Munich University of Technology

FLM

Institute of Fluid Mechanics

Hydraulic Machinery Department

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

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

Test Cases, Applications and Real Time Design Dipl.-Ing. Susanne Thum Dipl.-Ing. Thomas Lepach

Munich University of Technology

FLM

Institute of Fluid Mechanics

Hydraulic Machinery Department

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

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

Munich University of Technology

FLM

Institute of Fluid Mechanics

Hydraulic Machinery Department

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

Analytical Test Functions - Rosenbrock-Function Z

General Rosenbrock-Function :


F(X ) =

n

∑

[100(X

i = 2 j −1

)

2 + (1 − X i ) i +1 − X i 2

2

]

X

2D-Rosenbrock-Function:

(

) + (1 − X )

2 2

F ( x1 , x2 ) = 100 X 2 − X 1

1000

2

1

δF = −400 X 1 (X 2 − X 12 ) − 2(1 − X 1 ) δX 1 δF = 200(X 2 − X 12 ) δX 2

Z

Derivates: 500

0 -1 -1

Y 0

0 1

1

X

Y

FLM

Start :

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

F(−1,−1) = 404

Optimum : F(1,1)

= 0.0

1.5

HOOKE&JEEVES FSQP DONLP2

1

Optimum

0.5

− 3 ≤ x1 ≤ 3

x2

Constraints: − 3 ≤ x2 ≤ 3

0 -0.5

Start

-1

Number of Function calls: FSQP: 83 DONLP: 346 Hooke&Jeeves: 296

-1.5 -1.5

-1

-0.5

0 x1

0.5

1

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

1.5

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Objective Function and Optimisation Variables versus the Number of Iterations 1

DONLP2 FSQP HOOKE&JEEVES

0.8 0.6

350 300

0.2

250

0

200

0

50

100

150

200

250

300

150

350

DONLP2 FSQP HOOKE&JEEVES

1.5 1 0.5 0

-0.5

100 50 0

2

DONLP2 FSQP HOOKE&JEEVES

0.4

Optimisatation Variables

Objective Function

400

-1

-1.5

0

50

100

150

200

Iterations

250

300

350

-2

0

50

100

150

200

Iterations

250

300

350

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Laminar 2-D Diffusor Relative Length:

LD =

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

lD = 7.56 A1

A2 AR = A1

lD

Reynolds Number:

u1 ⋅ A1

ν

= 200

α y







1 ⋅u ⋅ 1− 2 AR 2 2 1

u1 x



ρ



p2 − p1 

ηD =

A1

Efficiency :

A2

Re =

Institute of Fluid Mechanics

FLM



Munich University of Technology Hydraulic Machinery Department

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

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

Starting Point Golden Section: Hooke&Jeeves: FSQP:

8 30 50

0.028

FSQP Hooke&Jeeves GoldenSection

0.02 0.018065 0.0180586 0.01807

FSQP Hooke&Jeeves GoldenSection

0.026

Optimisation Variable A2

0.44

Objective Function (1-etaD)

0.414157 0.396316 0.396327 0.396314

0.024

0.42

0.022 0.02

0.4

0.018 0.016

0.38

0.014 0.012

0.36 0

5

10

15 20 25 30 35 Number of Iterations

40

45

50

0.01

0

5

10

15 20 25 30 35 Number of Iterations

40

45

50

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Laminar 2-D Diffusor Area Ratio:

Optimisation of the relative Diffusor Length LD with lD as optimisation variable:

A2 AR = = 1.68 A1

lD LD = A1

Reynolds Number:

u1 ⋅ A1

ν

= 200 α

Efficiency : A1

u1



x 



1 ⋅u ⋅ 1− 2 AR 2 2 1



ρ



p2 − p1

ηD =

y A2

Re =

lD

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Optimisation of the relative Diffusor Length LD with lD as Optimisation Variable Starting Point Golden Section: Hooke&Jeeves: FSQP: 0.45

Iterations

Objective (1-ηD) 0.446631 0.407421 0.407409 0.407393

10 28 19 0.11

Objective Function (1-etaD)

0.445

Optimisation Variable lD

FSQP Hooke&Jeeves GoldenSection

0.44

0.435

FSQP Hooke&Jeeves GoldenSection

0.1

0.09

0.43

0.08

0.425 0.42

0.07

0.415

0.06

0.41

0.405

Variable lD 0.05 0.077709 0.0772539 0.077723

0

5

10 15 20 Number of Iterations

25

30

0.05

0

5

10 15 20 Number of Iterations

25

30

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

AR =

Optimisation of the Diffusor Contour with respect to the Efficiency for:

A2 = 1.68 A1

AR = const

Relative Length:



lD = 7.56 A1



LD =

LD = const lD

Reynolds Number:

ν

α

= 200

y

Efficiency :

u1



1 AR2





x



2

⋅ u12 ⋅ 1 − 

ρ



ηD =

p2 − p1

A2

u1 ⋅ A1

A1

Re =

-



Area Ratio:



Laminar 2-D Diffusor

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

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

1 Parameter

2 Parameters

4 Parameters

6 Parameters

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

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Wall stress

Diffusor efficiency

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

30

2x/A1

Iterations

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Optimisation of a Francis Turbine with NQ41 using a Multi Level CFD Strategy Three CFD-Levels 1. 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 : NS3D 1 : 10 : 100

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Optimisation of a Francis Turbine with NQ41 RUN 1 / CFD - Level 1:

Objective: • Prescribed Pressure Distribution at Shroud c p = p − pva 1.4 2

H RInitial = 89.1 m H RRUN 1 = 86.5 m HRDesign = 81.8 m

η hInitial = 95.6% η hRUN 1 = 96.4%

2 ⋅ uref

RUN 1 Initial

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

cp

ρ

0

0.2

0.4

0.6 s/smax

0.8

1

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Free Design Parameters: • Leading and Trailing Edge in ϕ-direction (2 x 3 Parameters) • Inlet Angle (2 x 3 Parameters) 0 0.1

110

RUN 1 Initial

100 Blade Angle [degree]

0.2 Shroud

0.3 L

0.4 Mean

0.5 0.6 0.7

Hub

90 80 70 60 50

Hub

40 30

0.8 0.9

RUN 1 Initial

Mean

20 0.5

0.6

u

0.7

0.8

10

Shroud

0.5

0.6

u

0.7

0.8

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

RUN 2 / CFD - Level 1: Objectives: c • Outlet circumferential Velocity Distribution CU = u u1a • Head 0.4 RUN 1 RUN2 Required

0.3

H RRUN 1 = 86.5 m H RDesign = 81.8 m

ηhRUN 1 = 96.4% ηhRUN 2 = 96.7%

0.2 Cu

H RRUN 2 = 82.5 m

0.1 0

-0.1

0

0.2

0.4

s/smax

0.6

0.8

1

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Free Design Parameters : • Trailing Edge in ϕ-direction (3 Parameters) • Outlet Angle (3 Parameters) 0.1

100 Shroud

0.3 0.4

Mean

0.5 0.6

Hub

0.7

80 70 60 50

Hub

40

Mean

30

0.8 0.9

RUN 1 RUN 2

90

0.2

L

110

RUN 1 RUN 2

Blade Angle [degree]

0

Shroud

20 0.5

0.6

u

0.7

0.8

10

0.5

0.6

u

0.7

0.8

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Real Time Design Thomas Lepach

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Analytical Testfunctions - McGormick – Function F(x1,x2)

F ( x1 , x2 ) = sin( x1 + x2 )

120 100 + ( x1 − x2 ) 2 80 − 1.5 x1 + 2.5 x2 + 1 60 40 20 Constraints: 0 -20

− 1.5 ≤ x1 ≤ 4 − 3 ≤ x2 ≤ 3

-4 -3 -2 -1

Start : F (0,0) = 1

π

π

1 π Optimum : F (− + 0.5,− − 0.5) = − 3− 3 3 2 3

x1

0

1

2

3

4

5 -4

-3

-2

-1

0

1

2

3

4 x2

5

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Level Curves of the McGormick Function and Optimisation Process 3

2

Number of Function calls: 19 120 190

1

x2

FSQP: DONLP: Hooke:

FSQP DONLPP2 Hooke&Jeeves

Start

0

-1

Optimum

-2

-3

-1

0

1

x1

2

3

4

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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

Objective Function and Optimisation Variables versus the Number of Iterations 5 4

0

3

-0.5

2 1 0

-1 -1.5 -2

-1 -2

FSQP DONLP2 HOOKE&JEEVES

0.5

x1,x2

Zielfunktionswert

1

FSQP DONLP2 HOOKE&JEEVES

-2.5 0

20

40

60

80 100 120 140 160 180 200 Iterationen

-3

0

20

40 60

80 100 120 140 160 180 200 Iterationen

FLM

Institute of Fluid Mechanics



Munich University of Technology Hydraulic Machinery Department

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