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

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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

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

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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

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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

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

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

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

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

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Shape Optimization Problem Multi-phase process:

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

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Tangent Problem (forward linearization) Examine Individual Terms: –

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Sensitivity Analysis Tangent Problem: Adjoint Problem

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

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

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

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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

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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 2 nd order residual

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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

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

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

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

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

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Step 3: Matrix-Vector Product 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 AND/OR

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

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

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

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

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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

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

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

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

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

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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

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

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

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Drag Minization Problem Objective Function Decreases Monotonically –Corresponding decrease in Drag Coefficient –Lift Coefficient held approximately constant

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

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Drag Minimization Problem Surface Displacements = Design Variable Values –Smooth –Mostly on upper surface

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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

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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

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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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GUIDED PRACTICE for Example 3 5.5. 2 – 2 0 2 0 – 2 12 – 4 – 6 A = Use a graphing calculator to find the inverse of the matrix A. Check the result by showing.

GUIDED PRACTICE for Example 3 5.5. 2 – 2 0 2 0 – 2 12 – 4 – 6 A = Use a graphing calculator to find the inverse of the matrix A. Check the result by showing.

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