Rough surface generation

The characterization of a rough surface can be done with respect to its spatial frequency content. This can be transformed into a constructive method by using a sum of trigonometric functions similar to a Fourier series. Each term in such a sum represents a certain frequency of spatial oscillation.

Self-affine scaling

A broad-spectrum of natural surfaces shows an intrinsically multi-scale roughness. This scaling phenomenon is known as self-affine that differs from the self-similar fractal geometry by the anisotropric character of the scaling transformation. The self-affine scaling transformation for a height distribution \(h(x,y)\) can be resumed by,

\[\lambda^{H}h(\frac{x}{\lambda}, \frac{y}{\lambda}) = h(x,y)\]

where \(H\) is the Hurst exponent and \(\lambda\) is a scalar.

Power spectrum density

The characterization of surface roughness relies on the description of representative spatial frequencies. The 2D power spectral density (PSD) \(C^{2D}(\mathbf{q})\) is used to analyze a surface across a range of wave vectors \(\mathbf{q} = (q_x , q_y )\) and describes the contribution of the various spatial frequencies. It is expressed as the fast Fourier transform of the autocorrelation function \(\langle h(\mathbf{x})h(\mathbf{0})\rangle\) (with \(\mathbf{x} = (x,y)\)),

\[C^{2D}(\mathbf{q}) = \frac{1}{(2\pi)^2} \int \langle h(\mathbf{x})h(\mathbf{0}) \rangle e^{-i\mathbf{q}.\mathbf{x}} d\mathbf{x}\]

In the case of self-affine scaling, the PSD has a power-law dependence on the spatial frequency such as,

\[C^{2D}(\textbf{q}) \propto \textbf{q}^{-2-2H}\]

where \(H\) is related to the surface fractal dimension \(D_f\) by the relation \(D_f = 3 − H\).

Construction method

In Pyrough, rough surfaces are modeled in the real space using a sum of trigonometric functions where each term of the sum represents the contribution of a spatial frequency. In cartesian coordinates, the mathematical expression of spatial oscillations is given by,

\[cos(\mathbf{q}.\mathbf{x} + \phi) = cos[2\pi(\nu_x x + \nu_y y) + \phi]\]

where \(\nu_x\) and \(\nu_y\) are the the spatial frequencies along \(\vec{u_x}\) and \(\vec{u_y}\) directions respectively and \(\phi\) is the phase.

A discrete set of spatial frequencies \(\nu_x = a\) and \(\nu_y = b\) (where \(a\) and \(b\) are integers) is used to rationalize the range of investigated frequencies. \(A\) and \(B\) are defined as the respective high-frequency cutoffs for \(a\) and \(b\) so that \(a \in [-A;A]\) and \(b \in [-B;B]\). Thus, the shortest wavelengths are \(\lambda_{x,min} = \frac{1}{A}\) and \(\lambda_{y,min} = \frac{1}{B}\) along \(\vec{u_x}\) and \(\vec{u_y}\) directions, respectively. \(a\) and \(b\) can be positive or negative to ensure oscillations in both directions. A rough surface \(h(x,y)\) can be described by a sum of elementary waves as,

\[h(x,y) = \sum_{a=-A}^{A} \sum_{b=-B}^{B} \alpha_{a,b}cos[2\pi(a x + b y) + \phi]\]

where \(\alpha_{a,b}\) is the associated amplitude to each elementary wave.

Two more contributions are made in order to allow Pyrough to generate self-affine rough surfaces that are randomly perturbed. First, the phase is randomly perturbed using \(\phi = U(a,b)\) where \(U\) states for a uniform distribution on an interval of length \(\pi\). Also, random perturbations and self-affine aspects are implemented within \(\alpha_{a,b}\). One can for example choose \(\alpha_{a,b}\) as a zero-centered Gaussian distribution to get a smooth but random variation in amplitudes without constraining the magnitude i.e., \(\alpha_{a,b} = G(a,b){(a^2+b^2)}^{-(1+H)}\) where \(G\) states for a reduced centered normal law and \({(a^2+b^2)}^{-(1+H)}\) traduces the self-affine aspect of the surface. Finally, the construction of a randomly perturbed self-affine surface can be modeled as,

\[h(x,y) = C_1\sum_{a=-A}^{A} \sum_{b=-B}^{B} G(a,b) (a^2+b^2)^{-(1+H)} cos[2\pi(a x + b y) + U(a,b) ]\]

in which H allows to monitor the roughness degree and \(C_1\) is a normalization factor introduced to fit the surface heights to the sample dimensions.

Random numerical self-affine rough surfaces for various Hurst exponents. a) H=-0.4, b) H=-0.1, c) H=0.3, d) H=1.0, e) PSD of the various h profiles. For all calculations, A=50, B=50 and h(x,y) values are normalized between 1 and -1.