RoughSurface¶

Type:	section
Appearance:	optional
Excludes:	AxiconCut, EllipsoidCut, Pyramid, Python, SphereCut, Triangulation

Here, it is possible to define a texture as a synthetic randomized rough surface. This rough surface height profile $h(x, y)$ creation is based on a centralized Gaussian distribution in space,

$\begin{eqnarray*} g(x,y) = e^{-\frac{1}{2} \left( \frac{x}{L_x} \right )^2 -\frac{1}{2} \left( \frac{y}{L_y} \right )^2 }, \end{eqnarray*}$

with CorrelationLengths $L_x$ and $L_y$ which is randomly displaced while summed up and normalized. This process can be written in terms of the forward and backward Fourier transform (denoted by $(.)^{\wedge}$ and $(.)^{\vee}$ ) as follows:

$\begin{eqnarray*} h(x,y) = C \left( g^{\wedge}(k_x, k_y) e^{i \phi(k_x, k_y)} \right)^{\vee} \end{eqnarray*}$

The phase function $\phi(k_x, k_y)$ is randomly chosen, or by a deterministic pseudo random number generator when setting the seed for the random number generator (see RandomSeed).

The scaling factor $C$ is determined, so that the root mean square of the height profile function,

$\begin{eqnarray*} \mathrm{rms} = \lim_{l \rightarrow \infty} \sqrt{ \frac{\int_{-l}^{l} \int_{-l}^{l} h^2(x, y)\,\mathrm{d}x \, \mathrm{d}y}{\int_{-l}^{l} \int_{-l}^{l} 1\,\mathrm{d}x \, \mathrm{d}y} } \end{eqnarray*}$

equals the value passed in RMSHeight.

Note

In case of a periodic computational domain the rough surface is constructed in a periodified manner.