Skip to main content
SpringerLink
Log in

The optimal control of unsteady flows with a discrete adjoint method

  • Published:
Optimization and Engineering Aims and scope Submit manuscript

Abstract

This paper presents a general framework to derive a discrete adjoint method for the optimal control of unsteady flows. The complete formulation of a generic time-dependent optimal design problem is introduced and it is outlined how to derive the discrete set of adjoint equations in a general approach. Results are shown that demonstrate the application of the theory to the drag minimization of viscous flow around a rotating cylinder, and to the remote inverse design of laminar flow around the multi-element NLR 7301 configuration at a high angle of attack. In order to reduce the considerable computational costs of unsteady optimization, the use of bigger time steps over transitional or unphysical adjusting periods as well as omitting time steps while recording the flow solution are investigated and are shown to work well in practice.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson WK, Bonhaus DL (1999) Airfroil design on unstructured grids for turbulent flows. AIAA J 37(2):185–191

    Article  Google Scholar 

  • Broyden C (1970) The convergence of a class of double-rank minimization algorithms. J Inst Math Appl 6:76–90

    Article  MATH  MathSciNet  Google Scholar 

  • Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16(5):1190–1208

    Article  MATH  MathSciNet  Google Scholar 

  • Cox J, Brentner K, Rumsey C (1998) Computation of vortex shedding and radiated sound for a circular cylinder: subcritical to transcritical Reynolds numbers. Theor Comput Fluid Dyn 12:233–253

    Article  MATH  Google Scholar 

  • Duta M, Giles M, Campobasso M (2002) The harmonic adjoint approach to unsteady turbo machinery design. Int J Numer Methods Fluids 40:323–332

    Article  MATH  Google Scholar 

  • Fletcher R (1970) A new approach to variable metric algorithms. Comput J 13:317–322

    Article  MATH  Google Scholar 

  • Goldfarb D (1970) A family of variable metric updates derived by variational means. Math Comput 24:23–26

    Article  MATH  MathSciNet  Google Scholar 

  • He J-W, Glowinski R, Metcalfe R, Nordlander A, Periaux J (2000) Active control and drag optimization for flow past a circular cylinder. J Comput Phys 163:83–117

    Article  MATH  MathSciNet  Google Scholar 

  • Henderson R (1997) Nonlinear dynamics and patterns in turbulent wake transition. J Fluid Mech 352:65

    Article  MATH  MathSciNet  Google Scholar 

  • Homescu C, Navon I, Li Z (2002) Suppression of vortex shedding for flow around a circular cylinder using optimal control. Int J Numer Methods Fluids 38:43–69

    Article  MATH  Google Scholar 

  • Isono S, Zingg D (2004) A Runge-Kutta-Newton-Krylov algorithm for fourth-order implicit time marching applied to unsteady flows. In: AIAA, 2004, p. 0433

  • Jameson A (1995) Optimum aerodynamic design using control theory. In: Hafez M, Oshima K (eds) Computational fluid dynamics review. Wiley, New York, pp 495–528

    Google Scholar 

  • Jameson A, Pierce N, Martinelli L (1998) Optimum aerodynamic design using the Navier-Stokes equations. Theor Comput Fluid Dyn 10(1):213–237

    Article  MATH  Google Scholar 

  • Khorrami M, Berkman M, Choudhari M (2000) Unsteady flow computations of a slat with a blunt trailing edge. AIAA J 38(11):2050–2058

    Article  Google Scholar 

  • Lyrintzis AS (2003) Surface integral methods in computational aeroacoustics—from the (CFD) near-field to the (acoustic) far-field. Aeroacoust 2(2):95–128

    Article  Google Scholar 

  • Mani K, Mavriplis DJ (2007) An unsteady discrete adjoint formulation for two-dimensional flow problems with deforming meshes. In: AIAA, 2007, p. 60

  • Nadarajah S, Jameson A (2002) Optimal control of unsteady flows using a time accurate method. In: AIAA, 2002, p. 5436

  • Nadarajah S, Jameson A, Alonso J (2002) An adjoint method for the calculation of remote sensitivities in supersonic flow. In: AIAA, 2002, p. 0261

  • Nadarajah S, Jameson A, Alonso J (2006) An adjoint method for the calculation of remote sensitivities in supersonic flow. Int J Comput Fluid Dyn 20(2):61–74

    Article  MathSciNet  Google Scholar 

  • Nadarajah S, Soucy O, Balloch C (2007) Sonic boom reduction via remote inverse adjoint approach. In: AIAA, 2007, p. 56

  • Nemec M, Zingg D (2002) Newton-Krylov algorithm for aerodynamic design using the Navier-Stokes equations. AIAA J 40(6):1146–1154

    Article  Google Scholar 

  • Nemec M, Zingg D (2004) Multipoint and multi-objective aerodynamic shape optimization. AIAA J 42(6):1057–1065

    Article  Google Scholar 

  • Obayashi S (1997) Aerodynamic optimization with evolutionary algorithms. In: den Braembussche RAV, Manna M (eds) Inverse design and optimization methods. Lecture series 1997-05. Von Karman Institute for Fluid Dynamics, Brussels

    Google Scholar 

  • Pueyo A, Zingg D (1998) Efficient Newton-Krylov solver for aerodynamic computations. AIAA J 36(11):1991–1997

    Article  Google Scholar 

  • Pulliam TH (1986) Efficient solution methods for the Navier-Stokes equations. In: Lecture notes for the Von Karman institute for fluid dynamics lecture series

  • Rumpfkeil M, Zingg D (2007a) A general framework for the optimal control of unsteady flows with applications. In: AIAA, 2007, p. 1128

  • Rumpfkeil M, Zingg D (2007b) The remote inverse shape design of airfoils in unsteady flows. In: Proceedings of the 12th annual CASI aerodynamics symposium, Toronto, 2007, paper 318

  • Saad Y, Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869

    Article  MATH  MathSciNet  Google Scholar 

  • Shanno D (1970) Conditioning of quasi-Newton methods for function minimization. Math Comput 24:647–656

    Article  MathSciNet  Google Scholar 

  • Singer B, Guo Y (2004) Development of computational aeroacoustics tools for airframe noise calculations. Int J Comput Fluid Dyn 18(6):455–469

    Article  MATH  Google Scholar 

  • Singer B, Brentner K, Lockard D (2000) Computational aeroacoustic analysis of slat trailing-edge flow. AIAA J 38(9):1558–1564

    Article  Google Scholar 

  • Tatossian C, Nadarajah S (2007) Optimum shape design of helicopter rotors in forward flight via control theory. In: AIAA, 2007, p. 3951

  • Tokumaru P, Dimotakis P (1991) Rotary oscillation control of a cylinder wake. J Fluid Mech 224:77

    Article  Google Scholar 

  • van den Berg B (1979) Boundary layer measurements on a two-dimensional wing with flap. National Aerospace Lab, NLR TR 79009 U, Amsterdam, Jan

  • van der Vorst H (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13:631–644

    Article  MATH  Google Scholar 

  • von Karman T (1911) Ueber den mechanismus des widerstandes den ein bewegter koerper in einer fluessigkeit erzeugt. Nachr Ges Wiss Goettingen Math Phys Kl. 509–517

  • Williamson C (1989) Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J Fluid Mech 206:579

    Article  Google Scholar 

  • Zhu C, Byrd R, Lu P, Nocedal J (1997) Algorithm 78:L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. ACM Trans Math Softw 23(4):550–560

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Toronto, Institute for Aerospace Studies, Toronto, Ontario, Canada

    Markus P. Rumpfkeil & David W. Zingg

Authors
  1. Markus P. Rumpfkeil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David W. Zingg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus P. Rumpfkeil.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rumpfkeil, M.P., Zingg, D.W. The optimal control of unsteady flows with a discrete adjoint method. Optim Eng 11, 5–22 (2010). https://doi.org/10.1007/s11081-008-9035-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11081-008-9035-5

Keywords

Search

Navigation