Generalized Polynomial Chaos Expansion Research Articles

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.

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.

A stochastic Galerkin method with adaptive time-stepping for the Navier–Stokes equations

We study the time-dependent Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion, and we use the stochastic Galerkin method to extend the methodology from Kay et al. (2010) [21] into this framework. For the resulting stochastic problem, we explore the properties of the resulting stochastic solutions, and we also compare the results with that of Monte Carlo and stochastic collocation. Since the time-stepping scheme is fully implicit, we also propose strategies for efficient solution of the stochastic Galerkin linear systems using a preconditioned Krylov subspace method. The effectiveness of the stochastic Galerkin method is illustrated by numerical experiments.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Sensitivity Analysis of NVH Simulations with Stochastic Input Parameters for a Car Body

<div class="section abstract"><div class="htmlview paragraph">Uncertainties play a major role in vibroacoustics - especially in car body design in the preliminary development because of the overall spread in the production that should be covered with one simulation model. Therefore, we use uncertain input parameters to determine the stochastically distributed admittance of the car body before each part of the car is fully designed. To gain a stochastic result - the stochastically distributed admittance curve - we calculate a deterministic finite element simulation several times with sets of stochastically distributed input parameter values. To reduce simulation time and cost of the car model with many million degrees of freedom we focus on the uncertain parameters that show a significant influence on the admittance curve. It is therefore necessary to be able to accurately estimate for each parameter if its influence on the admittance of the car body plays a major role for the noise vibration harshness simulation. A sensitivity analysis describes the connection between model input and output and the influence of the input on the output. We conduct a two-step global sensitivity analysis which is based on the generalized polynomial chaos expansion to determine the sensitivity of the parameters. Since the less sensitive parameters hardly influence the admittance curve of the car body, we can simulate them as deterministic values. In the further research, we will focus on the most sensitive parameters.</div></div>

Nonlinear Vibrations of Simply Supported Cylindrical Panels with Uncertain Parameters: An Intrusive Application of the Generalized Polynomial Chaos Expansion

Nonlinear Vibrations of Simply Supported Cylindrical Panels with Uncertain Parameters: An Intrusive Application of the Generalized Polynomial Chaos Expansion

On distribution-based global sensitivity analysis by polynomial chaos expansion

This paper presents a novel distribution-based global sensitivity analysis based on the Kullback–Leibler divergence derived directly from generalized polynomial chaos expansion (PCE). The synergy between PCE and Gram–Charlier expansion is utilized for derivation of novel and computationally efficient sensitivity indices. In contrast to a standard procedure for estimation of higher statistical moments, this paper reviews standard linearization problem of Hermite and Jacobi polynomials in order to efficiently estimate skewness and kurtosis direclty from PCE. Higher statistical moments are used for an estimation of probability distribution by Gram–Charlier expansion, which is represented by derived explicit cumulative distribution function. The proposed sensitivity indices taking the whole probability distribution into account are calculated for several numerical examples of increasing complexity in order to present their possibilities. It is shown, that the proposed sensitivity indices are obtained without any additional computational demands together with Sobol indices, and thus can be easily used as complementary information for a complex sensitivity analysis or any decision making in industrial applications. Application of the proposed approach on engineering structure is presented in case of prestressed concrete roof girders failing shear. Moreover, the potential of the proposed approach for reliability-oriented sensitivity analysis is investigated in pilot numerical example.

A strategy to personalize a 1D pulse wave propagation model for estimating subject-specific central aortic pressure waveform

The central aortic pressure (CAP) provides insights into the prediction, prevention, diagnosis, and treatment of cardiovascular disease, but can't be directly measured non-invasively. Therefore, the development of a noninvasive CAP estimation method based on the non-invasively measured peripheral pressure waveform is critical for clinical decisions based on the CAP. Some existing widely applied methods, such as the generalized transfer function (GTF) method relating measured peripheral pressure to the CAP, do not or only partly account for inter-subject or intra-subject variability of the cardiovascular system. To overcome this pitfall, we propose a subject-specific central aortic pressure estimation method in this paper. The novel method presented can derive an accurate aortic pressure from the peripheral pressure based on an individualized pulse wave propagation model using a GTF method as a first guess. To develop a strategy to personalize a pulse wave propagation model one usually needs to optimize many input parameters. Therefore, we present a two-step approach with the screening method of Morris and the adaptive sparse generalized polynomial chaos expansion (agPCE) algorithm for the sensitivity analysis of the wave propagation model. First, for a-priori defined output of the model, a subset of important parameters is identified using the screening method of Morris. Next, a quantitative variance-based sensitivity analysis is performed using agPCE. This approach is applied to a 1D pulse wave propagation model to get the personalized parameters of the pulse wave propagation model for the estimation of a subject-specific central aortic pressure waveform and is validated with 26 patients. Compared with the GTF method, the proposed method showed better performance in estimating the central aortic pulse wave and predicting the parameters.

Reliability-based design optimization under dependent random variables by a generalized polynomial chaos expansion

Reliability-based design optimization under dependent random variables by a generalized polynomial chaos expansion

A Reliability-Based Formulation for Simulation-Based Control Co-Design Using Generalized Polynomial Chaos Expansion

Abstract Combined plant and control design (control co-design (CCD)) methods are often used during product development to address the synergistic coupling between the plant and control parts of a dynamic system. Recently, a few studies have started applying CCD to stochastic dynamic systems. In their most rigorous approach, reliability-based design optimization (RBDO) principles have been used to ensure solution feasibility under uncertainty. However, since existing reliability-based CCD (RBCCD) algorithms use all-at-once formulations, only most-probable-point (MPP) methods can be used as reliability analysis techniques. Though effective for linear/quadratic RBCCD problems, the use of such methods for highly nonlinear RBCCD problems introduces solution error that could lead to system failure. A multidisciplinary feasible (MDF) formulation for RBCCD problems would eliminate this issue by removing the dynamic equality constraints and instead enforcing them through forward simulation. Since the RBCCD problem structure would be similar to traditional RBDO problems, any of the well-established reliability analysis methods could be used. Therefore, in this work, a novel reliability-based MDF formulation of multidisciplinary dynamic system design optimization has been proposed for RBCCD. To quantify the uncertainty propagated by the random decision variables, Monte Carlo simulation has been applied to the generalized polynomial chaos expansion of the probabilistic constraints. The proposed formulation is applied to two engineering test problems, with the results indicating the effectiveness of both the overall formulation as well as the reliability analysis technique for RBCCD.

Identification of active magnetic bearing parameters in a rotor machine using Bayesian inference with generalized polynomial chaos expansion

Identification of active magnetic bearing parameters in a rotor machine using Bayesian inference with generalized polynomial chaos expansion

The Uncertainty Quantification for Parameters Optimization in SERF Atomic Magnetomer

The magnetometer is a multi-parameter system, parameters optimization is always the problem to improve the sensitivity of the atomic magnetometer. Multiple parameters hardly decide the optimized conditions in the experiments because of the influence of several parameters and parameters may interact with each other. Besides, the influence of parameters on the signal-to-noise ratio can hardly be analyzed theoretically due to the complex physical configuration. We used a generalized polynomial chaos expansion to construct an agent model for the SERF atomic magnetometer to replace its complex physical model, and used a variance-based sobol method to perform a global sensitivity analysis of the parameters in this paper. The mathematical method of uncertainty quantification is used to comprehensively analyze six parameters’ effect on SERF atomic magnetometer performance. The results show that the probe laser power has the greatest impact on the atomic magnetometer, and it is the parameter with the highest degree of linearity with the atomic magnetometer. In accordance with the results of the uncertainty quantification analysis, the parameters of the probe laser power of the SERF atomic magnetometer were experimentally optimized, and the performance of the magnetometer was significantly improved.

Comparison of quadrature and regression based generalized polynomial chaos expansions for structural acoustics

This work performs a direct comparison between generalized polynomial chaos (GPC) expansion techniques applied to structural acoustic problems. Broadly, the GPC techniques are grouped in two categories: , where the stochastic sampling is predetermined according to a quadrature rule; and , where an arbitrary selection of points is used as long as they are a representative sample of the random input. As a baseline comparison, Monte Carlo type simulations are also performed although they take many more sampling points. The test problems considered include both canonical and more applied cases that exemplify the features and types of calculations commonly arising in vibrations and acoustics. A range of different numbers of random input variables are considered. The primary point of comparison between the methods is the number of sampling points they require to generate an accurate GPC expansion. This is due to the general consideration that the most expensive part of a GPC analysis is evaluating the deterministic problem of interest; thus the method with the fewest sampling points will often be the fastest. Accuracy of each GPC expansion is judged using several metrics including basic statistical moments as well as features of the actual reconstructed probability density function.

Derivative-Free Observability Analysis of a Stochastic Dynamical System

This paper proposes a novel approach to analyze the observability of a stochastic linear or non-linear dynamical system. Unlike the traditional observability analysis approach, our approach is fully derivative-free, resulting in a low complexity and an easy implementation. Furthermore, it can not only analyze the observability of each system state, but also it can assess the effect of the observation and the system noise on the system observability, which is not considered by the traditional approach. Specifically, using the generalized polynomial chaos expansion, an observability-coefficient matrix is first calculated, which enables us to judge whether the system is observable or not. Then, the puny and brawny observability indexes are assessed to quantify the degree of observability. The equivalence between the conventional method and the proposed method for a continuous-time linear time-invariant system is proved in the appendix. The effectiveness of the proposed approach is mathematically proved and its good performance is demonstrated in several test cases.

Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM

Abstract We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces.

Uncertainty quantification in acoustics through generalized polynomial chaos expansions

Uncertainty quantification in acoustics through generalized polynomial chaos expansions

Robust design optimization under dependent random variables by a generalized polynomial chaos expansion

New computational methods are proposed for robust design optimization (RDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on a generalized polynomial chaos expansion (GPCE) for determining the second-moment statistics of a general output function of dependent input random variables, an innovative coupling between GPCE and score functions for calculating the second-moment sensitivities with respect to the design variables, and a standard gradient-based optimization algorithm, establishing direct GPCE, single-step GPCE, and multi-point single-step GPCE design processes. New analytical formulae are unveiled for design sensitivity analysis that is synchronously performed with statistical moment analysis. Numerical results confirm that the proposed methods yield not only accurate but also computationally efficient optimal solutions of several mathematical and simple RDO problems. Finally, the success of conducting stochastic shape optimization of a steering knuckle demonstrates the power of the multi-point single-step GPCE method in solving industrial-scale engineering problems.