Generator of a periodic random self-affine field

Generate periodic 1D/2D/3D self‑affine Gaussian random fields with a prescribed power‑law spectrum in Fourier space. Includes two variants: a "noisy" spectrum (filtered white noise) and an exact‑magnitude spectrum with random phases only. C++ and Python implementations.

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Project information

• Author: Vladislav A. Yastrebov
• Affiliation: CNRS, Mines Paris, Centre des Materiaux, Versailles, Paris
• Date: 2015-2025
• Licence: BSD 3-Clause

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Description & references

This repository implements (in C++ and Python) the Fourier‑space filtering approach introduced by:

• Hu, Y.Z.; Tonder, K. Simulation of 3‑D random rough surface by 2‑D digital filter in Fourier space. Int. J. Mach. Tools Manufact. 32 (1992) 83–90. DOI: 10.1016/0890-6955(92)90064-N

It has been used, e.g., in:

• Yastrebov, V.A.; Anciaux, G.; Molinari, J.F. The role of the roughness spectral breadth in elastic contact of rough surfaces. J. Mech. Phys. Solids 107 (2017) 469–493. DOI: 10.1016/j.jmps.2017.07.016, arXiv:1704.05650

Python also includes a generator with an idealized spectrum: Fourier magnitudes match the target exactly; only phases are random.

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Features

• Periodic Gaussian random fields in 1D/2D/3D
• Self‑affine spectrum with Hurst exponent
• Two methods:
  • Filtered noise: spectrum follows the target on average
  • Prescribed magnitudes: exact magnitudes, random phases (real field guaranteed via Hermitian symmetry)
• Configurable spectral band: $k_{\text{low}}, k_{\text{high}}$
• Optional PSD plateau for $0 \le k < k_{\text{low}}$
• C++ implementation and pure‑NumPy Python implementation

C++ usage

Build

make # build src/* into bin/SURFACE_GENERATOR

Run

./bin/SURFACE_GENERATOR k1 k2 H N seed rms Npdf plateau

if you do not provide arguments, it will print the arguments it needs.

Arguments

• [1] k1 (int) -- lower cutoff wavenumber
• [2] k2 (int) -- upper cutoff wavenumber
• [3] H (double) -- Hurst exponent, $0 < H < 1$
• [4] L (int) -- number of points per size (prefer powers of 2, i.e. 128, 256, 512, etc.)
• [5] s (int) -- seed for the random number generator
• [6] rms (double) -- standard deviation of heights
• [7] Npdf (int) -- number of bins in pdf data
• [8] if_plateau (bool) -- boolean argument determining whether the power spectral density has a plateau up to k1? (0 - non, 1 -yes)

Python usage

The Python code lives in python/RandomField.py with a simple test in python/test.py. Import the module (add the folder to PYTHONPATH or place it next to your script):

import numpy as np
import RandomField as rf

np.random.seed(42)
N = 1024

random_field = rf.periodic_gaussian_random_field(dim = 2, N = N, Hurst = 0.5, k_low = 4 / N, k_high = 128 / N)

ideal_random_field = rf.ideal_periodic_gaussian_random_field(dim = 2, N = N, Hurst = 0.5, k_low = 4 / N, k_high = 128 / N)

ideal_random_field_with_plateau = rf.ideal_periodic_gaussian_random_field(dim = 2, N = N, Hurst = 0.5, k_low = 4 / N, k_high = 128 / N, plateau = True)

API(Python)

periodic_gaussian_random_field(dim, N, Hurst, k_low, k_high, plateau=False, verbose=False)
    -> ndarray
  - Filtered white noise. The radially averaged PSD follows the target power law.

ideal_periodic_gaussian_random_field(dim, N, Hurst, k_low, k_high, plateau=False, verbose=False)
    -> ndarray
  - Exact Fourier magnitudes per bin (target spectrum), random phases only.

Parameters:

• dim (int): Dimension of the field (1, 2, or 3).
• N (int): number of points per side in the field (N x N for 2D, N x N x N for 3D).
• Hurst (float): Hurst exponent, in the range [0, 1].
• k_low (float): Lower bound of the wavenumber range (k_low > 0), given in $k1/L$.
• k_high (float): Upper bound of the wavenumber range (k_high < 0.5, which represents Nyquist frequency), given in $k2/L$.
• plateau (bool): If True, the power spectrum is flat up to k_low, otherwise all wavenumbers below k_low are set to zero.
• verbose (bool): If True, print the parameters used for generating the random field.

Examples

Random self-affine 2D field with a "noisy" spectrum with a plateau

Random self-affine 2D field with an "ideal" spectrum with a plateau

License

• C++ repository: BSD 3‑Clause.
• Python module: CC0