Appl ying to B ea m Ground St ruc ture 2

Lect ure m at erial fo r Topo logy O p tim iz ation Des ign #2

Contents

Basic Conc ept 1

Appl ying to B ea m Ground St ruc ture 2

As sembl y o f Sti ffn ess Ma tric es 3 Optimiz ation and Result

4

Boundar y Condi tions 5

Repeat un til convergence Basic Concept

Examp le: Des ig n th e s tron ges t stru ctu re with in a given mas s co nstr aint

Fini te e le m en t approxi m at ion

Ite ration process

Topologically opti m ized struc tur e

this lecture, we will carry out topology optimization with the Euler beam elements instead of inuum elements.

T op. Opt. with Beam Ground Bea m Structure

Let’s find an s trongest layout of th e s truc tur e us ing a bea m ground st ruc ture as a des ign do m ain.

Design Do m ain d iscre ti zed by a “ GROUND BEAM STRUCTURE ”

f f f

A ground beam structure is a frame structure with repeated cells that consists of beam elements with uniform cross-sectional dimensions.

RWe will use an Euler beam which has circular cross-sectional area with radius R.

u ; Ma trix W e will

Basic Concept

™ Desi gn var ia bles ƒ d ensity of fi n ite elem en ts, or b eam th ick n ess/rad ius: (t y p ical ly var ie s fr om 0 to 1 ) ™ Ob jec tiv e fu nc ti on ƒ For i n st ance , “ com pliance” , “ E igenfre que nc y” etc. ƒ Ob jec tiv e gradien t • S ensitivit y of th e obj ect ive fun ct ion corr espondin g to the change of th e d esign variables. ™ Constr ai nts ƒ Lim itatio n s given wh ile so lv ing t h e d esign pro b lem ƒ Con str ai nt gr adi en t • S ensitivit y of th e constr aint valu e corr esponding to th e ch ange of the design var iables . ™ Des ign v ar iable Up dat e by an Optimizat ion A lg or ith m Ingredi ents of topology opti mi za tion

Optimiz ation pr ocess

Fina l to po lo gy

Compliance

Formulation Aspect

1. Design variab les:

ii

rR γ = T h e d esi gn va ria bl es fo rm a vec tor c ons isti ng o f and re pr esent the r adii of be am elemen ts . Th us, th e r adii of be ams will b e op ti miz ed in thi s pr obl em, and the be am element with s m all val u es ar e tre ate d to be non -e xis ten t. I n other pr oble ms, the de si gn var iables can be eleme nt de nsi ty, or ele m ent connec tivi ty, etc.. ( = 1,2,...N B )

i γ

0 1

ε γ < ≤≤

m in () () c γ γγ == TT Fu u K u 2. Obje ct fu nction: C omplianc e () c γ

Th e s tron ger th e s tru ctu re b ec omes , th e les se r th e en er gy wi ll be s tored . Th erefore , com plian ce, or sto red s train en er gy , wil l b e min imiz ed .

1 Ne

e=

= KK A w h ere

F=Ku 2*Strain En er g y

f f f Formulation Aspect

*

1

g= ( ) 0 ,

iiii

AL V γ

−≤ ∑ 3. E xp licit Constrai nt: Mass Constraint

*

th

th

: giv en v o lu m e : area o f b eam L : l ength of be am

i

V Ai i ⎧ ⎪ ⎨ ⎪⎩ The to tal mass ( or v olume) of th e used beam elements is constrained:

V o lume of ith b e am elemen t

Ku = F m pli cit Constraint: ) Formulation Aspect

=2 e ee e

ee A g LL πγ γγ ∂ ∂ = ∂∂ 4.1 C on strai nt gradient:

e

g γ ∂ ∂ 4. Sensi tivi ty (or G rad ie nt of object ive func tion and constraint wrt d esign variables):

*

1

g= ( ) 0

eeei

AL V γ

− ≤ ∑ Formulation Aspect

/ e c γ ∂ ∂

= 2

e

c γ ∂ ′ ′′ ′ ′ =+ + + ∂

uK u u K u u K u u K u u K u

wh er e ,

ee

γ γ ∂ ∂ ′′== ∂ ∂ Ku Ku ()

c γ ∂ = ∂

uK u 4.1 S ensitivit y of Ob jective function : Formulation Aspect

thod 1 : D ir e c t Diffe re ntiation

Easy to ca lculate directl y Not effi cient to directly calcul ate Æ T h e c alcul a tion of u’ can be av oide d b y adjoint method .

To ob tai n

u , w e m u st di ffer en tiat e

()

e

γ ∂ ∂ Ku = F 0 ′ ′ + = Ku K u

solving f or ,

u

'1−

′ =− uK K u w h ic h can b e use d for the equatio n above Formulation Aspect

2

= - 2

=- 2

= -

c γ ∂ ′′ =+ ∂ ′ ′ + ′ ′ + ′

-1TT

TT-TT

T

uK u u K u

(K Ku ) K u u Ku uK K K u u K u uK u Thus

11

2222

222222

11

2222

222222

00 0 0

01 2 6 0 1 2 6

06 4 0 6 2

00 0 0

01 2 6 0 1 2 6

06 2 0 6 4

CC

CC L C C L

CL CL CL CL

CC

CC L C C L

CL CL CL CL − ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ − = ⎢ ⎥ −⎢ ⎥ ⎢ ⎥ −− − ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ K

ee

γγ ∂∂ → ∂∂

KK K= T K T = T T Formulation Aspect

123

wi th and A EE I CC LL == Ca lc ul ation of (e as y )

e

γ ∂ ′= ∂ K K

m L e ctu re a m The or y :

re

ii rR

γ

e=

= KK A

1611

6166 ii

i

ii

K K KK γγ

γ

γγ ∂ ∂⎡ ⎤ ⎢ ⎥ ∂∂ ⎢ ⎥ ∂ ⎢ ⎥ ∂ ⎢ ⎥ ∂∂⎢ ⎥ ⎢ ⎥ ∂∂ ⎣ ⎦ K = L

MO M

L Since the differentiation of a mat rix w ill lead to t h e differentiation of each elem ent:

() , () II A A γ γ = = and

fo r circular c ylinder:

3

2,

AI πγπ γ γγ ∂ ∂ == ∂∂

, 4

AI πγ πγ == Formulation Aspect

3232

223232

2

00 0 0

12 6 1 2 6 00

64 6 2 00

00 0 0

12 6 1 2 6 00

62 0

iiii

iiii

iiii

iiiie

ii

ii

iiiii

iiiii

ii

AA EE LL

II I I EE E E

LL L L

II I I EE E E

LL L L

AA EE

LL

II I I I EE E E

LL L L

I EE

LL γγ

γ γγ γ

γ γγ γ

γγ

γ γγ γ γ

γ ∂∂ − ∂∂ ∂∂ ∂ ∂ − ∂∂ ∂ ∂ ∂∂ ∂ ∂ − ∂∂ ∂ ∂ ′ ∂∂ − ∂∂ ∂ ∂∂ ∂ ∂ −− − ∂∂ ∂ ∂ ∂ ∂∂ ∂ K=

2

64 0

iii

II I EE LL γ γγ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ∂∂ − ⎢⎥ ∂∂ ∂ ⎢⎥⎣⎦ T h en w ill be as belo w :

′ K Formulation Aspect

Objective gradie nt can be obtained using s o-cale d the adjoint m ethod , w hic h is mo re e ffi cient than dire ct di ff erentiation.

First, w e modif y the equation by adding an equilibrium equat ion multip lied by a vector, λ . ( N ot e E quilibrium eq. = 0 Æ ac tuall y “ 0” added.)

Then w e differ en tiate the obje ctive fu nction: ) +

λ (K u -F

( ) )

eee

c γγ γ ⎡ ⎤ ∂ ∂+ ∂ ⎣ ⎦ == ∂∂ ∂

Fu Fu λ (K u -F c=

Fu

Wher e λ is a ve ct or to be adjust ed la ter to sim p lif y the se nsitivit y c alcu latio n . Formulation Aspect

Method 2: A d jo int M et h od f o r Sensiti v ity Calculation

TT

TT

()

( ) (* )

ee

c γ γγ γ γ

γγ ∂∂ ∂ ∂ ∂ =+ − − − ∂∂ ∂ ∂ ∂ ∂∂ =− − ∂∂

u λ Ku FK u F λ u λ K

uK F λ K λ u T h e equation in the previous page w ill be expanded as:

T

(o n ly d ep end s on t h e e- th el em en t)

eei

c γγ γ ∂∂ ∂ =− → − ∂∂ ∂

KK uu u u

eee

ei

c γγ ∂∂ =− ∂∂

K uu For e

elem ent,

e

ele m ent Formulation Aspect

Te

T o av oi d u / , s elect ed s u ch t h at 0 , . .,

ie γ ∂ ∂− = = λ F λ KK λ F

T h erefo re, ch oos e = λ u

Mechanism to push γ=0 or γ=1at the end of optimization?

u

f

γ

γ

121

γ γ

u

It can be observed that the displacement can be minimized when either of the beam element has the maximum value with another becoming zero, rather than two beams share the design variable. This can act as a penalization to push the design variables to have a value 0 or 1.

Formulation Aspect

5. Updateinput values

Optimiz ation process and result

1. Start

2. Get initial values

3-1. Solve FEM

3-2. Obtain objective,gradients and checkconstraint

4. Converged?

yes no

End 1. Set design domain, BCs, objective and constraint.

2. Decide initial values of design variables,γ

3. Use KU=F to obtain displacement vector, U. Calculate the needed values.

4. Check if oldnew

old

cc

c ε − <

5. Set reflecting gradient values new ←γγ

Optimiz ation pr ocess

Fina l to po lo gy

Appendix- top olog y op ti m iz ati on i n c on tinuous dom ain

mi n ( ) ( ) c

γγ ==

FU U K U

*

1

g= ( ) 0 ,

iii

vV γγ

−≤ ∑ KU = F Design the strong est st ruc tur e wi th li mi te d ma ss .

riables:

γ

0iγ=

Design variables forms a vector to indicate the density of the elements.

t function:

() c γ

The stronger the structure becomes, the lesser the energy will be stored. As in the previous page

traint:Since the volume is limited in this problem, the constraint willbe as following.

*

th

: giv en v o lu m e : v o lu me o f el emen t

V vi ⎧ ⎨ ⎩

Appendix- top olo gy op ti m iz ati on for c onti nuu m b od y

=

i

g v γ ∂ ∂

c γ ∂ ∂

( )

TTT )

(), iii

iiiii cγγγ

γγγγγ ⎡⎤∂∂∂⎣⎦==∂∂∂∂∂∂∂∂=−−−−=−∂∂∂∂∂ TTT

TT FUFU-U(KU-F

UUKUKFKUFUUUKUU

g γ ∂ ∂

Appen d ix- SIMP

method

Since the design variables ( ) vary continuously, i.e., not 0 or 1, there needs some interpolation when take intermediate values.

γ

Th e m at eri al p ro pert y a t each po in t o f d om ain is co n sid ered t o b e p ro p or tion al to pe na lize d e lem ent de nsi ty, w hic h c an be e xpr es se d as

neo

KK γ =

neo

M M γ = For Sti ffn ess Ma trix :

F or Mass Matrix :

E E γ =

Th eref ore, t h e o b ject g ra di en t d eriv ed p rev io u sly ca n b e ex p res se d a s:

T1Tni

ii

c n γ γγ

∂ ∂ =− =− ∂∂ K UU U K U γ