Classification of Optimization Problems

(1)

Dieter Bestle and Akin Keskin

Chair of Engineering Mechanics and Vehicle Dynamics Brandenburg University of Technology Cottbus

Some Applications of

Multicriterion Strategies in Mechanical Engineering

(2)

Classification of Optimization Problems

optimization

topology optimization

shape optimization

sizing

scalar optimization

multicriterion optimization

► scalarization

► hierarchization

unconstrained constrained )

(

min p

p

h

f

R

∈ I

) ( min p

p

f

∈ P

{ Î ^R ^h û ô }

P = p ∈ g ( p ) = 0 , h ( p ) ≤ 0 , p ≤ p ≤ p

(3)

What is Optimization Good For?

min

l P u

0

-

∈

 Ω 

 

Ω Ω

 

 

p

maximum stable speed range

0

: min . . max Re( ) 0 : min . . max Re( ) 0

l

j j u

j j

s t s t

λ λ

≤ Ω ≤ Ω

Ω < Ω ≤ Ω

Ω = Ω ≤

Ω = Ω >

{

2 3 4

}

, ,

min max ( ) , ( ) , ( )

x t

x t x t x t

γ

∆

ε minimum motion of inner balls

( )

min T

ρ ϕ

. . ( ) ˆ s t s T = s minimum time

• Designing Birthday Presents

(4)

What is Optimization Good For?

• Designing Birthday Presents

• Solving Problems of Existing Machines

ideal contact geometry

( )

max min

r z

ρ ψ

δ δ

 

 

 

. . tan

0

s t ϕ ≤ µ

ideal ring shape min , ,

^T

S w

γ γ +

r r r

0 0

. .

_r_k _k

/ ,

^r^k ^r^k

rk

s

S S G

t S

S γ

µ ⁻ ≤

=

S rk

D

H S S

r k

ϕ

ρ ψ ( ) δ r

δ z

N

T

(5)

What is Optimization Good For?

• Designing Birthday Presents

• Solving Problems of Existing Machines

• System Identification

• actuator behavior

(hydraulic, pneumatic, electro-mechanical)

• dynamic behavior of passive components

• system parameters

• control behavior

(6)

What is Optimization Good For?

• Designing Birthday Presents

• Solving Problems of Existing Machines

• System Identification

• Virtual Prototyping

• hardware-in-the-loop optimization

• mechatronics (control design)

• multibody systems (vibrations)

• multi-disciplinary optimization

(7)

{ Î ^R ^h û ô }

P = p ∈ g ( p ) = 0 , h ( p ) ≤ 0 , p ≤ p ≤ p scalar

optimization

vector

optimization

min f(p)

p ∈ P

min

p ∈ P

f ₁ (p)

. . .

f _n (p)

P

f ₁ f ₂

f(P)

1

2 multiple optimal compromises f(p)

Why Multi-criterion Optimization?

a technical point of view

p ₁

f

single optimal solution

(8)

scalar

optimization algorithm

Reduction Principles for Vector Optimization

vector optimization problem

scalar optimization problem

…

o b je c ti v e 1 o b je c ti v e n c o n s tr a in t 1 …

c o n s tr a in t m

• scalarization

(weighted obj., distance method, goal attainment)

• hierarchization

(hierarchical opt., compromise method)

• combination

(goal programming)

(9)

Example 1: Horizontal Platform Insulation

physical modeling

V

p

n

Q F

y A

Q

i

+ A

i

v

i

Q

a

+ A

a

v

a

p

L

V

p

n

Linearization

Equivalent Mechanical Model ( Q=0 )

₂

₂ D

₁ D

D

D D

D

u

G

x y = ∆ − ∆

∆

problem: increase of horizontal damping

( )

(

₁ ₂

)

2

2 1 0

1

2 2

2 2 2

1 1

1 1 1

0 1

,

p p q Q V

p p q y A V

V V p p n

A p p F

L L L

−

=

− +

−

=

−

=

−

=

−

=

ɺ ɺ ɺ

ɺ ɺ ɺ ɺ

( ) ^p ^A ^q ^y ^A ^y ^q ^Q ^dt

q p

A p F

L L

L

⁺ ^∆ ⁺ ^∆ ⁺ ^∆ ⁺ ∫

−

=

∆

=

∆

αβ α

αβ β

α ^ɺ

ɺ

1 1 0 0

0 1

( )

2 1 2

1 2

2 2

k k k

F F y k k y

d d

∆ + ɺ ∆ = ∆ + + ∆ ɺ

(10)

multi-measurement identification experimental

setup Simulink model

2

0

1 , ,

,

T i

F F

sim i meas i

where dt

T F meas i

ϕ

 

 

 

−

= ∫

0 ,

: = ∑

i

≥

i

w

u ϕ

objectives

1 ,

:

/ 1

0

 ≥



 



 −

= ∑ ^r

u

r r i

i

ϕ

optimization (Matlab)

F

meas

F

_sim

Example 1: Horizontal Platform Insulation

(11)

platform insulation

ˆ , ˆ

min 



 





℘

∈

f H

p

,

ˆ ˆ min 



 





℘

∈

T H

p

 





 





℘

∈

T f ˆ ˆ min

p D

M+F0ńg

F0

M

Ë₁

d₂ Ë₂ Dw Du_G

DxāDx^. Q

L R

k₁

d₂ k₂

Dw_L

M

Ë1

d2 Ë

2 Dw

Du_G

Dx_ā,ĂDx^.

Dw_R Du_G

Q k₂

d₂ k₁

passive

active

x D Q = ∆ ɺ

stream flow control

6 0 ≤ s

_BV

≤

bar 6 bar

4 ≤ p

₀

≤

[ ^p

₀

^s

_BV

]

^T

= p

0 ) Re( λ

_i

<

0.010 0.012

- ≤ D ≤

bar 6 bar

4 ≤ p

₀

≤

[ ^p

₀

^D ]

^T

= p

0 ) Re( λ

_i

<

bi-criterion problems

1 1.5 2 2.5 3 3.5 4

0 1 2 3 4 5

H^{^}

^f Ăā(Hz)

0 1 2 3 5

0.14 0.16 0.18 0.2 0.22 0.24 0.28 T^{^}Ăā(s)

1 1.5 2 2.5 3 3.5 4

0.14 0.16 0.18 0.2 0.22 0.24 0.28

H^{^} T^{^}Ăā(s)

3.26 3.27 3.28 0.201

0.202 0.203

passive

passive passive

active

( )

(

_i

)

i

T

i H i

H f

i H H

λ

ω ω

π ω

ω

ω ω

Re max ˆ 1

) ( max ˆ )

( :

2 ˆ / ˆ

) ( ˆ max

−

=

objectives to be minimized

Example 1: Horizontal Platform Insulation

(12)

f ₁ f ₂

f(P)

* j

f

i *

f

* *

: =   _i _j   F f f

Fβ β β β

n Fβ + t

n

single EP-solution

) ( p f n Fβ + t =

t

t P

min

∈ , p

s t . .

. const

= β

) (p f n Fβ + t =

t

t P

min

, , β p ∈

s t . .

1 ,

0 ∑ =

≥ _i

i β

β

knee search

Normal Boundary Intersection Approach

related to I. Das and J.E. Dennis

recursive knee search

(13)

2) lower and upper recursion limits

Recursive Knee Search

individual minima

• initial design initial design

knee search

global optimality local

optimality

stopping criteria

trade-off curve

2 λ / λ =

fulfilled

recursion loop

• stopping criteria

• bounds on knee search

( ) ( )

(0)

i

λ 1 ε rand 1,1

j i

∗ ∗ ∗

= +   + −   −

p p p p

( ^{0,1 ,} ) ^{0, min 1,} ¹ ¹

λ ε

λ

   

∈ ∈   −  

 

 

. . 0.2, 0.5

e g ε = λ =

| | t ≤ t , e g t . . = 0.1 1)

1)

2) β ∈ [ β

_min

, β

_max

], e g . . β ∈ [0.4, 0.6]

, ] 1 , [− 1

∈ t

• local optimality 1) solution feasible 2) local EP-optimality

3) limits on No. of non-successful restarts

Matlab applet

(14)

VIT-project (Virtual Turbomachinery, LuFo III)

Improvement of current Rolls-Royce compressor design process by tool integration and optimization

step 1: tool by tool analysis of current compressor design process step 2: automation and integration of

single analysis tools into an optimization environment step 3: partial automation of

multidisciplinary design process

Tool8

Tool7

Tool5 Tool1

Tool2

Tool3

Tool4 Tool6

iSight

the future: automation of whole multidisciplinary design process

Example 2: Compressor Design Process

(15)

Aerodynamics (Keskin)

c w

u

Meanline

Prediction Throughflow 2D Blading 3D Blading

Design 2D-Thermal Design and Stress (Otto)

High Cycle and Low Cycle

Fatigue

(16)

Meanline Prediction

annulus parameterization

0 5 10 15 20 25 30

4 6 8 10 12

Annulus Line

• classical

point-wise definition

• Reduced parameter definition

(Bézier-splines

with 4 control points)

0 10 20 30

5 6 7 8 9

Mid-Height Line

0 10 20 30

0 2 4 6

Thickness Line

2

,

2

,

3

,

3

,

2

,

2

,

3

,

3

T

x y x y x y x y

design parameters p =   b b b b t t t t  

Matlab applet

(17)

c, poly

max η

p

6 . 0

max _i ≤

i Ψ

1 . 1 max _I ^' ^R _, _i ≤

i

M

8 . 0 max _I ^S _, _i ≤

i

M

55 . 0 max _i ^R ≤

i

DF

55 . 0 max _i ^S ≤

i

DF

58 . 0 max _i ^R ≥

i

DH

58 . 0 max _i ^S ≥

i

DH 92

. 0

max _h _, _i ≤

i

C

27 .

, ≤ 0

S N E

_s

M ⁱ ^∈ { ¹ ^, ^… ^,N _s }

SM max

p

25 SM ≥ %

Meanline Prediction criteria

constraints

(18)

87.5 88 88.5 89 89.5 90 90.5

Pareto Opitimal Solutions for Meanline Annulus Modification

Datum, Human Designer iSight, MIGA, 4CP Pareto Optimal Design

Results for Bézier-splines with 4 control points

E3E

human design Meanline

Prediction

Polytropic Efficiency, η η η η

poly

[%]

S u rg e M a rg in , S M [ % ]

(19)

0 10 20 30 5

6 7 8 9

Mid-Height Line

0 10 20 30

0 2 4 6

Thickness Line

0 5 10 15 20 25 30

4 6 8 10 12

Annulus Line

4 control points allow

1 turning point, only !

E3E has two turning points

explanation for sub-optimal solution 1. parameterization too restrictive

5 control points Meanline

Prediction

(20)

poly

max η c, p

6 . 0

max _i ≤

i Ψ

1 . 1 max _I ^' ^R _, _i ≤

i

M

8 . 0 max _I ^S _, _i ≤

i

M

55 . 0 max _i ^R ≤

i

DF

55 . 0 max _i ^S ≤

i

DF

58 . 0 max _i ^R ≥

i

DH

58 . 0 max _i ^S ≥

i

DH 92

. 0

max _h _, _i ≤

i

C

27 .

, ≤ 0

S N E

_s

M ⁱ ^∈ { ¹ ^, ^… ^,N _s }

SM max

p

25 SM ≥ %

criteria

constraints

explanation for sub-optimal solution 2. constraints too restrictive Meanline

Prediction

0.93

0.285

(21)

Pareto Opitimal Solutions for Meanline Annulus Modification

Datum, Human Designer iSight, MIGA, 4CP

iSight, MIGA, 5CP Pareto Optimal Design

Matlab applet

Meanline Prediction

E3E

human design

Polytropic Efficiency, η η η η

poly

[%]

S u rg e M a rg in , S M [ % ]

(22)

E3E

max. efficiency E3E

max. surge margin Meanline

Prediction

(23)

Pareto Opitimal Solutions for Meanline Annulus Modification

iSight, NLPQL, 5CP Pareto Optimal Design

Meanline Prediction

E3E

human design

Polytropic Efficiency, η η η η

poly

[%]

S u rg e M a rg in , S M [ % ]

(24)

Pareto Opitimal Solutions for Meanline Annulus Modification

iSight, NLPQL, 5CP Pareto Optimal Design

SM increase 17.3%

η

_p

increase 0.17%

Meanline Prediction

E3E

human design Polytropic Efficiency, η η η η

poly

[%]

S u rg e M a rg in , S M [ % ]

(25)

pressure ratio

1 2 3 4 5 6 7 8 9

1 1.2 1.4 1.6 1.8

Parametric Stage Pressure Ratio Distribution

1 2 3 4 5 6 7 8 9

1 1.2 1.4 1.6 1.8

Applied Stage Pressure Ratio Distribution

2

,

2

,

3

,

3

,

4

,

4

,

2

,

2

,

3

,

3

,

4

,

4

,

x y x y x y x y x y x y

b b b b b b t t t t t t

=   p

1

,

2

,

2

,

3

,

3

,

4

,

4

,

5

^T

y x y x y x y y

p p p p p p p p 



Meanline Prediction

Annulus Line & Pressure Ratio Optimization

(26)

poly

c,

max η

p

6 . 0

max _i ≤

i Ψ

1 . 1 max _I ^' ^R _, _i ≤

i

M

8 . 0 max _I ^S _, _i ≤

i

M

55 . 0 max _i ^R ≤

i

DF

55 . 0 max _i ^S ≤

i

DF

58 . 0 max _i ^R ≥

i

DH

58 . 0 max _i ^S ≥

i

DH 93

. 0

max _h _, _i ≤

i

C

285 .

, ≤ 0

S N

E _s

M

M max S

p

{ ^, ^,N _s }

i ∈ 1 …

1 0 2 0

y

j

. ≤ p ≤ .

{ ¹ ^, ^, ⁵ }

j ∈ … c

max Π

p

criteria

constraints Meanline Prediction

25 SM ≥ %

ignored

Annulus Line & Pressure Ratio Optimization

(27)

21.5886 22.0712 22.5538 23.0363 23.5189 24.0015 24.4841 24.9667 25.4492 25.9318 26.4144 26.897

Π

ⁱ

*

Meanline Prediction

E3E

human design

Annulus Line & Pressure Ratio Optimization

Polytropic Efficiency, η η η η

poly

[%]

S u rg e M a rg in , S M [ % ]

(28)

Meanline Prediction

Annulus Line & Pressure Ratio Optimization

(29)

Comparison: some numbers

18.2h 839

(~8%) 10637

(~99%) 10768

NLPQL 5CP (Annulus)

94.1h 30.5h 19.3h total time

11909 (~28%) 42854

(~79%) 54000

MIGA 5CP (Annulus+PR)

5559 (~40%) 13743

(~86%) 16000

MIGA 5CP (Annulus)

4174 (~58%) 7166

(~90%) MIGA 4CP 8000

(Annulus)

feasible converged

function evals

17min 24

(~16%) 151

(100%) 151

NLPQL 5CP (Annulus) best efficiency Meanline

Prediction

(30)

Off-Design

Process Overview

Throughflow

(31)

-0.59293051 0.964143808

0.03 K 0.023 %

0.19 %

~2.1min 3.6 h

98 iSight (Case1)

1.01857 design parameter χ

0.83 K T criterion (∆T ≤ 1K)

-0.59 design parameter ∆D

0.53 % η criterion (∆η/η ≤ 1%)

0.22 % Π criterion (∆Π/Π ≤ 1%)

~21min time for one iteration

8 h overall opt. time

No. iterations (feasible) 23 human

eng.

min

D ,

T

χ

η η

∆

 ∆ 

 

 ∆Π Π 

 ∆ 

 

multi criteria problem

nonlinear programming

problem

^D

min u

∆ ,χ

where ( )

2 2

100 100

2

u η T

η

 ∆   ∆Π 

=  ⋅  +  ⋅  + ∆

 Π 

 

best fastest

-0.5 1.0 1.2 K 0.544 % 0.063 %

~2.1min 1.6 h

46 iSight (Case2)

0.95351 0.98 K -0.62655

0.162 % 0.032 %

~2.1min 0.76 h

22 iSight (Case3) Throughflow

eng.

(32)

20

02

20

40

40 40

40

40 60

60

60 60

60

60 80

80

80 80

80

0 10

0 10 100

100

0 12

0 12 120

120

0 14

0 14 140

140

0 16

160

0 16 160

160

0 18

0 18 180

180

0 20

0 20 200

200

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

A d d it io n a l D e v ia ti o n [d e g ] ∆ D

0.6 0.8 1.0 1.2 1.4

Pressure Loss Coefficient Factor χ [-]

0 20 40 60 80 100 120 140 160 180 200

Converged Solution

Engineer iSight

iSight

Throughflow

(33)

Conclusions

• by nature, technical design problems are multi-criterion optimization problems

• multi-criterion optimization cuts down costs by releasing human design engineer from time-consuming parameter studies without taking him off the decision process

Classification of Optimization Problems

Full text

Dieter Bestle and Akin Keskin

Chair of Engineering Mechanics and Vehicle Dynamics Brandenburg University of Technology Cottbus

Some Applications of

Multicriterion Strategies in Mechanical Engineering

Classification of Optimization Problems

optimization

topology optimization

shape optimization

sizing

scalar optimization

multicriterion optimization

► scalarization

► hierarchization

unconstrained constrained )

(

min p

p

f

R

∈ I

) ( min p

p

f

∈ P

{ I R h u o }

P = p ∈ g ( p ) = 0 , h ( p ) ≤ 0 , p ≤ p ≤ p

What is Optimization Good For?

min

-

 Ω 

 

Ω Ω

 

 

maximum stable speed range

: min . . max Re( ) 0 : min . . max Re( ) 0

s t s t

λ λ

Ω = Ω ≤

Ω = Ω >

{

}

min max ( ) , ( ) , ( )

x t x t x t

γ

ε minimum motion of inner balls

min T

ρ ϕ

. . ( ) ˆ s t s T = s minimum time

• Designing Birthday Presents

What is Optimization Good For?

• Designing Birthday Presents

• Solving Problems of Existing Machines

ideal contact geometry

max min

r z

δ δ

 

 

 

 

. . tan

s t ϕ ≤ µ

ideal ring shape min , ,

S w

γ γ +

r r r

. .

/ ,

s

S S G

t S

S γ

µ − ≤

=

S rk

D

H S S

{ Î ^R ^h û ô }

µ ⁻ ≤

{ Î ^R ^h û ô }

f ₁ (p)

f _n (p)

f ₁ f ₂

p ₁