Multipurpose Optimization with and withoutMultipurpose Optimization with and withoutUncertainties for Deformed Bodies andUncertainties for Deformed Bodies andStructural ElementsStructural Elements

(1)

University of Jyvaskyla , 2014

̈

University of Jyvaskyla , 2014

̈

Multipurpose Optimization with and without

Uncertainties for Deformed Bodies and

Structural Elements

A.V. Sinitsin, S.Yu. Ivanova, E.V. Makeev, and N.V. Banichuk

Ishlinsky Institute for Problems in Mechanics RAS

(2)

OUR TEAM - Laboratory of Mechanics and Structural Optimization

(3)

Pareto approch

J

(

h

)

=

{

J

₁

(

h

)

,. . . ,J

_i

(

h

)

,. .. ,J

_N

(

h

)

}

T

→

min

_{h∈ Λ}

h

h=argmin

_{h∈ Λ}

h

J

(

h

)

J

_C

(

h

)=

∑

C

_i

J

_i

(

h

)

→

min

_{h∈ Λ}

h

C

_i

≥0,

∑

C

_i

=1,i= 1, . .. ,N

(4)

Nash Approach

J

₁

₍

h

_1,

h

₂₎

,J

₂

₍

h

_1,

h

₂

₎

,h

₁

∈

Λ

_h

1

,h

2 ∈

Λ

h

2

Suppose h

₂

given

J

₁

₍

h

_1,

h

₂₎

→

min

_h

1

∈

Λ

_h

1

Optimal solution

h

₁

=argmin

_h

1

∈

Λ

_h

1

J

₁

₍

h

_1,

h

₂

₎

Suppose h

₁

given

J

₂

₍

h

_1,

h

₂

₎

→

min

_h

2

∈

Λ

_h

2

Optimal solution

h

=argmin

J

₍

h

₎

(5)

Uncertainty in the

problems of

multi-objective optimization:

objective optimization:

1. Guaranteed approach (minimax) -

implementation scenario "is based on worst

case";

2. Probabilistic approach (stochastic) -

applicable when available to the necessary

statistical data;

(6)

THE PROBLEM OF SUPERSONIC STRIKERS PENETRATION INTO DEFORMABLE MEDIA

Multi-layered nonhomogeneous plate and striker

1

1 ,

0 0,

i

n

(7)

DEFORMABLE MEDIA

Discrete family of given materials

( )

0

2 Piece-wise constant functions:

Dynamical hardness

Density

A x

−

(8)

DYNAMICAL HIGH SPEED PENETRATION OF RIGID AXISYMMETRIC

STRIKER

Resistance force

( )

,

( )

0

0 imp

dv

Mv

D x

v

dx

= −

= =

( )

2

(

( )

( ) ( )

2

)

0

2 D x

=

π

R A x

+

κ

A x v x

(9)

STRIKER IN GENERAL CASE

Resistance force in general case

( )

2

0

2

0

3

2

0 ₂

2

1 0,

, when 0

,

, when

,

, when

nose

lat

x

D x

D

x

D

x

B x

B x v

B x

r A x

A

yy d

A

yy

B x

r A x

d

y

x

x l

x

x l x

x

l x L

x

x l x

L

L x L l

η

π

η

π

η

∗∗ ∗ ∗∗ ∗

∗

∗∗

∗

∗∗

∗

∗∗

=

+

=

+

=

−

=

−

+

=

≤ <

= −

=

≤ ≤

= −

=

< ≤ +

∫

(10)

BALLISTIC LIMIT VELOCITY - BLV

Ballistic Limit Velocity

( )

0 :

if

is such that

0,

then 0

BLV

imp

n

_{x L}

imp

BLV

v

=

(11)

PENETRATION OF AXISYMMETRIC STRIKER

WITH SMALL OGIVE NOSE PART AND CYLINDRICAL REAR PART

Spaced nodes used for solution of the

initial-value problem

,

L

x

d

dx

(12)

BASIC RELATIONS

(

)

( )

0

2

1

2

0

1

0

2

1

2

2 ,

0 ,

,

0,1, 2,...,

1

2 j

j

n

_L

BLV

dv

v

d

v

j

n

A

R

A

M

v

ξ ξ

β

α

ξ

α

κ

π κ

β

+

=

+

= =

=

+

=

= + ∆

=

−





=  ÷





=

(13)

EVALUATION OF BALLISTIC LIMIT VELOCITY

Integration w.r.t. ξ

Relation between v

_j2

_{and v}

j+12

( )

1

2

1

2

1

1 j+1

1

2

1

0 ln

,

where

,

0,1,2,...,

1 exp

0,

j

n

BLV

v

j

n

v

α

µ

ξ ξ ξ

α

µ

β

ξ

µ

α

+



₊



=

≤ ≤



÷



₊

÷





=

∆

∆ =

−

=

−

=

− +

=

(14)

( )

(

)

( )

(

)

1

2

0

1

2 min

,

,...,

- total mass

...

- total thickness

h

v

M

h

L

v

BLV

n

L

M

L

n

J

J h

J

v

t

J

S k t x dx

J

L

∗

_∈Λ







=

_

_







= −

∆

∆ = ∆ ∆

∆

=

= = ∆ + ∆ + + ∆

∫

MULTIOBJECTIVE OPTIMIZATION PROBLEM

v-blv - ballistic velocity , delta – variable depended on the solution

(15)

( )

(

)

( )

(

)

(

( )

)

[

)

(

)

{

( )

}

1

2

1

0

2

1

0

2

0

2 ,

, ,

,...,

:

;

,

0,1,2,...,

1;

,

1,2,..., ;

:

0;

1,2,...,

n

i

h

_s

j

h

t

t A t x

A

A t x

A

x

x x

i

n

A

s

r

j

n

+

= ∆ = ∆ ∆

∆



=

∈





Λ =

_

=

−

∈

_



₌

_{∆ ∆ >}

₌







DESIGN VARIABLES

(16)

OPTIMIZATION IN PARETO SENSE

h

_*

=

arg min

_h∈Λ

h

J(h)

h

_*

is optimal if there is no other such that ̃h

J

_i

(̃

h)≤J

_i

(

h

_*

)

,i=μ, M , L

and most for one criteria compomemt

(17)

( )

0,

1 arg min

h

C

v v

M

L L

v

M

L

v

M

L

C

_h

C

J

C J

C

J h

_∗

J h

∈Λ

=

+

≥

+

=

METHOD OF OBJECTIVE WEIGHTING

(18)

(

)

(

)

1

2 ...

20 min

1 min

t

n

v

t

M

C

v v

M

v

M

_t

L

n

J

C J

C

J

C J

∈Λ

∆ = ∆ = = ∆ =

=





=

_

_

→





=

+

= −

+

→

OPTIMAL DESIGN BY GENETIC ALGORITHM Cv>0 summ cv=1

(19)

( )

6

2

3

0 ₁

_{2 1}

6

2

3

0 ₂

_{2 2}

6

2

3

0 ₃

_{2 3}

aluminum

1 A

350 10 N/m ,

A

2765Kg/m

soft steel

2 A

1850 10 N/m , A

7830Kg/m

copper

3 A

910 10 N/m ,

A

8920Kg/m

duraluminum

4 s

s

=

×

=

×

=

×

=

( )

6

2 ( )

3

0 ₁

_{2 1}

A

=

1330 10 N/m ,

×

A

=

2765Kg/m

OPTIMAL DESIGN BY GENETIC ALGORITHM

(20)

OPTIMAL DESIGN BY GENETIC ALGORITHM blv - ballisti limit velosity

(21)

(dependence ob different values of functional weights)

Materials distributions:

1) C

_M

=0-0.2

2) C

_M

=0.3

3) C

_M

=0.4

4) C

_M

=0.5

5) C

_M

=0.55

(22)

OPTIMAL DESIGN BY GENETIC ALGORITHM

Materials distributions:

6) C

_M

=0.57

7) C

_M

=0.58

8) C

_M

=0.59

9) C

_M

=0.6

10) C

_M

=0.605

(23)

OPTIMAL DESIGN BY GENETIC ALGORITHM

(24)

OPTIMAL DESIGN BY GENETIC ALGORITHM

(25)

OPTIMAL DESIGN BY GENETIC ALGORITHM

Dependence of the striker velocity on the depth of

penetration when C

_M

=0

(26)

SOME NOTES AND CONLUSIONS



Optimal layered structures are found in frame of

multiobjective problem statement.



Global optimal structures are defined taken into account

discrete set of available materials.



Defined optimal layered structures provide maximal

ballistic limit velocity (BLV) of rigid strikers and have

minimal total mass.



To solve this nonlocal optimization problem the

numerical method of genetic algorithm is effectively

used.



Entrance and exit effects for penetration in each layer of

optimizing structure are ignored in this investigation.

(27)

Contact Optimization Problems with Uncertainties

GEOMETRY OF THE CONTACT INTERACTION

Contact interaction under external loading

0

1

0

0 0,

0,

0 f

q

i N

i

q

i

f

q

i

q

=

Ω = Ω + Ω + Ω

Ω =

Ω

Ω = Ω

Ω =

Ω

Ω =

U

I

(28)

STATEMENT OF THE PROBLEM

Design variable and external forces

( )

( , ), ( , )

;

0, ( , )

( , , , ),

,

, , ;

1,2,...,

f

i

j

q

z

f x y

x y

z

x y

q

q x y

x y

j x y z i

N

ξ η

=

∈Ω =

∈∂Ω

=

∈Ω

=

(29)

STATEMENT OF THE PROBLEM

Boundary conditions and reactions

( )

(

)

(

)

(

)

0 0, ( , )

( , , , ),

,

( , , , )

,

( , , , )

,

( , , , )

f f f

xz

yz

zz

i

xz

x

yz

y

zz

z

q

f

x

f

y

f

x y

q x y

x y

P

p x y

d

M

yp x y

d

M

xp x y

d

σ

ξ η σ

ξ η

Ω

=

∈Ω

=

∈Ω

=

Ω

=

Ω

=

Ω

∫

(30)

STANDARD PUNCH AND ACTUAL PUNCH

Standard punch with plane bottom

0

0 ( , )

( , )

, ,

given coefficients

( , )

( , ) 0,

( , )

( , ),

( , )

( , ),

( , )

(

f

x

y

z

q

f

z

f x y

x

y

x y

q x y

x y

p x y

x y

u x y

v x y

v x

α

β

γ

α

β

γ

α

β

α β

γ

α β γ

α

β

γ

α

β

γ

α

β

=

= +

+

∈ Ω

−

=

∈ Ω

=

+

∈ Ω

=

+

=

+

0

0 , )

( , ),

( , )

( , ),

y

v x y

w x y

γ

α

β

γ

α

β

γ

+

=

+

Actual punch

( )

0 ( , )

0,

( , )

0,

( , )

( , , , ),

,

xz

yz

f

xz

yz

zz

i

xz

x

i

yz

y

i

zz

z

i

q

f

f x y

w

f x y

x y

q x y

x y

σ

ξ η

σ

ξ η

σ

ξ η

=

= =

∈ Ω

=

∈ Ω

=

∈ Ω

(31)

RECIPROCITY RELATION

The first system

0

0 ( , ),

( , )

, ,

( , )

1, 2,...,

f

i

q

w

f x y p

x y

u v w

x y

i

N

=

∈ Ω

=

The second system

(

)

0

1 ( , ),

( , )

, ,

( , )

f f i q

f

i

x

y

z

q

f

N

i

x

y

z

q

i

w

f x y p

x y

q q q

x y

p fd

f pd

u q

v q

w q d

Ω

= Ω

=

∈ Ω

Ω =

Ω +

+

Ω

∫

∑ ∫

(32)

RECIPROCITY THEOREM

(

)

(

)

(

)

0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

0

f f i q f f i q f f i f f N i i i i x y z q i f f N i i i i x y z q i f f N i i i i x y z q i

pd

p fd

u q

v q

w q d

xpd

p fd

u q

v q

w q d

ypd

p fd

u q

v q

w q d

α α α α β β β β γ γ γ γ

α

β

γ

Ω Ω = Ω Ω Ω = Ω Ω Ω = Ω





Ω −

Ω +







+

Ω +







+

Ω −

Ω +







+

Ω +







+

Ω −

Ω +







+

Ω =



∫

∑ ∫

∫

∑ ∫

∫

∑ ∫

( )

(

)

( )

(

)

( )

(

)

0

1

0

1

0

0 ,

,

f f f i q i q

f

q

f

q

j

f

x

f

y

f

N

i

q

x

y

z

q

i

N

q

i

x

y

z

q

i

N

q

i

P P f

P q

M

f

M q

j x y

P f

p fd

M

f

p fd

M

f

p fd

P q

u q

v q

w q d

M q

u q

v q

w q d

M q

u q

v q

w q d

α

γ

β

α

γ

Ω

= Ω

=

−

=

−

=

Ω

=

Ω

=

Ω

=

+

Ω

=

+

Ω

=

+

Ω

∫

∑ ∫

(33)

PROBABILISTIC APROACH

Probability densities and joint probability distribution functions

2 ( , )

0, ( , )

1 ( , )

( , ), ( , )

( , )

q

i

q

i

q

F

g

d

ξ η

_{ξ η}

π

ξ η

π

Ω

∂

=

∂ ∂

∉ Ω



= 

_{∈ Ω}



Ω =

∫

%

(34)

RANDOM LOAD

(

)

{

}

(

)

( ) ( )

1

2

1

2

1

2

1

2

1

2

1

2

2 , , ,

,

, ,

,

the only domain

,

0,

,

0,

,

1

1 ( )

,

( )

,

0,

,

0,

( )

j

q

q x y

j x y z

x

x x y

y y

g g

g

x

y

g

x

g

y

x

y

x

y

F

ξ

η

ξ

η

ξ

ξ η

ξ

η

ξ

η

ξ

η

ξ

η

ξ

=

Ω

≤ ≤

−

=



<



<



=

_

< <

=

_

< <

−



_>



_>



=

(

)

(

)

1

2

1

2

1

2

1

2

2 0,

,

0,

,

( )

,

1,

,

1,

ˆ ˆ

ˆ

Mathematical expectations ,

_x

,

_y

x

y

x

y

x

F

y

x

y

x

y

P M M

η

ξ

η

ξ

_ξ

_η

η

_η

ξ

η



<



<



₋



₋



_{< <}

₌



_{< <}



₋



₋



_>



_>



(35)

FORMULATION OF OPTIMIZATION PROBLEM

Minimizing functional and constraints

( )

2

1 min

2 ˆ

_,

ˆ

_,

ˆ

,

given problem parameters

:

f f

f

x

y

x

y

f

J

f d

S

f d

P P

M

P M M

x

y

a

ρ

Ω

∗

=

+ ∇

Ω ≈

≈

+

∇

Ω →

=

−

Ω +

≤

∫

Lagrange augmented functional

( )

(

)

( )

(

)

( )

(

)

2 0 0 0 0 2 2 2 0 2 2 2 0 2 2 2 2 2

2

1

2 cos

,

1

2 sin

,

1

f L f

J

f

p f

p f d

E

p r

a

r

Er

p r

a

r

Er

p r

a

r

x

y

α α β β γ γ α β γ

ρ

λ

π

ν

θ

π

ν

θ

π

ν

Ω



=

_

∇

−



−

_

Ω

=

−

=

−

=

−

=

+

∫

(36)

SOLUTION OF OPTIMIZATION PROBLEM

Optimality condition and

boundary condition

2

0

1

1 ( )

( , )

( ( , ))

_{r a}

0, 0

2 f

f

r

r r r

p r

f r

β

γ

α

β

γ

θ

λ

_θ

ρ

θ

₌

θ

π

∂

∆ ≡

+

=

∂

= −

−

= ≤ ≤

Bounded solution and

shape function

( , )

( )

( , )

( , ),

( )

( ),

( , )

( , ),

( , )

f r

r

f r

r

f r

r

α

β

γ

α

α α

β

β β

γ

γ γ

θ

λ χ

θ

λ χ

θ

λ χ

θ

=

+

=

(37)

SHAPE FUNCTIONS

Symmetric shape function

(

)

2

2 ( )

ln

1 E

a

r

a

r

a

α

χ

ρπ

ν



₊

₋





÷

=

_

−



_

÷

_÷

−

_

_

(38)

ANALYTICAL SOLUTION FOR

i=

β,γ

Finding of nonsymmetric

shape functions

2

0

1

2 ( )cos

( , )

( )cos

r

r r

r

rp r

r

β

α

β

χ

θ

ρ

χ

θ

χ

θ

∂

+

=

∂

= −

=

(

)

₍

₎

(

)

2

0

2

3

0

2

2 ₂

2

3

1

2 ,

1

2 ( )

3

2

4

2 ,

3

3 d

d

dr

r dr

r

K d

E

a

r

K

dr

K

C

r

a

r

D

K

Ka

C

a D

χ

_χ

ρ

π

ν

χ

ρ

+

−

=

−

=

−

= −

−

+

= −

=

(39)

SHAPE FUNCTIONS

Nonsymmetric shape function for i=

β

(

)

(

) (

)

3 0 2 2 2 _{2 2} 2

2 cos

( , )

( )cos

3 1

E

r

a a

r

a

r

β

θ

χ

θ

χ

θ

π

ν ρ





=

_

−

_

−

_

_

(40)

SHAPE FUNCTIONS

Nonsymmetric shape function for i=

γ

(

)

(

) (

)

3 0 2 2 2 _{2 2} 2

2 sin

( , )

( )sin

3 1

E

r

a a

r

a

r

γ

θ

χ

θ

χ

θ

π

ν ρ





=

_

−

_

−

_

_

(41)

OPTIMAL PUNCH SHAPES

Uniform distribution of probability density

(

2

)

2 2 max 2

,

1,

0.25

1 1.7148

0.5810

0.58 0.4796

x y

E

f

Kf

K

P

M

K

f

K

α α β β γ γ

π

ν

λ

ρ

λ

ρ

λ

ρ

∗ ∗ ∗

=

−

=

≈

=

≈

=

≈

%

(42)

OPTIMAL PUNCH SHAPES

Gauss distribution of probability density (x

₀

=3, y

₀

=0)

(

2

)

2 2 max 2

,

1,

0.25

1 1.7067

0.5745

0.58

x y

E

f

Kf

K

P

M

K

f

K

α α β β

π

ν

λ

ρ

λ

ρ

∗ ∗ ∗

=

−

=

≈

=

≈

%

(43)

SOME CONLUSIONS



Contact interaction of rigid punch and elastic foundation

has been investigated under conditions of incomplete

information regarding the external actions.



Shape optimization problem with uncertainties has been

formulated and studied in frame of probabilistic approach.



With application of Betti’s reciprocity theorem the original

probabilistic optimization problem has been simplified.



Optimal shapes of the punches have been found

analytically for different cases of the problem formulation.



Some peculiarities of the optimized model have been

studied.

(44)

Multipurpose Optimization with and

without Uncertainties

Mixed boundary value

problem

(

)

0

0 :

,

:

,

f

q

n

q

n

f

q

w

f x y

p

τ

σ

τ

µ

Ω = Ω + Ω + Ω

Ω

=

Ω

=

Ω

=

= +

Contact interaction of a moving rigid punch

and elastic half-space

Ω

₀

Stamp-shell

Deformable environ

Ω

0

Ω

q

Ω

q

Ω

_f

Ω

_f

Ω

0

Ω

0

Ω

0

V≠0

V=0

q

(45)

FUNCTIONAL CHARACTERISTICS

The total force of the contact

pressure

(

)

(

)

*

0 .

2 ( , )

,

( , )

,

f f f f

f

F

f

n

m

W

g

f

P

p x y d

M

xp x y d

J

p Vd

W

K p V

J

p p

d

τ

µ

Ω

=

Ω

=

Ω

=

+

Ω

=

−

Ω

∫

Power dissipation caused by

friction

The total moment

The wear rate of the material

The functional differences

*

{ ( , ) 0,

( , )

,

( , )

},

0,

0

f f

p

f

p

p x y

p x y d

P

xp x y d

M

P

M

Ω

∈Λ =

≥

∫

Ω =

∫

Ω =

>

(46)

Problem Formalization of Mutipurpose Punch Shape Optimization

,

*

,

{ ,

}

Найти функцию ( , )

( ) min ( )

при условии

{ ( , ) 0,

( , )

},

0,

0 Здесь оператор min рассматривается в смысле Паретто

arg min ( )

f f

T

g

F

w

f

p

f

J

J J J

f x y

J

J f

p

p x y

p x y d

P

xp x y d

M

P

M

f

J f

Ω

=

:

=

∈Λ =

≥

Ω =

>

=

∫

Find Function

under condition

(47)

Problem 1 (Finding )

( )

* * * *

{ ,

,

}

min

{ ( , ) 0,

( , )

,

( , )

},

0,

0 ,

,

p f f T g F W _p p f f

J

J J J

p

p x y

p x y d

P

xp x y d

M

P

M

p x y

w x y

f x y

w x y

∈Λ Ω Ω ∗ ∗ ∗ ∗

=

→

∈Λ =

≥

Ω =

>

⇒

=

∫

Decompositon

Problem 2 (Finding )

f x y

∗

(

,

)

(

,

)

p x y

_∗

(48)

Optimal distribution for a rectangular stamp

Optimal Pressure Distribution

( )

(

) ( )

( )

(

)

( )

{

}

( )

(

) (

)

0

2

0 ,

,

/ 2

,

/ 2

,

1 1 2

1 ,

,

2 y

C

f

C

CC

f

C

f

y

P

M

p x y

x

S

I

f x y

C Ax D x y

AD x y

C Ax D

x y

AD

x y

D

x y

P

M

С

A

S

I

E

κ

µκ

τ κ

ν

κ

π

∗

=

+

=

+

−





−

_

+

_

−

+

−

=

(49)

Shape Functions for Rectangular Shape

(50)

Pareto Optimization Front Construction

2

1

2

1

2

1 (

)

f

g

F

g

f

J

V

p d

V S

S

meas

κ

µ

κ

µ

Ω

=

−

=

Ω

=

Ω

∫

(51)

Optimal Rectangular Punch Shape

(52)

Dependance on Coefficient

С

_g

(53)

displacements of a free end of console beam

(54)

Multipurpose minimization of horizontal and vertical displacements of a

free end of console beam

(55)

free end of console beam

(56)

References

1. N. V. Banichuk, S. Yu. Ivanova. Optimization problems of contact mechanics. Mechanics

Based Design of Structures and Machines., 37(2):144–156, 2009.

2. N. V. Banichuk, S. Yu. Ivanova, E. V. Makeev. Finding of rigid punch shape and optimal

con-tact pressure distribution. Mechanics Based Design of Structures and Machines., 38(4):417–

429, 2010.

3. N. V. Banichuk, S. Yu. Ivanova, E. V. Makeev, A. V. Sinitsin. New approach for contact

optimization problems with friction and wear. In Proceedings of the XXXIX Summer

School-Conference APM (2011) (Advanced Problems in Mechanics) , pages 80–86, St.Petersburg,

2011.

4. N. V. Banichuk, S. Y. Ivanova, F. Ragnedda and M. Serra. Multiobjective Approach for

Op-timal Design of Layered Plates Against Penetration of Strikers. Mechanics Based Design of