Analiza wrażliwości ze względu na kształt dla pewnej klasy funkcjonałów w optymalizacji konstrukcji

ZESZY TY N A U K O W E P O L IT E C H N IK I ŚLĄSKIEJ 1994

Seria: M E C H A N IK A z. 116 Nr kol. 1231

M artin P. B E N D S 0 E In stitu te o f M ath em atics Technical U n iversity o f Denm ark Jan SO K O Ł O W SK I

System s R esearch In stitu te Polish A cad em y o f Sciences and IN R IA Lorraine, Francja

A N A L IZ A W R A Ż L IW O ŚC I ZE W ZG LĘD U N A KSZTA ŁT DLA P E W N E J K L A SY F U N K C JO N A Ł Ó W W O PTY M A LIZA C JI K O N ST R U K C JI

Streszczen ie. W pracy przedstaw iono zagadnienie analizy wrażliwości funkcjonałów , które w ystęp ują w zagadnieniach optym alnego projektow ania ze w zględ u na m inim aln ą p od atn ość przy jednoczesnej optym alizacji topologii i k szta łtu konstrukcji.

S H A P E S E N S IT IV IT Y AN ALYSIS O F O PT IM A L C O M PL IA N C E FU N C T IO N A L S

Sum m ary. T his paper deals w ith the shape design sen sitiv ity analysis of d om ain d epend en t fun ctionals th at arise in optim al com pliance design for sim u ltan eou s op tim iza tio n o f m aterial and structure.

F O R M E M PF L IN D L IC II K E IT SA N A L Y SE F Ü R EIN E G E W ISSE K LA SSE D E R F U N K T IO N A L E BEI D E R K O N ST R U K T IO N SO P T IM IE R U N G

Zusam m enfaßung. In der A rbeit wurde das Problem der Form em pfindli­

ch k eitsan alyse fü r die Funktionale, welche in den ein e op tim ale P rojektierung unter B each tu n g der m inim alen Nachgiebigkeit bei gleichzeitiger O ptim ierung der T op ologie und der Form der K onstruktion bezw eckten Aufgaben auftreten, d argestellt.

1. IN T R O D U C T IO N

In th is paper we stu d y the sh ap e design sensitivity analysis o f a structural op tim ization problem which encom passes th e design of structural m aterial, the design o f top ology and th e design o f shape.

38 M .P. Bendsge, J. Sokołowski

T h e stron g interrelation betw een th e fields of optim al design and m aterials scien ce has been un derlined in recent years and op tim al design w ith advanced m aterials and op tim al to p ology design usin g hom ogenization m eth od s have been th e su b ject o f in ten siv e research (B en dsp e and M ota Soares, 1993, P edersen, 1993), and th e w e ll- known lack o f ex isten ce to generalized sh ap e design problem s and th e resulting regularization has provided a natu ral m a th em a tic a l angle to th e sim ultaneous design o f m aterial and overall structure.

2. T H E IN N E R M A TER IA L D E SIG N PR O BLEM FO R F IX E D D O M A IN

In a recen t paper (B endspe et.a l, 1993) on th e o p tim ization o f structures, th e d istri­

bu tion o f m aterial as well as th e m aterial properties them selves have been considered as design variables. T h e goal o f th is stu d y is to form ulate a structural op tim iza tio n prob­

lem in a form th a t en com p asses th e design o f structural m aterial in a broad sen se, w hile encom passin g th e provision o f predicting th e structural topologies and sh apes associated w ith th e o p tim u m distrib u tion o f th e optim ized m aterials. T h is goal is accom p lish ed by rep resenting as design variables the m aterial properties in th e m ost general form p o ssi­

ble for a (lo ca lly ) linear ela stic continuu m nam ely as th e unrestricted set of elem en ts of p o sitiv e sem i-defin ite co n stitu tiv e tensors.

T h e problem we consider is an exten sion o f th e problem o f m inim izing th e com p lian ce of a stru ctu re m ad e o f a given m aterial, to the situ ation where th e m aterial properties th em selv es appear in th e role o f design variables. We form ulate th e problem for a reference dom ain fb

m in / • ud x + u d T ( x ) } (1)

su b ject to :

E > 0

f n * ( E ) d x < M

f n e(u): E : t ( u ) d x = f n E i j u t i j t u d x = i ( u ) for all v 6 U

W e take th e m in im ization over all p ositive, sem i-definite rigidity tensors E ^ u and use the integral over th e dom ain o f som e invariant $ ( £ ) o f th e rigidity tensor as th e m easure of cost. In (1) w e have w ritten th e equilibrium equation in th e weak form , using U to d en o te th e sp ace o f k in em atically adm issib le trial functions. N o te th at w e treat a single loading con d ition s, represented by th e specified b od y forces / and boundary traction P , and w e use th e com plian ce for this loading case as the ob jective for our op tim a l design problem .

N e x t, w e reform ulate problem (1) into a convenient form th at em p h asizes th e role of th e m aterial design:

Shape sensitivity analysis of optim al. 39

m ax m in i l ( i i , u ) (2)

rigidityE > 0 f n <HE)dx<M with

n(£,u)

= \ J

For ph ysical reasons, th e possible rigidity tensors in th e above design form ulation are restricted to th e set o f p o sitiv e sem i-d efin ite, sym m etric tensors and su itab le cost fun ctions m ust have th e property of frame indifference, for exam ple in term s o f invariants of th e co n stitu tiv e tensor itself:

Case A: $ A( E ) = pA = E ,

C ase B: $b(E) = pB = [EijtiEijkift (4)

i.e., respectively, th e trace and th e Frobenius norm of th e tensor E. For th e sake of sim p lifyin g th e derivation, w e have introduced the resource density fun ctions, p A and pb and w e ch oose to restate th e design problem in the form:

C ase A:

Case B:

max max

density p a rigidityE > 0

°<Pmin^PA^()™ax<oo

I d PAd x <M

*a(E )<p a

max max

d ensity p q rig id ityE > 0

®<Pmin—P B —Pm ax< 0°

/ f t PBd x <M

®b(e) Sp b

min fl(p.4, E, u) u£U

m m I l ( p A, E, u)

(

)

(

)

T h is separation o f th e design variables provides a separation betw een the properties o f th e tensor E th a t can be optim ized locally (at each point in the structure) and those th at m ust be treated as a distribu ted param eter problem over the full dom ain. In (5) and (6) we have introduced upper and lower bounds on the resource d en sities in order to ensure th a t th e problem is well posed.

W e can perform a sim plification o f problem s (5) and (6) by exchanging th e order of th e m in and m ax operation s in th e tw o inner problem and solve for E (see Bendspe et.

ah, 1993). T h e resultin g reduced problem is for both cost m easures of the form:

m axpec{m in y pc(u): e( u) dx — i ( u

}

(

)

w ith th e d en sity being restricted to the closed, convex and weak-* com pact constraint set g in ¿ °°(ft):

40 M.P. B endsge, J. Sokołowski

Q = { p 6 L°°(Cl)I[ p d x < M , 0 < p min

<

Jn

T h e design variables can be rem oved entirely from th e problem , and th e resulting problem b eco m es a n on -lin ear and non -sm ooth, convex, a n a ly sis-o n ly problem , for which ex isten ce o f solu tio n is assured, bu t for which solution s m ay not be unique.

W e can also g iv e problem (7) in its equivalent com plem entary energy form as

( 8 )

. f 1'

[

mi n< m i n / —a: c d x \ t

P€£? 1a 1■Jn P J J

diva —J a-n— P

and w e n o te th a t th e derivation of (7) and (8) both ex ten d readily to th e case o f a co n ta ct p rob lem , for w hich th e con tact condition is stated as a design in d ep en d en t, convex constraint on th e disp lacem en ts. It is th e problem (7) th at we in th e follow ing w ill analyse w ith resp ect to sh ap e variations o f th e reference dom ain.

3. T H E D IS P L A C E M E N T S B A S E D FO R M U LA TIO N

T h e follow in g d om ain fun ctional is considered:

J ( Q ) = m i n j ^ m a x { (^ e (u ):e (u ) - A)pmox, ( i e ( u ) : e ( u ) - \ ) p min}d x

~ L f u d x ~ L p ■ u d r ( x ) + a m ] (9)

w here u belongs to th e convex su b set o f th e S ob olev sp ace I I 1 (SI)3 defined by th e follow ing relation s (in clu d in g th e p ossib ility o f m echanical contact):

u = 0 on To u ■ n < 0 on P2

W e are in terested in th e sh ape derivative, whenever it ex ists, o f th e fun ctional J ( S \ ) , see e.g. Sokołowski and Zolesio, 1992, for a detailed definition and a descrip tion o f the m aterial d erivative m eth od .

To th is en d, a one p aram eter fam ily o f dom ains {Slt }, t 6 [0,<5), is defined as follow s.

fii = T,(Sl) for t£ [0,5)

dSlt = T,(dSl),

r [ = T .fR ) for r = 0 , 1 , 2 .

here Tt = T t( V ) : IR3 —» IR3 is a sm ooth transform ation given by a su fficien tly sm ooth vector field V w ith V ( t , x ) = ^j f ( x)

S hape sen sitiv ity analysis of optim al. 41

T h en , th e d om ain functional (10) defined in { f ii} , takes the form

(10)

w here u b elongs to th e convex su bset of th e Sobolev space defined by th e follow ing relations:

u = 0 on Fq u • n < 0 on Tj For any t > 0 a minimize!' is denoted by ( u t , A(), so

In order to evalu ate th e sh ape derivative d J ( Q ; V ) , the dom ain functional J ( C l t ) is transported to th e fixed dom ain fi. T h e follow ing notation is introduced, we refer the reader to Sokołowski and Zolesio, 1992, for details, in order to m ake th e functional as well as th e boundary conditions dependent of the dom ain fi only:

/ ' = ~DTi - f o T t , P ‘ = • D T t P o T t

u ‘ = II, O T[

2' = D T P 1 ■ u‘e‘ ( f i = \ { D ( D T t ■ <t>) ■ D T t~~l + ' D i p 1 ■ * ( D ( DT, ■ * ) ) }

7 (t) = det ( D T t ) u ( t ) = \ \ M ( T t ) - n U ,

M ( T t ) = At \. (DTt) ' D T p 1

furtherm ore, th e non penetration condition on Fj takes an equivalent form

(U)

z ‘ ■ n < 0 on Fj

T h e d om ain fun ctional is thus transported to th e fixed dom ain

J t ( Q ) = J ( f i i ) =

[ ^J

(

)

(13)

Jn

r,

^ c(tii) o Tt. e(ut ) o T, - X, o T, )pmin}- f ( t ) dx

[ f* ■ u li ( t ) d x - [ P ‘ ■ u ‘u ( t ) d r ( x ) + \ t 0 T , M

42 M.P. B endsge, J. Sokołowski

T h e transp orted d om ain functional can be evaluated for any value o f t £ [0 ,6) by a m in im ization procedure, i.e.

mi n i mi z e with respect to (v , A) the following f unct i onal

(i>,A) = [ / m a x { (^ e ‘(u):e'(t>) - \ ) p max, ( ^ e ‘( v ) : e ‘( v) - \ p min) } l ( t ) d x

L

+

su b ject to th e constraints

v £ v = 0 on To v ■ n < 0 on r 2

In th e follow in g w e assum e, for sim plicity of presentation, th at th e m inim izers of problem (13) are un ique, t 6 [0, ¿). If this is not th e case, we have to invoke th e procedure o f sectio n 3 in B endspe and Sokołowski, 1992, w ith each derivative b ein g given below . N ow , w ith th is assu m p tion , let (u , A) denotes th e unique m inim izer for th e fu n ction al (9).

T h e follow in g n ota tio n is introduced

fli = f i i( V ) = { i £ n | | e ( t i ) :

e(u)

e'(u):e(u) > 0 or ^c(u): e(u) — A > 0}

n 2 = n 2(v) = n\n,(v') Xi = characteristic function o f fi;

Xn = characteristic function o f fi, i.e.

, , _ 1 x 6 n 0 o t h e r w i s e

therefore X i(x ) + X2(x ) = Xfl(x ) a -e -

Here e' d en o tes th e derivative of e1 and w e have divided th e dom ain in tw o parts, d ep en d in g on th e size o f th e specific strain energy relative to th e m ultiplier A. W ith this n otation we can now sta te th e result on th e shape sensitivity:

T h e o r e m 1.

If the m i n i mi z e r s o f funct i onal (13) are continuous in / /* ( f l) 3 x IR with respect to the p a r a me t e r t at t = 0+ , then the Eulerian derivative o f the shape funct i onal (3. 1) takes the f ol lowi ng f o r m

Shape sen sitiv ity analysis of optim al.. 43

d J ( i l - V ) = l i m t^ Ą ( J ( n t) - J ( S l ) ) (15)

(16)

=

[ J

(17) + / m a x { [-e (ii): e(u) - A\pmax, [-e (u ): e(u) - A]pmin}-y'dx +

Jn - -

— / [ ( V / - V ) • u + f • D V ■ u + / • -f Jn

-

P D V - u

P

where w e have th at

«' W = i j P -4>) + ' ( D [ D V ■ 4,)) - £>«1 • D V - ’ D V ■ -D4>},

7' = divf/,

u / = d iv r V ’ Z)^

is th e tran sp ose o f the Jacobian m atrix function D<j> ;<j> 1R3 —* 1113 1/(011 = 2Zt 01 |1=0

We refer th e reader to Bendspe and Sokołowski (to appear) for a proof o f T heorem 1. W e remark th at under our assum p tion s, the Eulerian derivative o f the shape functional (9) can b e evalu ated in th e sam e way as for the m ultiple eigenvalues in M yslinski and Sokołow ski, 19S5, Sokołowski and Zolesio, 1992. If the m easure o f th e set

1

fio = {x £ ^ l ^ £(“ ): £(u ) - A = 0}

is not zero th en th e m apping V -* d J (D ; V ) fails to be linear, and only one sided directional derivatives exist.

44 M.P. B endspe, J. Sokołowski

4. C O N C L U SIO N S

W e have p resented here, on th e basis of B endspe and Sokołowski (to appear) a general fram ew ork for an in tegrated design analysis of shape and topology variations o f structures.

T h e to p o lo g y variations are treated as an inner problem , w ith sh ape variations as an outer level problem . T h e to p ology and m aterial design problem em ployed used a co m p letely free param etrization o f rigidity, bu t th e analysis ex ten d s readily to h om ogen ization based to p ology problem s w ith reduced design indep en dent problem s, as described in e.g. Allaire and K ohn, 1993, and Jog, Haber and B endspe, 1993. W e n ote th at th e integrated sh ape and to p o lo g y and m aterial design form ulation allow s for the treatm ent of problem s w ith sh ape d ep en d en t loadings, such as pressure loads; th ese problem s cannot be han dled by standard to p o lo g y and m aterial design m eth odology based on th e use o f a fixed reference dom ain.

T h e results on derivatives involves dom ain integrals and a com p u tation al im p lem en ta­

tion is straightforw ard. In order to preserve consistency w ith th e com p u tation al procedure for th e solu tion o f th e an alysis problem , it is recom m ended th at th e sen sitiv ity analysis is carried ou t for th e d iscretized problem sta tem en t before im plem entation. C om p u tation al exp erien ce and d eta ils on th e derivation for a discretized version o f th e problem are the su b jects o f a forth com in g study.

R E F E R E N C E S

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Recenzent: Prof. dr bab. inż. Tadeusz Burczyński W p łyn ęło do R edakcji w grudniu 1993 r.

Streszczenie

W pracy p od an o now e wyniki z zakresu analizy wrażliwości uzyskane dla optym alnych funkcjonałów p od atn ości ze w zględu na k ształt obszaru geom etrycznego. W yniki te po­

zw alają na w yk orzystan ie dla rozw iązyw ania zadań optym alizacji konstrukcji połączonych m etod obliczeniow ych optym alizacji topologii i optym alizacji kształtu. W szczególności um ożliw ia to rozw iązyw ania m etod am i optym alizacji topologii tych zadań op tym aln ego projektow ania, w których obciążenie zależy od geom etrii obszaru - np. ciśnien ie - co nie było do tej pory robione ze w zględu na brak uzasadnienia teoretycznego. W pracy podano w yniki dla przykładow ego zadania optym alizacji kształtu dla zagadnienia typu kontaktu ciała sp rężystego ze sztyw n ą przeszkodą.