recent advances in nonlinear response structural optimization using … · recent advances in...

1

Recent Advances in Nonlinear Response Structural Optimization Using the Equivalent Static Loads

Method

Oct. 28, 2014 VR&D 2014 Users Conference

Monterey, CA

Gyung-Jin Park Professor, Hanyang University

Ansan City, Korea

Optimization

2

• Problem formulation is important.

• Do we have to understand the details of the optimization theory?

• Various software systems with various algorithms are available.

UL

,...,10)(,...,10)( subject to

)( minimize toFind

bbb

bbb

b

≤≤

=≤==

∈

mjglih

fR

j

i

n

Linear Static Response Structural Optimization

3

• Popular

• Easy

• Well developed software systems are available.

UL

,...,10),()(: subject to

),( minimize to,Find

bbb

zbfzbKh

zbzb

≤≤

=≤=

∈∈

mjg

fRR

j

ln

Linear Static Response Structural Optimization

4

• Size Optimization: The FEM data are fixed.

• Shape Optimization: Node and element data of FEM analysis are changed

during optimization.

• Topology Optimization: Material distribution is optimized.

Not – Linear Static Response Structural Optimization

5

(1) Dynamic Response Optimization

(2) Structural Optimization for Multibody Dynamic Systems

(3) Structural Optimization for Flexible Multibody Dynamic Systems

(4) Nonlinear Static Response Structural Optimization

(5) Nonlinear Transient Response Structural Optimization

fzbKzbMh =+ )()(:

fzzbKh =),(:

system dynamicmultibody ofequation Governing:h

fzzbKzzbMh =+ ),(),(:

systems dynamicmultibody flexible ofequation Governing:h

The Equivalent Static Loads Method

6

Not a Linear Response Analysis

Linear Static Optimization

Disp. Field

Multiple Loading Conditions

Equivalent Static Loads

Updated design

Analysis Domain Design Domain

• This method has been developed for not-linear static response structural optimization.

• Analysis is performed in the analysis domain.

• Equivalent loads are calculated.

• Linear response optimization is performed using the equivalent static loads in the design domain.

• The process proceeds in a cyclic manner.

Contents

7

(1) Linear Dynamic Response Optimization

(2) Structural Optimization for Multibody Dynamic Systems

(3) Structural Optimization for Flexible Multibody Dynamic Systems

(4) Nonlinear Static Response Structural Optimization

(5) Nonlinear Transient Response Structural Optimization

8

Nonlinear Static Response

Optimization Using Equivalent Loads

(NROEL) – Installed in NASTRAN

Nonlinear Response Optimization

9

General Formulation

,m,ig

f

i 1;0)()(subject to)(minimize to

Find

=≤=

zb,fzzb,K

zb,b

NROEL

10

Definition: An Equivalent Load is a load in a linear static system that makes an identical response to that in a nonlinear system.

Design-Oriented Loads

displacement stress etc.

Responses in optimization formulation

Flow of NROEL

11

START

? No

Nonlinear Analysis fzzb,K =NN )(

Calculate Equivalent Loads

NLeq zKf =

Update Design

Linear Response Optimization

,m,ig

f

Li

eqLL

L

1;0)()(subject to

)(minimize toFind

=≤

=

zb,fzbK

zb,b

END

Optimization process using equivalent loads

ε≤− − )1()( keq

keq ff

k=0 k=k+1

The one for stress constraints is separately defined.

Shape Optimization 1

12

11.22E+6 N

b1

Geometric and Material Nonlinearity -Linear hardening

E = 200.0 GPa σy = 300.0 Mpa Eh = 50.0 GPa

Shape change – using domain element

2. A plate


2001;00.1.350/)(subject to

Massminimize tochange) shape(Find 321

,,j

, b, bb

j

NN

=≤−=

σfzzb,K

b2

b3

b1

b2 b3

<D.V. and perturbation of the shape >

<Applied loads and boundary conditions>

Shape Optimization 1

13

Results of Optimization Initial design

Conventional method

NROESL Initial design

Con. Meth. (215)

NROEL (7)

Large Scale Size Optimization 1

14

Only Geometric Nonlinearity E = 68.9 GPa

Loading and Boundary Conditions

4. 280 shell structure


( )( )315,,1005.0

280,,100.1.60/)(subject to

Massminimize to)280,,1(Find

=≤

=≤−=

=

j

i

it

j

i

NN

i

δ

σfzzb,K

NRO using EL Abaqus 6.4 – Optistruct 7.0

Large Scale Size Optimization 1

15

Results of Optimization

< Design history graph > < Optimum thickness contour >

Optimization using the conventional method is fairly expensive.

0.5kN 0.5kN

0.5kN 0.5kN

16

Shape Optimization with Linear Contact

6. A Ring: Problem Definition

Problem model

- Loading condition: The forces are applied at the elements of the upper parts.

- The element property: PSOLID

- The total number of elements: 672 ( 64 CPENTA + 608 CHEXA)

- Only the boundary nonlinearity is considered.

- NASTRAN is used for contact analysis and linear response optimization.

- NASTRAN DMAP is utilized for calculating the equivalent loads.

Solver: SOL 101

- Linear contact: Linear analysis + Nonlinear contact parameters

cΓcontact boundaries ( )

17

Design condition

Design perturbation vectors

- Perturbation vectors are utilized for shape change in shape optimization.

- Each arrow is a perturbation vector.

- Seven design variables are selected based on the perturbation vectors.


b1

b2 b2

b1 b3 b3

b4 b4

b5 b5

b6 b6

b7

)672,,1(0KPa0.2subject tomassmin. to

ring) theof(shape,,,Find 721

=≤− i

bbb

iσ

Formulation

18

Optimization result – shape change


Initial shape Optimum shape

19

Nonlinear Dynamic Response Optimization Using Equivalent Static

Loads – Installed in GENESIS and OptiStruct

NDROESL

20

Equivalent Static Loads for Displacement

App

lied

forc

e

Time Time step: Load set:

0t 1t 2t nt0s 1s 2s ns

)t()t()t()t( NNN f)zzK(b,zM =+

)t()t( NLz zKf =eq

)s()s( zLL eqf(b)zK =

Nonlinear transient analysis

Transformation to equivalent loads

Linear analysis with ESLs

Linear response optimization

)t(Nz

st →

Find

to minimize

subject to

0zz ≤− allowableL )s(

b)z(b L,F

)s()s(LL eqf(b)zK =

newb

Flow of NDROESL

21

START

? No

Nonlinear Dynamic Analysis

)()()( tt(t) NNN fzzb,KzM(b) =+

Calculate Equivalent Static Loads

)()( tt NLeq zKf =

Update Design

Linear Response Optimization

,m,igqttt

f

Li

eqLL

L

1;0)(

,,1);()()(subject to)(minimize to

Find

=≤

==

zb,fzbK

zb,b

END

Optimization process using equivalent static loads

ε≤− − )1()( keq

keq ff

k=0 k=k+1

Nonlinear Transient Size Optimization

22

Loading and Boundary Conditions

1. 160 shell structure


( )205,,1005.0

)(subject toMassminimize to

)160,,1(Find

=≤

=+

=

j

it

j

NNN

i

δ

fzzb,KzM(b) NRO using ESL

Abaqus 6.4 – Optistruct 7.0

Time Loading time: 0.01 sec Total analysis time: 2.0 sec

Only Geometric Nonlinearity

Nonlinear Transient Size Optimization

23

Results of Optimization

< Design history graph >

< Optimum thickness contour >

Initial

Optimum

< Tip displacement >

Roof Crush Optimization

24

3. Roof crush problem - Ford Explorer Model: developed by the GWU

Finite Element Model

· Number of Parts : 394

· Number of Elements : 432,596

· Number of Nodes : 431,629

· Number of total DOFs : 2,589,774

FMVSS 216 Standard (Roof crush resistance)

The current FMVSS 216 standard requires that a passenger car roof withstand a load of 1.5 times the vehicle’s unloaded weight in kilograms multiplied by 9.8 or 22,240 Newton’s, whichever is less, to either side of the forward edge of the vehicle’s roof with no more than 127 mm of crush.

25


Definition of design variables

DV 1: Thickness of A-Pillar (t1)

DV 2: Thickness of B-Pillar (t2)

DV 3: Thickness of Roof-Rail (t3)

< Roof Structure >

DV 1

DV 3

DV 2

26


0.23dv6.0.022dv.60.021dv.60

67.5ms)(t 0.0 crush roof theof distance127mm)(subject to

massminimize to )3,2,1(Find

step

≤≤≤≤≤≤

=≤−=+

=

fzzb,KzM(b) NNN

i it

Modify

Modified formulation for the ESL method

.023dv6.0.022dv.60.021dv.60

127mm) crush (roof 0.0forcewallrigidweight1.65)(subject to

massminimize to)3,2,1(Find

≤≤≤≤≤≤

=≥−×=+

=

fzzb,KzM(b) NNN

i it

Roof Crush Optimization Using RSM and ESL

27

Response Surface Method · Software : LS-DYNA 971, LS-OPT · Linear + Interaction terms are used for RSM. · D-Optimal method as the sampling method is used. · The number of experimental points is eight. · Nonlinear analysis time : about 30 hours (1CPU) for a full car model · The CPU time per 1 iteration : about 240 hours

· LS-DYNA 971 is used for the roof crush analysis. · DMAP of NASTRAN 2006 is used for the calculation of equivalent static loads. · NASTRAN 2006 is used for linear static optimization using equivalent static loads. · Nonlinear analysis time : about 30 hours for a full car model · Linear optimization time : about 6 hours · The CPU time per 1 cycle : about 36 hours

Equivalent Static Loads Method

* Equipment of solver : HP-UX Itanium II (4CPU)

28

Results

4 Number of iterations

0.6 mm 0.6 mm 1.0 mm DV 3

3.329 kg 3.346 kg 4.481 kg Mass

5 33 Number of nonlinear analyses

5 Number of cycles

0.96 mm 0.6 mm 1.1 mm DV 2 0.86 mm 1.16 mm 1.2 mm DV 1

ESL result RSM result Initial model

+0.7% -11.2% Constraint violation

Roof Crush Optimization Using RSM and ESL

Total CPU time (1CPU) 990 hours 180 hours

29

Nonlinear Dynamic Response Topology Optimization Using the

Equivalent Static Loads

30

Issues of Nonlinear Dynamic Response Topology Optimization

Example of unstable elements under large deformation

• Low-density elements appear during and even after the optimization process.

• Low-density elements cause excessive mesh distortion.

• This phenomenon leads to many Newton-Raphson iterations or divergence in the numerical analysis.

F1=0.03

F2=0.02 E=1

E=10-3

10

10

Negative

area

(a) Finite element model (b) Deformed shape

* Mesh distortion problem

31

Issues of Nonlinear Dynamic Response Topology Optimization

• The purpose of topology optimization

Maximization of the stiffness of the structure = Minimization of the compliance

• The general objective function for linear topology optimization fTz

• When topology optimization in the time domain is performed, the objective functions are as follows:

1) The weighted summation compliance

2) The weighted summation compliance near the peaks

* Definition of the objective function

( ) luωl

uuuu ...,,1=;∑

1=

Tzf

( ) puωp

uuuu ...,,1=;∑

1=

Tzf : the number of time steps in the time domain

: the weighting factor

: the magnitude of the dynamic load vector at the uth time step

: the displacement vector of the uth time step

: the number of time steps near the peaks

p

ωl

u

u

u

zf

32

ESLSO

* ESLSO for nonlinear dynamic response topology optimization

)(=)())(,(+)()(+)()(

NNN

NN

ttttt

fzzbKzbCzbM

Nonlinear dynamic analysis

)()(=)( NL tseq zbKfCalculate ESLs

( )

1≤≤<0

≤

)(=)()(s.t.

min. to

...,,1=;Find

min

1=

LL

1=

T

∑

∑

i

n

iii

eq

p

uuuu

i

bb

Vbv

uu

ω

nib

fzbK

zf

Linear topology optimization

Transform into transformation variables

Start End

Update design

( ) 32)1()(countif εεαα ×− − nk

ik

i ≤≥

k=k+1

No

Yes

>≤

=1

1

when1when0

εε

αi

ii b

b

33

Example (I): A Plate Fixed along Both Ends

* Problem definition

• Information of the problem

- The load duration time: 0.1 sec.

- The maximum magnitude of a dynamic load: 10 kN

- Commercial software: ABAQUS (analysis), GENESIS (optimization), NASTRAN (ESLs) 1.6 m

0.2 m

x

y

F(t)

Problem description

Dynamic load profile Bilinear elastoplastic stress-strain curve

E = 210 GPa

Eh = 105 GPa

34

Example (I): A Plate Fixed along Both Ends

* Optimization results

Linear optimization

Nonlinear dynamic optimization

considering geom.

nonlinearity

considering mat.

nonlinearity

considering geom. & mat.

nonlinearities

No. of iterations 19 - - -

No. of cycles - 3 4 3

No. of nonlinear

dynamic analyses - 3 4 3

Optimization results for a plate fixed at both ends

a) Linear static topology optimization

c) Material nonlinearity

b) Geometric nonlinearity

d) Material and geometric nonlinearity

A B

B A

35

Example (II): A Crash Box for Crashworthiness

* Background

• Crash box

- Location: Between the bumper rail and the side rail

- Role: Preventing the transmitted impact energy to the vehicle body by absorbing the energy in the event of a crash.

• RCAR (Research Council for Automobile Repairs): International organization that works toward reducing insurance costs by improving automotive damageability, repairability, safety and security.

Radiator

Bumper rail

Side rail

Crash box

Rigid wall

15 km/h

40% offset

Overall width

Head lamp

10˚

RCAR test conditions of the frontal structure

36


* Problem definition

• Geometric, material and contact nonlinearities are considered.

• Design domain: Only the crash box

• The objective function: Maximizing the strain energy of the crash box at some time steps near the end time of the impact.

Side rail Crash

box

x

y

15 km/h

Rigid

wall

d1 d2

Commercial software: LS-DYNA (analysis), GENESIS (optimization), NASTRAN (ESLs)

Finite element model of the crash box

1≤≤<0

≤

≤

%50×≤

...,,1=);(=)()(s.t.

energystrain max. to

6804...,,1=;Find

min

allowable,22

allowable,11

6804

1=ref

LL

1=

∑

∑

i

iii

eq

q

ss

i

bb

dd

dd

Vbv

qsss

ib

fzbK

Formulation

37


* Optimization results

• Optimization process converges in 7 cycles.

• Part A: Primary contacted part when the crash box is impacted on the rigid wall

• Part B: Increasing the absorbed energy of the crash box

y

z

x

(b) Isometric view

A

B

y

z A B

(a) Side view

Optimization results of the crash box

Original design

38

Sheet Metal Forming with a Tapered Square Cup

Geometric description of the tooling for the oblique square cup

• The tooling: the die, the punch and the blank holder • Blank holding force: 100 kN • Stroke of the punch: 40 mm (-z direction) • Wrinkling occurrence part: the side-wall • Reason of wrinkling occurrence at the side-wall - The geometry of the wall is not constrained by the die and the punch.

x

y

Die

114.3

Punch

R31.75

R50.8

152.4

y

z R6.35

R6.35

Punch

Die

Blank holder

Blank

* Formulation

0.10tosubject

)700...,,2,1(minimizeto)21...,,2,1(Find

≤

==

j

j

i

D

jsib

• bi: the scale factors for the perturbation vectors

• sj: standard deviation

• i: the perturbation vector number

• j: the sampling node number at the side-wall

• Dj: the distance between sampling nodes and the reference surface

• The reference surface is made on the assumption that the wrinkling disappeared.

( )2

1700

1

26991,where

∑ −==j

jj DDs

The initial blank shape and perturbation vectors

Reference surface

Sampling node

D Reference

surface

The distance between the sampling nodes and reference surface


39

* Results

• The ranges of each design variable are changed after the second cycle.

• The move limit strategy is used.

• Objective function: 0.4781 0.2741 (convergence criteria: 2.0%)


40

* Results

Shape of blank after the sheet metal

forming

Split plane – height 20 mm Standard deviation Clip (+) Clip (-)

Initial model

0.4781

Optimum model

0.2741

x y

x y

x y

x y

y z

x

y z

x


41

Optimization of the Prefrom Shape for H Shape Forging

42

* Problem information

unfilled area

flash

• Model: ¼ H shape forging • The type of element: the axe-symmetric 2D solid element • The unfilled area: low quality of the product • The flash: cost is high because of material loss • For reduction of the unfilled area and flash, the optimization of the preform shape is needed

43

* Formulation

• Design variable: Shape change of the preform bi : The scale factors for the perturbation vectors • Objective function: Reduction of the unfilled area

Yj : The mean value of the sample nodes in the top corner (y-direction) • Constraint: Removal the flash

mh : The mean distance between the sample nodes and the target line (x-direction) Sh : The standard deviation of the sample nodes (x-direction)

)21...,,3,2,1(2.02.00.0subject to

)28...,,3,2,1(minimize to)3,2,1(Find

=≤≤≤

=−=

hSm

jYib

h

h

j

i

b1

b3

x

y

b2


44

* Result

• Objective function: -32.892 -33.925

• Constraint violation: 175.4% 0%

• Number of cycles: 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-34.0-33.8-33.6-33.4-33.2-33.0-32.8-32.6-32.4-32.2

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

Infin

ite n

orm

Obj

ectiv

e fu

nctio

n .

Cycle number

History of the objective function and the infinite norm of the design variable variation vector

Objective function Infinite norm of the design variable variation vector


45

* Result

No unfilled area and flash

74

35

70

40 43

Analysis Result

Initial Model

Optimum Model


46

ESLSO Software

47

Software Development

• Software is developed using C and C++ on the Windows system.

• ESLSO software system has been developed based on the theory of ESLSO.

• Linear dynamic, nonlinear static and nonlinear dynamic response optimization using ESLs are supported in the software system.

• Ls-DYNA and Nastran can be utilized for finite element analysis while linear static response optimization using Nastran, Genesis and OptiStruct.

• ESLs for displacement, stress and/or strain constraints are included in the current software system.

48

Flow of the Software System

User input

Linear dynamic analysis NASTRAN

Nonlinear dynamic analysis NASTRAN, ABAQUS, LS-DYNA

Calculating ESLs for

displacement responses




stress responses


strain responses




stress responses


strain responses

Size optimization NASTRAN, GENESIS, OptiStruct

Shape optimization NASTRAN, GENESIS, OptiStruct

Topology optimization NASTRAN, GENESIS, OptiStruct

Satisfy the convergence criterion?











Yes Update design

No Output

Nonlinear static analysis NASTRAN, ABAQUS

49

Current development of ESLSO

a

b

c

Automobile Crash Optimization

50

Optimization of a Frontal Structure • Formulation

• Design variables

• The size of the bumper: width: 1127 mm, length: 762 mm, height: 412 mm

• Initial velocity of the frontal structure: 8 km/h

• In optimization, inertia relief is used.

,a ,c

,b ,c

Find ( 1,...,28)to minimize mass

subject to 142.5( )

142.5( )

0.7 mm 2.5mm

i

x x

x x

i

b i

mm

mm

b

δ δ

δ δ

=

− ≥

− ≥

≤ ≤


51

Frontal Structure Optimization • Results

• Objective function: 16.16 kg 9.98 kg

• The total number of nonlinear analyses: 21

• The total number of cycles: 21

• The total CPU time per one cycle: 9 minutes

7.08.09.0

10.011.012.013.014.015.016.017.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

mas

s

Cycle No.

Objective Function


52

Side Impact Optimization 1. Initial velocity of the barrier : 50 km/h 2. Rating (Good): The distance between B-pillar point of maximum intrusion and the center line of the seat > 125mm


53

Side Impact Optimization • Model information

- TOYOTA YARIS [National Crash Analysis Center, NCAC]

- No. of elements: 977,810

- CPU time : 8 hours (LS-DYNA R5)


54

Side Impact Optimization

B pillar,max

Find : 1, ,10to minimize masssubject to 125mm

it i

δ −

=

≥

• Formulation (using ESLs)

DV 2

DV 1 DV 3

DV 9

DV 4 DV 10

DV 6 DV 7

DV 8 DV 5

In optimization, we do not have boundary conditions – inertia relief

Disassembled view


55

Roof Crush Optimization • Formulation (FMVSS216 condition)

1 2 3

roof

Find , ,to minimize masssubject to 1.5 unloaded vehicle's weight (displacement =127mm)rigidwall

t t t

F ≥ ×

Modify

• Formulation (using ESLs) conventional method 1 2 3

max critical time step

eq

Find , , (thickness)to minimize masssubject to 127mm (at t )

t t t

d ≤

=Kd f

DV 1

DV 3

DV 2

Displacement (mm) R

igid

wal

l for

ce (N

) 127mm

1.5ｘunloaded vehicle’s weight

Rigid wall force constraints

Displacement constraints


56

Roof Crush Optimization • Formulation (using ESLs) current method

1 2 3 1 2 3

max critical time steps

eq,

eq,

Find , , , , , (thickness,force)to minimize masssubject to 127mm (at t )

f 1.5 unloaded vehicle's weight

f ( node on the contact surface)j

j

t t t p p p

d

j

≤

≥ ×

=∑

Displacement (mm)

Rig

id w

all f

orce

(N)

127mm

1.5ｘunloaded vehicle’s weight 1p

2p

3p

virtual beams (ESLs)

eq, 1 1

eq 2 2

eq,

ff

f

j i

j i

jn n i

c pc p

c p

= = =

feq,j is proportional to the size of pi.

ESLs

e

e

cn

p

∆p

e: unit direction vector

= feq, j

DV grid (perturbation vector)

eq,f 1.5 unloaded vehicle's weightj ≥ ×∑

cn: distributed force

57

Design optimization of the substructure (Superelement) method

Finite element model

Substructure Method Problems

< Design area: residual >

< Non-design area: superelement > < Finite element model >

• Nonlinear dynamic response optimization

• Generation of substructures: LS-DYNA, MSC.Nastran

• Nonlinear dynamic analysis: LS-DYNA, MSC.Nastran

• Linear static response optimization: MSC.Nastran

58

Optimization process of the substructure method

Formulation

( )( ) ( ) ( )( )

( )N N N N N

N

N

L U

Find to minimize

subject to , , ,

, 0 ( , ) 0; 1, , ( , ) 0; 1, ,

n

i

j

Rf

t t t

th i pg j q

∈

+ +

− =

= =≤ =

≤ ≤

bb

Mz b Cz b K z b z

F bb zb z

b b b


Divide the structure into a design area and a non-design area

Generate the substructures for the non-design area

Nonlinear dynamic analysis using the substructure and the residual part

Calculate the equivalent static loads for the residual part

Linear static response optimization for the residual part

Converge ?

Yes. No.

Updated design

variables

END

< Design area >

< Non-design area >

59


If there is no boundary condition in the design area, the inertia relief can be utilized.

< Finite element model >

Formulation

Node 2021

Node 7278

Find 𝑏𝑖 ( i = 1,2,…,10)

to minimize Weight [kg]

subject to 70 ≤ 𝛿𝑥,𝑁𝑁𝑁𝑁#7278 − 𝛿𝑥,𝑁𝑁𝑁𝑁#2021 ≤ 80 [mm]

1.2 ≤ 𝑏𝑖 ≤ 3.5 [mm] (i=1~6)

1.6 ≤ 𝑏𝑖 ≤ 3.5 [mm] (i=7~10)

< Deformed> < Initial >

Simultaneous Optimization of Control and Structural Systems

60

Formulation Since the conventional equivalent static loads method cannot handle the control forces, a new

method is developed.

st

st

st 0

st

LB UBst, st, st,

Find , ( )tominimize ( , ( ), ( ))subject to ( , ( ), ( )) ( )

0 ( 1,2, , )( , ( ), ( ))

0 ( , 1, , )

( 1,2, , )

n

F

j

q q q

R tf t t

t t t t tj m

g t tj m m m

b b b q n

∈

= < <′= =

′ ′≤ = +≤ ≤ =

b ub u z

z h b u z

b u z

0

0 st 0 st( , ( ), ) ( , ( ), ( ), )dft

f ft

f G t t F t t t t= + ∫b z b u z

st st v A 0 1( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( , , , )lt t t t t t t t t+ + + = + =M b z C b H z K z f u

A

ph

vh

xy

z

)(tf

( )u t

B

A st p( ( ) )= +K K b H


61

Equivalent Static Loads Method Analysis domain Design domain

+ = eqf

Equivalent static loads

New design variables The external loads are the functions of

design variables.

st st v A

0 1

( ) ( ) ( ( ) ) ( ) ( ) ( ) ( )( , , , )l

t t t t tt t t t

+ + + = +

=

M b z C b H z K z f u

T Tst v p

Tst v p 1 2

A eq1 eq2 eq3

v p

LB UB

Find where [ , , , ]

to minimize ( , , , , , , ) where [ , , , ]

subject to ( ) ( ) ( ) ( 0,1, , )

( , , , , , , ) 0 ( 1,2, , )

( 1, 2, , )

n

l

w

q q q

R h h

f h h u u us s s s l

g h h w m

q n

∈ =

=

= + + =

≤ =

≤ ≤ =

b b b u

b u z z z uK z f f f

b u z z z

b b b

{ } [ ] [ ]

eq A

st st v

eq1 eq2 eq3

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

s t

t t t t t

s s s

=

= − + + − + = + +

f K z

f M b z C b z H z u

f f f

xy

z

A

{ }st st( ) ( ) ( ) ( ) ( )t t t− −f M b z C b z

eq1fB

v ( )t−H z

A eq2fB +

( )tu

A eq3fB


62

An external load can be expressed by a shape variable of a virtual beam and a distributed force. It is the same as the hydroforming case.

Design domain

T Tst v p

Tst v p 1 2

A eq1 eq2 eq3

v p

LB UB

Find where [ , , , ]

tominimize ( , , , , , , ) where [ , , , ]

subject to ( ) ( ) ( ) ( 0,1, , )

( , , , , , , ) 0 ( 1,2, , )

( 1,2, , )

n

l

w

q q q

R h h

f h h u u us s s s l

g h h w m

q n

∈ =

=

= + + =

≤ =

≤ ≤ =

b b b u

b u z z z uK z f f f

b u z z z

b b b

B

( )sud

d

A

( )ccz ⋅)(A s

vhc

t, s 43210 sssss

u1 u2

u3 u4

u0

( )u t


63

Optimization process using equivalent static loads for structural and control systems

No


64

Example: Single degree of freedom linear impact absorber

{ }p v

12.0 2 2 2

0.0

p

v

Find , , ( 1,2, ,100)

tominimize 100.0 1.0 0.005

subject to 0.6mm 0.6mm0.100kN/mm 1.000kN/mm

0.300kN ms/mm 1.194kN ms/mm0 2.000kN

i

i

h h u i

J x x u dt

xh

hu

=

= + +

− ≤ ≤≤ ≤

⋅ ≤ ≤ ⋅

≤ ≤

∫

M

Initial

velocity

u

x

0%

5%

10%

15%

20%

25%

30%

35%

40%

75

85

95

105

115

125

135

0 1 2 3 4 5 6

Con

stra

int v

iola

tion

Obj

ectiv

e fu

nctio

n

Cycle (#)

Objective function & constraint violationObjective function Constraint violation

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 3 6 9 12Dis

plac

emen

t (m

m)

Time (ms)

Displacementinitial (#0) optimum (#6)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 3 6 9 12

forc

e (k

N)

Time (ms)

Actuator force, u(t)initial (#0) optimum (#6)

hp

hv

Results: hp 0.597→1.000kN/mm, hv 0.597→0.686kN·ms/mm, objective function 82.9→115.3


65

Example: Cantilevered beam (30-DOF): Mass minimization b1 b2 b3 b4 b5 h p v

z,tip

1

Controller1

Controller2

Controller1 Controller2

b h

p

Find , , , , , , , , ( 1,2, 100)

tominimize masssubject to 0.02m 0.02m

6Hz30.0N1000.0N

1000.0N0.005m , 0.1m5.0N / m 10000.0

it t t t t t h h u i

fFFF F

t th

δ

=

− ≤ ≤

≥≤

≤

+ ≤

≤ ≤

≤ ≤

v

N / m

120.0N s / m 1000.0N s / mh⋅ ≤ ≤ ⋅

Controller1 p a v a

Controller2

where,

i

F h z h z

F u

= ⋅ + ⋅

=

1 2 3 4 5 6 7 8 9 10

a b

① ②

10m

Concentrated mass=200kg

P1(t)

0.50m

0.50m

th

tb

0%

5%

10%

15%

20%

25%

30%

0

500

1000

1500

2000

2500

3000

0 1 2 3 4

Con

stra

int v

iola

tion

(%)

Obj

ectiv

e fu

nctio

n (k

g)

Cycle (#)


-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0 1 2 3 4 5

Dis

plac

emen

t (m

)

Time (s)


0

200

400

600

800

1000

1200

0 1 2 3 4 5

Act

uato

r fo

rce

(N)

Time (s)


Results: hp 2000→1912N/m, hv 500→398N·s/m, objective function 2491→770kg

x

z


66

Example: Cantilevered beam (30-DOF): Control energy minimization

1 2 3 4 5 6 7 8 9 10

a b

① ②

10m

Concentrated mass=200kg

P1(t)

0.50m

0.50m

th

tb

Results: hp 2000→998N/m, hv 500→591N·s/m, objective function 2644→1322N·s

b1 b2 b3 b4 b5 h p v

tip

1

b h

p

v

Find , , , , , , , , ( 1,2, 100)

tominimize control energy (N s)subject to 0.03m 0.03m

6Hzmass 650kg0.005m , 0.1m5.0N / m 10000.0N / m

120.0N s / m 1000.0N s / m

it t t t t t h h u i

f

t th

h

δ

=

⋅− ≤ ≤

≥≤≤ ≤

≤ ≤

⋅ ≤ ≤ ⋅

{ }5

p a v a0controlenergy ( ) ( ) ( )h z t h z t u t dt= ⋅ + ⋅ +∫

x

z

0%

10%

20%

30%

40%

50%

60%

0

500

1000

1500

2000

2500

3000

0 1 2 3

Con

stra

int v

iola

tion

(%)

Obj

ectiv

e fu

nctio

n (N

·s)

Cycle (#)


-0.050

-0.040

-0.030

-0.020

-0.010

0.000

0.010

0 1 2 3 4 5

Dis

plac

emen

t (m

)

Time (s)


-500

0

500

1000

1500

2000

2500

0 1 2 3 4 5

Act

uato

r fo

rce

(N)

Time (s)


Earthquake Problems

67

Optimization of a structure that has moving boundary conditions

1. Define the reference model that has rigid body motion. 2. The relative displacement can be calculated from the dynamic analyses of the original model

and the reference model. 3. The ESLs for displacement are generated from relative displacement: structural deformation.

Reference model

ANd

Aid

A1d

0d

BNd

Bid

B1d

0d

Original model Relative displacement

0

r1d

rid

rNd

− =θ

θ

68

Optimization of the tube hydroforming process • Nonlinear dynamic response optimization

• Nonlinear dynamic analysis: LS-DYNA

• Linear static analysis: GENESIS

• Linear static optimization: GENESIS

Formulation

( )t,bF

( )t,bP

Die

Die

tube

( )t,bF

( )t,bF

( ),tbF

( )( )( ) ( ) ( )( ) ( )

( ) ( )( )

0

N N N N N

N

L U

Find

to minimize ,

subject to , , , ,

, , 0

( , , ) 0; 1, ,

n

l

R

f

t t t t

t t

g t l q

∈

+ +

− − =

≤ =

≤ ≤

b

h b h

Mz b Cz b K z b z b

F b P b

b z bb b b

• : axial force

• : pressure

• : initial thickness

• : thickness after forming

( )t,bF

( )t,bP

0h

( )bh

• Tube hydroforming

- Metal-forming process

- Uses pressurized fluid

- The pressures and axial forces are defined in the time domain.

- The quality of the formed material is determined by the external forces.

Hydroforming Problems

69

Definition of ESLs for tube hydroforming

• Governing equation of tube hydroforming analysis

• Equivalent static loads

( ) ( ) ( )( ) ( ) ( ) ( )tttttt ,,,,,, NNNNN bPbFbzbzKbzCbzM +=++

• Virtual model (using the virtual Young’s modulus)

iensionlessfvon,iL

ensionlessfN,if

i

hEE dim,

_

dim,*

σ≡

where : element number

: Young’s modulus of the FE model

: dimensionless form of the von Mises stress from linear analysis with ESLs

: dimensionless form of the thickness from nonlinear analysis

: final time step

i

iEensionlessf

von,iLdim,

_σ

ensionlessfN,i

h dim,

f

( ) ( ) ( )

( )( ) ( )

( ) ( )( ) ( )( )

∂

∂+

∂

∂−=

∂

∂+

∂

∂−=

+≡

bbzK

bbzK

bzKb

bfb

bfbf

bfbfbfff

f

ff

f

fff

N

*

LN

*

LN

*

L

*

eq

*

eq*

eq

*

variableeq,

*

consteq,

*

eq


70


• Equivalent static loads in linear static response optimization

( ) ( ) ( )

( )( ) ( )

( ) ( )( ) ( )( )

∂

∂+

∂

∂−=

∂

∂+

∂

∂−=

+≡

bbzK

bbzK

bzK

bbf

bbf

bf

bfbfbf

f*f*f*

*f*f

*f

*f*f*f

NLNLNL

eqeq

eq

variableeq,consteq,eq

<The distributed axial load; j-d.o.f >

1 1

( )( )

∂

∂−

bbf

bf*

eq*

eq

f

f

( )

∂

∂

bbf j

f |*

eq

( ) ( )b

bbf

bbf

δ∂

∂+

∂

∂ jf

jf || *

eq

*

eq

bδ

Tube Virtual beam

xy

z

A

( )bf *fvariableeq,

( )bf *fconsteq,


71


• Linear static response optimization

- Formulation: The external forces are functions of design variables.

- The points of the profiles are design variables.

- Basis functions can be used for the expression of the axial forces and pressures.

- and are design variables.

- The number of design variables can be reduced.

( )( )( ) ( )( )( )

UL

L

eqLL

0

1 0

0 subject to minimize to Find

bbbbzb

bfbzKhbh

b

≤≤

=≤

=−

∈

q,,l;,g

,fR

*fl

*f*f*

n

( ) ( )( ) ( )∑=

∑=

=

=

npk di

nfk di

uBu

uBu

1 ,i

1 ,i

PP

FF ( )otherwise

if 01 1

1+<≤

= ii,i

uuu,,

uB

( ) ( ) ( )uBuuuuuB

uuuuuB d,i

idi

did,i

idi

id,i 11

11

1−+

++

+−

−+ −−

+−

−=

ia ic

<Force profile> t

1t 2t 3t

( )tF

ft1−ft1b2b

3b

1−fb fb

<Pressure profile> t

1t 2t 3t

( ) tP

ft1−ft

1+fb 2+fb 3+fb

12 −fb fb2

=i

i

at

iF

=i

i

ct

iP


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