# A Discrete Adjoint-Based Approach for Optimization Problems on 3D Unstructured Meshes - PowerPoint PPT Presentation

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

9
• 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
• Objective function definition

12
Tangent Problem (forward linearization)
• Examine Individual Terms

13
Sensitivity Analysis
• Tangent Problem

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

15
• 1 Objective flow sensitivity
• 3Objective sens. wrt mesh
• 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
• 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
• 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
• 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
• 1 Objective flow sensitivity
• 3Objective sens. wrt mesh
• 5 Final sensitivity

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

28
• 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
• 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
• 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