A COUPLED-ADJOINT METHOD FOR HIGH-FIDELITY AERO …aero-comlab.stanford.edu/Jmartins/Doc/Defense2002.pdfHIGH-FIDELITY AERO-STRUCTURAL OPTIMIZATION ... Analysis Optimizer Structural

A COUPLED-ADJOINT METHOD FORHIGH-FIDELITY AERO-STRUCTURAL

OPTIMIZATION

Joaquim Rafael Rost Avila Martins

Department of Aeronautics and AstronauticsStanford University

Ph.D. Oral Examination, Stanford University, September 2002 1

Outline

• Introduction

– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods

• Complex-step derivative approximation

• Coupled-adjoint method

– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations

• Results

– Aero-structural sensitivity validation– Optimization results

• Conclusions


High-Fidelity Aerodynamic Shape Optimization

• Start from a baseline geometry providedby a conceptual design tool.

• High-fidelity models required for transonicconfigurations where shocks are present,high-dimensionality required to smooththese shocks.

• Accurate models also required forcomplex supersonic configurations, subtleshape variations required to takeadvantage of favorable shock interference.

• Large numbers of design variables andhigh-fidelity models incur a large cost.


Aero-Structural Aircraft Design Optimization

• Aerodynamics and structures are coredisciplines in aircraft design and are verytightly coupled.

• For traditional designs, aerodynamicistsknow the spanload distributions that leadto the true optimum from experience andaccumulated data. What about unusualdesigns?

• Want to simultaneously optimize theaerodynamic shape and structure, sincethere is a trade-off between aerodynamicperformance and structural weight, e.g.,

Range ∝ L

Dln(

Wi

Wf

)


The Need for Aero-Structural Sensitivities

OptimizationStructural

OptimizationAerodynamic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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)

Student Version of MATLAB

Aerodynamic Analysis

Optimizer

Structural Analysis

• Sequential optimization does not lead tothe true optimum.

• Aero-structural optimization requirescoupled sensitivities.

• Add structural element sizes to the designvariables.

• Including structures in the high-fidelitywing optimization will allow largerchanges in the design.


Introduction to Optimization

x1

x2

g

Iminimize I(x)

x ∈ Rn

subject to gm(x) ≥ 0, m = 1, 2, . . . , Ng

• I: objective function, output (e.g. structural weight).

• xn: vector of design variables, inputs (e.g. aerodynamic shape); boundscan be set on these variables.

• gm: vector of constraints (e.g. element von Mises stresses); in generalthese are nonlinear functions of the design variables.


Optimization Methods

• Intuition: decreases with increasing dimensionality.

• Grid or random search: the cost of searching the designspace increases rapidly with the number of design variables.

• Genetic algorithms: good for discrete design variablesand very robust; but infeasible when using a large numberof design variables.

• Nonlinear simplex: simple and robust but inefficient formore than a few design variables.

x1

x2

g

I

• Gradient-based: the most efficient for a large number ofdesign variables; assumes the objective function is “well-behaved”.


Gradient-Based Optimization: Design Cycle

x

Analysis

Sensitivity

Calculation

Optimizer

x = x+�x

• Analysis computes objective function andconstraints (e.g. aero-structural solver)

• Optimizer uses the sensitivity informationto search for the optimum solution(e.g. sequential quadratic programming)

• Sensitivity calculation is usually thebottleneck in the design cycle.

• Accuracy of the sensitivities is important,specially near the optimum.


Sensitivity Analysis Methods

• Finite Differences: very popular; easy, but lacks robustness andaccuracy; run solver Nx times.

dfdxn≈ f(xn + h)− f(x)

h+O(h)

• Complex-Step Method: relatively new; accurate and robust; easy toimplement and maintain; run solver Nx times.

dfdxn≈ Im [f(xn + ih)]

h+O(h2)

• Algorithmic/Automatic/Computational Differentiation: accurate;ease of implementation and cost varies.

• (Semi)-Analytic Methods: efficient and accurate; long developmenttime; cost can be independent of Nx.


Research Contributions


– Contributed new insights to the theory of this approximation, includingthe connection to algorithmic differentiation.

– Automated the implementation of the complex-step method for Fortrancodes.

– Demonstrated the value of this method for sensitivity validation.


– Developed the general formulation of the coupled-adjoint formultidisciplinary systems.

– Implemented this method in a high-fidelity aero-structural solver.– Demonstrated the usefulness of the resulting framework by performing

aero-structural optimization.


The History of the Complex-Step Method

• Lyness & Moler, 1967:nth derivative approximation by integration in the complex plane

• Squire & Trapp, 1998:Simple formula for first derivative.

• Newman, Anderson & Whitfield, 1998:Applied to aero-structural code.

• Anderson, Whitfield & Nielsen, 1999:Applied to 3D Navier-Stokes with turbulence model.

• Martins et al., 2000, 2001:Automated Fortran and C/C++ implementation. 3D aero-structuralcode.


Finite-Difference Derivative Approximations

From Taylor series expansion,

f(x+ h) = f(x) + hf ′(x) + h2f′′(x)2!

+ h3f′′′(x)3!

+ . . . .

Forward-difference approximation:

⇒ df(x)dx

=f(x+ h)− f(x)

h+O(h).

f(x) 1.234567890123484

f(x+ h) 1.234567890123456

∆f 0.000000000000028

x x+h

f(x)

f(x+h)


Complex-Step Derivative Approximation

Can also be derived from a Taylor series expansion about x with a complexstep ih:

f(x+ ih) = f(x) + ihf ′(x)− h2f′′(x)2!− ih3f

′′′(x)3!

+ . . .

⇒ f ′(x) =Im [f(x+ ih)]

h+ h2f

′′′(x)3!

+ . . .

⇒ f ′(x) ≈ Im [f(x+ ih)]h

No subtraction! Second order approximation.


Simple Numerical Example

Step Size, h

Norm

alized E

rror

,

e

Complex-Step

Forward-Difference

Central-Difference

Estimate derivative atx = 1.5 of the function,

f(x) =ex√

sin3x+ cos3x

Relative error defined as:

ε =

∣

∣

∣f ′ − f ′ref∣

∣

∣

∣

∣

∣f ′ref

∣

∣

∣


Algorithmic Differentiation vs. Complex Step

Look at a simple operation, e.g. f = x1x2,

Algorithmic Complex-Step∆x1 = 1 h1 = 10−20

∆x2 = 0 h2 = 0f = x1x2 f = (x1 + ih1)(x2 + ih2)∆f = x1∆x2 + x2∆x1 f = x1x2−h1h2 + i(x1h2 + x2h1)df/dx1 = ∆f df/dx1 = Im f/h

Complex-step method computes one extra term.

• Other functions are similar:

– Superfluous calculations are made.– For h ≤ x× 10−20 they vanish but still affect speed.


Implementation Procedure

• Cookbook procedure for any programming language:

– Substitute all real type variable declarations with complexdeclarations.

– Define all functions and operators that are not defined for complexarguments.

– A complex-step can then be added to the desired variable and thederivative can be estimated by f ′ ≈ Im[f(x+ ih)]/h.

• Fortran 77: write new subroutines, substitute some of the intrinsicfunction calls by the subroutine names, e.g. abs by c abs. But ... needto know variable types in original code.

• Fortran 90: can overload intrinsic functions and operators, includingcomparison operators. Compiler knows variable types and chooses correctversion of the function or operator.

• C/C++: also uses function and operator overloading.


Fortran Implementation

• complexify.f90: a module that defines additional functions andoperators for complex arguments.

• Complexify.py: Python script that makes necessary changes to sourcecode, e.g., type declarations.

• Features:

– Compatible with many platforms and compilers.– Supports MPI based parallel implementations.– Resolves some of the input and output issues.

• Application to aero-structural framework: 130,000 lines of code in 191subroutines, processed in 42 seconds.

• Tools available at: http://aero-comlab.stanford.edu/jmartins/


Outline

• Introduction

– High-fidelity aircraft design– Aero-structural optimization– Optimization and sensitivity analysis methods



– Sensitivity equations for multidisciplinary systems– Lagged aero-structural adjoint equations

• Results

– Aero-structural sensitivity validation– Optimization results

• Conclusions


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.


Sensitivity Equations

xn

Rk = 0

yiI

Total sensitivity of the objective function:

dIdxn

=∂I

∂xn+∂I

∂yi

dyidxn

.

Total sensitivity of the governing equations:

dRkdxn

=∂Rk∂xn

+∂Rk∂yi

dyidxn

= 0.


Solving the Sensitivity Equations

Solve the total sensitivity of the governing equations

∂Rk∂yi

dyidxn

= −∂Rk∂xn

.

Substitute this result into the total sensitivity equation

dIdxn

=∂I

∂xn− ∂I

∂yi

− dyi/ dxn︷︸︸︷

[

∂Rk∂yi

]−1∂Rk∂xn

,

︸︷︷︸

−Ψk

where Ψk is the adjoint vector.


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:

dIdxn

=∂I

∂xn+ Ψk

∂Rk∂xn

The partial derivatives are inexpensive, since they don’t require the solutionof the governing equations.


Aero-Structural Adjoint Equations

xn

Ak = 0 Sl = 0

wi

uj

I

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

Rk′ =[

AkSl

]

, yi′ =[

wiuj

]

, Ψk′ =[

ψkφl

]

.

Flow variables, wi, five for each grid point.

Structural displacements, uj, three for each structural node.


Aero-Structural Adjoint Equations

∂Ak∂wi

∂Ak∂uj

∂Sl∂wi

∂Sl∂uj

T[

ψkφl

]

= −

∂I∂wi∂I∂uj

.

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

• ∂Ak/∂uj: wing deflections cause the mesh to warp, affecting theresiduals.

• ∂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 overwhich drag is integrated; for stresses, related to σm = Smjuj.


Lagged Aero-Structural Adjoint Equations

Since the factorization of the complete residual sensitivity matrix isimpractical, 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 forcingterm that takes the coupling into account.

System is solved iteratively, much like the aero-structural analysis.


Total Sensitivity

The aero-structural sensitivities of the drag coefficient with respect to wingshape perturbations are,

dIdxn

=∂I

∂xn+ ψk

∂Ak∂xn

+ φl∂Sl∂xn

.

• ∂I/∂xn: CD changes when the boundary over which the pressures areintegrated is perturbed; stresses change when nodes are moved.

• ∂Ak/∂xn: the shape perturbations affect the grid, which in turn changesthe residuals; structural variables have no effect on this term.

• Sl/∂xn: shape perturbations affect the structural equations, so this termis equal to ∂Klj/∂xnuj − ∂fl/∂xn.


3D Aero-Structural Design Optimization Framework

• Aerodynamics: FLO107-MB, aparallel, multiblock Navier-Stokesflow solver.

• Structures: detailed finite elementmodel with plates and trusses.

• Coupling: high-fidelity, consistentand conservative.

• Geometry: centralized databasefor exchanges (jig shape, pressuredistributions, displacements.)

• Coupled-adjoint sensitivityanalysis: aerodynamic andstructural design variables.


Sensitivity of CD wrt Shape

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

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

dCD /

dxn

Avg. rel. error = 3.5%

Complex step, fixed displacements

Coupled adjoint, fixed displacements

Complex step

Coupled adjoint


Sensitivity of CD wrt Structural Thickness


-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

dCD /

dxn


Complex step

Coupled adjoint


Structural Stress Constraint Lumping

To perform structural optimization, we need the sensitivities of all thestresses 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− σmσyield

≥ 0,

using the Kreisselmeier–Steinhauser function

KS (gm) = −1ρ

ln

(

∑

m

e−ρgm)

,

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


Sensitivity of KS wrt Shape


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dKS

/ dx

n


Complex, fixed loads

Coupled adjoint, fixed loads

Complex step

Coupled adjoint


Sensitivity of KS wrt Structural Thickness


-10

0

10

20

30

40

50

60

70

80

dKS

/ dx

n


Complex, fixed loads

Coupled adjoint, fixed loads

Complex step

Coupled adjoint


Computational Cost vs. Number of Variables

0 400 800 1200 1600 2000Number of design variables (Nx)

0

400

800

1200

Nor

mal

ized

tim

e

Complex step

2.1

+ 0.

92 N

x

Finite difference

1.0 + 0.38 N x

Coupled adjoint

3.4 + 0.01 Nx


Computational Cost Breakdown

2:40:64

1:20

0:60

< 0:001

0:01Nx

@Ak

@wi

k = �@I

@wi

�@Sl

@wi

~�l

@Sl

@uj�l = �

@I

@uj�@Ak

@uj~ k

dI

dxn= @I

@xn+ k

@Ak

@xn+ �l

@Sl@xn


Supersonic Business Jet Optimization Problem

Natural laminar flowsupersonic business jet

Mach = 1.5, Range = 5,300nm1 count of drag = 310 lbs of weight

Minimize:

I = αCD + βW

where CD is that of the cruisecondition.

Subject to:

KS(σm) ≥ 0where KS is taken from a maneuvercondition.

With respect to: external shapeand internal structural sizes.


Baseline Design

CD = 0.007395

Weight = 9,285 lbs

0.5 1.4 0.0 1.0

Von Mises stresses (maneuver)Surface density (cruise)


Design Variables

0.5 1.4

9 bumps along fuselage axis

10 skin thickness groups

Total of 97 design variables

10 Hicks-Henne bumps

Twist

6 de�ning airfoilsLE camber

TE camber


Aero-Structural Optimization Convergence History

0 10 20 30 40 50Major iteration number

3000

4000

5000

6000

7000

8000

9000W

eigh

t (lb

s)

Weight

60

65

70

75

80

Dra

g (c

ount

s)

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

0.1

0.2

KS

KS


Aero-Structural Optimization Results

0.5 1.4 0.0 1.0

Surface density (cruise) Von Mises stress (maneuver)

CD = 0.006922

Weight = 5,546 lbs


Comparison with Sequential Optimization

CD(counts)

σmaxσyield

Weight(lbs)

Objective

All-at-once approachBaseline 73.95 0.87 9, 285 103.90Optimized 69.22 0.98 5, 546 87.11Sequential approachAerodynamic optimization

Baseline 74.04Optimized 69.92

Structural optimizationBaseline 0.89 9, 285Optimized 0.98 6, 567

Aero-structural analysis 69.95 0.99 6, 567 91.13


Conclusions

• Shed a new light on the theory behind the complex-step method.

• Showed the connection between complex-step and algorithmicdifferentiation theories.

• The complex-step method is excellent for validation of more sophisticatedgradient calculation methods, like the coupled-adjoint.

Step Size, h

No

rm

alize

d E

rro

r

,e

Complex-Step

Forward-Difference

Central-Difference


-0.006

-0.004

-0.002

0

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


Conclusions

• Developed the general formulation for a coupled-adjoint method formultidisciplinary systems.

• Applied this method to a high-fidelity aero-structural solver.

• Showed that the computation of sensitivities using the aero-structuraladjoint is extremely accurate and efficient.

• Demonstrated the usefulness of the coupled adjoint by optimizing asupersonic business jet configuration.

0 400 800 1200 1600 2000Number of design variables (Nx)

0

400

800

1200

Nor

mal

ized

tim

e

Complex step

2.1

+ 0.

92 N

x

Finite difference

1.0 + 0.38 N x

Coupled adjoint

3.4 + 0.01 Nx


Long Term Vision

Continue work on a large-scale MDO framework for aircraft design

ConceptualDesign

DetailedDesign

PreliminaryDesign

CentralDatabase

CAD

Discretization

Multi-DisciplinaryAnalysis

Optimizer

Aerodynamics

Structures

Propulsion

Mission


Acknowledgments


Acknowledgments


Acknowledgments



CFD and OML grids

v

u

CFD mesh point

OML point


Displacement Transfer

u1;2;3

u4;5;6

u7;8;9

�ri

rigid link

associated point

OML point


Mesh Perturbation

Baseline

1

2, 3

4

Perturbed


Load Transfer

v

u

CFD mesh point

OML point


Aero-Structural Iteration

CFD

CSM

Load

tran

sfer

Dis

plac

emen

t tra

nsfe

ru(0)

= 0

w(0)= w1

u(1)

1

2

35

4

w(N)


A COUPLED-ADJOINT METHOD FOR HIGH-FIDELITY AERO …aero-comlab.stanford.edu/Jmartins/Doc/Defense2002.pdfHIGH-FIDELITY AERO-STRUCTURAL OPTIMIZATION ... Analysis Optimizer Structural

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