Non-Gaussian Surface Simulations

Author: Sebastian Korsak

🌄 Gaussian Self-Affine Surface Generator

This Python module generates 2D Gaussian random surfaces with spatial correlations, based on a prescribed roughness exponent and correlation lengths.
It implements a spectral method using FFT-based synthesis and supports anisotropic scaling.

Adapted from the method described in:

Yang et al., CMES, vol.103, no.4, pp.251–279, 2014

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🔧 Function

SAimage_fft_2(N, rms, skewness, kurtosis, corlength_x, corlength_y, alpha, non_Gauss=False)

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📥 Parameters

Parameter	Type	Description
N	int	Size of the generated surface (NxN)
rms	float	Target root-mean-square roughness
corlength_x	float	Correlation length along the x-axis
corlength_y	float	Correlation length along the y-axis
alpha	float	Roughness exponent (also known as Hurst exponent)
non_Gauss	bool	Set to False to generate Gaussian surface only
skewness, kurtosis	float	Ignored if non_Gauss=False

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🧠 Mathematical Model

The spatial autocorrelation function is modeled as:

R(tx, ty) = rms² · exp( - ( sqrt((tx/ξx)² + (ty/ξy)²) )^(2α) )

Where:

• ξx, ξy: correlation lengths in x and y directions
• α: roughness exponent (e.g., α = 0.5 for Brownian motion)

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📈 Output

Returns a 2D NumPy array:

surface = SAimage_fft_2(..., non_Gauss=False)

• Gaussian-distributed height values
• RMS roughness equal to rms
• Zero mean
• Spatial correlations matching ksix, ksiy, and alpha

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🌍 Visualization Example

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

z = SAimage_fft_2(N=256, rms=1.0, skewness=0, kurtosis=3,
                  corlength_x=20, corlength_y=20, alpha=0.8, non_Gauss=False)

X, Y = np.meshgrid(np.arange(z.shape[0]), np.arange(z.shape[1]))
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, z, cmap='viridis', edgecolor='none')
plt.title('Gaussian Self-Affine Surface')
plt.show()

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⚠️ Notes

• This is a purely Gaussian surface generator (zero skewness, kurtosis = 3).
• If you need non-Gaussian statistics, set non_Gauss=True and provide skewness and kurtosis. See the full version with rank-order mapping for that.
• Spatial correlation is implemented via FFT filtering based on the Wiener-Khinchin theorem.

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📜 References

• Yang, F., et al. Statistical generation of 3D rough surfaces with arbitrary correlation, CMES, vol. 103, no. 4, 2014.
• Persson, B.N.J. Theory of rubber friction and contact mechanics.

👥 Credits

• Python translation by Max Pierini @ EpiData.it
• Ported from original MATLAB toolbox by Dave (2021)
• Matlab code from dr. Vasilis Costantoudis.
• Extended and maintained by Sebastian Korsak.
• Correction of non gaussian surfaces with copilot.

Results