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An efficient adjoint computation for flow control problems

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  • pg 1
									             An efficient adjoint computation for
                   flow control problems
                              Denise Holfeld1 , Andrea Walther2


         1   Institut für Wissenschaftliches Rechnen, TU Dresden
               2 Institut für Mathematik, Universität Paderborn




                     9th AD Workshop 2009, Sophia-Antipolis       SFB 609



Denise Holfeld (TU Dresden)                                                 1 / 17
                                          Outline


1   Motivation and examples

2   Semtex

3   Adjoint calculation

4   Numerical results

5   Outlook



    Denise Holfeld (TU Dresden)   Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   2 / 17
                                Motivation and examples



                                            Motivation
DFG SFB 609:
Electromagnetic flow control for laminar, transient, and turbulent flows




  Denise Holfeld (TU Dresden)           Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   3 / 17
                                Motivation and examples



                                            Motivation
DFG SFB 609:
Electromagnetic flow control for laminar, transient, and turbulent flows




         Simulation? Optimal control? Coupling with experiments?



  Denise Holfeld (TU Dresden)           Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   3 / 17
                                Motivation and examples




                                            Motivation
coupling advanced methods of physical numerical models and
methods of mathematical control
        improvement of flow qualities

example I




              [T. Albrecht, J. Stiller, Proc. 3rd iTi Conf. Turbulence, 2008]


  Denise Holfeld (TU Dresden)           Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   4 / 17
                                                                         Details of the EM force
                                    Transition delay using EM forces
                                        Control examples
                               Motivation and of separated flows          The process of transition to turbulence
                                                                         Optimization problem
                                            Other fields of research




           Transition to turbulence in a flat plate BL
                  linear (2D)    non-linear (3D)                                                                           δ1
                                                                       Λ-vortices                   Ω-vortices
           U∞       Tollmien-Schlichting waves




       0         Recritical                                                                                        Retransition   Re
             stable unstable                                           Blasius
                                                                       Rec=519
                                                                                                            laminar turbulent
   0                                                                                                         cf ,lam cf ,turb
                                                                y
                     u(t) - umean




                                                                               u
example II


 Denise Holfeld (TU Dresden)              Adjoint computation for flow control                           Sofia-Antipolis, 26.11.2009     5 / 17
                                                                         Details of the EM force
                                    Transition delay using EM forces
                                        Control examples
                               Motivation and of separated flows          The process of transition to turbulence
                                                                         Optimization problem
                                            Other fields of research




           Transition to turbulence in a flat plate BL
                  linear (2D)    non-linear (3D)                                                                           δ1
                                                                       Λ-vortices                   Ω-vortices
           U∞       Tollmien-Schlichting waves




       0         Recritical                                                                                        Retransition   Re
                                                                       Blasius
                                                                       Rec=519
                                                                       exponential                          laminar turbulent
   0                                                                   Rec=47120                             cf ,lam cf ,turb
                                                                y
                     u(t) - umean




                                                                               u
example II


 Denise Holfeld (TU Dresden)              Adjoint computation for flow control                           Sofia-Antipolis, 26.11.2009     6 / 17
                                        Semtex




                                        Semtex

• numerical simulation of fluid dynamics
• solve time dependent Navier-Stokes equations

                              δu
                                 +    p=ν             u − (u ·         )u + q
                              δt
• spectral element method
    • high order finite element technique
    • combination of geometric flexibility of finite elements and
      high accuracy of spectral methods
• using:
    • parametrically mapped quadrilateral elements
    • Gauss-Lobatto-Legendre ’nodal’ shape function basis
    • continuous Galerkin projection


Denise Holfeld (TU Dresden)      Adjoint computation for flow control      Sofia-Antipolis, 26.11.2009   7 / 17
                                         Semtex



example I




                [T. Albrecht, J. Stiller, Proc. 3rd iTi Conf. Turbulence, 2008]

  • geometric flexibility: grid with 1757 square elements
  • spectral accuracy: within each element 12 x 12 = 144 GLL
                                  grid points used for Lagrange interpolation
 Denise Holfeld (TU Dresden)      Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   8 / 17
                                              Semtex



                            Modifications in Semtex
definition of a functional

  example I: to minimize aerodynamic resistance
      minimize the momentum thickness using force q

                                                                            yup




                                                                            ydown



                           N    tn      yup
                                               u      u                             2
         J(x, q) =                                (1 − )dy dt + µ q                     → min!
                                               u0     u0
                          n=1 tn−1    ydown

         x = (u, v ) u0 inflow velocity in horizontal direction
  Denise Holfeld (TU Dresden)        Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   9 / 17
                                       Semtex




Flow control
optimization by using body forces
        high dimensional optimal control problem
        calculate gradients by using adjoint informations
         to reduce the effort

Adjoint calculation
different techniques: algorithmic differentiation
                      discretization of continuous adjoint
                      hand coded
                      ...



  Denise Holfeld (TU Dresden)   Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   10 / 17
                              Adjoint calculation




         Gradient Calculation (Time Integration)
1   Forward integration (flow simulation)
    xi+1 = Fi (xi , qi ), i = 1, . . . , l
    with xi ∈ Rn state, qi control at time ti
    number of time steps l known

2   Evaluation of target function J(x, q)

3   Reverse integration:
           ¯ ¯
    xi−1 = Fi (xi , xi−1 , qi−1 , qi ), i = l, . . . , 1
    ¯
4   Gradient calculation


Denise Holfeld (TU Dresden)      Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   11 / 17
                                Adjoint calculation




                              Structure exploitation

• target function
             modified often
                 ADOL-C
• Navier-Stokes solver/SEMTEX
             constant parts
                  hand coded adjoints
• for nonlinear parts
             storage of states from forward integration
                   optimal checkpointing




Denise Holfeld (TU Dresden)        Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   12 / 17
                                 Numerical results




                                Numerical results
example I: simulation of flows with Re = 1000:
  • spectral element mesh with 1757 elements
  • polynomial order N = 12

        acceptable level of divergence errors

example II: 620 elements

                                                              example I            example II
execution time of one time step:                                1.80s               0.237s
execution time of one adjoint time step:                          3.34s              0.595s
!including the time step
                                                 ratio:          1.85                  2.51
  Denise Holfeld (TU Dresden)      Adjoint computation for flow control    Sofia-Antipolis, 26.11.2009   13 / 17
                                                  Numerical results


Comparison of flow simulation runtime with adjoint calculation
with 500 steps for 20, 30, 40, 50 and 60 checkpoints

                                          4.5


                                          4.4


                                          4.3
                      runtime grad(f)/f

                                          4.2


                                          4.1


                                           4


                                          3.9


                                          3.8


                                          3.7


                                          3.6


                                          3.5
                                             10   20           30         40         50      60           70

                                                                # checkpoints



  Denise Holfeld (TU Dresden)                          Adjoint computation for flow control        Sofia-Antipolis, 26.11.2009   14 / 17
                                                 Numerical results



for stationary flow: comparison of the number of iterations by using
different numbers of timesteps for adjoint calculation\optimization
                                      90


                                      80


                                      70


                                      60
                       # iterations


                                      50


                                      40


                                      30


                                      20


                                      10


                                       0
                                                 50              100           150          200          250

                                           # steps for adjoint calc. and optimization
necessary time for optimization over 10 timesteps: 98s
                                   100 timesteps: 75s
  Denise Holfeld (TU Dresden)                         Adjoint computation for flow control         Sofia-Antipolis, 26.11.2009   15 / 17
                                    Numerical results


values of |grad(f )|, target function and the parameter (penetration
depth) during optimization over 15000 timesteps
                       5
                                                                                 |grad(f)|
                                                                                 target func
                      4.5                                                        parameter


                       4


                      3.5


                       3


                      2.5


                       2


                      1.5


                       1


                      0.5


                       0
                            0   2            4            6            8    10                 12

                                                 iteration step

  Denise Holfeld (TU Dresden)         Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   16 / 17
                                       Outlook




                                       Outlook
Goal: adjoint-based control of turbulent flows
     using electromagnetic fields
Applications: aerodynamics, crystal growth
Done:
  • extension of a target function for optimization
  • inclusion of hand coded adjoints and AD applications
  • coupling with optimal checkpointing
  • coupling with gradient based optimzation

Then:
  • optimal control and experiments
  • development of optimization strategies for turbulent flows

  Denise Holfeld (TU Dresden)   Adjoint computation for flow control   Sofia-Antipolis, 26.11.2009   17 / 17

								
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