need and use of multiobjective optimization in large …gaggle/s05/mop/...need and use of...

Need and Use of Multiobjective Optimization in Large Scale Engineering Applications

N. Kroll

DLR (German Aerospace Center)Institute of Aerodynamics and Flow Technology

Braunschweig

Outline

Role of CFD in Aircraft Design

Shape optimization examples, requirements

Multiobjective optimizationexamples, need for research

Conclusion

Need and Use of Multiobjective Optimization in Aircraft Design

Role of Computational Fluid Dynamics (CFD) in Aerodynamic Aircraft Design

Objectives of CFD• detailed analysis of complex flow fields• cost efficient configuration studies• shape optimization• extrapolation of wind tunnel results

to free flight conditions

engineintegration

high lift

Requirements on CFD• high level of physical modeling

– compressible flow– transonic flow– laminar - turbulent flow – high Reynolds numbers (60 million)– large flow regions with flow separation – steady / unsteady flows

• complex geometries• short turn around time

Use of CFD in Aerodynamic Aircraft Design

Consequences for “Detailed Design Phase”solution of 3D compressible Reynolds averaged Navier-Stokes equations turbulence models based on transport equations (2 – 6 eqn)models for predicting laminar-turbulent transition flexible grid generation techniques with high level of automation(block structured grids, overset grids, unstructured/hybrid grids)link to CAD-systemsefficient algorithms (multigrid, grid adaptation, parallel algorithms...)large scale computations ( ~ 10 - 25 million grid points)

Use of CFD in Aerodynamic Aircraft Design

DLR Software MEGAFLOW

MEGAFLOWNational CFD software dedicated for aerodynamic aircraft applications which

allows Navier-Stokes computations for 3D complex configurations at cruise and high-lift conditionsestablishes numerical flow simulation as a routinely used tool at DLR andin German aircraft industryCFD kernel for multidisciplinary simulation and optimizationserves as a development platform for universities

MEGAFLOW Software

Structured RANS solver FLOWer

block-structured grids moderate complex configurationsfast algorithms (unsteady flows)design option

Unstructured RANS solver TAU

hybrid grids very complex configurationsgrid adaptation fully parallel software

Physical model→ 3D compressible Navier-Stokes equations→ arbitrarily moving bodies→ steady and time accurate flows→ state-of-the-art turbulence models (RSM)

Reynolds-Averaged Navier-Stokes Solver FLOWer

Numerical algorithms→ 2nd order finite volume discretization

(cell centered & cell vertex option)→ central and upwind schemes→ multigrid→ implicit treatment of turbulence equations→ implicit schemes for time accurate flows→ preconditioning for low speed flow→ vectorization & parallelization→ adjoint solver

Grid strategy→ block-structured grids→ discontinuous block boundaries→ overset grids (Chimera)→ deforming grids

Physical model→ 3D compressible Navier-Stokes equations→ arbitrarily moving bodies→ steady and time accurate flows→ state-of-the-art turbulence models

Reynolds-Averaged Navier-Stokes Solver TAU

Numerical algorithms→ 2nd order finite volume discretization

based on dual grid approach→ central and upwind schemes→ multigrid based on agglomeration → implicit schemes for time accurate flows→ preconditioning for low speed flow→ optimized for cash and vector processors→ MPI parallelization

Grid strategy→ unstructured/hybrid grids→ semi-structured sublayers→ overset grids (Chimera)→ deforming grids → grid adaptation (refinement, de-refinement)

Fluid / Structure Coupling

CADpreparation

Initialmesh generation

Flow computation

Structurecomputation

Mesh deformation /mesh generation

(Visualization)

Post processing / Analysis

Time stepping-loop

RWTH AachenFLOWer-Code

Megaliner landing configurationInfluence of nacelle strakes

Complex High- Lift Configurations

BrodersenTAU computations

grid generation: 9.5 millon points3 days

one calculation: 24hrs on 64 Proc. PC-Cluster

Aerodynamic Shape design

analysis of different geometries

inverse design (ill-posed problem)

numerical shape optimization

single discipline optimizationsingle point multi point

multidisciplinary optimizationsingle/multi point, single objectivesingle/multi point, multi objective

„ Winglet“ „ Shark“

Approach

Inverse Design

start configurationDLR design

Objective: drag rise improvementConfiguration: F/D 728/928 (n=1.0g)

Application• airfoil, isolated wings• wing/body • isolated and integrated nacelles

differencetarget /calculated

pressure ∆CP

flow analysisFlower, TAU

geometry/grid modification

- target pressuredistribution

- initial geometry Z

inversedesign module

∆CP ∆Z

designconverged

designed geometry

No

Streit

Aerodynamic Shape Optimization

setting optimization, M=0.197, Re=3.52x106

maxaC

evaluation ofcost function

analysisFlower, TAU

geometry/grid generation

optimizedgeometry

input- cost function- constraints- start geometry

input- cost function- constraints- start geometry

change designparameters

optimizer- gradient based

adjoint solver- Extrem, Simplex

quality factor

fulfilled

Numerical Optimization

Wild

Numerical Optimization of High-Lift 3-Element Airfoilsingle point, single discipline, single objective

Application• drag optimization for 3-element airfoil• take-off configuration

(M∞=0.2, Re=3.52x106)

Cost function:• minimum drag with constant lift

and constraint pitching momentDesign parameters (12)

• element position & deflection• element-size variations

0 1 2 3 4 5 66.50

7.00

7.50

8.00

8.50

9.00

optimization cycles

Fobj(X) = 100⋅CD

Testcase:DRA NHLP L1T2

CL = const. = 3,77Re = 3,56⋅106

-Cm ≤ -Cm,start

optimized:∆CD = -20,815 %

design parameter:element deflections

Ma∞ = 0,2

cutout geometries

Computational effort: 50 hrs, 4 procs NEC SX5.

~ 24 hrs, 12 procs PC-Cluster

Wild

Geometric constraintsh minimum wing thickness distribution

along the spanwise directionh minimum fuselage radius

Drag reduction at constant lifth Mach number = 2.0h lift coefficient = 0.1207

Design variables h Fuselage 10 parametersh twist deformation: 10 parametersh camberline (8 sections): 32 parametersh thickness (8 sections) : 32 parametersh angle of attack: 1 parameter .

85 parameters

CamberlineThickness

Optimization of SCT Configuration

Approachh inviscid flow computationh Euler adjoint for calculation of flow sensitivitiesh 230.000 points

Brezillion, Gauger

14.6 Drag Counts


optimized geometry

baseline geometry

Brezillion, Gauger

Wing section and pressure distribution

η=0.24

η=0.49

η=0.92


Objective function

4 Reduction of drag in 2 design points

Design points

4 1 : M∞=0.734, CL = 0.80 , α = 2.8°, Re=6.5x106, xtrans=3%, W1=2

4 2 : M∞=0.754, CL = 0.74 , α = 2.8°, Re=6.2x106, xtrans=3%, W2=1

Constraints

4 No lift decrease, no change in angle of incidence

4 Variation in pitching moment less than 2% in each point

4 Maximal thickness constant and at 5% chord more than 96% of initial

4 Leading edge radius more than 90% of initial

4 Trailing edge angle more than 80% of initial

Multipoint airfoil optimization RAE2822

),(2

1iid

ii MCWI α∑

=

=

Brezillion

Parameterization4 20 design variables changing camberline, Hicks-Henne functions

Optimization strategy4 Constrained SQP, Synaps Pointer Pro; W1=2, W2=1

4 Navier-Stokes solver FLOWer, Baldwin/Lomax turbulence model

4 Gradients provided by FLOWer Adjoint, based on Euler equations

ResultsPt α Mi Clt Cdt (.10-4) Cl Cdt (.10-4) ∆Cd/Cdt ∆Cl/Clt ∆Cm/Cmt

1 2.8 0.734 0.811 197.1 0.811 135.5 -31.2% 0% +1.6%

2 2.8 0.754 0.806 300.8 0.828 215.0 -27.4% +2.7% +2.0%


),(2

1iid

ii MCWI α∑

=

=

Brezillion

1. design point 2. design point

shape geometry


Remark: weights W1 and W2 not necessarily known

),(2

1iid

ii MCWI α∑

=

=

Brezillion

Requirementsmulti-point design, multi-objective optimization, multidisciplinary optimization

large number of design variables

physical and geometrical constraints

complex configurations

efficient parametrization techniques based on CAD model meshing & mesh deformation techniques ensuring grid quality

compressible Navier-Stokes equations with accurate models for turbulence and transition efficient flow solversefficient and robust optimization algorithmsautomatic framework

Aerodynamic Shape Optimization

Multidisciplinary OptimizationMotivation

Natural laminar flow supersonicbusiness jet configuration(pics: Alonso et al., AIAA-2002-5402)

Motivation

Wing deflection up to 7% of wing span!

Deflected aerodynamicoptimal shape can beworse than the initial …

Boeing 737Boeing 737--800 at ground and in cruise (Ma = 0.76)800 at ground and in cruise (Ma = 0.76)

Multidisciplinary Optimization

Aero-Structure MDO - Example

FWW

CCR

D

L

−∝ ln

)1(0 ksWW λ+=

∑ −=

n

nks ))exp(ln(1

0

0

σσσβ

βλ=0.2 , σ0 =30.000 and β=40

Range R:

Weight W:

Fuel Weight F

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−+

+=

0

1

1ln

WFks

ksCC

D

L

λ

λ

Single objective !

A. Fazzolari

AMP wing

240 design variables(control points free form deformation)

Ma=0.78alpha=2.83

Range maximization byconstant lift

Aero-Structure MDO - Example

∆CD = -25 %

∆ks = -10 %

∆R = +37 %

feasible direction method

Single Criterion Optimization

Status:• efficient gradient based optimization available • determination of gradients/sensitivities basedon adjoints

• “accurate” approximation of Hessian often not possible• problem: treatment of constraints

I∇

start geometryx0

ψ

x0

w

xn+1

k-loop

k-loop

Adjoint Based Optimization

min Ι (w,x)s.t. R(w,x)=0

optimizationstrategy

RANS solverR(wk,xn)=0

gradient

∫=∇V

mn

m dVxiI ))(,(w,)( δψ

Adjoint solverR*(w,ψk,xn)=0

dim x = M

n-loop n=1,…,N

m-loopm=1,…,M

Lx∇

start geometryx0

ψk+1

x0

wk+1

xk+1

primal updatewk+1=wk-∆t·R(wk,xk)

gradient

∫ ++=∇V

mkkk

mx dVxlL ))(,,(w)( 11 δψ

k-loop dual update

ψk+1= ψk-∆t·R*(wk+1,ψk,xk)

111 LxRBLBtxx w

T

kxkkk ∇⎟

⎠⎞

⎜⎝⎛

∂∂

−∇∆−= −−+

m-loopm=1,…,M

design update

Bk – BFGS updatesof reduced Hessian Lxx

University of Trier

Simultaneous Pseudo-Time stepping -One Shot Approach

Schulz et al.

Simultaneous Pseudo-Time stepping -One Shot Approach University of Trier

Optimization problem• drag reduction for RAE 2822 • inviscid flow• M=0.73, a=20

Tools• FLOWer• FLOWer adjoint

Schulz et al.

Optimization at the cost of 4 flow simulations!

A

TAD

S

TS

uR

uC

uR ψψ ~⎟

⎠⎞

⎜⎝⎛

∂∂

−∂

∂=⎟

⎠⎞

⎜⎝⎛

∂∂

xu

uC

xw

wC

xC

dxdC DDDD

∂∂

∂∂

+∂∂

∂∂

+∂

∂=

xR

xR

xC

dxdC ST

SAT

ADD

∂∂

−∂

∂−

∂∂

= ψψ

S

TSD

A

TA

wR

wC

wR ψψ ~⎟

⎠⎞

⎜⎝⎛

∂∂

−∂

∂=⎟

⎠⎞

⎜⎝⎛

∂∂

0=AR

0=−= fKuRS

Aerodynamics, e.g Euler Eqn.:

Structure:

K: Symmetric stiffness matrixf: Aerodynamic forceu: Displacement vectorx: Vector of Design variables

Coupled Aero-Structure Adjoint

Adjoint Gradient:

Aero/Structure Adjoint System:

Conventional Gradient:

::

S

A

ψψ Aerodynamic Adjoint

Structure Adjoint~: Lagged ...

Coupled Aero-Structure Adjoint

AMP wing

240 design variables(control points free form deformation)

Ma=0.78alpha=2.83

Drag reduction byconstant lift

baseline

optimized

A. Fazzolari

Multiobjective Optimization

In aircraft design very often a single objective does notadequately represent the design problemVector of objective functions must be traded off (Aircraft design: Maximize lift/drag & minimize structural weight)The relative importance of the objectives is not generally knownuntil the system’s capabilities are determined and the tradeoffs are understood

Multi-objective strategies are need which are reliable, efficient and which offers the engineer to express preferences throughoutthe optimization cycle Multi-objective optimization leads to a set of compromised solutions


Problem description

nRXXF

∈

)(min

0)(0)(

≤=

XHXG

k

i

NkMi

,....1,....1

==

subject to constraints

mn RRF →:

Remark• F is vector• if any of components of F(X) are competing, no unique optimum


F1

F2x2

x1

F

nRX ∈=Ω

0)(0)(

≤=

XHXG

k

i

feasible region

subject to

Ω

Λ


Concept of Pareto optimality

DefinitionΩ∈*X

X∆Ω∈∆+ )*( XX

*)()*(*)()*(XFXXF

xFXXF

jj

ii

<∆+≤∆+ mi ,....,1=

A point is a local Pareto optimal point if for someneighborhood of *X there does not exist a such that

and

for some j

F1

F2Λ

Pareto front

How to determine Pareto points• Genetic algorithms• Weighted Sum Strategy• Normal Boundary Intersection method• …….

Shape optimization of wing plan form• flow condition: M = 0.85, a = 1°• inviscid flow (Euler)• computational mesh: 630.000 nodes • multi objective optimization:

- maximize lift and minimize drag

Multiobjective Optimization - Example

EADS-M: Deister

• design parameters: - sweep angle (range: -60° to +60°)- half span (range: 0.750 [m] to 1.250 [m])- aspect ratio (defined by const. wing plan area constraint)- taper ratio (range: 0.2 to 0.8)

• design constraints:- pitching moment restricted to range –0.025 to +0.0001

A

B

C

EADS-M: Deister

Genetic Algorithm

Wing plan form optimization

Pareto Front

sweep = 20°aspect r. = 6.6taper r. = 0.58half span = 1.250 [m]



lift = 0.064drag = 0.0089moment = - 0.022glide ratio = 7.2

lift = 0.048drag = 0.0018moment = - 0.025glide ratio = 26.7

lift = 0.031 drag = 0.00087moment = - 0.024glide ratio = 35.6

A B

CFlow

Flow

EADS-M: Deister


Shape optimization of wing plan form

EADS-M: Deister

Computational effort:

- for one design evaluation: 50 min (4 XEON 2.6 GHz processors)

• complete mesh generation time: approx. 15 min.• complete flow simulation time: 35 min.

- 12 concurrent design evaluations using 4 processors each - 30 design generations

- all together 360 design evaluations in less than 25 h

but just 5 design variables and inviscid flow !


Alternatives ????

Test case definition : 2 variables - 2 objectives

a = 0.5sin(1) - 2.0cos(1) + 1.0sin(2) - 1.5cos(2)

b = 0.5sin(x) - 2.0cos(x) + 1.0sin(y) - 1.5cos(y)

c = 1.5sin(1) - 1.0cos(1) + 2.0sin(2) - 0.5cos(2)

d = 1.5sin(x) - 1.0cos(x) + 2.0sin(y) - 0.5cos(y)

Obj1 = -(1+(a-b)2 + (c-d)2)

Obj2 = -(x+3) 2 - (y+1) 2

x,y ∈[-π,π]

Two objective functions to be maximised


Academic test case

Visualisation of the design space (101x101 evaluations)

Obj1 vs. Obj2 Obj1 and Obj2 in the design space

Pareto front


Academic test case

Multi-objective optimisation using Genetic Algorithm

4 Latin Square DOE

4 Population size of 64 individuals(i.e. 64 concurrent evaluations could be performed in parallel)

4 6 generations

4 383 evaluations

Almost the complete

Pareto front is captured

within a single run !

Global Optimumfrom amono-obj. form.(obtained afterpost processing)


Academic test case

Object 1

Obj

ect2

-20 -15 -10 -5 0-30

-25

-20

-15

-10

-5

0

Pareto-Front, GABFGS, cos(a)*Obj1+sin(a)*Obj2Dennis&Das, Optima SubproblemeDennis&Das, Pareto-Frontcomputed


Academic test case

Dennis & Das:• use warm start

for sub problems(starting point, gradients)

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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

f1

f2

-1 -0.8 -0.6 -0.4 -0.2 0

-1

-0.8

-0.6

-0.4

-0.2

0

Exact Pareto FrontPareto MOGA IIPareto NBI+

ZDT1 Test CaseNBIPareto front discretised with 62 points

MOGA IIPopulation = 100Generation = 10010.000 EvaluationsPareto front discretised with 129 points


30 design variables

Concluding Remarks

Shape optimization is essential for future aircraft design

Shape design requires multidisciplinary optimization(>>> multiobjective optimization, Pareto front)

“Efficient” methods for single objective optimization are available or under development (adjoint sensitivities,Newton-type methods, one shot methods,,,)

Multidisciplinary optimization based on high-fidelity CFDsolvers not practical, appropriate deterministic methodsare missing

need and use of multiobjective optimization in large …gaggle/s05/mop/...need and use of...

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