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.

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 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 correlation 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., \(\alpha_{a,b} = G(a,b){(a^2+b^2)}^{-(1+\eta)}\) with \(G\) states for a reduced centered normal law and \({(a^2+b^2)}^{-(1+\eta)}\) expresses the power-law dependency of spatial frequencies involving \(\eta\), a generalized roughness coefficient that scales with the Hurst coefficient \(H\) (\(\eta = (H-1)/2\)). Finally, the construction of a randomly perturbed surface can be modeled as,

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

in which eta allows to monitor the roughness degree i.e., self-affine surfaces for \(-0.5<\eta<0\) or non-stationnary ones for \(\eta>0\), and \(C_1\) is a normalization factor introduced to fit the surface heights to the sample dimensions.

Rough surfaces computed using Pyrough for various roughness exponent \(\eta\), a):math:eta`=-0., b) :math:eta`=0.25, c) \(\eta`=0.5, d) :math:\)eta`=0.75, 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.