An estimation method of fractal parameters on rough surfaces based on the exact spectral moment using artificial neural network

Abstract

Fractal parameters (FP) significantly influence contact mechanics characteristics, such as contact stiffness, friction and wear. The existing methods liking the power spectral density (PSD), structure function (SF), autocorrelation function (ACF), box-counting (Box) and roughness length (RMS) methods are limited by the identification accuracy of FP (fractal dimension D and fractal roughness G). These methods are not suitable for global D interval (Kulesza et al., 2014; Feng et al., 2018), thereby resulting in large estimation errors (error (D) exceeds 40 %, and the value of G is incorrect). Thus, in this paper, a neural network FP estimation method based on the exact spectral moment is proposed. The main contribution of this paper is to establish the mapping relationship between the exact spectral moment and FP through the neural network. Firstly, the exact spectral moments, m0, m2 and m4, are derived via the differentiability of the series Weierstrass-Mandelbrot (WM) function in the finite interval. Secondly, a series of spectral moment parameter correspondence tables are generated according to the provided ideal fractal parameters. Then, the spectral moment is taken as the input layer and fractal parameters as the output layer, after which the BP neural network is optimized using the NSGA-II algorithm. Moreover, the mapping relationship between the FP and the spectral moment is established, thus obtaining the Fractal parameters estimation neural network (FPENN). The FP estimation model is trained by a relatively large amount of data and packaged to form a brand-new FP estimation method. Finally, the effectiveness of the proposed method is demonstrated by comparison with the existing methods. The results show that the relative error of D is <0.1 %, and the relative error of G is <25 %.