HIGH-FIDELITY AERO-STRUCTURAL DESIGN OF COMPLETE …aero-comlab.stanford.edu/aa200b/lect_notes/dlr2002.pdfHigh-Fidelity Aerodynamic Shape Optimization † Start from a baseline geometry

HIGH-FIDELITY AERO-STRUCTURAL DESIGN OFCOMPLETE AIRCRAFT CONFIGURATIONS

Juan J. AlonsoDepartment of Aeronautics & Astronautics

Stanford [email protected]

DLR - Institute for Aerodynamics and Flow TechnologyBraunschweig, December 16, 2002

DLR Lecture, December 2002 1

Outline

• Introduction

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

• Coupled-adjoint method

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

• Results

– Aero-structural sensitivity validation– Optimization results

• Conclusions & future work


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.


Quiet Supersonic Platform (QSP) Program

• Range = 6,000 nmi

• Cruise Mach No. = 2.4

• TOGW = 100,000 lbs

• Payload = 20,000 lbs

• Initial Overpressure =0.3 psf

• Swing-wing concept

Gulfstream Aerospace Corporation QSJ Configuration



• Is this combination of requirements achievable? Can we actually do this?This is a set of conflicting requirements: the airplane may not “close”.

• Classical sonic boom minimization theory says that ∆p ≡ W

L32.

• What is the necessary aircraft length? Can we achieve this with ourtarget TOGW?

• At Stanford, we have decided to focus on

– Using aerodynamic shape optimization to take advantage of thenonlinear interactions between shock waves and expansions andproduce shaped booms.

– Using Multidisciplinary Design Optimization (MDO) methods tominimize the weight of the airframe.



Ground Plane

Mid Field

CFD Far Field

Near Field

Method for Computing Ground Boom

Signatures

Symmetry Plane View of CFD Computation


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

��

��

�

� minimize 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.

• Evolutionary/Genetic algorithms: good for discretedesign variables and very robust; are they feasible whenusing a large number of design variables?

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

��

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�

• Gradient-based: the most efficient for a large numberof design variables; assumes the objective function is“well-behaved”. Convergence only guaranteed to a localminimum.


Gradient-Based Optimization: Design Cycle

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*,+-*/.102*

• 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, particularlyfor large dimensional design spaces.

• 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.

df

dxn≈ f(xn + h)− f(x)

h+O(h).

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

df

dxn≈ 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.


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

aliz

ed E

rror,

e

Complex-StepForward-DifferenceCentral-Difference

Estimate derivative atx = 1.5 of the function,

f(x) =ex

√sin3x + cos3x

.

Relative error defined as:

ε =

∣∣∣f ′ − f ′ref

∣∣∣∣∣∣f ′ref

∣∣∣.


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.


Outline

• Introduction




• Results




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

��

��

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.


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.


Adjoint Sensitivity Equations

Solve the adjoint equations

∂Rk

∂yiΨk = − ∂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 solutionof the governing equations.


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 (Euler).

Structural displacements, uj, six for each structural node.


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 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,it is 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

∂wiψk = − ∂I

∂wi︸︷︷︸Aerodynamic adjoint

−∂Sl

∂wiφ̃l,

∂Sl

∂ujφl = − ∂I

∂uj︸︷︷︸Structural adjoint

−∂Ak

∂ujψ̃k,

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 and stresses withrespect 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 areintegrated is perturbed; stresses change when nodes are moved.

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

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


Outline

• Introduction




• Results




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 known method to calculate this matrix of sensitivities efficiently.

Therefore, lump stress constraints

gm = 1− σm

σyield≥ 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

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

With respect to: external shapeand internal structural sizes.


Baseline Design

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Design Variables

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I J K L MNPO Q RTS U S VXW U Y Z [\U ] ^

_ `.a b c�d e f

g�hTi j k�l m n


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

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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


Outline

• Introduction




• Results




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 and spacecraftdesign

ConceptualDesign

DetailedDesign

PreliminaryDesign

CentralDatabase

CAD

Discretization

Multi-DisciplinaryAnalysis

Optimizer

Aerodynamics

Structures

Propulsion

Mission


Direct Interfaces to CAD

• Make use of existing CADpackages to control geometrymanipulation.

• Use CAD packages forstructural definition as well.

• Preferrably, CAD-independentimplementation.

• Master model for aircraftshape and structure geometryis defined parameterically.

• Aircraft model has over 5,000parameters that can becontrolled.


Parallel/Distributed CAD Geometry Server

Arbitrary Network Interconnect (Ethernet)

SYN107−MB−AE SYN107−MB−AESYN107−MB−AE SYN107−MB−AE SYN107−MB−AE



CAD Model CAD Model CAD Model

PVM Session

Task 1

ProE

Task 2 Task N

AEROSURF MASTER PROGRAM

MPI COMMUNICATION LAYER FOR SOLVER AND OPTIMIZER

ProE

ProE

v_1, v_2, ..., v_N v_1, v_2, ... , v_N v_1, v_2, ... , v_N

Parallel Computer 1

Parallel Computer 2

• Completely distributed parallelsystem.

• CAD packages run on onecomputer, simulation/designon another.

• Geometry regeneration iscompletely parallel.

• PVM is used for distributedframework.

• Concept of Geometry Serverthat can be usedsimultaneously with multipleapplications.


Future Work

A lot of work remains to be done for truly high-fidelity MDO environmentsto be realized: it is not simply a matter of development. In particular,

• Extension of coupled-adjoint procedure to multiple disciplines.

• Multilevel optimization strategies with variable fidelity approach.

• Combination of optimization strategies (evolutionary + gradient) forproblems with difficult design spaces.

• Algorithms for coordination in the design of multiple disciplines.

• Introduction of aeroelastic constraints (divergence, flutter) with bothlow- and high-fidelity methods into the design framework.

are some of the areas that we are currently pursuing.

See http://aero-comlab.stanford.edu for more details.



CFD and OML grids

v

u

CFD mesh point

OML point


Displacement Transfer

��

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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 tr

ansf

er

Dis

pla

cem

ent

tran

sfer

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1

2

35

4

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Documents

HIGH-FIDELITY AERO-STRUCTURAL DESIGN OF COMPLETE …aero-comlab.stanford.edu/aa200b/lect_notes/dlr2002.pdfHigh-Fidelity Aerodynamic Shape Optimization † Start from a baseline geometry