Adaptive Aerodynamic Design Optimization for Navier-Stokes Using Shape Derivatives with Discontinuous Galerkin Methods

  • L. Kaland
  • M. SonntagEmail author
  • N. R. Gauger
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 36)


We state and analyze one-shot optimization methods in a function space setting for optimal control problems, for which the state equation is given in terms of a fixed-point equation. Further, we concentrate on the application of a design optimization problem incorporating the solution of the compressible Navier-Stokes equations using a discontinuous Galerkin method. For the given primal fixed-point solver an appropriate adjoint solver is constructed. For the following design update we compute the shape derivative analytically based on the weak formulation of the governing equations. The primal, adjoint and design updates are performed in a one-shot manner, i.e., the corresponding equations are not fully solved, instead only a few iteration steps are performed. Finally, we add an additional adaptive step. During the optimization routine we refine or coarsen the grid to obtain a better accuracy.


One-shot method Function space analysis Design optimization Adaptivity Shape derivative 



The authors gratefully acknowledge the support of BMBF Grant 03MS632D/ “DGHPOPT”.


  1. 1.
    Biros G, Ghattas O (2005) Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: the Krylov-Schur solver. SIAM J Sci Comput 27:687–713CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ziems JC, Ulbrich S (2011) Adaptive multilevel inexact SQP methods for PDE-constrained optimization. SIAM J Optim 21:1–40CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ito K, Kunisch K, Schulz V, Gherman I (2010) Approximate nullspace iterations for KKT systems. SIAM J Matrix Anal A 31:1835–1847CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Hamdi A, Griewank A (2010) Properties of an augmented Lagrangian for design optimization. Optim Methods Softw 25:645–664CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gauger NR, Griewank A, Hamdi A, Kratzenstein C, Özkaya E, Slawig T (2012) Automated extension of fixed point PDE solvers for optimal design with bounded retardation. In: Leugering G et al (eds) Constrained optimization and optimal control for partial differential equations. Springer, Basel, pp 99–122Google Scholar
  6. 6.
    Hazra SB, Schulz V, Brezillon J, Gauger NR (2005) Aerodynamic shape optimization using simultaneous pseudo-timestepping. J Comput Phys 204:46–64CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kaland L (2013) The one-shot method: function space analysis and algorithmic extension by adaptivity. PhD thesis, RWTH Aachen University, Aachen, Germany, in preparationGoogle Scholar
  8. 8.
    Kaland L, De Los Reyes JC, Gauger NR (2013) One shot methods in function space for PDE-constrained optimal control problems. Optim Methods Softw doi: 10.1080/10556788.2013.774397
  9. 9.
    Hartmann R, Held J, Leicht T, Prill F (2010) Discontinuous Galerkin methods for computational aerodynamics 3D adaptive flow simulation with the DLR PADGE code. Aerosp Sci Technol 14:512–519CrossRefGoogle Scholar
  10. 10.
    Kuhn M (2011) Herleitung von Formableitungen aerodynamischer Zielgrößen für die laminaren kompressiblen Navier-Stokes Gleichungen und deren Implementierung in diskontinuierlichen Galerkin-Verfahren. Masters thesis, Humboldt-Universität zu Berlin, Berlin, GermanyGoogle Scholar
  11. 11.
    Hartmann R (2008) Numerical analysis of higher order discontinuous Galerkin finite element methods. In: Deconinck H (ed) VKI LS 2008-08: CFD— ADIGMA course on very high order discretization methods, 13–17 October 2008, Von Karman Institute for Fluid Dynamics, Rhode Saint Genèse, BelgiumGoogle Scholar
  12. 12.
    Sokolowski J, Zolésio J-P (1992) Introduction to shape optimization. Springer, BerlinCrossRefzbMATHGoogle Scholar
  13. 13.
    Delfour MC, Zolésio J-P (2001) Shapes and geometries. Society for Industrial and Applied Mathematics, PhiladelphiazbMATHGoogle Scholar
  14. 14.
    Schmidt S (2010) Efficient large scale aerodynamic design based on shape calculus. PhD thesis, University of Trier, Trier, GermanyGoogle Scholar
  15. 15.
    Castro C, Lozano C, Palacios F, Zuazua E (2007) Systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids. AIAA 45:2125–2139CrossRefGoogle Scholar
  16. 16.
    Ito K, Kunisch K, Gunther G, Peichl H (2008) Variational approach to shape derivatives. ESAIM Control Optim Calc Var 14:517–539CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Sonntag M, Gauger NR, Schmidt S (2013) Shape derivatives for the compressible Navier-Stokes equations in variational form. J Comput Appl Math (submitted)Google Scholar
  18. 18.
    Hartmann R (2005) Discontinuous Galerkin methods for compressible flows: higherorder accuracy, error estimation and adaptivity. In: Deconinck H, Ricchiuto M (eds.), VKI LS 2006-01:CFD-Higher Order Discretization Methods, Nov. 14-18, 2005, Von KarmanInstitute for Fluid Dynamics, Rhode Saint Genèse, BelgiumGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Computational Mathematics Group, Department of Mathematics and CCESRWTH Aachen UniversityAachenGermany

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