The Use of Fractional Brownian Motion in the Generation of Bed Topography for Bodies of Water Coupled with the Lattice Boltzmann Method
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The Use of Fractional Brownian Motion in the Generation of Bed Topography for Bodies of Water Coupled with the Lattice Boltzmann Method

Authors: Elysia Barker, Jian Guo Zhou, Ling Qian, Steve Decent

Abstract:

A method of modelling topography used in the simulation of riverbeds is proposed in this paper which removes the need for datapoints and measurements of a physical terrain. While complex scans of the contours of a surface can be achieved with other methods, this requires specialised tools which the proposed method overcomes by using fractional Brownian motion (FBM) as a basis to estimate the real surface within a 15% margin of error while attempting to optimise algorithmic efficiency. This removes the need for complex, expensive equipment and reduces resources spent modelling bed topography. This method also accounts for the change in topography over time due to erosion, sediment transport, and other external factors which could affect the topography of the ground by updating its parameters and generating a new bed. The lattice Boltzmann method (LBM) is used to simulate both stationary and steady flow cases in a side-by-side comparison over the generated bed topography using the proposed method, and a test case taken from an external source. The method, if successful, will be incorporated into the current LBM program used in the testing phase, which will allow an automatic generation of topography for the given situation in future research, removing the need for bed data to be specified.

Keywords: Bed topography, FBM, LBM, shallow water, simulations.

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References:


[1] Mandelbrot. B. B, Mandelbrot. B. B, The fractal geometry of nature (Vol. 1) 1982. New York: WH freeman.
[2] A. M. Astakhov, V. A. Avetisov, S. K. Nechaev, and K. E. Polovnikov, Fractal Dimension Meets Topology: Statistical and Topological Properties of Globular Macromolecules with Volume Interactions, Macromolecules, vol. 54, no. 3, pp. 1281–1290, Feb. 2021, doi: 10.1021/acs.macromol.0c01717.
[3] A. N. Kolmogorov, Wienersche spiralen und einige andere interessante Kurven in Hilbertscen Raum, C. R. (doklady), 1940. Acad. Sci. URSS (N.S.) 26 115–118. MR0003441
[4] M. S. Taqqu, Benoit Mandelbrot and Fractional Brownian Motion, Statistical Science, vol. 28, no. 1, Feb. 2013, doi: 10.1214/12-sts389.
[5] B. B. Mandelbrot and J. w. van Ness. Fractional Brownian motions, fractional noises and applications, 1968, SIAM Rev. 10 422–437. MR0242239
[6] F. Family and T. Vicsek, Dynamics of Fractal Surfaces, 1991 ISBN 981-02-0720-4 page 45
[7] H. Zahouani, R. Vargiolu, and J.-L. Loubet, Fractal models of surface topography and contact mechanics, Mathematical and Computer Modelling, vol. 28, no. 4, pp. 517–534, 1998, doi: https://doi.org/10.1016/S0895-7177(98)00139-3.
[8] J. van Lawick van Pabst and H. Jense, Dynamic Terrain Generation Based on Multifractal Techniques, in High Performance Computing for Computer Graphics and Visualisation, 1996, pp. 186–203.
[9] A. Cristea and F. Liarokapis, Fractal Nature - Generating Realistic Terrains for Games. 1-8. 10.1109/VS-GAMES.2015.7295776. (2015).
[10] Indigo Renderer, 2004, Advanced Fractional Brownian Motion Noise, https://www.indigorenderer.com/indigo-technical-reference/indigo-shaderlanguage- reference/built-functions-%E2%80%93-procedural-noise-fun-0, 16/03/2023
[11] B. T. Milne, The utility of fractal geometry in landscape design, Landscape and Urban Planning, vol. 21, no. 1, pp. 81–90, 1991, doi: https://doi.org/10.1016/0169-2046(91)90034-J.
[12] G. Franceschetti, D. Riccio, Editor(s): G. Franceschetti, D. Riccio, Scattering, Natural Surfaces, and Fractals, Academic Press, 2007, Pages 1-19, ISBN 9780122656552, https://doi.org/10.1016/B978-012265655-2/50001-5. (https://www.sciencedirect.com/science/article/pii/B9780122656552500015)
[13] D. Della-Bosca, D. Patterson, S. Costain, (2014) Fractal Complexity in Built and Game Environments. In: Pisan Y., Sgouros N.M., Marsh T. (eds) Entertainment Computing – ICEC 2014. ICEC 2014. Lecture Notes in Computer Science, vol 8770. Springer, Berlin, Heidelberg.
[14] N. Enriquez, A simple construction of the fractional Brownian motion, Stochastic Processes and their Applications, vol. 109, no. 2, pp. 203–223, 2004, doi: https://doi.org/10.1016/j.spa.2003.10.008.
[15] A. Patrice, and F. Sellan. The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F. Sellan and Y. Meyer: Remarks and Fast Implementation. Applied and Computational Harmonic Analysis 3, no. 4 (October 1996): 377–83. https://doi.org/10.1006/acha.1996.0030.
[16] B. Jean-Marc, G. Lang, G. Oppenheim, A. Philippe, S. Stoev, and M. S. Taqqu. Generators of Long-Range Dependent Processes: A Survey. In Theory and Applications of Long-Range Dependence, edited by Paul Doukhan, Georges Oppenheim, and Murad S. Taqqu, 579–623. Boston: Birkhauser, 2003.
[17] A. Hayes, Autoregressive Integrated Moving Average (ARIMA),2010, https://www.investopedia.com/terms/a/autoregressive-integrated-movingaverage- arima.asp
[18] MathWorks. 2022. wfbm: Fractional Brownian Motion synthesis.
[online] https : //uk.mathworks.com/help/wavelet/ref/wfbm.html#mw3fcd5614− 1ace − 4b39 − bcf0 − a9b10550eae6
[Accessed 1 September 2022].
[19] N. Clifford, N. Wright, GL. Harvey, A. Gurnell, O. Harmar, P. Soar. (2010). Numerical Modeling of River Flow for Ecohydraulic Applications: Some Experiences with Velocity Characterization in Field and Simulated Data. Journal of Hydraulic Engineering. 136. 1033-1041. 10.1061/(asce)hy.1943-7900.0000057.
[20] Zhou, J, Lattice Boltzmann Methods for Shallow Water Flows. Springer-Verlag 2004
[21] Zhou, J. G., An Elastic-Collision Scheme for Lattice Boltzmann Methods, International Journal of Modern Physics C, vol. 12, no. 3, pp. 387–401, 2001. doi:10.1142/S0129183101001833.