Non-intrusive Generalized Polynomial Chaos Research Articles

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.

Nonlinear stochastic dynamics of a rub-impact rotor system with probabilistic uncertainties

This paper presents a stochastic model for performing the uncertainty and sensitivity analysis of a Jeffcott rotor system with fixed-point rub-impact and multiple uncertain parameters. A probabilistic nonlinear formulation is developed based on the combination of the harmonic balance method and an alternate frequency/time procedure (HB-AFT) in stochastic form. The non-intrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is employed to represent the propagation of uncertainties on rotor dynamics. In conjunction with the path continuation scheme and the Floquet theory, the developed model enables one to expediently evaluate the uncertainty bounds and probability density functions (PDFs) on periodic solution branches and associated stabilities. A global sensitivity analysis is then carried out by evaluating Sobol’s indices from gPCE to quantitatively ascertain the relative influence of different stochastic parameters on vibrational behaviors and conditions for the occurrence of rub-impact. The efficiency of the proposed algorithm for nonlinear stochastic dynamics of rub-impact rotors is validated with Monte Carlo simulation. Parametric studies are finally carried out to investigate the effects of multiple random parameters on the probabilistic variability in nonlinear responses of rub-impact rotors, which reveals the necessary to consider input uncertainties in analyses and designs to ensure the sustainable system performance.

Numerical investigation of skewed spatially evolving mixing layers

The sensitivity of turbulent dynamics in spatially evolving mixing layers to small skew angles is investigated via direct numerical simulation. Angle is a measure of the lack of parallelism between the two asymptotic flows, whose interaction creates the turbulent mixing region. The analysis is performed considering a large range of values of the shear intensity parameter . This two-dimensional parameter space is explored using the results of a database of 18 direct numerical simulations. Instantaneous fields as well as time-averaged quantities are investigated, highlighting important mechanisms in the emergence of turbulence and its characteristics for this class of flows. In addition, a stochastic approach is used in which and are considered as random variables with a given probability distribution. The response surfaces of flow statistics in the parameter space are built through non-intrusive generalized polynomial chaos. It is found that variations of the parameter have a primary effect on the growth of the mixing region. A secondary effect associated with is observed as well. Higher values for the skew angle are responsible for a rapid increase in growth of the inlet structures, enhancing the development of the mixing region. The impact on the turbulence features and, in particular, on the Reynolds stress tensor is also significant. A modification of the normalized diagonal components of the Reynolds stress tensor due to is observed. In addition, the interaction between the parameters and is here the governing element.

Nonlinear Dynamic Behaviour Analysis of a Clutch System with Uncertainties Using Polynomial Chaos and the Constrained Harmonic Balance Method

The study of the nonlinear dynamic behaviour of friction systems in general and of clutch systems in particular remains an open problem. Noise and vibrations induced by friction in the sliding phase of a clutch are very sensitive to design parameters. The latter have significant dispersions. In the study of the system stability, the problem is not only to know if the parameter values lead to the appearance of unstable equilibrium points; the real challenge lies in estimating the vibration levels when such unstable equilibrium points occur. This estimation is analyzed using the limit cycles. This article aims to study the ability of robust approaches based on developments in nonintrusive generalized polynomial chaos and a constrained harmonic balance method to estimate the vibration levels through the limit cycles of a clutch system in the presence of uncertainty. The purpose is to provide a low-cost, high precision approach, compared to the classic Monte Carlo method.

Application of Generalized Polynomial Chaos for Quantification of Uncertainties of Time Averages and Their Sensitivities in Chaotic Systems

In this paper, we consider the effect of stochastic uncertainties on non-linear systems with chaotic behavior. More specifically, we quantify the effect of parametric uncertainties to time-averaged quantities and their sensitivities. Sampling methods for Uncertainty Quantification (UQ), such as the Monte–Carlo (MC), are very costly, while traditional methods for sensitivity analysis, such as the adjoint, fail in chaotic systems. In this work, we employ the non-intrusive generalized Polynomial Chaos (gPC) for UQ, coupled with the Multiple-Shooting Shadowing (MSS) algorithm for sensitivity analysis of chaotic systems. It is shown that the gPC, coupled with MSS, is an appropriate method for conducting UQ in chaotic systems and produces results that match well with those from MC and Finite-Differences (FD).

Stability analysis of a clutch system with uncertain parameters using sparse polynomial chaos expansions

In vehicle transmission systems, frictional forces acting during the sliding phase of the clutch engagement may produce unwanted vibrations. The prediction of the stability of a clutch system remains however a laborious task, as the parameters which have the highest impact on the stability, such as the friction law or the damping, lead to significant dispersions and must be considered as uncertain in such studies. Non-intrusive generalized polynomial chaos (gPC) expansions have already been used in this context. However, the number of deterministic model evaluations (i.e. the computational cost) required to compute the PC coefficients becomes prohibitive for large numbers of uncertain parameters. The sparse polynomial chaos, recently developed by Blatman and Sudret, may overcome this issue. In this paper, the method has been applied to the stability analysis of a clutch system owning up to eight uncertain parameters. Comparisons with the reference Monte Carlo method and classic full PC expansions show that sparse PC expansions allow substantial computational cost reductions while ensuring a high accuracy of the results.

A principled approach to conductivity uncertainty analysis in electric field calculations

Uncertainty surrounding ohmic tissue conductivity impedes accurate calculation of the electric fields generated by non-invasive brain stimulation. We present an efficient and generic technique for uncertainty and sensitivity analyses, which quantifies the reliability of field estimates and identifies the most influential parameters. For this purpose, we employ a non-intrusive generalized polynomial chaos expansion to compactly approximate the multidimensional dependency of the field on the conductivities. We demonstrate that the proposed pipeline yields detailed insight into the uncertainty of field estimates for transcranial magnetic stimulation (TMS) and transcranial direct current stimulation (tDCS), identifies the most relevant tissue conductivities, and highlights characteristic differences between stimulation methods. Specifically, we test the influence of conductivity variations on (i) the magnitude of the electric field generated at each gray matter location, (ii) its normal component relative to the cortical sheet, (iii) its overall magnitude (indexed by the 98th percentile), and (iv) its overall spatial distribution. We show that TMS fields are generally less affected by conductivity variations than tDCS fields. For both TMS and tDCS, conductivity uncertainty causes much higher uncertainty in the magnitude as compared to the direction and overall spatial distribution of the electric field. Whereas the TMS fields were predominantly influenced by gray and white matter conductivity, the tDCS fields were additionally dependent on skull and scalp conductivities. Comprehensive uncertainty analyses of complex systems achieved by the proposed technique are not possible with classical methods, such as Monte Carlo sampling, without extreme computational effort. In addition, our method has the advantages of directly yielding interpretable and intuitive output metrics and of being easily adaptable to new problems.

Robustness of stochastic expansions for the stability of uncertain nonlinear dynamical systems – Application to brake squeal

Abstract This paper is devoted to the prediction and analysis of brake squeal under random uncertainty. This problem, which is a particular application of a wider issue, namely the stability of random parameter-dependent (RPD) nonlinear dynamical systems, is undertaken by using the non-intrusive generalized polynomial chaos (GPC) and Wiener-Haar expansions. The main objective is to assess the capacities of these meta-models within this framework. A reduced nonlinear non-equally damped, iso-damped and non-damped, disc/pad models are considered in this perspective in order to analyze the robustness of the proposed meta-models with respect to perfect and non-perfect mode coalescence. It turns out that the Wiener-Haar meta-model shows a more robust performance than GPC expansion and consequently offers a more reliable tool for the nonlinear stability analysis and thus for the prediction of brake squeal under parameter uncertainty.

Active Fault Diagnosis for Stochastic Nonlinear Systems: Online Probabilistic Model Discrimination

Reliable and timely diagnosis of system faults under uncertainties is imperative for safe, reliable, and profitable operation of technical systems. This paper presents an input design method for active fault diagnosis for nonlinear systems that are subject to probabilistic model uncertainty and stochastic disturbances, and are under operational constraints. A computationally efficient sample-based method is presented for joint propagation of model uncertainty and stochastic disturbances using non-intrusive generalized polynomial chaos and unscented transformation. A tractable sample-based distance measure, inspired by the k-nearest neighbors algorithm, is used for fault diagnosis, which seeks to discriminate between probabilistic predictions of the model hypotheses for normal and faulty operation. Simulation results on a benchmark bioreactor case study demonstrate the effectiveness of the proposed input design method for reliable fault diagnosis under uncertainty through online model discrimination.

Uncertainty Analysis in Lorentz Force Eddy Current Testing

The paper addresses the analysis of uncertainties in the framework of the nondestructive evaluation technique Lorentz force eddy current testing. A non-intrusive generalized polynomial chaos expansion is used in order to quantify the impact of multiple unknown input parameters. In this context, the statistics of the velocity and the conductivity of the specimen as well as the magnetic remanence and the lift-off distance of the permanent magnet are determined experimentally and modeled as $\beta $ -distributed and uniform distributed random variables. The results are compared with Monte Carlo simulations and showed errors <0.2%. Furthermore, the numerically predicted force profiles are validated with experiments. A sensitivity analysis by means of the Sobol decomposition revealed that the magnetic remanence and the lift-off distance contribute to more than 90% of the total variance of the resulting Lorentz force profile and should be considered first to improve reproducibility.

A Goal-Oriented Reduced Basis Methods-Accelerated Generalized Polynomial Chaos Algorithm

The nonintrusive generalized polynomial chaos (gPC) method is a popular computational approach for solving partial differential equations with random inputs. The main hurdle preventing its efficient direct application for high-dimensional input parameters is that the size of many parametric sampling meshes grows exponentially in the number of inputs (the “curse of dimensionality''). In this paper, we design a weighted version of the reduced basis method (RBM) for use in the nonintrusive gPC framework. We construct an RBM surrogate that can rigorously achieve a user-prescribed error tolerance and ultimately is used to more efficiently compute a gPC approximation nonintrusively. The algorithm is capable of speeding up traditional nonintrusive gPC methods by orders of magnitude without degrading accuracy, assuming that the solution manifold has low Kolmogorov width. Numerical experiments on our test problems show that the relative efficiency improves as the parametric dimension increases, demonstrating the potential of the method in delaying the curse of dimensionality. Theoretical results as well as numerical evidence justify these findings.

Non-intrusive generalized polynomial chaos for the robust stability analysis of uncertain nonlinear dynamic friction systems

This paper is devoted to the stability analysis of uncertain nonlinear dynamic dry friction systems. The stability property of dry friction systems is known to be very sensitive to the variations of friction laws. Moreover, the friction coefficient admits dispersions due to the manufacturing processes. Therefore, it becomes necessary to take this uncertainty into account in the stability analysis of dry friction systems to ensure robust predictions of stable and instable behaviors. The generalized polynomial chaos formalism is proposed to deal with this challenging problem treated in most cases with the prohibitive Monte Carlo based techniques. Two equivalent methods presented here combine the non-intrusive generalized polynomial chaos with the indirect Lyapunov method. Both methods are shown to be efficient in the estimation of the stability and instability regions with high accuracy and high confidence levels and at lower cost compared with the classic Monte Carlo based method.

A stochastic surrogate model approach applied to calibration of unstable fluid flow experiments

We propose a stochastic approach for calibration of mixing zone lengths in shock tube experiments. The methodology relies on taking into account uncertain initial data propagated through the basic multifluid Euler equations. In this work, the initial interface position is supposed uncertain, modeled by a stochastic process. The size of the mixing zone is then defined as the support of the probability density function of the stochastic process. This time dependent probability density function is estimated with non-intrusive generalized Polynomial Chaos, its support being in this case cheaply evaluated. This methodology relies on the application of an ergodic principle (Wiener, 1938) and generalizes linear perturbations analysis. It is applied in this Note to the calibration of several experimental results.