Polynomial chaos expansion in structural dynamics: Accelerating the convergence of the first two statistical moment sequences
Introduction
The modelling of uncertain structures is a research field that has rapidly developed recently. A common objective is to estimate the statistics of the stochastic response of a structure. The oldest method is Monte Carlo simulation (MCS) [1], [2], [3], [4], which has the drawback of very slow convergence. This problem becomes even more pronounced for structural dynamic problems, where response statistics is required for various time instants and frequency ranges [5], [6]. For efficient computation of the response statistics, an approach to obtain some kind of response surface in the space of random parameters is necessary. There are two broad ways of obtaining response surfaces. Perturbation based methods (lower-order Taylor approximation) [7] and the Neumann expansion method, [8], [9] are generally computationally efficient, however, depending on applications, may not give accurate results if the uncertainties are large. Polynomial Chaos (PC) expansion [10], [11], [12], High Dimensional Model Reduction (HDMR) [13], [14], [15] and Gaussian Process (GP) Emulators [16], [17], [18] have been developed for any level of uncertainty. Consequently these latter approximations have gained significant interest in recent years. This paper concerns Polynomial Chaos Expansion (PCE) for multi-degree of freedom damped dynamic systems with multi-parametric uncertainties.
Polynomial Chaos Expansion belong to the class of spectral methods where a Galerkin type of error minimization approach is generally employed in the space of random variables. Computation cost of the approach can be high if number of random variables are high. Several novel approaches have been developed to address the issue of computational cost, see for example [19], [20], [21], [22], [23]. In addition to the computational cost arising due to large random dimension, the issue of convergence of the series can be of concern in certain cases. Field and Grigoriu [24], [25] studied the convergence properties of this PCE. Recently Keshavarzzadeh et al. [26] proposed a transformation approach to enhance the convergence of PCE in the context of elliptic problems.
Recent work [27], [28] has showed that the convergence may be very slow when the statistics of the response is estimated by PCE for dynamic problems. Other studies [5], [6] also show that care must be taken while directly extending concepts, results and computational methods based on static problems to dynamic problems. In particular, very slow convergence occurs for the response of a dynamical system submitted to harmonic forces around some critical frequencies (e.g. eigenfrequencies, critical rotating speed in rotordynamics) that are of great interest in the design of structures where large oscillations may arise. Hence the acceleration of the convergence of the statistics provided by a PCE is of great interest.
The Padé approximant [29], [30] method is one of the most popular procedures to accelerate the convergence of a sequence defining a function. However, using the Padé approximant expression will produce a rational function of a random variable, which makes the analytical derivation of the moments of the response difficult. An alternative approach to tackle this problem is to work directly on the moments evaluated from the PC expansion; from the moments, it is possible to define sequences whose convergence may be accelerated. Several acceleration methods exist [29], [31], such as Aitken׳s method, the recursive Aitken׳s method, and Shank׳s method [26]. The aim of this paper is to understand how these methods perform in the case of steady-state response of non-proportionally damped dynamic systems. A frequency-domain approach is adopted.
First the steady-state response of a random dynamical system with multiple random variables is obtained by a PCE; this is an extension to the case presented in [27], which involved a single random variable only. Then a dynamical system with a single random variable is studied to highlight the specific features due to the use of a PCE. In Section 3 Aitken׳s method is presented. Finally the methods developed in this paper are illustrated with two degrees of freedom uncertain linear dynamical system with one and two random variables. Both intrusive and non-intrusive methods have advantages and drawbacks. The non-intrusive approaches, such as black box methods, do not require any change in the standard structural mechanics codes, but they require a high number of function evaluations. The intrusive methods are closer to the initial problem, but require modification of standard codes. An intrusive method is used in this study to determine the PCE coefficients to address the modification of the initial problem.
Section snippets
Response of a random dynamical system
Consider an n degrees of freedom (dof) dynamical system described by its mass, damping and stiffness matrices, , and . The forces acting on this system are described by force vector , and denotes the response vector, which is a solution of
The stiffness matrix is assumed to be uncertain and given bywhere and ξi is a zero-mean random variable. is the deterministic part of the stiffness matrix and () is the stiffness
Convergence acceleration of a PCE: Aitken׳s transformation and its generalization
The objective is to estimate the first two moments of the response from several predicted responses evaluated with a relatively low PC order. Suppose that denotes the approximation of with a PCE of order n, and is the corresponding ith moment, for i=1 or 2. The PCE may be used in several ways to calculate the first two moments of the response. First, a Monte-Carlo simulation (MCS) can be applied to Eq. (18) to estimate the probability density function and then to
Application to two degrees of freedom system
MCS and PCE will be used to evaluate the mean and the standard deviation of the response, X, for the discrete spring, mass and damper system shown in Fig. 1 [27]. In this example, the spring stiffnesses are equal but uncertain, i.e. , and hence a single uncertain parameter is considered. The stiffness k is assumed to be random and given byThus, stiffness matrix, , mean stiffness matrix, , mass matrix, and damping matrix, , are
Conclusion
The PCE is a very efficient way to estimate the dynamic response of a system, except around the deterministic eigenfrequencies, where a very high PC order is required. Numerical issues arise in the calculation of the PCE coefficients for a high PC order, where factorial numbers exceed the computer precision and the systems involve a high number of equations. These issues have been solved in [27], [28] but the PCE is still not very efficient around the deterministic eigenfrequencies, where
Acknowledgment
J.-J. Sinou acknowledges the support of the Institut Universitaire de France.
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