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Sensitivity Analysis and Computations of the Time Relaxation Model

Published online by Cambridge University Press:  09 January 2015

Monika Neda ,
Fran Pahlevani  and
Jiajia Waters
Monika Neda
Affiliation:
Department of Mathematics, University of Nevada, Las Vegas, NV 89154, USA
Fran Pahlevani*
Affiliation:
Department of Mathematics, Penn State University-Abington College, PA 19001, USA
Jiajia Waters
Affiliation:
Department of Mathematics, University of Nevada, Las Vegas, NV 89154, USA
*
*Email:monika.neda@unlv.edu(M. Neda), fxp10@psu.edu(F. Pahlevani), wangj16@unlv.nevada.edu(J. Waters)
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Abstract

This paper presents a numerical study of the sensitivity of a fluid model known as time relaxation model with respect to variations of the time relaxation coefficient χ. The sensitivity analysis of this model is utilized by the sensitivity equation method and uses the finite element method along with Crank Nicolson method in the fully discretization of the partial differential equations. We present a test case in support of the sensitivity convergence and also provide a numerical comparison between two different strategies of computing the sensitivity, sensitivity equation method and forward finite differences.

Keywords

65M6065M1276D99Sensitivity analysistime relaxation modelNavier-Stokes equationsfinite element method

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Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 1 , February 2015 , pp. 89 - 115
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Adams, N.A. and Stolz, S., Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, 2001.Google Scholar
[2]Adams, N.A. and Stolz, S., A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., 178 (2001), pp. 391–426.Google Scholar
[3]Anitescu, M. and Layton, W.J., Sensitivities in large eddy simulation and improved estimates of turbulent flow junctionals, J. Comput. Phys., 178 (2001), pp. 391-426.Google Scholar
[4]Anitescu, M., Pahlevani, F. and Layton, W. J., Implicit for local effects and explicit for nonlocal effects is unconditionally stable, Electronic Transactions of Numerical Analysis, 18 (2004), pp. 174–187.Google Scholar
[5]Borggaard, J. and Burns, J., A sensitivity equation approach to shape optimization in fluid flows, Flow Control, (1995), pp. 49–78.CrossRefGoogle Scholar
[6]Borggaard, J., Burns, J., Cliff, E. and Gunzburger, M., Sensitivity calculations for a 2D, inviscid, supersonic forebody problem, Siam, (1993), pp. 14–24.Google Scholar
[7]Borggaard, J., Pelletier, D. and Turgeon, E., A continuous sensitivity equation method for flows with temperature, Siam, (1993), pp. 14–24.Google Scholar
[8]Borggaard, J., Pelletier, D. and Turgeon, E., A continuous sensitivity equation method for flows with temperature dependent properties, Proceedings of the 8th A-IAA/USAF/NASA/ISSMO Symposium on Multidisciplanary Analysis and Design, 1993, 2481.Google Scholar
[9]Borggaard, J., Pelletier, D. and Turgeon, E., Sensitivity and uncertainty analysis for variable property flows, Proceedings of the 39th AIAA Aerospace Sciences Meeting and Exhibit, 1993, 0140.Google Scholar
[10]Borggaard, J. and Burns, J., A PDE sensitivity equation method for optimal aerodynamic design, J. Comput. Phys., 136 (1997), pp. 366–384.Google Scholar
[11]Borggaard, J. and Verma, A., Solutions to continuous sensitivity equations using automatic differentiation, Siam J. Sci. Comput., 22 (2001), pp. 39–62.CrossRefGoogle Scholar
[12]Borggaard, J. and Verma, A., On efficient solutions to the continuous sensitivity equation using automatic differentiation, Siam J. Sci. Comput., 22(1) (2000), pp. 39–62.CrossRefGoogle Scholar
[13]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994.Google Scholar
[14]De, D.Hannasch, S., Neda, M. and Nikonova, E., Numerical analysis and computations of a higher accuracy time relaxation fluid flow model, Int. J. Comput. Math., (2008), pp. 1–22.Google Scholar
[15]Dunca, A. and Epshteyn, Y., On the Stolz-Adams deconvolution LES model, Siam J. Math. Anal., 2006.CrossRefGoogle Scholar
[16]Eithier, C. and Steinman, D., Exact fully 3d Navier-Stokes solutions for benchmarking, Int. J. Numer. Methods Fluids, 19 (1994), pp. 369–375.CrossRefGoogle Scholar
[17]Ervin, V. J., Layton, W. J. and Neda, Monika, Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Model., 4 (2007), pp. 648–670.Google Scholar
[18]Green, A. E. and Taylor, G. I., Mechanism of the production of small eddies from larger ones, Proceeding of Royal Society A, 158 (1937), pp. 499–521.Google Scholar
[19]Godfrey, A. and Cliff, E., Direct calculation of aerodynamic force derivatives: a sensitivity equation approach, Proceedings of the 36th AIAA Aerospace Sciences Meeting and Exhibit, (1998), 0393.CrossRefGoogle Scholar
[20]Guenanff, R., Non-Stationary Coupling of Navier-Stokes/Euler for the Generation and Radiation of Aerodynamic Noises, PhD thesis, Universite Rennes, Rennes, France, 2004.Google Scholar
[21]Guermond, J., Stabilization of Galerkin approximations of transport equations by subgrid modeling, M2AN, 33 (1999), pp. 1293–1316.CrossRefGoogle Scholar
[22]Gunzburger, M., Perspectives in Flow Control and Optimization, SIAM publication, 2003.Google Scholar
[23]Gunzburger, M., Sensitivities, adjoints and flow optimization, Int. J. Numer. Methods Fluids, 31 (1999), pp. 53–78.Google Scholar
[24]Averick, B., More, J., Bischof, C., Carle, A. and Griewank, A., Computing large sparse Jacobian matrices using automatic differentiation, Siam J. Sci. Comput., 15(2) (1994), pp. 285–294.Google Scholar
[25]Griewank, A., A mathematical view of automatic differentiation, Acta Numer., 12 (2003), pp. 321–398.Google Scholar
[26]Hecht, F., Pironneau, O. and Oshtsuka, K., Software Freefem++, 2003, http://www.freefem++.org.Google Scholar
[27]Heywood, J. G. and Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem, Part IV: error analysis for the second order time discretization, Siam J. Numer. Anal., 2 (1990), pp. 353–384.Google Scholar
[28]Hovland, P., Mohammadi, B. and Bischof, C., Automatic differentiation and Navier-Stokes computations, Comput. Methods Optimal Design Control, 24 (1998), pp. 265–284.Google Scholar
[29]Abdel-Khalik, H., Hovland, P., Lyons, A., Stover, T. and Utke, J., A low rank approach to automatic differentiation, Advances in Automatic Differentiation, 64 (2008), pp. 55–65.Google Scholar
[30]Labovsky, A., Layton, W., Manica, C., Neda, M. and Rebholz, L., The stabilized, extrapolated trapezoidal finite element method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 198 (2009), pp. 958–974.Google Scholar
[31]Layton, W., Introduction to Finite Element Methods for Incompressible, Viscous Flows, Siam publications, 2008.Google Scholar
[32]Layton, W., Manica, C., Neda, M. and Rebholz, L., Numerical analysis of a high accuracy Leray-deconvolution model of turbulence, Numer. Methods Partial Differential Equations, 24 (2008), pp. 555–582.Google Scholar
[33]Layton, W. and Rebholz, L., Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis, Springer Lecture Notes in Mathematics, 2012.Google Scholar
[34]Layton, W. and Neda, M., Truncation of scales by time relaxation, Technical Report (www.math.pitt.edu/techrep/0515.pdf), 2005.Google Scholar
[35]Liang, B. and Mahadevan, S., Error and uncertainty quantification and sensitivity analysis in mechanics computational models, Int. Uncertainty J. Quantification, 2 (2011), pp. 147–161.Google Scholar
[36]Neda, M., Discontinuous time relaxation method for the time dependent Navier-Stokes equations, Adv. Numer. Anal., 419021 (2010), pp. 1–21.Google Scholar
[37]Neda, Monika and Nikonova, Elena, Sensitivity computations of time relaxation model with an application in cavity computation, World Academy of Science, Engineering and Technology, 79 (2011), pp. 542–545.Google Scholar
[38]Neda, M., Pahlevani, F. and Waters, J., Sensitvity computations of the Leray-α model, Contemp. Math. Recent Adv. Sci. Comput. Appl., (2013), 586.Google Scholar
[39]Pahlevani, F., Sensitivity computations of eddy viscosity models with an application in drag computation, Int. J. Numer. Methods Fluids., 52-4 (2006), pp. 381–392.Google Scholar
[40]Pahlevani, F. and Davis, L., Parameter Sensitivity of an eddy viscosity model: analysis, computation and its application to quantifying model reliability, Int. Uncertainty J. Quantification, 3(5) (2013), pp. 397–419.CrossRefGoogle Scholar
[41]Sagaut, P. and Lê, T., Some investigations of the sensitivity of large eddy simulation, Technical Report (Onera), 1997.Google Scholar
[42]Stanley, L. and Stewart, D., Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods, SIAM publications, Philadelphia, 2002.Google Scholar
[43]Taylor, G. I., On decay of vortices in a viscous fluid, Philadelphia Magazine, 46 (1923), pp. 671–674.Google Scholar