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New progress in fractal research for intelligent manufacturing

Fractal is a nonlinear research approach that has been developed to study irregular geometric features. The fractal concept was proposed by mathematician Benoît B. Mandelbrot to describe rough and fragmented geometries. The universality of investigated objects indicated the interest-oriented characteristics of fractal as a fundamental research method, and it has been valued in various fields for decades. On the other hand, the fractal method has brought important practical value through its applications, which have been widely and effectively used in mechanical engineering, physics, materials, biology, and other fields. For example, in the literature about technologies such as machining and thin film growth, surfaces with fractal characteristics were reported. Furthermore, fractal simulation methods have also been applied to numerous types of research including establishing fractal models of rough surfaces, contact mechanisms of rough surfaces, temperature distribution in friction surfaces, and thin film growth mechanisms. However, traditional algorithms generally exhibit issues of poor accuracy or efficiency in the calculation of fractal dimensions.

To address the challenges of the aforementioned issues, researchers from the Division of Advanced Manufacturing at Tsinghua SIGS, including Feng FENG, Xiang QIAN, Xinghui LI, and Min ZHANG, have carried out a collaborative study on the fractal method and its applications in recent years. Roughness Scaling Extraction (RSE), an algorithm that demonstrates both high accuracy and efficiency, was proposed to calculate fractal dimension based on a single image of surface topography or a single curve of time-series signal. According to the artificial surfaces with ideal fractal dimensions generated by the Weierstrass-Mandelbrot function, the mean relative error of the RSE algorithm could be lower by an order of magnitude relative to the traditional algorithms such as autocorrelation function, structure function, and box-counting and power spectral density methods. These results were published in Fractals, Fractal and Fractional, and Applied Surface Science.

The transcending fractal behaviors of roughness scaling characteristics in vibration signals and surface topographies found in the study

Recently, the Division of Advanced Manufacturing achieved further progress in fractal research. The wide existence of a continuous variation of roughness scaling characteristics across fractal and non-fractal was found by using RSE algorithms in various investigations, including thin-film surfaces, machined surfaces, machine tool vibration signals, electroencephalogram signals, and artificial surfaces/curves. In contrast, the traditional fractal algorithms cannot effectively characterize such a phenomenon. Therefore, a new method was developed to effectively characterize the complexity of signals and surfaces. Moreover, based on the actual phenomenon, a description called a transcending fractal was used. The findings in this study exhibited that the complexity could be continuously quantified across fractal and non-fractal, which was verified by the validity of transcending fractal behavior in the surface topographies and signal analysis. The proposed method could be promising to be widely utilized in the field of intelligent manufacturing technology, and help to improve the non-linear characteristic analysis of various data types, including in-situ monitoring of machining signals and surface analysis of machined workpieces.

These findings were published in Chaos, Solitons & Fractals in an article entitled “An indicator to quantify the complexity of signals and surfaces based on scaling behaviors transcending fractal.” The first author of the article is Zhiwei LI, a PhD student in the Department of Mechanical Engineering, Tsinghua University. The corresponding author is Associate Professor Feng FENG in the Division of Advanced Manufacturing, Tsinghua SIGS.

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Written by Feng Feng

Edited by Alena Shish & Yuan Yang