RoughSurfaceΒΆ

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.