COMPLETE CONFIGURATION AERO-STRUCTURAL OPTIMIZATION USING A COUPLED - - PowerPoint PPT Presentation

COMPLETE CONFIGURATION AERO-STRUCTURAL OPTIMIZATION USING A COUPLED SENSITIVITY ANALYSIS METHOD

Joaquim R. R. A. Martins Juan J. Alonso James J. Reuther Department of Aeronautics and Astronautics Stanford University

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 1

Outline

• Introduction

– High-fidelity aircraft design optimization – The need for aero-structural sensitivities – Sensitivity analysis methods

• Theory

– Adjoint sensitivity equations – Lagged aero-structural adjoint equations

• Results

– Optimization problem statement – Aero-structural sensitivity validation – Optimization results

• Conclusions

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 2

High-Fidelity Aircraft Design Optimization

• Start from a baseline geometry provided

by a conceptual design tool.

• Required for transonic configurations

where shocks are present.

• Necessary for supersonic, complex

geometry design.

• High-fidelity analysis needs high-fidelity

parameterization, e.g. to smooth shocks, favorable interference.

• Large number of design variables and

complex models inccur a large cost.

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Aero-Structural Aircraft Design Optimization

• Aerodynamics and structures are core

disciplines in aircraft design and are very tightly coupled.

• For traditional designs, aerodynamicists

know the spanload distributions that lead to the true optimum from experience and accumulated data. What about unusual designs?

• Want to simultaneously optimize the

aerodynamic shape and structure, since there is a trade-off between aerodynamic performance and structural weight, e.g., Range ∝ L D ln Wi Wf

• 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002

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The Need for Aero-Structural Sensitivities

Optimization Structural Optimization Aerodynamic

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Spanwise coordinate, y/b Lift Aerodynamic optimum (elliptical distribution) Aero−structural optimum (maximum range)

Aerodynamic Analysis Optimizer Structural Analysis

• Sequential optimization does not lead to

the true optimum.

• Aero-structural optimization requires

coupled sensitivities.

• Add structural element sizes to the design

variables.

• Including structures in the high-fidelity

wing optimization will allow larger changes in the design.

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Methods for Sensitivity Analysis

• Finite-Difference:

very popular; easy, but lacks robustness and accuracy; run solver Nx times. df dxn ≈ f(xn + h) − f(x) h + O(h)

• Complex-Step Method: relatively new; accurate and robust; easy to

implement and maintain; run solver Nx times. df dxn ≈ Im [f(xn + ih)] h + O(h2)

• (Semi)-Analytic Methods: efficient and accurate; long development

time; cost can be independent of Nx.

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Objective Function and Governing Equations

Want to minimize scalar objective function, I = I(xn, yi), which depends on:

• xn: vector of design variables, e.g. structural plate thickness.
• yi: state vector, e.g. flow variables.

Physical system is modeled by a set of governing equations: Rk (xn, yi (xn)) = 0, where:

• Same number of state and governing equations, i, k = 1, . . . , NR
• Nx design variables.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 7

Sensitivity Equations

• ✁

Total sensitivity of the objective function: dI dxn = ∂I ∂xn + ∂I ∂yi dyi dxn . Total sensitivity of the governing equations: dRk dxn = ∂Rk ∂xn + ∂Rk ∂yi dyi dxn = 0.

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Solving the Sensitivity Equations

Solve the total sensitivity of the governing equations ∂Rk ∂yi dyi dxn = −∂Rk ∂xn . Substitute this result into the total sensitivity equation dI dxn = ∂I ∂xn − ∂I ∂yi

− dyi/ dxn

• ∂Rk

∂yi −1 ∂Rk ∂xn ,

• −Ψk

where Ψk is the adjoint vector.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 9

Adjoint Sensitivity Equations

Solve the adjoint equations ∂Rk ∂yi Ψk = − ∂I ∂yi . Adjoint vector is valid for all design variables. Now the total sensitivity of the the function of interest I is: dI dxn = ∂I ∂xn + Ψk ∂Rk ∂xn The partial derivatives are inexpensive, since they don’t require the solution

• f the governing equations.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 10

Aero-Structural Adjoint Equations

Two coupled disciplines: Aerodynamics (Ak) and Structures (Sl). Rk′ = Ak Sl

• ,

yi′ = wi uj

• ,

Ψk′ = ψk φl

• .

Flow variables, wi, five for each grid point. Structural displacements, uj, three for each structural node.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 11

Aero-Structural Adjoint Equations

   ∂Ak ∂wi ∂Ak ∂uj ∂Sl ∂wi ∂Sl ∂uj   

T

ψk φl

• = −

  ∂I ∂wi ∂I ∂uj   .

• ∂Ak/∂wi: a change in one of the flow variables affects only the residuals
• f its cell and the neighboring ones.
• ∂Ak/∂uj:

wing deflections cause the mesh to warp, affecting the residuals.

• ∂Sl/∂wi: since Sl = Kljuj − fl, this is equal to −∂fl/∂wi.
• ∂Sl/∂uj: equal to the stiffness matrix, Klj.
• ∂I/∂wi: for CD, obtained from the integration of pressures; for stresses,

its zero.

• ∂I/∂uj: for CD, wing displacement changes the surface boundary over

which drag is integrated; for stresses, related to σm = Smjuj.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 12

Lagged Aero-Structural Adjoint Equations

Since the factorization of the complete residual sensitivity matrix is impractical, decouple the system and lag the adjoint variables, ∂Ak ∂wi ψk = − ∂I ∂wi

• Aerodynamic adjoint

−∂Sl ∂wi ˜ φl, ∂Sl ∂uj φl = − ∂I ∂uj

• Structural adjoint

−∂Ak ∂uj ˜ ψk, Lagged adjoint equations are the single discipline ones with an added forcing term that takes the coupling into account. System is solved iteratively, much like the aero-structural analysis.

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Total Sensitivity

The aero-structural sensitivities of the drag coefficient with respect to wing shape perturbations are, dI dxn = ∂I ∂xn + ψk ∂Ak ∂xn + φl ∂Sl ∂xn .

• ∂I/∂xn: CD changes when the boundary over which the pressures are

integrated is perturbed; stresses change when nodes are moved.

• ∂Ak/∂xn: the shape perturbations affect the grid, which in turn changes

the residuals; structural variables have no effect on this term.

• Sl/∂xn: shape perturbations affect the structural equations, so this term

is equal to ∂Klj/∂xnuj − ∂fl/∂xn.

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3D Aero-Structural Design Optimization Framework

• Aerodynamics: FLO107-MB, a

parallel, multiblock Navier-Stokes flow solver.

• Structures: detailed finite element

model with plates and trusses.

• Coupling: high-fidelity, consistent

and conservative.

• Geometry: centralized database

for exchanges (jig shape, pressure distributions, displacements.)

• Coupled-adjoint sensitivity

analysis: aerodynamic and structural design variables.

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Sensitivity of CD wrt Shape

1 2 3 4 5 6 7 8 9 10 Design variable, n

• 0.006
• 0.004
• 0.002

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 dCD / dxn Complex step, fixed displacements Coupled adjoint, fixed displacements Complex step Coupled adjoint

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Sensitivity of CD wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20 Design variable, n

• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 dCD / dxn Complex step Coupled adjoint

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Structural Stress Constraint Lumping

To perform structural optimization, we need the sensitivities of all the stresses in the finite-element model with respect to many design variables. There is no method to calculate this matrix of sensitivities efficiently. Therefore, lump stress constraints gm = 1 − σm σyield ≥ 0, using the Kreisselmeier–Steinhauser function KS (gm) = −1 ρ ln

• m

e−ρgm

• ,

where ρ controls how close the function is to the minimum of the stress constraints.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 18

Sensitivity of KS wrt Shape

1 2 3 4 5 6 7 8 9 10 Design variable, n

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dKS / dxn Complex, fixed loads Coupled adjoint, fixed loads Complex step Coupled adjoint

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 19

Sensitivity of KS wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20 Design variable, n

• 10

10 20 30 40 50 60 70 80 dKS / dxn Complex, fixed loads Coupled adjoint, fixed loads Complex step Coupled adjoint

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 20

Computational Cost vs. Number of Variables

400 800 1200 1600 2000 Number of design variables (Nx) 400 800 1200 Normalized time Complex step 2.1 + 0.92 × Nx Finite difference 1.0 + 0.38 × Nx Coupled adjoint 3.4 + 0.01 × Nx

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 21

Supersonic Business Jet Optimization Problem

Natural laminar flow supersonic business jet Mach = 1.5, Range = 5,300nm 1 count of drag = 310 lbs of weight Minimize: I = αCD + βW where CD is that of the cruise condition. Subject to: KS(σm) ≥ 0 where KS is taken from a maneuver condition. With respect to: external shape and internal structural sizes.

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 22

Baseline Design

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 23

Aero-Structural Optimization Convergence History

10 20 30 40 50 Major iteration number 3000 4000 5000 6000 7000 8000 9000 Weight (lbs) Weight 60 65 70 75 80 Drag (counts) Drag

• 1.2
• 1.1
• 1
• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 KS KS

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Aero-Structural Optimization Results

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization Atlanta, GA, September 2002 25

Comparison with Sequential Optimization

CD (counts)

σmax σyield

Weight (lbs) Objective All-at-once approach Baseline 73.95 0.87 9, 285 Optimized 69.22 0.98 5, 546 87.12 Sequential approach Aerodynamic optimization Baseline 74.04 Optimized 69.92 Structural optimization Baseline 0.89 9, 285 Optimized 0.98 6, 567 Aero-structural analysis 69.95 0.99 6, 567 91.14

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Conclusions

• Presented a framework for high-fidelity aero-structural design.
• The computation of sensitivities using the aero-structural adjoint is

extremely accurate and efficient.

• Demonstrated the usefulness of the coupled adjoint by optimizing a

supersonic business jet configuration with external shape and internal structural variables.

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