×

A non-intrusive reduced-order modeling for uncertainty propagation of time-dependent problems using a B-splines Bézier elements-based method and proper orthogonal decomposition: application to dam-break flows. (English) Zbl 1524.76130

Summary: A proper orthogonal decomposition-based B-splines Bézier elements method (POD-BSBEM) is proposed as a non-intrusive reduced-order model for uncertainty propagation analysis for stochastic time-dependent problems. The method uses a two-step proper orthogonal decomposition (POD) technique to extract the reduced basis from a collection of high-fidelity solutions called snapshots. A third POD level is then applied on the data of the projection coefficients associated with the reduced basis to separate the time-dependent modes from the stochastic parametrized coefficients. These are approximated in the stochastic parameter space using B-splines basis functions defined in the corresponding Bézier element. The accuracy and the efficiency of the proposed method are assessed using benchmark steady-state and time-dependent problems and compared to the reduced-order model-based artificial neural network (POD-ANN) and to the full-order model-based polynomial chaos expansion (Full-PCE). The POD-BSBEM is then applied to analyze the uncertainty propagation through a flood wave flow stemming from a hypothetical dam-break in a river with a complex bathymetry. The results confirm the ability of the POD-BSBEM to accurately predict the statistical moments of the output quantities of interest with a substantial speed-up for both offline and online stages compared to other techniques.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

CUDA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bērziņš, A.; Helmig, J.; Key, F.; Elgeti, S., Standardized non-intrusive reduced order modeling using different regression models with application to complex flow problems (2020), arXiv preprint
[2] Sanderse, B., Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods, J. Comput. Phys., 421, Article 109736 pp. (2020) · Zbl 07508361
[3] Kalinina, A.; Spada, M.; Vetsch, D. F.; Marelli, S.; Whealton, C.; Burgherr, P.; Sudret, B., Metamodeling for uncertainty quantification of a flood wave model for concrete dam breaks, Energies, 13, 14, 3685 (2020)
[4] Zokagoa, J.-M.; Soulaïmani, A., A pod-based reduced-order model for uncertainty analyses in shallow water flows, Int. J. Comput. Fluid Dyn., 32, 6-7, 278-292 (2018) · Zbl 07474471
[5] Georgaka, S.; Stabile, G.; Star, K.; Rozza, G.; Bluck, M. J., A hybrid reduced order method for modelling turbulent heat transfer problems, Comput. Fluids, Article 104615 pp. (2020) · Zbl 1502.65087
[6] Sirovich, L., Turbulence and the dynamics of coherent structures. i. Coherent structures, Q. Appl. Math., 45, 3, 561-571 (1987) · Zbl 0676.76047
[7] Chatterjee, A., An introduction to the proper orthogonal decomposition, Curr. Sci., 808-817 (2000)
[8] Zokagoa, J.-M.; Soulaïmani, A., A pod-based reduced-order model for free surface shallow water flows over real bathymetries for Monte-Carlo-type applications, Comput. Methods Appl. Mech. Eng., 221, 1-23 (2012) · Zbl 1253.76070
[9] Fang, F.; Pain, C. C.; Navon, I.; Elsheikh, A. H.; Du, J.; Xiao, D., Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods, J. Comput. Phys., 234, 540-559 (2013) · Zbl 1284.65132
[10] Couplet, M.; Basdevant, C.; Sagaut, P., Calibrated reduced-order pod-Galerkin system for fluid flow modelling, J. Comput. Phys., 207, 1, 192-220 (2005) · Zbl 1177.76283
[11] Willcox, K., Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Comput. Fluids, 35, 2, 208-226 (2006) · Zbl 1160.76394
[12] Ballarin, F.; Manzoni, A.; Quarteroni, A.; Rozza, G., Supremizer stabilization of pod-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng., 102, 5, 1136-1161 (2015) · Zbl 1352.76039
[13] Guo, M.; Hesthaven, J. S., Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput. Methods Appl. Mech. Eng., 341, 807-826 (2018) · Zbl 1440.65206
[14] Walton, S.; Hassan, O.; Morgan, K., Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions, Appl. Math. Model., 37, 20-21, 8930-8945 (2013) · Zbl 1426.76576
[15] Xiao, D.; Yang, P.; Fang, F.; Xiang, J.; Pain, C. C.; Navon, I. M., Non-intrusive reduced order modelling of fluid-structure interactions, Comput. Methods Appl. Mech. Eng., 303, 35-54 (2016) · Zbl 1425.74167
[16] Xiao, M.; Breitkopf, P.; Coelho, R.; Knopf-Lenoir, C.; Sidorkiewicz, M.; Villon, P., Model reduction by cpod and kriging: application to the shape optimization of an intake port, Struct. Multidiscip. Optim., 41, 4, 555-574 (2010/04) · Zbl 1274.90365
[17] Xiao, D.; Fang, F.; Buchan, A.; Pain, C.; Navon, I.; Muggeridge, A., Non-intrusive reduced order modelling of the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 293, 522-541 (2015) · Zbl 1423.76287
[18] Hesthaven, J. S.; Ubbiali, S., Non-intrusive reduced order modeling of nonlinear problems using neural networks, J. Comput. Phys., 363, 55-78 (2018) · Zbl 1398.65330
[19] Zhiwei, S.; Chen, W.; Zheng, Y.; Junqiang, B.; Zheng, L.; Qiang, X.; Qiujun, F., Non-intrusive reduced-order model for predicting transonic flow with varying geometries, Chin. J. Aeronaut., 33, 2, 508-519 (2020)
[20] Guo, M.; Hesthaven, J. S., Data-driven reduced order modeling for time-dependent problems, Comput. Methods Appl. Mech. Eng., 345, 75-99 (2019) · Zbl 1440.62346
[21] Swischuk, R.; Mainini, L.; Peherstorfer, B.; Willcox, K., Projection-based model reduction: formulations for physics-based machine learning, Comput. Fluids, 179, 704-717 (2019) · Zbl 1411.65061
[22] Wang, Q.; Hesthaven, J. S.; Ray, D., Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem, J. Comput. Phys., 384, 289-307 (2019) · Zbl 1459.76117
[23] Jacquier, P.; Abdedou, A.; Delmas, V.; Soulaïmani, A., Non-intrusive reduced-order modeling using uncertainty-aware deep neural networks and proper orthogonal decomposition: application to flood modeling, J. Comput. Phys., 424, Article 109854 pp. (2021) · Zbl 07508459
[24] McKay, M. D.; Beckman, R. J.; Conover, W. J., Comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 2, 239-245 (1979) · Zbl 0415.62011
[25] Ghanem, R. G.; Spanos, P. D., Stochastic finite element method: response statistics, (Stochastic Finite Elements: a Spectral Approach (1991), Springer), 101-119
[26] Sudret, B., Polynomial chaos expansions and stochastic finite element methods, (Risk and Reliability in Geotechnical Engineering (2014)), 265-300
[27] Hijazi, S.; Stabile, G.; Mola, A.; Rozza, G., Non-intrusive polynomial chaos method applied to full-order and reduced problems in computational fluid dynamics: a comparison and perspectives, (Quantification of Uncertainty: Improving Efficiency and Technology (2020), Springer), 217-240 · Zbl 1455.62159
[28] Raisee, M.; Kumar, D.; Lacor, C., A non-intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition, Int. J. Numer. Methods Eng., 103, 4, 293-312 (2015) · Zbl 1352.80004
[29] Raisee, M.; Kumar, D.; Lacor, C., Non-intrusive uncertainty quantification by combination of reduced basis method and regression-based polynomial chaos expansion, (Uncertainty Management for Robust Industrial Design in Aeronautics (2019), Springer), 169-184
[30] Sun, X.; Pan, X.; Choi, J.-I., A non-intrusive reduced-order modeling method using polynomial chaos expansion (2019), arXiv preprint
[31] El Moçayd, N.; Mohamed, M. S.; Ouazar, D.; Seaid, M., Stochastic model reduction for polynomial chaos expansion of acoustic waves using proper orthogonal decomposition, Reliab. Eng. Syst. Saf., 195, Article 106733 pp. (2020)
[32] Sun, X.; Choi, J.-I., Non-intrusive reduced-order modeling for uncertainty quantification of space-time-dependent parameterized problems, Comput. Math. Appl., 87, 50-64 (2021) · Zbl 1524.76230
[33] Abdedou, A.; Soulaimani, A., A non-intrusive b-splines Bézier elements-based method for uncertainty propagation, Comput. Methods Appl. Mech. Eng., 345, 774-804 (2019) · Zbl 1440.65028
[34] Abdedou, A.; Soulaïmani, A.; Tchamen, G. W., Uncertainty propagation of dam break flow using the stochastic non-intrusive b-splines Bézier elements-based method, J. Hydrol., 590, Article 125342 pp. (2020)
[35] Liang, Y.; Lee, H.; Lim, S.; Lin, W.; Lee, K.; Wu, C., Proper orthogonal decomposition and its applications—part i: theory, J. Sound Vib., 252, 3, 527-544 (2002) · Zbl 1237.65040
[36] Borden, M. J.; Scott, M. A.; Evans, J. A.; Hughes, T. J., Isogeometric finite element data structures based on Bézier extraction of nurbs, Int. J. Numer. Methods Eng., 87, 1-5, 15-47 (2011) · Zbl 1242.74097
[37] Hughes, T. J., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2012), Courier Corporation
[38] Cottrell, J. A.; Hughes, T. J.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009
[39] Scott, M. A.; Borden, M. J.; Verhoosel, C. V.; Sederberg, T. W.; Hughes, T. J., Isogeometric finite element data structures based on Bézier extraction of t-splines, Int. J. Numer. Methods Eng., 88, 2, 126-156 (2011) · Zbl 1242.65243
[40] Hosder, S.; Walters, R.; Balch, M., Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables, (48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2007)), 1939
[41] Hosder, S.; Walters, R., Non-intrusive polynomial chaos methods for uncertainty quantification in fluid dynamics, (48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (2010)), 129
[42] Burgers, J. M., A mathematical model illustrating the theory of turbulence, (Advances in Applied Mechanics, vol. 1 (1948), Elsevier), 171-199
[43] Ahmed, S. E.; San, O.; Rasheed, A.; Iliescu, T., A long short-term memory embedding for hybrid uplifted reduced order models, Physica D, 409, Article 132471 pp. (2020) · Zbl 1505.76071
[44] San, O.; Maulik, R.; Ahmed, M., An artificial neural network framework for reduced order modeling of transient flows, Commun. Nonlinear Sci. Numer. Simul., 77, 271-287 (2019) · Zbl 1479.76082
[45] Maleewong, M.; Sirisup, S., On-line and off-line pod assisted projective integral for non-linear problems: a case study with Burgers’ equation, Int. J. Math. Comput. Phys. Electr. Comput. Eng., 5, 7, 984-992 (2011)
[46] Delmas, V.; Soulaïmani, A., Multi-gpu implementation of a time-explicit finite volume solver for the shallow-water equations using cuda and a cuda-aware version of openmpi (2020), arXiv preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.