A CONTINUOUS VARIATION OF ROUGHNESS SCALING CHARACTERISTICS ACROSS FRACTAL AND NON-FRACTAL PROFILES

Zhiwei Li, Xiang Qian, Feng Feng, Timing Qu, Yousheng Xia and Wenmeng Zhou
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Zhiwei Li: Laboratory of Intelligent Manufacturing and Precision, Machining, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, P. R. China†Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, P. R. China
Xiang Qian: ��Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, P. R. China
Feng Feng: Laboratory of Intelligent Manufacturing and Precision, Machining, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, P. R. China†Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, P. R. China
Timing Qu: ��Department of Mechanical Engineering, Tsinghua University, Beijing 100084, P. R. China
Yousheng Xia: Laboratory of Intelligent Manufacturing and Precision, Machining, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, P. R. China†Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, P. R. China
Wenmeng Zhou: Laboratory of Intelligent Manufacturing and Precision, Machining, Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, P. R. China†Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 05, 1-10

Abstract: In this study, the scaling characteristics of root-mean-squared roughness (Rq) was investigated for both fractal and non-fractal profiles by using roughness scaling extraction (RSE) method proposed in our previous work. The artificial profiles generated through Weierstrass–Mandelbrot (W–M) function and the actual profiles, including surface contours of silver thin films and electroencephalography signals, were analyzed. Based on the relationship curves between Rq and scale, it was found that there was a continuous variation of the dimension value calculated with RSE method (DRSE) across the fractal and non-fractal profiles. In the range of fractal region, DRSE could accurately match with the ideal fractal dimension (FD) input for W–M function. In the non-fractal region, DRSE values could characterize the complexity of the profiles, similar to the functionality of FD value for fractal profiles, thus enabling the detection of certain incidents in signals such as an epileptic seizure. Moreover, the traditional methods (Box-Counting and Higuchi) of FD calculation failed to reflect the complexity variation of non-fractal profiles, because their FD was generally 1. The feasibility of abnormal implementation of W–M function and the capability of RSE method were discussed according to the analysis on the properties of W–M function, which would be promising to make more understandings of the nonlinear behaviors of both theoretical and practical features.

Keywords: Scaling Characteristics; Fractal Dimension; Roughness; Weierstrass–Mandelbrot Function; Non-fractal (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501097

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