1. July 11, 2006 Comparison of Exact and Approximate Adjoint for Aerodynamic Shape Optimization ICCFD 4 July 10-14, 2006, Ghent Giampietro Carpentieri and Michel J.L. van Tooren Delft University of Technology Barry Koren Centre for Mathematics and Computer science, Amsterdam

2. July 11, 2006 Introduction Flow solver Adjoint solver Gradient computation Shape Optimization

3. July 11, 2006 Median-dual discretization Control volume for node i On each control volume/node  ( + BC) N nodes, semi-discrete form  Conserved variables vector Residual vector ( ) DUAL OF THE MESHMESH

4. July 11, 2006 MUSCL reconstruction on each edge Primitive variables reconstruction at edge mid-point: Least-squares or Green-Gauss gradient Numerical flux: Roe’s approximate Riemann solver 2 nd order accuracy: evaluate flux with reconstructed variables Venkatakrishnan’s limiter

5. July 11, 2006 Adjoint equations Shape parameter Functional ( e.g. lift,drag ) State of system ( e.g. residuals) Gradient/sensitivity computed as: : adjoint variables, obtained from adjoint equation:

6. July 11, 2006 Discrete Adjoint for MUSCL scheme Reconstructed left and right states Second order fluxes Three vectors of length E, the number of edges (N is number of nodes) Dummy matrix ( E x N ) Diagonal matrices, differentiated flux ( E x E ) Reconstruction matrices ( N x E ) Chain rule + transposition  Dependence residual vector on conservative variables: To compute consider:

7. July 11, 2006 At each time step linear system is solved iteratively : Time marching flow/adjoint equations Flow equations  Adjoint equations  Backward Euler scheme: Symmetric Gauss-Seidel preconditioner (Matrix-free) is used

8. July 11, 2006 Geometric sensitivities Mesh coordinates Mesh metrics Boundary deformations Limiter vector Gradient vector To compute consider: Coordinates depend on shape parameter: Residuals depend on coordinates: Each term is computed using source code generated by Automatic Differentiation tool Tapenade Chain rule 

9. July 11, 2006 Shape parameterization and mesh deformation Chebyschev polynomials used to parameterize shape of airfoil Mesh deformations computed with spring analogy solved by Jacobi iterations. Boundary deformation implies mesh update is stiffness of edge ij, inverse of edge length

10. July 11, 2006 Shape optimization Objective function, scaled drag coefficient Relative maximum thickness constraint Upper nose radius constraint Lower nose radius constraint Trailing edge angle constraint Lift equality constraint Minimize function with equality and inequality constraints and bounds on variables

11. July 11, 2006 Optimization Algorithms It is necessary to use constrained minimization techniques ! Unconstrained minimization techniques that treat constraints as penalty terms could be used. However, they are ill-conditioned and inaccurate. Two algorithms used in this work: Sequential Quadratic Programming (SQP) Search direction found by solving sub-problem with quadratic objective and linearized constraints. Line search is performed using Lagrangian function. Hessian of Lagrangian updated by BFGS (or other) formulas. Sequential Linear Programming (SLP) Method of centers is used. Hypersphere fitting into linearized design space found by solving Linear Programming sub-problem. Design updated by moving in the center of the sphere. No second order information is collected.

12. July 11, 2006 Exact and approximate Discrete Adjoint Edge-based assembly Exact Discrete Adjoint Approximations

13. July 11, 2006 Edge-based assembly Matrix-vector products with transposed residual Jacobians are assembled directly on edges similarly to the residuals assembly: Two loops on the edges

14. July 11, 2006 Differentiation of flux and reconstruction Roe’s flux Jacobian  Reconstruction matrix  Five matrices (M) come from differentiation of. Reconstruction contribution amounts to two transformation matrices and a diagonal matrix which contains limiter and gradient derivatives. EXACT ADJOINT !

15. July 11, 2006 Approximations Approximation 1  neglect differentiation of limiter Approximation 2  neglect differentiation of Roe matrix Differentiation of limiter is complicated due to construction phase (muldi-dimensional limiter) Differentiation of Roe matrix is very difficult, symbolic differentiation is used. For both approximations, compared to exact adjoint, a relative error of 0.1-2.5% in computed gradient is found.

16. July 11, 2006 Approximations Approximation 3  neglect reconstruction operator Ignoring reconstruction operator makes implementation of adjoint trivial. Only simple loop on edges is required. Error in computed gradient increases to 10-30%. Two loops on the edges One loop on the edges

17. July 11, 2006 Optimization test cases NACA64A410 (SQP) RAE2822 (SQP) NACA0012 (SLP)

18. July 11, 2006 NACA64A410  =0, Mach =0.75 Pressure contours Drag (scaled) vs gradient iterations Initial values Lift Relative max thickness Upper nose radius Lower nose radius Trailing edge angle

19. July 11, 2006 NACA64A410  =0, Mach =0.75 EXACT ( 19 gradients )APPROX 1 ( 17 gradients ) APPROX 2 ( 34 gradients )APPROX 3 ( stalled )

20. July 11, 2006 NACA64A410  =0, Mach =0.75 LiftThicknessNose U.Nose L.TE Angle EXACT -9.9x10 -6 < 10 -8 -0.724 -0.126 -0.350 APPROX 1 4.6x10 -7 < 10 -8 -0.782 -0.172 -0.032 APPROX 2 7.3x10 -6 < 10 -8 -0.693-4.1x10 -7 -0.3 Constraint values show that airfoils satisfy design problem accurately Lift constraint h = -9.9x10 -6 means that final Lift coefficient is: CL = (1 + h) CL 0 = (1 - 9.9x10 -6 ) CL 0 Thickness constraint is always critical

21. July 11, 2006 MESH 2 : 30000 nodesMESH 1 : 12000 nodes NACA64A410  =0, Mach =0.75 CHECK IF MESH 1 IS CAPABLE OF CAPTURING WEAK SHOCK Mach number distributions do not change on second mesh EXACT APPROX 1 APPROX 2

22. July 11, 2006 NACA64A410  =0, Mach =0.75 Mach number Three airfoils have differences in geometry of order 10 -3

23. July 11, 2006 RAE2822  =2, Mach =0.73 Lift Relative max thickness Upper nose radius Lower nose radius Trailing edge angle Pressure contoursDrag (scaled) vs gradient iterations Initial values

24. July 11, 2006 RAE2822  =2, Mach =0.73 EXACT ( 15 gradients )APPROX 1 ( 20 gradients ) APPROX 2 ( 10 gradients )APPROX 3 ( stalled)

25. July 11, 2006 RAE2822  =2, Mach =0.73 LiftThicknessNose U.Nose L.TE Angle EXACT -1.3x10 -6 < 10 -8 -0.416 -3.347 10 -8 APPROX 1 -4.4x10 -6 < 10 -8 -0.481 -1.964 9x10 -8 APPROX 2 -6.2x10 -7 < 10 -8 -0.589 -0.292 < 10 -8 Constraint values show that airfoils satisfy design problem accurately Thickness and trailing edge angle constraints are critical for the 3 airfoils

26. July 11, 2006 RAE2822  =2, Mach =0.73 Differences in geometry of order of 10 -3

27. July 11, 2006 NACA0012  =2, Mach =0.75 Lift Relative max thickness Upper nose radius Lower nose radius Trailing edge angle Pressure contours Drag (scaled) vs gradient iterations

28. July 11, 2006 NACA0012  =2, Mach =0.75 EXACT, APPROX 1, APPROX 2 APPROX 3 LiftThicknessNose U.Nose L.TE Angle EXACT -4.8x10 -6 -0.012 -0.04 -4.17 -0.44 APPROX 1 1.5x10 -6 -0.013 -0.04 -4.12 -0.432 APPROX 2 3.6x10 -6 -0.013 -0.036 -4.22 -0.438

29. July 11, 2006 NACA0012  =2, Mach =0.75 Differences in y-coordinates are of order 10 -4 only

30. July 11, 2006 Conclusions and future work Adjoint codes with approximation in the differentiation of fluxes and reconstruction operator, approximations 1 and 2, can be effective for shape optimization; When approximations are used, at least with SQP algorithm, the optimization can converge to different airfoils. The SLP algorithm has appeared to be insensitive to the approximations and converged to a unique airfoil; When the reconstruction operator is ignored, approximation 3, the adjoint code is not effective. The optimization with SQP and SLP algorithms stall and shock-waves are not removed completely from the airfoil.

31. July 11, 2006 Thank you for your interest