DETC2003/DAC TOPOLOGY OPTIMIZATION OF HEAT-RESISTANT STRUCTURES

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1 Procdings of DETC 03 AME 003 Dsign Enginring Tchnical Confrncs Chicago, llinois, ptmbr -6, 003 DETC003/DAC TOPOLOGY OPTMZATON OF HEAT-RETANT TRUCTURE Aljro R Diaz * Dpartmnt of Mchanical Enginring Michigan tat Univrsity East Lansing M 4884 diaz@gr.msu.du Andr Bnard Dpartmnt of Mchanical Enginring Michigan tat Univrsity East Lansing M 4884 bnard@gr.msu.du ABTRACT Th stard problm of finding th optimal layout of structural matrial associatd with maximum stiffnss is xpd to includ considration of thrmal critria. Th problm is posd as a thr-phas layout problm whr th phass includ an insulating or fir rtardant matrial an unknown distribution of hat sourcs, in addition to th structural matrial. Th modl usd is simpl, yt rsults suggst that th introduction of masurs to control th tmpratur in th structur whn subjctd to significant hat transfr rats can rsult in layouts that diffr substantially from solutions whr thrmal issus ar ignord. Kywords: multiphysics topology dsign optimization, topology optimization, layout optimization. NTRODUCTON Th topic of intrst in this work is th dsign protction of structurs xposd to a significant hat transfr rat such as during a fir. t is assumd hr that th hat cratd by this fir is larg can damag th structur. Thus, dsign considrations should includ masurs to protct th structur from hat, in addition to th mor common critria rlatd to structural prformanc, such as stiffnss or strngth. Th problm is cast in th fashion of a stard topology optimization problm, whr th shap of th structur is rprsntd as a matrial proprty,. Howvr, in this cas, in addition to th layout of structural matrial, th optimization problm sks optimal distributions of fir protctiv matrial that surrounds but dos not stiffn th structur. Firs typically involv two stps: ignition flam. Flam propagation, which oftn follows ignition, occurs by hating of th adjacnt matrials to th point whr th volatils rlasd by th dgradation of th solid matrials can b usd as ful. Th mchanisms associatd with th rlas of th volatils ar quit complx involv two distinct procsss: ful gnration in th solid phas combustion in th gas phas. Radiation from th flam, conduction or convction through th gas from th flam conduction through th solid ar thr important mchanisms -bsids xtrnal sourcs- that control th rat of flam sprad Modling of flam propagation is vry complx. As a first approximation, th prsnc of a fir its ffct on a structur can b vry crudly approximatd by rplacing th burning mdium with a mdium with a larg uniform hat sourc. This would rprsnt, for xampl, a fir that occupis all th availabl spac but dos not burn th insulation. Th ffcts of radiation convction ar nglctd initially rplacd by a conduction hat transfr procss with hat sourc. Th ffct of th hat sourc is to crat tmpratur profils in th mdium that approximat th tmpratur profils of a hatd mdium with convction. This is th xtnt of th thrmal analysis prformd hr. n addition, th ffct of rising tmpraturs on th structur is dalt with only - implicitly, by constraining th maximum allowabl tmpratur in th structur. Thus, assuming that this bound is kpt sufficintly low allows us to trat th structur as an lastic mdium whos proprtis rmain unchangd dcoupld from th hat transfr problm. Th shortcomings of such a * Corrsponding author Copyright 003 by AME

2 modl ar rcognizd, but this approach should b sufficint to gain insights as a first stp towards a mor comprhnsiv approach. THE OPTMZATON PROBLEM Undr th assumptions statd in th prvious sction th optimal layouts of structur fir-rtardant matrials ar controlld by two d-coupld quations of lasticity stady-stat conductiv hat transfr. For simplicity of xposition w limit our attntion to problms in two spatial dimnsions dfind ovr a dsign domain of prscribd shap. n our formulation is dcomposd into disjoint rgions occupid by structur fir-protctiv matrial (insulation). A third rgion, disjoint from th othr two, is ncssary to modl th sourc of hat. W labl ths thr rgions, H (Figur ) whr = H H = is th portion of occupid by th structur. Matrial hr has high stiffnss high conductivity. is th portion of occupid by fir-protctiv matrial. Matrial hr has low stiffnss low conductivity. H is th portion of filld by th hat sourc. This modls th fir. Figur. Partition of th dsign domain. Th matrial in has th proprtis k if x kx ( ) = k if x E if x Ex ( ) = 0 if x Th proprtis of th rfrnc matrials, k, k, E ar prscribd. k k ar matrics, rspctivly, th H H () () conductivity of th structur th insulating matrial, whr k > k k is low. E is th lastic tnsor of th structur which, in this discussion, is assumd to b isotropic. n addition, modling of th hat sourc is introducd by dfining ( ) q if x H q0 x = 0 if x q is th strngth of th hat sourc in H. Th objctiv of th layout optimization problm is to find an optimal partition of into, H. Analysis Undr th statd assumptions th simplifid modl of th physical problm consists of th following d-coupld quations of stady-stat conductiv hat transfr lasticity: Hat Conduction Th tmpratur distribution satisfis whr a T T q T T V 0 T(, ) = ( ) T ( ) a T is th usual bi-linar form, T T at( T, T ) = kij d x x qt ( ) is qt ( ) = qtd 0 i j (3) T V 0 T ( ) Th spatial distribution of matrial is controlld via in (5), th conductivity of th matrial, q 0 in (6), th strngth of th hat sourc. n our problm, this hat sourc is th mchanism through which w modl th fir. Finally, V T0 ( ) is usd to rprsnt kinmatic admissibility (for simplicity, only homognous boundary conditions ar considrd, Fig. (a)). Plan Elasticity Th displacmnt fild u V 0 u ( ) satisfis whr a ( u, u) = f( u) u V ( ) u u a u is th bi-linar form in lasticity, au ( u, u) = Eε( u) ε( u) d (8) f( u) is k ij (4) (5) (6) (7) f ( u) = tudγ (9) t Γ Copyright 003 by AME

3 t Hr t is a prscribd traction applid on Γ, a portion of th boundary of. Th spatial distribution of matrial is controlld through E in (8), th lastic tnsor. Kinmatic admissibility is introducd via V 0 u ( ) whr, again, only homognous boundary conditions ar considrd (Fig. (b)). Prformanc Functions Optimization Formulation Th potntial impact of th tmpratur ris on th structur is controlld by limiting th avrag tmpratur in th structur to b blow a prscribd limit. This is don through th following constraint: m m T d T mas( ) for intrgr m or, mor simply, m m ( T T ) d 0 (6) (a) (b) Figur. llustration of th boundary conditions associatd with th structural (a) thrmal (b) problms. Th thr-phas MP modl To complt th formulation of th optimization problm w us a thr-phas MP (implifid Matrial with Pnalization) modl (.g., s,3,4 ). Th controls ar ρ : (0,] ρ : [0,]. W lt p p kx ( ) = [ ρ k + ( ρ ) k] 0 Ex ( ) = ρ E () p whr p> is th pnalty paramtr, q ( x) = ρ ( ρ )q Not from ()-(3) (0)-() that = x T is a prscribd tmpratur valu (hr T > 0 if q > 0 ). As th control ρ acts as a mmbrship function for th fuzzy st, constraint (6) is rplacd in computations by ( m m T T ) ρ d 0 (7) Th rst of th optimization problm is stard. Th objctiv is to minimiz th man complianc of th structur. soprimtric constraints limit th total structural insulation rsourcs. This complts th formulation of th optimization problm, which is: Find ρ that ρ Minimiz f( u ) (8) ubjct to (0) ( m m T T ) ρd 0 () (3) ( ρ ( x))( ) = x (4) ρ( x)( ρ( x)) = x H ρ ρ Thus, in a 0- solution ( tak only 0 or valus ) Computations = indicats that x for any ρ = 0 = 0 indicats that x = 0 ( x) ρ indicats that = x H (5) ρ d r mas( ) ( ρ )( ρ ) d r mas( ) 0< ρ ρ 0 ρ min n addition, u satisfis quilibrium (7) (dfind ovr th whol domain with E as in () ), T satisfis (4) r r ar prscribd prcntags of th domain occupid, rspctivly, by structural matrial insulation. For th problm to mak sns on nds r + r <. Th lasticity hat transfr quations ar modld using a stard finit lmnt mthod (4-nod squar lmnts ar usd in both cass). Th controls ρ ρ ar discrtizd using lmnt-wis constant functions. A stard adjoint variabl snsitivity analysis is prformd to comput gradints of th tmpratur constraint, which is nforcd lmnt-wis (a) (b) (c) (d) 3 Copyright 003 by AME

4 at th cntr of ach lmnt. To illustrat, with m= th discrtizd constraint (8) (a) taks th form tmpratur rlatd considrations. (Not that issus rlatd to tmpratur inducd strsss ar ignord hr!) = v g ( t T T) ρ (9) insulatd F 0 whr T is th (4x) vctor of tmpraturs at th nods in lmnt, is th valu of ρ at lmnt ρ v = 4(,,,) t. Th adjoint problm associatd with this function is Kλ = V V = ρ v (0) whr K is th (finit lmnt) stiffnss matrix associatd with th hat conduction problm V is obtaind from th assmbly Th gradint (snsitivity vctor) of g is computd from whr dg dρ t t λ p λ p dρ () dg = = () q q p v K T p v (3) = = ρ ρ ρ q = qρ ( ρ ) is th strngth of th hat sourc in lmnt. Th xplicit dpndnc of th lmnt stiffnss matrix K on ρ is asily availabl from th MP modl quations (0). Finally, problm (8) is solvd using th mthod of moving asymptots of vanbrg 5 a mshindpndnc filtr from 6 is usd. EXAMPLE Th following xampls illustrat th ffct of introducing tmpratur-rlatd considrations into th problm. n all cass w us isotropic matrials with k = k = 0. κ = µ = 0.4 ρ = 0.0 min κ µ ar, rspctivly, th bulk shar moduli of th matrial. Th dsign domain is rctangular, discrtizd using 3x0 squar lmnts of unit sid. Th total amount of structural matrial is controlld by r = 0.35 in constraint (8) (b). Loads kinmatic boundary conditions ar as shown in Fig. 3. n th absnc of th tmpratur constraint, this problm corrsponds to th vry wll known 8-bar truss problm that was bn xhaustivly studid in th litratur. Th load F 0 is adjustd so that f*, th man complianc of th optimal structur, quals in th absnc of tmpratur constraints. Th layout of this structur is shown in Fig. 4. Th xampls ar mant to illustrat how this solution is affctd by latd insu insulatd U=0 Figur 3. Boundary conditions usd in Exampls. sub-domain of that is not occupid by ithr structur or insulation (i.., H ) is subjctd to a hat sourc of strngth q =. This modls th bhavior of th systm whn a fir fills all availabl spac. Boundary conditions ar as shown in Fig. 3. Th hat gnratd within H can only scap through th right sid of th rctangl, whr th tmpratur is fixd at T=0. cool wall: T=0 Figur 4. olution to Exampls without tmpratur constraints. f* =. Exampl : Effct of T n this st of solutions w wish to invstigat how th optimal layout changs whn incrasingly tightr rstrictions ar imposd on th structur by rducing th accptabl (man) tmpratur valut. n this xampl th total amount of availabl insulation is controlld by =0.0 in (8) (c). Th r 4 Copyright 003 by AME

5 aras whr th structur is forcd to li away from th cool wall. Figur 6 shows a typical distribution of th hat sourc ( H ). Thr w apprciat that in this modl in th optimal solution th hat sourc can b nar th structur, as long as this happns in aras nar th cool wall, whr th hat can b dissipatd mor quickly. (a) T =0 f*=.09 (b) (a). (b) H Figur 6. Optimal layout of structural matrial distribution of hat for T =60. (c) T =00 f*=.3 (d) Th introduction of th tmpratur constraint has th ffct of rducing th stiffnss of th optimal structur. For xampl, whn th bound on th avrag tmpratur is T =60, th complianc of th rsulting structur (Fig. 5 ()) is incrasd by a factor of.4 from th complianc of th structur computd without a tmpratur constraint (Fig 4). n rturn, th man tmpratur is dcrasd by 53%. Th tmpratur distribution on th structurs in Figurs 4 5() is shown in Fig. 7. () T =60 f*=.4 (f) Figur 5. Optimal layout of structural matrial insulation for diffrnt targt man tmpraturs T. A squnc of solutions for dcrasing valus of T is shown in Fig. 5. Th figur shows that, for high valus of T, th optimal layout rsmbls th layout in Fig. 4 (no hat gnratd). Howvr, as th targt tmpratur is dcrasd, incrasing amounts of structural matrial ar shiftd towards th cool wall, in an attmpt to kp th structur coolr. For vry low valus of th targt tmpratur, tmpratur considrations dominat th complianc can bcom larg. Th insulating matrial is typically lumpd btwn th structur th sourc of hat, with highr concntrations in (a) No tmpratur constraint (b) T =60 Figur 7. Tmpratur distribution for layouts ignoring (a) considring (b) thrmal ffcts ( T =60). 5 Copyright 003 by AME

6 Exampl : Effct of Amount of nsulation n this xampl w invstigat th ffct of incrasing th amount of availabl insulation. With a largr amount of insulation availabl ( r = 0.35 ) th structur is mor protctd mor of th structural matrial can b placd away from th cool wall (Fig. 8). Rsults with incrasing amounts of availabl insulation follow similar trnds as squncs of rsults for incrasing th targt tmpratur. Notic that in our formulation, incrasing th siz of but kping th siz of constant has th ffct of rducing th amount of gnratd hat, as th siz of H is rducd. shown for T =60 T =0 in Fig. 0. n this cas th hat cannot b dissipatd as fficintly, as it has to travl across th whol of to rach th cool wall. Thrfor, vn though mor insulation is availabl, th avrag tmpraturs ar highr. As a rsult, th layout of th structur for a givn tmpratur targt T rsmbl solutions obtaind for a distributd hat sourc (Exampl ) but with for a lowr T. hatd wall: q=q 0 insulatd F 0 cool wall : T=0 insulatd U=0 Figur 9. Boundary conditions usd in Exampl 3. (a) for r = 0.0 (b) for r = 0.35 (a) for T =60 (b) for T =0 Figur 0. Optimal layout of structural matrial insulation for boundary conditions as in Fig. 9. CONCLUON (c) for r = 0. 0 (d) for r = 0.35 Figur 8. Optimal layout of structural matrial insulation for T =60 diffrnt amounts of insulation. Exampl 3: Effct of Boundary Conditions Hr w modify th initial assumptions by kping th hat sourc outsid th dsign domain (i.., stting ρ 0 ) imposing a prscribd hat sourc along th lft wall ( Fig. 9). 0 Hat strngth q is adjustd to that th total amount of hat is th sam as in th prvious xampls. n this modl th insulation fills whatvr spac is not occupid by th structur, turning th problm into a two-phas dsign. Th solution is W hav shown that it is possibl to introduc considration of masurs to protct a structur from hat into th stard topology optimization problm with a rlativly straight forward xtnsion. Th modl usd is admittdly vry simpl, yt it shows that th introduction of thrmal critria can significantly chang th natur of th optimal layout. ACKNOWLEDGMENT Th implmntation of th mthod of moving asymptots usd hr was providd by Prof. Kristr vanbrg from th Dpartmnt of Mathmatics at KTH in tockholm. W thank Prof. vanbrg for allowing us to us his program. 6 Copyright 003 by AME

7 REFERENCE Bndsø MP. Optimal shap dsign as a matrial distribution problm. truct. Optim;990; :93-0. Bndsø MP. And igmund, O., Topology Optimization pringr, Brlin, 003. Rozvany GN, Zhou M, Birkr T. Gnralizd shap optimization without homognization. truct. Optim. 99; 4: Mljnik HP, chirrmachr R. An nginring approach to optimal matrial distribution shap finding. Comp. Mth. Appl. Mch. Eng.993; 06:-6. vanbrg K. Th mthod of moving asymptots. nt. J. Num. Mth. Eng.987; 4, igmund, O, Ptrsson, J. Numrical nstabilitis in Topology Optimization: A urvy on Procdurs Daling with Chckrboards, Msh-Dpndncis Local Minima. truct. Optim., 998; 6: Copyright 003 by AME