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FINITE ELEMENT COMPUTATIONS OF TIME RELAXATION FOR FLOW ENSEMBLES

Authors:

Monika Neda1

, Jiajia Waters2

1Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154‐4020, USA
2Los Alamos National Laboratory, T‐3 Solid Mechanics and Fluid Dynamics, Los Alamos, New Mexico, USA

Received: 17 May 2016
Accepted: 15 June 2016
Available online: 30 June 2016

Abstract:

Numerical continuous finite element computations of a fluid flow modelling based on ensemble technique are presented herein. This ensemble algorithm is based on time relaxation. At each time step, it requires the storage of a single coefficient matrix with multiple right‐hands sides which corresponds to each ensemble member. Finite element convergence results for the time relaxation ensemble algorithm are presented and tested in a numerical experiment. This numerical experiment supports the theoretical finite element results. The time relaxation efficiency was shown in the second cavity benchmark problem.

Keywords:

Fluid dynamics, partial differential equations, finite element, simulation

References:

[1] N. Jiang, W. Layton, An algorithm for fast calculation of flow ensembles, International Journal for Uncertainty Quantification, 4 (2014), 273‐301.
[2] M. Leutbecher, T. N. Palmer, Ensemble forecasting, J. Comp. Phys. 227 (2008), 3515‐ 3539.
[3] C. Webster, G. Zhang, M. Gunzburger, An adaptive wavelet stochastic collocation method for irregular solutions of Stochastic Partial Diferential Equations, ORNL Tech. Rep., ORNL/TM‐ 2012/186, 2012.
[4] O. Le Matre, O. Knio, H. Najm, R. Ghanem, A stochastic projection method for fluid flow I. Basic formulation, J. Comput. Phys. 173 (2001), 481‐511.
[5] P. Harasim, On the worst scenario method: Application to a quasilinear elliptic 2D‐ problem with uncertain coeficients, Appl. Math., 56(5) (2011), 459‐480.
[6] L. G. Stanley, D. L. Stewart, Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods, SIAM, Philadelphia, 2002.
[7] A. Saltelli, K. Chan, E. Scott eds., Sensitivity Analysis, Chichester, NY, Wiley, 2000.
[8] W. Oberkampf, J. Helton, Evidence theory for engineering applications, In Engineering Design Reliability Handook, ed. E Nikolaidis, D Ghiocel, S Singhal, pp. 130. Boca Raton, FL, CRC, 2005.
[9] Z. Toth, E. Kalnay, Ensemble Forecasting at NMC: The Generation of Perturbations, Bull. Amer. Meteor. Soc., 74(12), 1993, 2317‐2330.
[10] W. J. Martin, M. Xue, Initial condition sensitivity analysis of mesoscale forecast using very‐large ensembles, Mon. Wea. Rev., 134 (2006), 192‐207.
[11] A. Takhirov, M. Neda, J. Waters, Time relaxation algorithm for ow ensembles, Numerical Methods for Partial Diffierential Equations, 32 (2016), 757‐777.
[12] N.A. Adams, S. Stolz, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids, 2 (1999), 1699‐1701.
[13] N.A. Adams, S. Stolz, Deconvolution methods for subgrid‐scale approximation in large eddy simulation, in: R.T. Edwards (Ed.), Modern Simulation Strategies for Turbulent Flow, Edwards, Philadelphia, 2001, p.21.
[14] N.A. Adams, S. Stolz, L. Kleiser, Approximate deconvolution model for large eddy simulation with application to wall‐bounded ows, Phys. Fluids, 13(4) (2001), 997‐1015.
[15] W. Layton, M. Neda, Truncation of scales by time relaxation, J. Math. Anal. Appl., 325 (2007), 788‐807.
[16] C. D. Pruett, T. B. Gatski, C. E. Grosch, and W. D. Thacker, The temporally filtered Navier‐ Stokes equations: Properties of the residual stress, Phys. Fluids, 15(8) (2003), 2127‐2140.
[17] S. Dee, D. Hannach, M. Neda, E. Nikonova, Numerical Analysis and Computations of a High Accuracy Time Relaxation Fluid Flow Model, Int. J. Comp. Math., 89 (2012), 2353‐ 2373.
[18] M. Neda, Discontinuous Time Relaxation Method for the Time Dependent Navier‐ Stokes Equations, Adv. Numer. Anal., (2010), Article ID 419021, doi:10.1155/2010/419021.
[19] L. Berselli, T. Iliescu, W. Layton, Mathematics of large eddy simulation of turbulent ows, Springer, 2006.
[20] W. Layton, L. Rebholz, Approximate Deconvolution Models of Turbulence, Springer, 2012.
[21] V. Girault, P. A. Raviart, Finite element methods for Navier‐Stokes equations: theory and algorithms, Springer, 1986.
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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0)

Volume 10
Number 3
September 2025

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Last Edition

Volume 10
Number 3
September 2025

How to Cite

M. Neda, J. Waters, Finite Element Computations of Time Relaxation for Flow Ensembles. Applied Engineering Letters, 1(2), 2016: 51-56.

More Citation Formats

Neda, M., & Waters, J. (2016). Finite Element Computations of Time Relaxation for Flow Ensembles. Applied Engineering Letters, 1(2), 51-56.

Neda, Monika, and Jiajia Waters. “Finite Element Computations of Time Relaxation for Flow Ensembles.“ Applied Engineering Letters, vol. 1, no. 2, 2016, pp. 51-56.

Neda, Monika, and Jiajia Waters. 2016. “Finite Element Computations of Time Relaxation for Flow Ensembles.“ Applied Engineering Letters, 1 (2): 51-56.

Neda, M. and Waters, J. (2016). Finite Element Computations of Time Relaxation for Flow Ensembles. Applied Engineering Letters, 1(2), pp. 51-56.

FINITE ELEMENT COMPUTATIONS OF TIME RELAXATION FOR FLOW ENSEMBLES

Authors:

Monika Neda1

, Jiajia Waters2

1Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154‐4020, USA
2Los Alamos National Laboratory, T‐3 Solid Mechanics and Fluid Dynamics, Los Alamos, New Mexico, USA

Received: 17.05.2016.
Accepted:15.06.2016.
Available online: 30.06.2016.

Abstract:

Numerical continuous finite element computations of a fluid flow modelling based on ensemble technique are presented herein. This ensemble algorithm is based on time relaxation. At each time step, it requires the storage of a single coefficient matrix with multiple right‐hands sides which corresponds to each ensemble member. Finite element convergence results for the time relaxation ensemble algorithm are presented and tested in a numerical experiment. This numerical experiment supports the theoretical finite element results. The time relaxation efficiency was shown in the second cavity benchmark problem.

Keywords:

Fluid dynamics, partial differential equations, finite element, simulation

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0)

Volume 10
Number 3
September 2025

Loading

Last Edition

Volume 10
Number 3
September 2025