Computational Mechanics

, Volume 43, Issue 6, pp 839–846 | Cite as

Optimal control of thermal fluid flow using automatic differentiation

  • Ayako KamikawaEmail author
  • Mutsuto Kawahara
Original Paper


The aim of this study is to present a method for numerical optimal control of thermal fluid flow using automatic differentiation (AD). For the optimal control, governing equations are required. The optimal controls that have been previously presented by the present authors’ research group are based on the Boussinesq equations. However, because the numerical results of these equations are not satisfactory, the compressible Navier–Stokes equations are employed in this study. The objective is to determine whether or not the temperature at the objective points can be kept constant by imposing boundary conditions and by controlling the temperature at the control points. To measure the difference between the computed and target temperatures, the square sum of these values is used. The objective points are located at the center of the computational domain while the control points are at the bottom of the computational domain. The weighted gradient method that employs AD for efficiently calculating the gradient is used for the minimization. By using numerical computations, we show the validity of the present method.


Optimal control Automatic differentiation Finite element method Thermal fluid flow Compressible flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Maruoka A, Marin M, Kawahara M (1998) Optimal Control in Navier–Stokes Equations. Int J Comp Fluid Dyn, vol 9, No. 3–4, Sp. No. 3, pp 313–322Google Scholar
  2. 2.
    Marin M, Kawahara M (1998) Optimal control of vorticity in Rayleigh–Bernard convection by finite element method. Commun Numer Methods Eng 1(1): 9–22CrossRefMathSciNetGoogle Scholar
  3. 3.
    Hatanaka K, Kawahara M (1991) A fractional step finite element method for conductive–convective heat transfer problems. Int J Numer Methods Heat Fluid Flow 1: 77–94CrossRefGoogle Scholar
  4. 4.
    Burns JA, King BB, Rubio D (1998) Feedback control of a thermal fluid using state estimation. Int J Comp Fluid Dyn 11(1–2): 93–112zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ravindran SS (1998) Numerical solutions of optimal control for thermally convective flows. Int J Numer Methods Fluid 25(2): 205–223CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ito K, Ravindran SS (1998) Optimal control of thermally convected fluid flows. SIAM J Sci Comp 19(6): 1847–1869zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Okumura H, Kawahara M (2003) A new stable bubble element for incompressible fluid flow based on a mixed Petrov-Glerkin finite element formulation. Int J Comp Fluid Dyn 17(4): 275–282zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Okumura H, Kawahara M (2000) Shape optimization of body located in incompressible Navier–Stokes flow based on optimal control theory. Comp Methods Eng Sci 1(2): 71–77MathSciNetGoogle Scholar
  9. 9.
    Okumura H, Kawahara M (2000) Shape optimization for the Navier–Stokes equations based on optimal control theory. Eur Cong Comp Methods Appl Sci Eng, ECCOMAS, pp 1–12Google Scholar
  10. 10.
    Matsumoto J, Kawahara M (2001) Shape identification for fluid–structure interaction problem using improved bubble element. Int J Comp Fluid Dyn 15(1): 33–45zbMATHGoogle Scholar
  11. 11.
    Hughes TJR, Tezduyar TE (1984) Finite element methods for first order hyperbolic systems with particular emphasis on the compressible Euler equations. Comp Methods Appl Mech Eng 45: 217–284zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Le Beau GJ, Ray SE, Aliabadi SK, Tezduyar TE (1993) SUPG finite element computation of compressible flow with the entropy and conservation variables formulations. Comp Methods Appl Mech Eng 104: 397–422zbMATHGoogle Scholar
  13. 13.
    Tezduyar TE (2004) Finite element method for fluid dynamics with moving boundaries and interfaces, Chap.17. In: Stein E, De Borset R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3, Fluids. Wiley, New YorkGoogle Scholar
  14. 14.
    Tezduyar TE, Senga M (2006) Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Comp Methods Appl Mech Eng 195: 1621–1632zbMATHMathSciNetGoogle Scholar
  15. 15.
    Pironneau O (2001) Automatic differentiation and domain decomposition for optimal shape design. International series, mathematical sciences and applications, vol 16. Computational methods for control applications, pp 167–178Google Scholar
  16. 16.
    Takahashi Y, Kawahara M (2005) Optimal control of fluid force around a circular cylinder located in incompressible viscous flow using automatic differentiation. Int J Comp Fluid Dyn 19(1): 31–36zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Civil EngineeringChuo UniversityTokyoJapan

Personalised recommendations