Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T02:25:06.061Z Has data issue: false hasContentIssue false
  • English
  • Français

Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

Published online by Cambridge University Press:  15 June 2009

ALEXANDER HAY ,
JEFFREY T. BORGGAARD  and
DOMINIQUE PELLETIER
ALEXANDER HAY*
Affiliation:
Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
JEFFREY T. BORGGAARD
Affiliation:
Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
DOMINIQUE PELLETIER
Affiliation:
Département de Génie Mécanique, Ecole Polytechnique de Montréal, Montréal, QC H3C3A7, Canada
*
†Email address for correspondence: hay@vt.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 629 , 25 June 2009 , pp. 41 - 72
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Antoulas, A. C. 2005 Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM.CrossRefGoogle Scholar
Antoulas, A. C., Sorensen, D. C. & Gugercin, S. 2001 A survey of model reduction methods for large-scale systems. Contemp. Math. 280, 193–219.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
Bangia, A. K., Batcho, P. F., Kevrekidis, I. G. & Karniadakis, G. E. 1997 Unsteady two-dimensional flows in complex geometries: comparative bifurcation studies with global eigenfunction expansions. SIAM J. Sci. Comput. 18, 775–805.CrossRefGoogle Scholar
Bergmann, M., Cordier, L. & Brancher, J.-P. 2005 Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model. Phys. Fluids 17 (9), 097101:1–21.CrossRefGoogle Scholar
Borggaard, J. & Burns, J. 1997 A PDE sensitivity equation method for optimal aerodynamic design. A PDE sensitivity equation method for optimal aerodynamic design 136 (2), 367–384.Google Scholar
Borggaard, J., Hay, A. & Pelletier, D. 2007 Interval-based reduced-order models for unsteady fluid flow. Interval-based reduced-order models for unsteady fluid flow 4 (3–4), 353–367.Google Scholar
Couplet, M., Basdevant, C. & Sagaut, P. 2005 Calibrated reduced-order POD-Galerkin system for fluid flow modeling. Calibrated reduced-order POD-Galerkin system for fluid flow modeling 207 (1), 192–220.Google Scholar
Davis, R. W. & Moore, E. F. 1982 A numerical study of vortex shedding from rectangles. J. Fluid Mech. 116, 475–506.CrossRefGoogle Scholar
Deane, E. A., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3 (10), 2337–2354.CrossRefGoogle Scholar
Fahl, M. 2000 Trust-region methods for flow control based on reduced order modelling. PhD thesis, Universität Trier, Germany.Google Scholar
Fox, R. L. & Kapoor, M. P. 1968 Rates of change of eigenvalues and eigenvectors. AIAA J. 6 (12), 2426–2429.CrossRefGoogle Scholar
Franke, R., Rodi, W. & Schoaconung, B. 1990 Numerical calculation of laminar vortex-shedding flow past cylinders. J. Wind Engng Ind. Aerodyn. 35, 237–257.CrossRefGoogle Scholar
Galletti, B., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161–170.CrossRefGoogle Scholar
Ganapathysubramanian, S. & Zabaras, N. 2004 Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Comput. Meth. Appl. Mech. Engng 193, 5017–5034.CrossRefGoogle Scholar
Graham, W. R., Peraire, J. & Tang, K. T. 1999 a Optimal control of vortex shedding using low order models. Part 1. Open-loop model development. Intl J. Numer. Methods Engng 44 (7), 945–972.3.0.CO;2-F>CrossRefGoogle Scholar
Graham, W. R., Peraire, J. & Tang, K. T. 1999 b Optimal control of vortex shedding using low order models. Part 2. Model-based control. Intl J. Numer. Methods Engng 44 (7), 973–990.3.0.CO;2-F>CrossRefGoogle Scholar
Hay, A., Borggaard, J. & Pelletier, D. 2008 On the use of sensitivity analysis to improve reduced-order models. In 4th AIAA Flow Control Conference, Seattle, Washington, AIAA-2008–4192.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University.CrossRefGoogle Scholar
Hotelling, H. 1933 Analysis of a complex of statistical variables with principal components. J. Educat. Psych. 24, 417–441.CrossRefGoogle Scholar
Hristova, H., Etienne, S., Pelletier, D. & Borggaard, J. 2006 A continuous sensitivity equation method for time-dependent incompressible laminar flows. A continuous sensitivity equation method for time-dependent incompressible laminar flows 50 (7), 817–844.Google Scholar
Ilinca, F., Pelletier, D. & Hay, A. 2008 First- and second-order sensitivity equation methods for value and shape parameters. First- and second-order sensitivity equation methods for value and shape parameters 57 (9), 1349–1370.Google Scholar
Ito, K. & Ravindran, S. 1996 Reduced basis method for flow control. Tech Rep. CRSC-TR96-25, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC.Google Scholar
Karhunen, K. 1946 Zur Spektraltheorie stochastischer Prozesse. Zur Spektraltheorie stochastischer Prozesse 37.Google Scholar
Kunisch, K. & Volkwein, S. 1999 Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theor. Appl. 102, 345–371.CrossRefGoogle Scholar
Kunisch, K., Volkwein, S. & Xie, L. 2004 HJB-POD-based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3 (4), 701–722.CrossRefGoogle Scholar
Lancaster, P. 1964 On eigenvalues of matrices dependent on a parameter. Numer. Math. 6, 377–387.CrossRefGoogle Scholar
Lehmann, O., Luchtenburg, D. M., Noack, B. R., King, R., Morzyński, M. & Tadmor, G. 2005 Wake stabilization using POD Galerkin models with interpolated modes. In 44th IEEE Conference on Decision and Control (CDC) and European Control Conference (ECC), Sevilla, Spain.Google Scholar
Lieu, T., Farhat, C. & Lesoinne, M. 2006 Reduced-order fluid/structure modeling of a complete aircraft configuration. Comput. Meth. Appl. Mech. Engng 195, 5730–5742.CrossRefGoogle Scholar
Loève, M. 1955 Probability Theory. Van Nostrand.Google Scholar
Lorenz, E. N. 1956 Empirical orthogonal functions and statistical weather prediction. Tech Rep. M.I.T.Google Scholar
Ma, X. & Karniadakis, G. E. 2002 A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190.CrossRefGoogle Scholar
Morzyński, M., Afanasiev, K. & Thiele, F. 1999 Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Meth. Appl. Mech. Engng 169, 161–176.CrossRefGoogle Scholar
Morzyński, M., Stankiewicz, W., Noack, B. R., King, R., Thiele, F. & Tadmor, G. 2007 Continuous mode interpolation for control-oriented models of fluid flows. In Active Flow Control (ed. King, R.), pp. 260–278. Springer-Verlag.CrossRefGoogle Scholar
Murthy, D. V. & Haftka, R. T. 1988 Derivatives of eigenvalues and eigenvectors of a general complex matrix. Intl J. Numer. Meth. Engng 26, 293–311.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 a A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 b A low-dimensional Galerkin method for the three-dimensional flow around a ciruclar cylinder. Phys. Fluids 6, 124–143.CrossRefGoogle Scholar
Okajima, A. 1982 Strouhal number of rectangular cylinders. J. Fluid Mech. 123, 379–398.CrossRefGoogle Scholar
Pelletier, D., Hay, A., Etienne, S. & Borggaard, J. 2008 The sensitivity equation method in fluid mechanics. The sensitivity equation method in fluid mechanics 17 (1–2), 31–61.Google Scholar
Peterson, J. S. 1989 The reduced basis method for incompressible viscous flow calculations. The reduced basis method for incompressible viscous flow calculations 10 (4), 777–786.Google Scholar
Rowley, C. W. & Williams, D. R. 2006 Dynamics and control of high-Reynolds-number flow over open cavities. Annu. Rev. Fluid Mech. 38, 251–276.CrossRefGoogle Scholar
Saha, A. K., Biswas, G. & Muralidhar, K. 2003 Three-dimensional study of flow past a square cylinder at low Reynolds numbers. Intl J. Heat Fluid Flow 24, 54–66.CrossRefGoogle Scholar
Seyranian, A. P., Lund, E. & Olhoff, N. 1994 Multiple eigenvalues in structural optimization problems. Struct. Optim. 8, 207–227.CrossRefGoogle Scholar
Sirisup, S. & Karniadakis, G. E. 2004 A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194, 92–116.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I. Coherent structures. Turbulence and the dynamics of coherent structures. Part I. Coherent structures 45 (3), 561–571.Google Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1997 Numerical simulation of unsteady low-Reynolds number flow around rectangular cylinders at incidence. J. Wind Engng Ind. Aerodyn. 69–71, 189–201.CrossRefGoogle Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1998 Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition. Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition 26 (1), 39–56.Google Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1999 Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers. Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers 11 (2), 189–201.Google Scholar