Treating Uncertainties in Multibody Dynamic Systems using a Polynomial Chaos Spectral Decomposition

ABSTRACT This study addresses the critical need for computational tools to model complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. Polynomial chaos has been used extensively to model uncertainties in structural mechanics and in fluids, but to our knowledge they have yet to be applied to multibody dynamic simulations. We show that the method can be applied to quantify uncertainties in time domain and frequency domain

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

Parameter Estimation for Mechanical Systems Using an Extended Kalman Filter

This paper proposes a new computational approach based on the Extended Kalman Filter (EKF) in order to apply the polynomial chaos theory to the problem of parameter estimation, using direct stochastic collocation. The Kalman filter formula is used at each time step in order to update the polynomial chaos of the uncertain states and the uncertain parameters. The main advantage of this method is that the estimation comes in the form of a probability density function rather than a deterministic value, combined with the fact that simulations using polynomial chaos methods are much faster than Monte Carlo simulations. The proposed method is applied to a nonlinear four degree of freedom roll plane model of a vehicle, in which an uncertain mass with an uncertain position is added on the roll bar. A major drawback was identified: the EKF can diverge when using a high sampling frequency, which might prevent the use of enough data to obtain accurate results when a low sampling frequency is necessary. When applying the polynomial chaos theory to the EKF, numerical errors can accumulate even faster than in the general case due to the truncation in the polynomial chaos expansions, which is illustrated on a simple example. An alternative EKF approach which consists of applying the filter formula on all the observations at once usually yields better results, but can still sometimes fail to produce very accurate results. Therefore, using different sampling rates in order to verify the coherence of the results and comparing the results to a different approach is strongly recommended

Sensitivity analysis and model order reduction for random linear dynamical systems

Abstract We consider linear dynamical systems defined by differential algebraic equations. The associated input-output behaviour is given by a transfer function in the frequency domain. Physical parameters of the dynamical system are replaced by random variables to quantify uncertainties. We analyse the sensitivity of the transfer function with respect to the random variables. Total sensitivity coefficients are computed by a nonintrusive and by an intrusive method based on the expansions in series of the polynomial chaos. In addition, a reduction of the state space is applied in the intrusive method. Due to the sensitivities, we perform a model order reduction within the random space by changing unessential random variables back to constants. The error of this reduction is analysed. We present numerical simulations of a test example modelling a linear electric network

Dynamic reliability analysis using the extended support vector regression (X-SVR)

© 2019 Elsevier Ltd For engineering applications, the dynamic system responses can be significantly affected by uncertainties in the system parameters including material and geometric properties as well as by uncertainties in the excitations. The reliability of dynamic systems is widely evaluated based on the first-passage theory. To improve the computational efficiency, surrogate models are widely used to approximate the relationship between the system inputs and outputs. In this paper, a new machine learning based metamodel, namely the extended support vector regression (X-SVR), is proposed for the reliability analysis of dynamic systems via utilizing the first-passage theory. Furthermore, the capability of X-SVR is enhanced by a new kernel function developed from the vectorized Gegenbauer polynomial, especially for solving complex engineering problems. Through the proposed approach, the relationship between the extremum of the dynamic responses and the input uncertain parameters is approximated by training the X-SVR model such that the probability of failure can be efficiently predicted without using other computational tools for numerical analysis, such as the finite element analysis (FEM). The feasibility and performance of the proposed surrogate model in dynamic reliability analysis is investigated by comparing it with the conventional ε-insensitive support vector regression (ε-SVR) with Gaussian kernel and Monte Carlo simulation (MSC). Four numerical examples are adopted to evidently demonstrate the practicability and efficiency of the proposed X-SVR method

Efficient uncertainty quantification in aerospace analysis and design

The main purpose of this study is to apply a computationally efficient uncertainty quantification approach, Non-Intrusive Polynomial Chaos (NIPC) based stochastic expansions, to robust aerospace analysis and design under mixed (aleatory and epistemic) uncertainties and demonstrate this technique on model problems and robust aerodynamic optimization. The proposed optimization approach utilizes stochastic response surfaces obtained with NIPC methods to approximate the objective function and the constraints in the optimization formulation. The objective function includes the stochastic measures which are minimized simultaneously to ensure the robustness of the final design to both aleatory and epistemic uncertainties. For model problems with mixed uncertainties, Quadrature-Based and Point-Collocation NIPC methods were used to create the response surfaces used in the optimization process. For the robust airfoil optimization under aleatory (Mach number) and epistemic (turbulence model) uncertainties, a combined Point-Collocation NIPC approach was utilized to create the response surfaces used as the surrogates in the optimization process. Two stochastic optimization formulations were studied: optimization under pure aleatory uncertainty and optimization under mixed uncertainty. As shown in this work for various problems, the NIPC method is computationally more efficient than Monte Carlo methods for moderate number of uncertain variables and can give highly accurate estimation of various metrics used in robust design optimization under mixed uncertainties. This study also introduces a new adaptive sampling approach to refine the Point-Collocation NIPC method for further improvement of the computational efficiency. Two numerical problems demonstrated that the adaptive approach can produce the same accuracy level of the response surface obtained with oversampling ratio of 2 using less function evaluations. --Abstract, page iii

A new hybrid uncertainty optimization method for structures using orthogonal series expansion

© 2017 Elsevier Inc. This paper proposes a new hybrid uncertain design optimization method for structures which contain both random and interval variables simultaneously. The optimization model is formulated with the feasible robustness and the reliability of the worst scenario. The hybrid uncertainty is quantified by using the orthogonal series expansion method that integrates the Polynomial Chaos (PC) expansion method and the Chebyshev interval method within a uniform framework. The design sensitivity of objective and constraints will be developed to greatly facilitate the use of gradient-based optimization algorithms. The numerical results show that this method will be more possible to seek the feasible solution

Uncertainty quantification in mooring cable dynamics using polynomial chaos expansions

Mooring systems exhibit high failure rates. This is especially problematic for offshore renewable energy systems, like wave and floating wind, where the mooring system can be an active component and the redundancy in the design must be kept low. Here we investigate how uncertainty in input parameters propagates through the mooring system and affects the design and dynamic response of mooring and floaters. The method used is a nonintrusive surrogate based uncertainty quantification (UQ) approach based on generalized Polynomial Chaos (gPC). We investigate the importance of the added mass, tangential drag, and normal drag coefficient of a catenary mooring cable on the peak tension in the cable. It is found that the normal drag coefficient has the greatest influence. However, the uncertainty in the coefficients plays a minor role for snap loads. Using the same methodology we analyze how deviations in anchor placement impact the dynamics of a floating axi-symmetric point-absorber. It is shown that heave and pitch are largely unaffected but surge and cable tension can be significantly altered. Our results are important towards streamlining the analysis and design of floating structures. Improving the analysis to take into account uncertainties is especially relevant for offshore renewable energy systems where the mooring system is a considerable portion of the investment