Fractal characteristic evaluation and interpolation reconstruction for surface topography of drilled composite hole wall

In this paper,an improved fractal interpolation model is proposed to reconstruct the surface topography of composite hole wall.This model adopts the maximum positive deviations and maximum negative deviations between the measured values and trend values to determine the contraction factors.Hole profiles in 24 directions are measured.Fractal parameters are calculated to evaluate the measured surface profiles.The maximum and minimum fractal dimension of the hole wall are 1.36 and 1.07,whereas the maximum and minimum fractal roughness are 4.05 x 10-5 and 4.36 x 10-10 m,respectively.Based on the two-dimensional evaluation results,three-dimensional surface topographies in five typical angles(0°,45°,90°,135°,and 165°)are reconstructed using the improved model.Fractal parameter Ds and statistical parameters Sa9 Sq,and Sz are used to evaluate the reconstructed surfaces.Average error of Ds,Sa,Sq,and Sz between the measured surfaces and the reconstructed surfaces are 1.53%,3.60%,5.60%,and 9.47%,respectively.Compared with the model in published literature,the proposed model has equal reconstruction effect in relatively smooth surface and is more advanced in relatively rough surface.Comparative results prove that the proposed model for calculating contraction factors is more reasonable.

Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.