test cases, applications and real time...

Munich University of Technology

Institute of Fluid Mechanics Hydraulic Machinery Department�

o. Prof. Dr.-Ing. habil. Rudolf SchillingFLM

ERCOFTAC- Introductory Course to Design Optimisation, April 2nd, 2003

Garching, Munich

Test Cases, Applications and Real Time Design

Dipl.-Ing. Susanne ThumDipl.-Ing. Thomas Lepach




Contents

• Analytical Test Functions– 2D-Rosenbrock - Function

• 2D - Laminar Diffusor– Optimisation of Parameter A2/A1

– Optimisation of Parameter LD/A1

– Optimisation of the Diffusor - contour

• Francis Turbine FT40• Real Time Design

– 3D -Pump Impeller




0

500

1000

Z

-1

0

1

X

-1

0

1

Y

X Y

Z

Analytical Test Functions - Rosenbrock-Function -

( ) ( )[ ]∑−=

+ −+−=n

jiiii XXXXF

12

2221 1100)(

�

( ) ( )21

22221 1100),(

1XXXxxF −+−=

( ) ( )

( )212

2

12

1211

200

12400

XXXF

XXXXXF

−=

−−−−=

δδδδ

General Rosenbrock-Function :

2D-Rosenbrock-Function:

Derivates:




-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

x2

x1

HOOKE&JEEVESFSQP

DONLP2

Number of Functioncalls:FSQP: 83DONLP: 346Hooke&Jeeves: 296

Start

Optimum

Level curves of the 2-D Rosenbrock- Functionand Optimisation Process of the 3 Optimisation Algorithms

33

33

2

1

≤≤−≤≤−

x

x

0.0)1,1(F:Optimum404)1,1(F:Start

==−−

Constraints:




0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350

DONLP2FSQP

HOOKE&JEEVES

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350

DONLP2FSQP

HOOKE&JEEVES

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200 250 300 350

DONLP2FSQP

HOOKE&JEEVES

Iterations Iterations

Opt

imis

atat

ion

Var

iabl

es

Obj

ectiv

eFu

nctio

n

Objective Function and Optimisation Variables versus the Number of Iterations




Laminar 2-D Diffusor

Relative Length: Optimisation of the Area Ratio AR, with A2 as Optimisation Variable:

Reynolds Number:

Efficiency :

1

2

AA

AR =56.71

==Al

L DD

200Re 11 =⋅=ν

Au

��

�

��

�

−⋅⋅

−=

221

12

11

2 R

D

Au

ppρ

η

A1 A2

lD

x

y

α

u1


Institute of Fluid Mechanics Hydraulic Machinery Department


Optimisation of the Area Ratio AR, with A2 as Optimisation VariableIterations Objective (1-ηD) Variable A2

Starting Point 0.414157 0.02Golden Section: 8 0.396316 0.018065Hooke&Jeeves: 30 0.396327 0.0180586 FSQP: 50 0.396314 0.01807

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

0 5 10 15 20 25 30 35 40 45 50

Opt

imis

atio

n V

aria

ble

A2

Number of Iterations

FSQPHooke&JeevesGoldenSection

0.36

0.38

0.4

0.42

0.44

0 5 10 15 20 25 30 35 40 45 50

Obj

ectiv

e Fu

nctio

n (1

-eta

D)




Institute of Fluid Mechanics Hydraulic Machinery Department


Laminar 2-D Diffusor

Area Ratio: Optimisation of the relative DiffusorLength LD with lD as optimisation variable:

Reynolds Number:

Efficiency :

68.11

2 ==AA

AR

1Al

L DD =

200Re 11 =⋅=ν

Au

�

�

�

�

−⋅⋅

−=

221

12

11

2 R

D

Au

ppρ

η A1

A2

lD

x

y

α

u1




Optimisation of the relative Diffusor Length LD with lD as OptimisationVariable Iterations Objective (1-ηD) Variable lDStarting Point 0.446631 0.05Golden Section: 10 0.407421 0.077709Hooke&Jeeves: 28 0.407409 0.0772539FSQP: 19 0.407393 0.077723

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0 5 10 15 20 25 30

Opt

imis

atio

n V

aria

ble

lD



0.405

0.41

0.415

0.42

0.425

0.43

0.435

0.44

0.445

0.45

0 5 10 15 20 25 30

Obj

ectiv

e Fu

nctio

n (1

-eta

D)






Laminar 2-D DiffusorArea Ratio: Optimisation of the Diffusor Contour

�

-

�

with respect to the Efficiency for:

Relative Length:

Reynolds Number:

Efficiency :

68.11

2 ==AA

AR

56.71

==Al

L DD

200Re 11 =⋅=ν

Au

��

�

��

�

−⋅⋅

−=

221

12

11

2 R

D

Au

ppρ

η

A1

A2

lD

x

y

α

u1

constL

constA

D

R

==

� �




Optimisation of the Diffusor Contour with a different Number of freeParameters (Control Points)

1 Parameter

2 Parameters

4 Parameters

6 Parameters

Objective:(1-ηD)=0.378215−> ηD = 62.18 %




30

Development of the Wall Stress and Diffusor Efficiency versus theNumber of Iterations for different Number of Control Points

2x/A1Iterations

Diff

usor

effic

ienc

y

Wal

l stre

ss




Optimisation of a Francis Turbine with NQ41 using a Multi Level CFD StrategyThree CFD-Levels1. Level: EQ3D

– Calibrated and very fast computing– 100 times faster than NS3D

2. Level: E3D– 3D structure of the flow on a very coarse computational mesh– 10 times faster than NS3D

3. Level: NS3D– „fine tuning“ of the geometry– Evaluation of the losses

EQ3D : E3D : NS3D1 : 10 : 100




-0.4-0.2

00.20.40.60.8

11.21.4

0 0.2 0.4 0.6 0.8 1

RUN 1Initial

s/smax

c p

RUN 1 / CFD - Level 1:Objective:• Prescribed Pressure Distribution

at Shroud

= 89.1 m

= 86.5 m

= 81.8 m

= 95.6%

= 96.4%

Optimisation of a Francis Turbine with NQ41

InitialRH

1RUNRHDesignRH

Initialhη

1RUNhη

2

2 ref

vap

u

ppc

⋅

−= ρ




Free Design Parameters:• Leading and Trailing Edge in ϕ-direction(2 x 3 Parameters)

• Inlet Angle(2 x 3 Parameters)

10

20

30

40

50

60

70

80

90

100

110

0.5 0.6 0.7 0.8

RUN 1Initial

Bla

de A

ngle

[deg

ree]

u

Hub

Mean

Shroud

10

20

30

40

50

60

70

80

90

100

110

0.5 0.6 0.7 0.8

RUN 1Initial

Bla

de A

ngle

[deg

ree]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 0.5 0.6 0.7 0.8

RUN 1

u

L

Initial

Hub

Mean

Shroud




= 86.5 m

= 82.5 m

= 81.8 m

= 96.4%

= 96.7%

RUN 2 / CFD - Level 1:Objectives:• Outlet circumferential Velocity Distribution • Head

1RUNRH

2RUNRHDesignRH

1RUNhη

2RUNhη

-0.1

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1

RUN 1RUN2Required

s/smax

Cu

a

uU u

cC

1

=




Free Design Parameters :• Trailing Edge in ϕ-direction(3 Parameters)

• Outlet Angle (3 Parameters)

L

u

Hub

Mean

Shroud

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.90.5 0.6 0.7 0.8

RUN 1RUN 2

u

Bla

de A

ngle

[deg

ree]

Hub

Mean

Shroud

10

20

30

40

50

60

70

80

90

100

110

0.5 0.6 0.7 0.8

RUN 1RUN 2




Real Time DesignThomas Lepach




Analytical Testfunctions - McGormick – Function -

15.25.1

)(

)sin(),(

21

221

2121

++−−+

+=

xx

xx

xxxxF

33

45.1

2

1

≤≤−≤≤−

x

x

33

21

)5.03

,5.03

(:

1)0,0(:πππ −−=−−+−

=

FOptimum

FStart

Constraints:

-4 -3 -2 -1 0 1 2 3 4 5x1 -4-3

-2-1

01

23

45

x2

-200

20406080

100120

F(x1,x2)




-3

-2

-1

0

1

2

3

-1 0 1 2 3 4

x2

x1

FSQPDONLPP2

Hooke&Jeeves

Start

Optimum

Number of Functioncalls:

FSQP: 19DONLP: 120Hooke: 190

Level Curves of the McGormick Function and Optimisation Process




-2

-1

0

1

2

3

4

5

0 20 40 60 80 100 120 140 160 180 200

Zie

lfun

ktio

nsw

ert

Iterationen

FSQPDONLP2

HOOKE&JEEVES

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140 160 180 200

x1,x

2

Iterationen

FSQPDONLP2

HOOKE&JEEVES

Objective Function and Optimisation Variables versus the Number of Iterations

test cases, applications and real time...

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