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

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A Discrete Adjoint-Based Approach for Optimization Problems on 3D Unstructured Meshes Dimitri J. Mavriplis Department of Mechanical Engineering – PowerPoint PPT presentation

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Title: 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
  • 3Objective sens. wrt mesh
  • 4 Mesh adjoint
  • 5 Final sensitivity

16
Flow Tangent/Adjoint ProblemStep 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 qw
    for 1st order

21
Storage Requirements
  • Reconstructed from preconditioner
    P1st 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
  • 3Objective 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
  • Mach0.75, Incidence1o , Re3M

33
Drag Minimization Problem
  • Mach0.75, Incidence1o , CL0.673
  • Convergence lt 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 lt 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
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