An analysis of generated fractal and measured rough surfaces

This work studies the fractal dimensions of rough surfaces as calculated by several existing methods on measured and generated surface profiles. Two methods for generating rough surfaces are used in this work. The first one is to reconstruct the rough surface through the inverse Fourier transform based on a prescribed Power Spectrum Density (PSD) and the other one is using the Weierstrass-Mandelbrot (W-M) function. The fractal dimension values of all the rough surfaces are calculated by four different methods, namely, (1) the box-counting method, (2) the roughness-length method, (3) the power spectral density method and (4) the variogram method. Then the results from these four methods are compared. Since fractal surfaces are always clarified either as self-similar (the scaling ratio is the same in all directions) or as self-affine (scaling ratio varies in prescribed fashion over scales), it can be found that the fractal dimension values are not the same after analyzing the generated self-similar rough surfaces by these two methods. The fractal dimension values for real rough surfaces, as well as other parameters, are also calculated by four different methods. The analysis indicates that real rough surfaces are not easily represented as perfect fractals as researchers and engineers often assume.