An Optimal Control Method for Reconstruction of Topography in Dam-Break Flows

Authors: Alia Alghosoun, Nabil El Moçayd, Mohammed Seaid

Abstract:

Modeling dam-break flows over non-flat beds requires an accurate representation of the topography which is the main source of uncertainty in the model. Therefore, developing robust and accurate techniques for reconstructing topography in this class of problems would reduce the uncertainty in the flow system. In many hydraulic applications, experimental techniques have been widely used to measure the bed topography. In practice, experimental work in hydraulics may be very demanding in both time and cost. Meanwhile, computational hydraulics have served as an alternative for laboratory and field experiments. Unlike the forward problem, the inverse problem is used to identify the bed parameters from the given experimental data. In this case, the shallow water equations used for modeling the hydraulics need to be rearranged in a way that the model parameters can be evaluated from measured data. However, this approach is not always possible and it suffers from stability restrictions. In the present work, we propose an adaptive optimal control technique to numerically identify the underlying bed topography from a given set of free-surface observation data. In this approach, a minimization function is defined to iteratively determine the model parameters. The proposed technique can be interpreted as a fractional-stage scheme. In the first stage, the forward problem is solved to determine the measurable parameters from known data. In the second stage, the adaptive control Ensemble Kalman Filter is implemented to combine the optimality of observation data in order to obtain the accurate estimation of the topography. The main features of this method are on one hand, the ability to solve for different complex geometries with no need for any rearrangements in the original model to rewrite it in an explicit form. On the other hand, its achievement of strong stability for simulations of flows in different regimes containing shocks or discontinuities over any geometry. Numerical results are presented for a dam-break flow problem over non-flat bed using different solvers for the shallow water equations. The robustness of the proposed method is investigated using different numbers of loops, sensitivity parameters, initial samples and location of observations. The obtained results demonstrate high reliability and accuracy of the proposed techniques.

Keywords: Optimal control, ensemble Kalman Filter, topography reconstruction, data assimilation, shallow water equations.

References:

[1] H. Abida. Identification of compound channel flow parameters. Journal of Hydrology and Hydromechanics, 57:172–181, 2009.
[2] S. Barth´el´emy, S. Ricci, O. Pannekoucke, O. Thual, and P. Malaterre. Emulation of an Ensemble Kalman Filter algorithm on a flood wave propagation model. Hydrology and Earth System Sciences Discussions, 10:6963–7001, 2013.
[3] F. Benkhaldoun, S. Sari, and M. Seaid. A flux-limiter method for dam-break flows over erodible sediment beds. Applied Mathematical Modelling, 36(10):4847–4861, 2012.
[4] F. Benkhaldoun and M. Seaid. A simple finite volume method for the shallow water equations. J. Comp. Applied Math, 234:58–72, 2010.
[5] F. Bouchut. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkh¨auser, Basel, 2004.
[6] W. Cheng and G. Liu. Analysis of non-linear channel friction inverse problem. Frontiers of Architecture and Civil engineering in China, 2:205–210, 2007.
[7] G. Evensen. The Ensemble Kalman Filter: Theoretical formulation and practical implementation. Ocean Dynamics, 53(4):343–367, 2003.
[8] M. Le Gal, D. Violeau, R. Ata, and X. Wang. Shallow water numerical models for the 1947 gisborne and 2011 tohoku-oki tsunamis with kinematic seismic generation. Coastal Engineering, 139:1–15, 2018.
[9] P. Garcia-Navarro, A. Fras, and I. Villanueva. Dam-break flow simulation: some results for one-dimensional models of real cases. Journal of Hydrology, 216(3-4):227–247, 1999.
[10] C. Heining and N. Askel. Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability. Physics and Fluids, 21:321–331, 2009.
[11] R. Hilldale and D. Raff. Assessing the ability of airborne liDAR to map river bathymetry. Earth Surf.Process, 33:773–783, 2008.
[12] A. Humberto, E. Schubert, and F. Sanders. Two-dimensional, high-resolution modeling of urban dam-break flooding: A case study of baldwin hills, california. Advances in Water Resources, 32(8):1323–1335, 2009.
[13] E. LeMeur, O. Gagliardini, T. Zwinger, and J. Ruokolainen. Glacier flow modelling: A comparison of the shallow ice approximation and the full-stokes solution. Comptes Rendus Physique, 5:709–722, 2004.
[14] F. Lu, M. Morzfeld, X. Tu, and A. Chorin. Limitations of Polynomial Chaos Expansions in the bayesian solution of inverse problems. Journal of Computational Physics, 282:138–147, 2015.
[15] S. Mart´ınez-Aranda, J. Murillo, and P. Garc´ıa-Navarro. A 1d numerical model for the simulation of unsteady and highly erosive flows in rivers. Computers & Fluids, 181:8–34, 2019.
[16] G. Michael and P. Malanotte-Rizzoli. Data Assimilation in metrology and oceanography. Advances in Geophysics, 33:141–266, 1991.
[17] N. El Moc¸ayd. La d´ecomposition en polynˆome du chaos pour l’am´elioration de l’assimilation de donn´ees ensembliste en hydraulique fluviale. PhD thesis, 2017.
[18] H. Nhuyen and J. Fenton. Identification of roughness in open channels. Advances in Hydro-science and engineering, 3:111–119, 2004.
[19] H. Oubanas, I. Gejadze, M. Pierre-Olivier, and M. Franck. River discharge estimation from synthetic SWOT-type observations using variational Data Assimilation and the full Saint-Venant hydraulic model. Journal of Hydrology, 559:638–647, 2018.
[20] R. Ramesh, B. Datta, S. Bhallamudi, and A. Narayana. Optimal estimation of roughness in open-channel flows. Journal of hydraulic engineering, 126:3–13, 2000.
[21] P. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43:357–372, 1981.
[22] H. Roux and D. Dartus. Sensitivity analysis and predictive uncertainty using inudation observations for parameter estimation in open-channel inverse problem. J. Hydraul.Eng, 9:134–541, 2008.
[23] M. Seller. Substrate design of reconstruction from free surface data for thin film flows. Phys. Fluids, 7:206–217, 2016.
[24] J. Simon, J. Jeffrey, and K. Uhlmann. New extension of the Kalman Filter to nonlinear systems. In Ivan Kadar, editor, Signal Processing, Sensor Fusion, and Target Recognition VI, volume 3068, pages 182 – 193. International Society for Optics and Photonics, SPIE, 1997.
[25] O. Soucek and Z. Martince. Iterative improvement of the shallow-ice approximation. Journal of Galciology, 54:812–822, 2008.
[26] Y. Spitz, J. Moisan, M. Abbott, and J. Richman. Data Assimilation and a pelagic ecosystem model: parameterization using time series observations. Journal of Marine Systems, 16(1):51–68, 1998.
[27] J. Stoker. Water waves. Interscience Publishers, Inc, New York, 1986.
[28] C. Vreugdenhil. Numerical Method for Shallow Water Flow. Kluwer Academic, Dordsecht, 1994.
[29] R. Westaway, S. Lane, and D. Hicks. The development of an automated correction procedure for digital photogrammetry for the study of wide, shallow, gravel-bed rivers. Earth surfaces processes and land forms, 25:209–225, 2000.