HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF A SUPERSONIC - - PowerPoint PPT Presentation

HIGH-FIDELITY AERO-STRUCTURAL DESIGN OPTIMIZATION OF A SUPERSONIC BUSINESS JET

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

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 1

Outline

• Introduction

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

• Theory

– Adjoint sensitivity equations – Lagged aero-structural adjoint equations

• Results

– Aero-structural sensitivity validation – Optimization results

• Conclusions and Future Work

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 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.

• Gradient-based optimization is the most

efficient and requires accurate sensitivity information.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 3

Aero-Structural Aircraft Design Optimization

• Aerodynamics traditionally has used a

shape corresponding to the flying shape

• f the wing, assuming that shape can be

reproduced.

• Wing shape depends on aerodynamic

solution, so need to couple aerodynamic and structural analyses to obtain the solution, specially for unusual designs.

• Want to optimize the structure as well,

since there is a trade-off between aerodynamic performance and structural weight: Range ∝ L D ln Wi Wf

• AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 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.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 5

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 subject to CL = CLT KS ≥ 0 xS ≥ xSmin Lump stress constraints gi = 1 − σiV M σyield ≥ 0, using the Kreisselmeier-Steinhauser function KS (gi(x)) = −1 ρ ln

• i

e−ρgi(x)

• .

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 6

Methods for Sensitivity Analysis

• Finite-Difference:

very popular; easy, but lacks robustness and accuracy; run solver n times. f ′(x) ≈ f(x + h) − f(x) h + O(h)

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

implement and maintain; run solver n times. f ′(x) ≈ Im [f(x + ih)] h + O(h2)

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

time; cost can be independent of n.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 7

Objective Function and Governing Equations

Want to minimize scalar objective function, I = I(x, y), which depends on:

• x: vector of design variables, e.g. structural plate thickness.
• y: state vector, e.g. structural displacements.

Physical system is modeled by a set of governing equations: R (x, y (x)) = 0, where:

• Same number of state and governing equations, nR
• nx design variables.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 8

Variational Equations

R =0 x I y

Total variation of the objective function: δI = ∂I ∂xδx + ∂I ∂yδy. Variation of the governing equations, δR = ∂R ∂x δx + ∂R ∂y δy = 0.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 9

Adjoint Sensitivity Equations

Since the variation of the governing equations must be zero, we can add it to the total variation of the objective, δI = ∂I ∂xδx + ∂I ∂yδy + ψT ∂R ∂x δx + ∂R ∂y δy

• ,

where ψ is a vector of arbitrary components known as adjoint variables. Re-arrange terms, δI = ∂I ∂x + ψT ∂R ∂x

• δx +

∂I ∂y + ψT ∂R ∂y

• δy.

If term in blue were zero, term in red would represent the total variation of the objective with respect to the design variables.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 10

Adjoint Sensitivity Equations

Since the adjoint variables are arbitrary, we can find a set such that, ∂R ∂y T

(nR×nR)

ψ

• (nR×1)

= −∂I ∂y

T (nR×1)

Adjoint valid for all design variables. Now the total sensitivity of the objective is: dI dx

• (1×nR)

= ∂I ∂x

• (1×nx)

+ ψT

• (1×nR)

∂R ∂x

• (nR×nx)

The partial derivatives are inexpensive, since they don’t require the solution

• f the governing equations.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 11

Aero-Structural Adjoint Equations

R =0 x I w

A

R =0 u

S

Two coupled disciplines: Aerodynamics (RA) and Structures (RS). R = RA RS

• ,

y = w u

• and ψ =

ψA ψS

• Flow variables, w, five for each grid point.

Structural displacements, u, three for each structural node.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 12

Aero-Structural Adjoint Equations

    ∂RA ∂w ∂RA ∂u ∂RS ∂w ∂RS ∂u    

T

• ψA

ψS

• = −

    ∂I ∂w ∂I ∂u    

• ∂RA/∂w: a change in one of the flow variables affects only the residuals
• f its cell and the neighboring ones.
• ∂RA/∂u:

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

• ∂RS/∂w: since RS = Ku − f, this is equal to −∂f/∂w.
• ∂RS/∂u: equal to the stiffness matrix, K.
• ∂I/∂w: for CD, obtained from the integration of pressures; for KS, its

zero.

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

which drag is integrated; for KS, related to σ = Su.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 13

Lagged Aero-Structural Adjoint Equations

Since the factorization of the complete residual sensitivity matrix is impractical, decouple the system and lag the adjoint variables, ∂RA ∂w T ψA = − ∂I ∂w

• Aerodynamic adjoint equations

− ∂RS ∂w T ˜ ψS ∂RS ∂u T ψS = −∂I ∂u

• Structural adjoint equations

− ∂RA ∂u T ˜ ψA. 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.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 14

Total Sensitivity

The aero-structural sensitivities of the drag coefficient with respect to wing shape perturbations are, dI dx = ∂I ∂x + ψT

A

∂RA ∂x + ψT

S

∂RS ∂x .

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

integrated is perturbed; stresses change when nodes are moved.

• ∂RA/∂x: the shape perturbations affect the grid, which in turn changes

the residuals; structural variables have no effect on this.

• ∂RS/∂x: shape perturbations affect the structural equations, so this is

∂K/∂x · u − ∂f/∂x.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 15

3D Aero-Structural Design Optimization Framework

• Aerodynamics: SYN107-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.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 16

Sensitivity of CD wrt Shape

1 2 3 4 5 6 7 8 9 10 −8 −6 −4 −2 2 4 6 8 10 x 10

−3

Shape variable, xj d CD / d xA Coupled adjoint Complex step

• Avg. rel. error = 3.5%

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 17

Sensitivity of CD wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20 −0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 d CD / d xS Structural variable, xj Coupled adjoint Complex step

• Avg. rel. error = 1.6%

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 18

Sensitivity of KS wrt Shape

1 2 3 4 5 6 7 8 9 10 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 Shape variable, xj d CD / d xA Coupled adjoint Complex step

• Avg. rel. error = 2.9%

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 19

Sensitivity of KS wrt Structural Thickness

11 12 13 14 15 16 17 18 19 20 −10 10 20 30 40 50 60 70 80 d CD / d xS Structural variable, xj Coupled adjoint Complex step

• Avg. rel. error = 1.6%

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 20

Rigid Wing Aerodynamic Optimization Results

SYMBOL SOURCE

Rigid Baseline Rigid, constrained, aero only design

ALPHA

3.440 4.113

CD

0.00743 0.00700

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONS NLF AERODYNAMIC DESIGN

MACH = 1.500 , CL = 0.100

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

8.6% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

24.3% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

40.8% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

58.9% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

76.5% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

91.5% Span

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 21

Aeroelastic Wing Aerodynamic Optimization Results

SYMBOL SOURCE

Baseline Aero-elastic Aero-elastic, Constrained, aero only design

ALPHA

3.330 4.095

CD

0.00741 0.00701

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONS NLF AEROELASTIC DESIGN - OML ONLY

MACH = 1.500 , CL = 0.100

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

8.6% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

24.3% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

40.8% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

58.9% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

76.5% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

91.5% Span

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 22

Simplified Optimization Problem

minimize I = αCD + βW + γ max(0, −KS)2 subject to CL = CLT xS ≥ xSmin α = 104, β = 3.226 × 10−3, γ = 103

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 23

Aero-Structural Optimization Convergence History

2 4 6 8 10 12 80 100 120 140 160 180 200 220 2 4 6 8 10 12 70 70.5 71 71.5 72 72.5 73 73.5 74 74.5 75 2 4 6 8 10 12 3000 3200 3400 3600 3800 4000 4200 4400 4600 Objective function Weight (lbs) Drag counts Iteration number

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 24

Aero-Structural Optimization Results

SYMBOL SOURCE

Aero-elastic Baseline Aero-elastic, Aero-structural design

ALPHA

3.330 3.689

CD

0.00741 0.00685

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONS NLF FULL AERO-STRUCTURAL DESIGN

MACH = 1.500 , CL = 0.100

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

8.6% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

24.3% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

40.8% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

58.9% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

76.5% Span

0.2 0.4 0.6 0.8 1.0

• 0.25
• 0.15
• 0.05

0.05 0.15 0.25 Cp X / C

91.5% Span

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 25

Aero-Structural Optimization Results

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 26

Conclusions

• Presented a framework for high-fidelity aero-structural design.
• The sensitivities given by the aero-structural adjoint were extremely

accurate when compared to the reference results.

• The coupled-adjoint method was shown to be extremely efficient, making

viable design optimization with respect to hundreds of design variables.

• Demonstrated the usefulness of the coupled-adjoint by optimizing a

supersonic business jet configuration with aerodynamic and structural variables.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 27

Future Work

• Implement multiple load cases, each one with a KS adjoint.
• Use the flow solver in Navier-Stokes mode with a full-configuration mesh.
• Use a better structural model and a more established structural solver.
• Explore the design space for the supersonic business jet configuration

and provide useful results to industry.

AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO, April 2002 28