A Combined Nonstationary Kriging and Support Vector Machine Method for Stochastic Eigenvalue Analysis of Brake Systems

Abstract

This paper presents a new metamodel approach based on nonstationary kriging and a support vector machine to efficiently predict the stochastic eigenvalue of brake systems. One of the difficulties in the mode-coupling instability induced by friction is that stochastic eigenvalues represent heterogeneous behavior due to the bifurcation phenomenon. Therefore, the stationarity assumption in kriging, where the response is correlated over the entire random input space, may not remain valid. In this paper, to address this issue, Gibb's nonstationary kernel with step-wise hyperparameters was adopted to reflect the heterogeneity of the stochastic eigenvalues. In predicting the response for unsampled input, the support vector machine-based classification is utilized. To validate the performance, a simplified finite element model of the brake system is considered. Under various types of uncertainties, including different friction coefficients and material properties, stochastic eigenvalue problems are investigated. Through numerical studies, it is seen that the proposed method improves accuracy and robustness compared to conventional stationary kriging.Carlo simulation (MCS), a sensitivity-based approach, polynomial chaos expansion (PCE), and kriging (Gaussian process). MCS [8] is a sampling-based technique which is considered the most robust method for uncertainty quantifications. However, since a large number of samples must be drawn to obtain sufficient accuracy, the computational cost becomes prohibitive. The sensitivity-based method [9] is a low-order Taylor expansion method that approximates the solution near the input parameters. Although this method is computationally efficient, the obtained solution will be inaccurate under large parameter variabilities.PCE and kriging are metamodel (surrogate model) -based techniques that replace the original model with an easy-to-evaluate function. PCE [10], also known as a spectral approach, is a regression-based methods that approximates the model by multivariate polynomials. Kriging [11] is an interpolation-based approach in which the model is assumed to be a realization of a Gaussian process. Since metamodel-based methods can be applied to various levels of uncertainty, numerous studies have utilized them to solve stochastic eigenvalue problems in brake systems [12][13][14][15].However, despite previous studies, it remains challenging to apply PCE and kriging to brake systems with the mode-coupling phenomenon. Since eigenvalues exhibit nonsmooth behavior around the bifurcation point, PCE is not adequate for capturing local solution accuracy. Several studies have tackled this issue by applying multi-element PCE [16], a wavelet basis [17]. Although kriging interpolates each sample and reflects the local characteristics, to the authors' knowledge, previous studies have all been based on the stationarity assumption, which means that the smoothness of response is entirely correlated throughout the random input space. Therefore, for mode-coupling problems where the stationarity a...

Active Learning-Based Kriging Model with Noise Responses and Its Application to Reliability Analysis of Structures

“…It is particularly effective in capturing intricate relationships within data and is widely used in various fields for tasks such as predictive modeling and optimization. Lee and Park [22] introduced a metamodeling approach, combining nonstationary Kriging and a support vector machine to predict the stochastic eigenvalues of brake systems. In the approach, Gibbs's nonstationary kernel was used with step-wise hyperparameters to account for the heterogeneity of the stochastic eigenvalues.…”