A Discrete Adjoint-Based Approach for Optimization Problems on 3D Unstructured Meshes

1. A Discrete Adjoint-Based Approach for Optimization Problems on 3D Unstructured Meshes Dimitri J. Mavriplis Department of Mechanical Engineering University of Wyoming Laramie, WY

2. Motivation • Adjoint techniques widely used for design optimization • Enables sensitivity calculation at cost independent of number of design variables • Continuous vs. Discrete Adjoint Approaches • Continuous: Linearize then discretize • Discrete: Discretize then Linearize

3. Motivation • Continuous Approach: • More flexible adjoint discretizations • Framework for non-differentiable tasks (limiters) • Often invoked using flow solution as constraint using Lagrange multipliers

4. Motivation • Discrete Approach: • Reproduces exact sensitivities of code • Verifiable through finite differences • Relatively simple implementation • Chain rule differentiation of analysis code • Transpose these derivates • (transpose and reverse order) • Includes boundary conditions

5. Discrete Adjoint Approach • Relatively simple implementation • Chain rule differentiation of analysis code • Enables application to more than just flow solution phase • Nielsen and Park: “Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design”, AIAA 2005-0491: Reno 2005. • Generalize this procedure to multi-phase simulation process

6. Generalized Discrete Sensitivities • Consider a multi-phase analysis code: • L = Objective(s) • D = Design variable(s) • Sensitivity Analysis • Using chain rule:

7. Tangent Model • Special Case: • 1 Design variable D, many objectives L • Precompute all stuff depending on single D • Construct dL/dD elements as:

8. Adjoint Model • Special Case: • 1 Objective L, Many Design Variables D • Would like to precompute all left terms • Transpose entire equation:

9. Adjoint Model • Special Case: • 1 Objective L, Many Design Variables D • Would like to precompute all left terms • Transpose entire equation: precompute as:

10. Shape Optimization Problem • Multi-phase process:

11. Tangent Problem (forward linearization) • Examine Individual Terms: • : Design variable definition (CAD) • : Objective function definition

12. Tangent Problem (forward linearization) • Examine Individual Terms:

13. Sensitivity Analysis • Tangent Problem: • Adjoint Problem

14. Tangent Problem • 1: Surface mesh sensitivity: • 2: Interior mesh sensitivity: • 3: Residual sensitivity: • 4: Flow variable sensitivity: • 5: Final sensitivity

15. Adjoint Problem • 1: Objective flow sensitivity: • 2: Flow adjoint: • 3:Objective sens. wrt mesh: • 4: Mesh adjoint: • 5: Final sensitivity:

16. Flow Tangent/Adjoint Problem:Step 2 or 4 • Storage/Inversion of second-order Jacobian not practical • Solve using preconditioner [P] as:

17. Flow Tangent/Adjoint Problem • Solve using preconditioner [P] as: • [P] = First order Jacobian • Invert iteratively by agglomeration multigrid • Only Matrix-Vector products of dR/dw required

18. Second-Order Jacobian • Can be written as: • q(w) = 2nd differences, or reconstructed variables • Evaluate Mat-Vec in 2 steps as: • Mimics (linearization) of R(w) routine Reconstruction 2nd order residual

19. Second-Order Adjoint • Can be written as: • q(w) = 2nd differences, or reconstructed variables • Evaluate Mat-Vec in 2 steps as: • Reverse (linearization) of R(w) routine

20. Memory Savings Store component matrices But: q=w for 1st order

21. Storage Requirements • Reconstructed from preconditioner [P]=1st order • Trivial matrix or reconstruction coefficients • Symmetric Block 5x5 (for art. dissip. scheme) Store or reconstruct on each pass (35% extra memory)

22. Mesh Motion • Mesh motion: solve using agg. multigrid • Mesh sensitivity: solve using agg. multigrid • Mesh adjoint: solve using agg. multigrid

23. Modular Multigrid Solver • Line-Implicit Agglomeration Multigrid Solver used to solve: • Flow equations • Flow adjoint • Mesh Adjoint • Mesh Motion • Optionally: • Flow tangent • Mesh sensitivity

24. Step 3: Matrix-Vector Product AND/OR • dR/dx is complex rectangular matrix • R depends directly and indirectly on x • R depends on grid metrics, which depend on x • Mat-Vec only required once per design cycle

25. Tangent Problem • 1: Surface mesh sensitivity: • 2: Interior mesh sensitivity: • 3: Residual sensitivity: • 4: Flow variable sensitivity: • 5: Final sensitivity

26. Adjoint Problem • 1: Objective flow sensitivity: • 2: Flow adjoint: • 3:Objective sens. wrt mesh: • 4: Mesh adjoint: • 5: Final sensitivity:

27. Step 3: Tangent Model • Linearize grid metric routines, residual routine • Call in same order as analysis code

28. Step 3: Adjoint Model • Linearize/transpose grid metric routines, residual routine • Call in reverse order

29. General Approach • Linearize each subroutine/process individually in analysis code • Check linearization by finite difference • Transpose, and check duality relation • Build up larger components • Check linearization, duality relation • Check entire process for FD and duality • Use single modular AMG solver for all phases

30. General Duality Relation • Analysis Routine: • Tangent Model: • Adjoint Model: • Duality Relation: • Necessary but not sufficient test • Check using series of arbitrary input vectors

31. Drag Minimization Problem • DLR-F6 Wing body configuration • 1.12M vertices, 4.2M cells

32. Drag Minimization Problem • DLR-F6 Wing body configuration • Mach=0.75, Incidence=1o , Re=3M

33. Drag Minimization Problem • Mach=0.75, Incidence=1o , CL=0.673 • Convergence < 500 MG cycles, 40 minutes on 16 cpus (cluster)

34. Drag Minimization Problem • Adjoint and Tangent Flow Models display similar convergence • Related to flow solver convergence rate • 1 Defect-Correction Cycle : 4 (linear) MG cycles

35. Drag Minimization Problem • Mesh Motion and Adjoint Solvers Converge at Similar Rates • Fast convergence (50 MG cycles) • Mesh operations < 5% of overall cpu time

36. Drag Minization Problem • Smoothed steepest descent method of Jameson • Non-optimal step size • Objective Function Decreases Monotonically

37. Drag Minization Problem • Objective Function Decreases Monotonically • Corresponding decrease in Drag Coefficient • Lift Coefficient held approximately constant

38. Drag Minimization Problem • Substantial reduction in shock strength after 15 design cycles • CD: 302 counts  288 counts : -14 counts • Wave drag

39. Drag Minimization Problem • Surface Displacements = Design Variable Values • Smooth • Mostly on upper surface

40. Drag Minimization Problem • Total Optimization Time for 15 Design Cycles: 6 hours on 16 cpus of PC cluster • Flow Solver: 150 MG cycles • Flow Adjoint: 50 Defect-Correction cycles (x 4 MG) • Mesh Adjoint: 25 MG cycles • Mesh Motion: 25 MG cycles

41. Conclusions • Given multi-phase analysis code can be augmented be discrete adjoint method • Systematic implementation approach • Applicable to all phases • Modular and verifiable • Mimics analysis code at all stages • No new data-structures required • Minimal memory overheads (50% over implicit solver) • Demonstrated on Shape Optimization • Exendable to more complex analyses • Unsteady flows with moving meshes • Multi-disciplinary

42. Future Work • Effective approach for sensitivity calculation • Investigate more sophisticated optimization strategies • Investigate more sophisticated design parameter definitions and ways to linearize these (CAD based) • Multi-objective optimizations in parallel • Farming out multiple analyses simultaneously