Disk Harmonics for Analysing Curved and Flat Self-affine Rough Surfaces and the Topological Reconstruction of Open Surfaces

Abstract

When two bodies get into contact, only a small portion of the apparent area is actually involved in producing contact and friction forces, because of the surface roughnesses. It is therefore crucial to accurately describe the morphology of rough surfaces for instance by extracting the fractal dimension and the so-called Hurst exponent which is a typical signature of rough surfaces. This can be done using harmonic decomposition, which is easy for periodic and nominally flat surfaces since Fourier transforms allow fast and reliable decomposition. Yet, it remains a challenging task in the general curved and non-periodic cases, where more appropriate basis functions must be used. In this work, disk harmonics based on Fourier-Bessel basis functions are employed for decomposing open single-edge genus-0 surfaces (no holes) as a practical and fast alternative to characterise self-affine rough surfaces with the power Fourier-Bessel spectral density. An analytical relationship between the power spectrum density decay and the Hurst exponent is derived through an extension of the Wiener-Khinchin theorem, in the special case where surfaces are assumed self-affine and isotropic. Finally, this approach is demonstrated to successfully measure the fractal dimension, and the Hurst exponent, without introducing typical biases coming from basis functions boundary conditions, surface discretisation or curvature of the surface patches. This work opens the path for contact mechanics studies based on the Fourier-Bessel spectral representation of curved and rough surface morphologies. All implementation details for this method are available under GNU LGPLv3 terms and conditions.

Figures (21)

• Figure 1: A planar unit disk $\mathbb{D}$ ($\rho(\phi) \leq 1 ~\forall \phi \in [0, 2\pi]$) that is centred about the pole in a polar coordinate system. The angle $\phi$ is measured counter-clockwise from the polar axis in radian.
• Figure 2: A planar disk area-preserving parameterisation $\varphi:\mathcal{S} \to \mathbb{D}$ of a benchmark face named Sophie (the mesh was retrieved from Riemannmapper). A) The input surface mesh $\mathcal{S}$. B) The initial flattening map $g:\mathcal{S} \to \mathbb{D}$. C) The final area-preserving map $\varphi:\mathcal{S} \to \mathbb{D}$ given by $\varphi = \tilde{h} \circ g$, where $\tilde{h}$ is the updated map after enforcing the bijectivity of the density-equalising map $h$.
• Figure 3: The normalised Fourier-Bessel basis functions $\mathfrak{R}(D^{k}_{m} (\rho, \phi) )$.
• Figure 4: The normalised Fourier-Bessel basis functions $\mathfrak{R}(D^{k}_{m} (\rho, \phi) )$. The complex part of the base functions is identical to the real ones except that they are rotated by $\pi/(2|m|)$ about the $z$-axis.
• Figure 5: Fitting the first degree ellipsoidal cap (FDEC) defined by an expansion at $k = 1$. A) We show how the FDEC, reconstructed at $k=1$, represents an ellipsoidal cap fitted over the input surface. B) The size of the cap is determined by $a$, representing the half-major axis, $b$, representing the half-minor axis, and $c$, representing the ellipsoidal cap depth, where $|c| \leq |b| \leq |a|$. Inset (B) was retrieved from Shaqfa et al. shaqfa2021b.