenous input (NARX) is coupled with it to avoid response degeneration in delayed time domain. The temperature-dependent material properties of composite lamina can be random and correlated, however the classical gPC expansion method is unable to accommodate the correlated random parameters. To address these challenges, the NARX-gPC expansion method is modified to conduct stochastic dynamic analysis of composite plates using correlated random material properties of the composite lamina in varying thermal environments. The study indicates that the correlation has a significant effect on the stochastic dynamic response at low and high temperatures for cross-ply laminates in comparison with angle-ply laminates, which is not captured through uncorrelated random material properties. The prediction efficiency of the developed surrogate by using the adaptive NARX-gPC expansion method is established even though it is trained using data in a shorter time domain, thereby computational performance is enhanced. Additionally, this surrogate is implemented effectively for higher order randomness in the input parameters.

Surrogate recycling for structures with spatially uncertain stiffness

This study expands the existing methods for non-destructively identifying the spatially varying material properties of a structure using modal data. It continues a recently published approach to this inverse problem that employed Bayesian inference in conjunction with the Karhunen-Loève expansion to solve it. Here, we present two developments. Firstly, eigenvectors are used instead of eigenvalues, improving the results significantly. Secondly, a generalized polynomial chaos surrogate accelerates the inversion procedure. Finally, we develop a methodology for reusing the surrogate model across inversion tasks. We demonstrate the efficacy and efficiency of this methodology via the field of additive manufacturing and the fused deposition modeling process. The good results promise profound computational cost saving potential for large-scale applications.

Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification

We consider the Least Squares (LSQ) regression method for Uncertainty Quantification (UQ) using generalised polynomial chaos (gPC) and augment the linear system with the gradient of the Quantity of Interest (QoI) with respect to the stochastic variables. The gradient is computed very efficiently for all variables from the adjoint system of equations. To minimise the condition number of the augmented LSQ system, an effective sampling strategy of the stochastic space is required. We compare two strategies. In the first, we apply pivoted QR decomposition to the standard LSQ matrix and evaluate both the QoI and its gradient at the sample points identified. In the second strategy, we apply QR decomposition directly to the augmented matrix. We find that the first strategy is more efficient in terms of accuracy vs number of evaluations. We call the new approach sensitivity-enhanced generalised polynomial chaos, or se-gPC, and apply it to several test cases including an aerodynamic case with 40 stochastic parameters. The method can produce accurate estimations of the statistical moments using a small number of sampling points. The computational cost scales as ∼mp−1, instead of ∼mp of the standard LSQ formulation, where m is the number of stochastic variables and p the chaos order. The solution of the adjoint system of equations is implemented in many computational mechanics packages, thus the infrastructure exists for the application of the method to a wide variety of engineering problems.

Safety Assessment and Uncertainty Quantification of Electromagnetic Radiation from Mobile Phones to the Human Head

With the rapid development of the mobile communication technology, the design of mobile phones has become more complex, and research on the electromagnetic radiation from mobile phones that reaches the human head has become important. Therefore, first of all, a model of mobile phone daily use was established. Then, based on the established simulation model, the safety of human head exposure to mobile phones was evaluated. The generalized polynomial chaos (gPC) method was used to establish a proxy model of the specific absorption rate (SAR) of the human head at different frequencies to perform a parameter uncertainty quantification (UQ). Finally, the Sobol method was used to quantify the influence of relevant variables on the SAR. The simulation results showed that the gPC method can save time and cost while ensuring accuracy, and the SAR value is greatly influenced by the electromagnetic materials of the mobile phone shell. Combined with the above analysis, this paper can provide reasonable suggestions for the design of mobile phone electromagnetic materials.

Deterministic and Probabilistic Approaches to Model and Update Dynamic Systems

Updating simulation models, based on flight test or other data, or creating a model using only experimental data is a necessary process that can be costly and laborious. Simulation updates are necessary for high-fidelity simulations used in analysis, control development, handling qualities prediction, fleet training, etc. This paper presents methods to update aircraft models based on deterministic and probabilistic approaches. The deterministic approach is based on a collection of signal processing, statistical analysis, and maximum likelihood estimators embodied in the Algorithms to Update Simulation Parameters with Experimental Data tool. Mathematically sound and statistically rigorous techniques are used to calculate updates of current model parameters and suggest new terms capturing unmodeled dynamics, thereby improving correlation with observed vehicle response. The probabilistic approach is based on generalized polynomial chaos theory included in the Algorithms for Uncertainty Representation and Analysis (AURA) tool. The AURA tool allows the user to update probabilistic models based on experimental data, formulated as a Bayesian inference problem. This paper will present the methodology for both deterministic and probabilistic approaches and provide examples of their application to different aircraft designs, including tilt-rotor and rotary-wing unmanned air system vehicles. The results for both deterministic and probabilistic modeling approaches demonstrate excellent results. The probabilistic approach is novel in its expressive power to propagate and update probabilistic models and provides users with an intuitive and powerful modeling approach.

Statistical laws of a one-dimensional model of turbulent flows subject to an external random force

This paper provides mathematical and numerical analysis of a one-dimensional model of turbulent flow generating the anomalous cascade of the inviscid conserved quantity. The model is based on the generalized Constantin–Lax–Majda–DeGregorio (gCLMG) equation with viscous dissipation under a large-scale forcing. Suppose that the forcing function and the initial data are random variables defined on a certain probability space. Then, the equation is regarded as a random partial differential equation. We prove the global existence of a unique solution to the gCLMG equation, from which a stochastic process is defined. In addition, by approximating the solutions numerically by Galerkin approximation of random variables with generalized polynomial chaos, we confirm the existence of a steady distribution. We find that the steady distribution reproduces qualitatively the same cascades of the energy and the enstrophy spectra as those of a turbulent flow generated by randomly moving pulse (Matsumoto and Sakajo 2016 Phys. Rev. E 93 053101). We also investigate the structure functions, showing intermittency.

Uncertainty analysis for cylindrical structure vibration according to generalized polynomial chaos method

Aiming to gain an accurate prediction model of the vibro-acoustic problem for the vibrating structure systems, the uncertainty vibration analysis for the cylindrical shell structure considering construction factors is investigated. The generalized polynomial chaos (GPC) method is used to gain the analytical solution of the free vibration for the cylindrical shell structure, in which the construction uncertainty factors, comprising the material uncertainty parameter and construction geometric uncertainty parameter, are considered. By combining the collocation method and generalized polynomial chaos method, the influence of the uncertainty construction factor on the free vibration properties for the cylindrical shell structure is verified by a numerical analysis. The results provide technical support for the design and construction of the vibrating structure for vibration and noise reducing.

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.

Effect of stochastic deformation on the vibration characteristics of a tube bundle in axial flow

Design evaluation of nuclear components using numerical methods has typically been focused on nominal geometries and conditions. In reality, the geometry of reactor components and operating conditions can differ from the design on the drawing table. Increasing understanding and safety of nuclear energy systems requires investigating these more realistic conditions as well. An example is the bow of fuel assemblies, a deformation that occurs after some residence time as a consequence of thermal and irradiation effects. A paradigm shift from purely deterministic simulations toward simulations involving stochastic processes can help to meet this demand.One of the aspects in the design assessment of nuclear reactors is flow-induced vibration (FIV). A lot of experimental effort has already been allocated to FIV phenomena in the past, and more recently also numerical approaches have been applied. However, it was not clear to what extent vibration characteristics would change for a geometry different from the design drawing. In this study the effect of a random bow deformation on the vibration behavior of a flexible tube in a rigid bundle is investigated using fluid–structure interaction simulations. As the (stressless) geometry presumably has a stochastic nature, the modal characteristics should be treated accordingly.For this uncertainty quantification of the modal characteristics, the generalized Polynomial Chaos method was applied. First, only the steady flow through the bundle was considered, without simulating the vibration. This sufficiently reduced the computational cost to apply the Monte Carlo method, which was used to validate the more efficient Polynomial Chaos method. This latter method was found adequate to predict the stochastic modal characteristics of the deformed, flexible tube.