Continuum structural topological optimization with dynamic stress response constraints

Transcription

1 Contnuum tuctual topologcal optmzaton wth dynamc te epone contant Bn Xu 1*, Le Zhao 1 and Wenyu L 1 1 School of Mechanc, Cvl Engneeng & Achtectue, Nothweten Polytechncal Unvety, X an 717, PR Chna Abtact Ste elated poblem of geat mpotance to tuctue optmzaton, howeve,mot wok wee pad attenton to tatc te contaned topology optmzaton,only a lmted numbe of wok have been devoted to the topology optmzaton of the tuctue wth dynamc te epone equement unde andom vbaton.th pape deal wth topology optmzaton of contnuum tuctue wth dynamc te epone contant. Baed on the ICM method and the b-dectonal evolutonay tuctual optmzaton method,a method fo the topology optmzaton of contnuum tuctue wth dynamc te epone contant lmt popoed, The topology optmzaton model wth the objectve functon beng the tuctual weght and the contant functon beng tuctual dynamc te epone unde andom vbaton peented n th pape,in ode to geatly educe the entvty cot of dynamc te epone, an optmzaton model wth educed dynamc te epone contant bult,beng ncopoated wth the atonal appoxmaton fo mateal popete(ramp, an vayng dynamc te epone lmt cheme,a tut egon cheme, and an effectve local te appoach lke the qp appoach to eolve the te ngulaty phenomenon.theefoe, a et of quadatc appoxmate functon fo tuctual dynamc te epone contant ae fomed,fnally,the popoed algothm, ntegated wth the dual theoy, to olve the optmzaton poblem.a ee of numecal example, whch ae unde dffeent knd of exctaton, dffeent bounday condton and dffeent model, ae peented to demontate the feablty and effectvene of the popoed appoach. Keywod:topology optmzaton,dynamc te epone, te contant, RAMP,MMA appoxmaton,dual theoy

2 1.Intoducton In ecent yea,tuctual optmzaton ubjected to te contant of geat mpotance n advance engneeng applcaton,mot attenton ha been pad to tatc te contant poblem, Matteo Bugg et.al [1]dealt wth the mpoton of local te contant n topology optmzaton by an appoach called a the qp appoach, Bugg and Duynx et.al [] popoed a electon method leadng to futhe educton n the numbe of actve contant, Paí et.al [3] nvetgated a technque of block aggegaton by ung moe global contant to educe the numbe of contant nvolved n a ngle global functon, The adaptve nomalzaton cheme popoed by Le et al.[4] wok well n contollng the maxmum te, Recently, a ee of vaed contant lmt wee ntoduced and a knd of new tuctual topologcal optmzaton method wth dffeent chaactetc equement,wa popoed by Rong et al. [5], baed on the atonalappoxmaton fo mateal popete (RAMP and the dual theoy. Howeve, few attempt have been made to ncopoate andom dynamc epone contant, Shu et.al [6] popoed level et baed topology optmzaton method fo mnmzng fequency epone. Rong et.al [7-8] ued ESO method and SQP method to obtan the optmal topology of the contnuum tuctue unde andom exctaton. Yang et al.[9] tuded the topology optmzaton unde tatc load and naow-band andom exctaton, whee the tatc and dynamc epone analye wee poceed ndependently wthout any upepoton. Zhang et al. [1] nvetgated the optmal placement of the component and the confguaton of the uppotng tuctue wthn a pedefned degn doman to mpove the tuctual tatc and andom dynamc epone multaneouly. Whle thee optmzaton wok ae not devoted to olve dynamc te epone contant lmt poblem. To mpove the dynamc chaactetc of the tuctue, t much mpotant and neceay to conde the dynamc te epone contant lmt n topology optmzaton.hee, a method fo the topology optmzaton of contnuum tuctue wth dynamc te epone contant lmt popoed n th pape, The topology optmzaton model wth the objectve functon beng the tuctual weght and the contant functon beng tuctual dynamc te epone unde andom vbaton peented. The followng ecton of th pape ae oganzed a follow: Secton fomulate the dynamc te epone unde andom exctaton. Secton 3 addee element topologcal vaable and ntepolaton functon ued n th pape. The topologcal optmzaton wth dynamc te epone contant lmt fo mnmzng the tuctual weght, ntoduced n Secton 4. Secton 5 peent the entvty analy on the dynamc te epone wth epect to the degn vaable, The Secton 6 ntoduce eveal numecal example to dcu the effectvene of the popoed method. Concludng emak ae fnally dawn n Secton 7.

3 .The dynamc te epone The Von Me te n the thee-dmenonal pace can be defned a follow: 3( (1 k k k k k k k vm x y z x y x z y z xy xz yz Hee, { } {,,,,, } epeent the te vecto of the k-th element of the k k k k k k k T x y z xy xz yz tuctue,equaton(1 can be expeed a: vm { } A{ } Tace{ A { }{ } } ( k T T whee A a ymmetc matx,and A Theefoe, the mean quae epone of the Von Me te can be wtten a: T E( k ({ } { } ( { { } k { } kt } { { } k { } kt vm E A E Tace A Tace AE } (3 Whee { } k kt E { } the covaance matx of the te vecto { }k.in a fnte element analy,the te vecto { } k of the k-th element can be expeed a: N 1 h k k k k k k { } [ D] [ B] h { u} S { u} (4 N h h1 whee [ D ] k the elatc matx of the k-th element, [ B ] k h the tan matx at the h-th Gua ntegaton pont of the k-th element, N h the total numbe fo the Gua ntegaton pont,and {} u k nodal dplacement vecto of the k-th element.then the covaance matx E { } k { } kt can be expeed a: k kt k k k k T E { } { } E ({ S} { u} ({ S} { u} k k kt kt E ({ S} { u} { u} { S} { S} E[{ u} { u} ]{ S} k k kt kt (5

4 k kt whee E[{ u} { u } ] the the covaance matx of the nodal dplacement vecto {} u k. The fnte element equaton fo a tuctual dynamc poblem can be expeed a: Whee[ M ], [ C ] and [ K ] ae the N [ M ]{ Y} [ C]{ Y} [ K]{ Y} { f ( t} (6 N ma. Dampng, and tffne matce of the tuctue epectvely. I the numbe of degee of feedom. {} Y, { Y } and { } ae the dplacement,velocty and acceleaton vecto, { f( t } the N 1 whte noe andom extenal exctaton vecto,whch loaded n the fom of pectum.it can be aumed that powe pectal denty matx of the extenal exctaton vecto { f( t } can be expeed a: ( S (7 Sf whee S N N em-potve eal ymmetc matx, S would be a dagonal matx f all degee of feedom ae unelated,.e., S S, S ( j. I II j The coepondng undamped fee vbaton equaton of (6 can be wtten a: [ M ]{ Y} [ K]{ Y} (8 Aume that the oluton fo equaton (8 n t,then the chaactetc vecto equaton fo the equaton (8 can be expeed a: ([ K] I[ M] (9 Y In equaton (9, and ae the -th ccle natual fequency and mode hape.we conde that the mode hape can be nomalzed wth epect to the ma matx and atfy equaton (1 unde the popotonal dampng. T { } [ M ]{ } I T { } [ K]{ } dag( T { } [ C]{ } dag( (1 Whee { } { 1,,,, N} and the dampng ato of the -th mode,aume that { Y} { y} (11 Equaton(6 can be wtten a: ( ( { } ( ( ydag ydag y f t F t (1

5 Then the coelaton matx of mode epone y and y can be expeed a: C d a (13 whee 1, T 1, d { } S{ } (14a a, b, e (14b 1, Q 1 a e g a b, g Q (14c 3.Element topologcal vaable and ntepolaton functon A topologcal optmzaton fo mnmzng the tuctual weght whle atfyng dynamc te epone contant lmt bult n th pape,in the topologcal optmzaton wth dynamc epone contant,numecal example how that the oluton fo dynamc epone value and t entvty would be wong becaue of ntepolaton cheme wthout enough accuacy,thee ntepolaton cheme,uch a Sold Iotopc Mateal wth Penalty(SIMP,whch cannot acheve the matchng punhment between the tffne and ma matx,and lead to localzed mode, educe the accuacy of modal analy. Refeng to the Ratonal Appoxmaton fo Mateal Popete(RAMP and the ICM (Independent, Contnuou and Mappng method,the wegth.tffne matx,te and ma matx of an element ae ecognzed by followng functon,epectvely whee w f ( w, K f ( K, w K [ S] k f ( [ S] M f ( M (15a M W (, fk( 1 (1, f ( M I, f M ( I (15b f w I whee ae the weght,tffne matx,te and ma matx of the -th element,epectvely,and w, K, [ S ] and M ae ognal weght,ognal tffne and ma matx of the -th element,epectvely.in th wok, w M 1.5, 5 and 6 ae ued.due to the atonal functon n the RAMP ntepolaton cheme, the devatve of the functon n t zeo even when tend to zeo,whch can pove the effect of weak mateal on tffne,and pevent the localzed mode.

6 4.Optmzaton poblem tatement The topologcal optmzaton model fo mnmzng the tuctual weght wth dynamc te epone contant lmt can be fomulated a follow: mn : W t..:, (, U vm vm (16 Fo the pupoe of obtanng pedomnantly black-and-whte degn n th pape, the vay dynamc epone contant method adopted,then,equaton(16 can be tanfeed nto equaton (17: mn : f w (1 w ( k1, U max ( vm, ( vm, 1,, Q t..: 1 1 Q Q Q 1 1 Q Q ( k1, U ( vm, ( vm, 1 Q 1 (17 Whee a weghtng expeence paamete; a vayng contant lmt,and ( k U ( 1, vm, expeed a: ( mn( (,( ( (.( ( ( k ( k U ( k ( k U ( k1, U vm, vm, L vm, vm, vm, vm, vm, ( k ( k U ( k ( k U ( vm, mn( ( vm,,( L( vm, ( vm,.( vm, ( vm, ( (18 Whee a dynamc epone lmt change facto, L a elaxaton coeffcent fo ( 1, the dynamc epone contant, ( k U ae vaed accodng to equaton (18at each oute loop teaton tep, vm, ( k the dynamc te epone of the -th ( vm, element of the tuctue at the k-th loop teaton tep. 5.Sentvty analy on the dynamc te epone Aume that x 1,the entvty of the contant functon wth epect to x can be expeed a: k kt ( vm, k k E[{ u} { u} ] kt k C kt tace( A{ S} { S} tace( A{ S} { S} x x x (19

7 ( C da ( ( x x x ( d ( a ( a d da x x x ( n d S x 1 j j a x x x b x x x e x x x a b a e g a b Q a e x x x x x (1 to the entvte of the ccle natual fequency and the mode hape wth the epect x can be epectvely wtten a: 1 T k [ K] [ M] k k k x k x x ( T [ K] [ M] k k p N x x k p k j j bkp p bkp k p x p1 1 T [ M ] k k k p x (3 [ K] (1 [ M ] [ K], [ M ] 1 x ((1 x x x (4 In ode to deal wth checkboad poblem of topologcal optmzaton,the appoach popoed by Sgmund and Peteon adopted to edtbute the entvty of the dynamc te epone. Then the dynamc te contant expeon n equaton (17 can be appoxmated by the one-ode appoxmaton functon,and the objectve functon n equaton (17 can be appoxmated by t econd-ode appoxmaton functon, and olvng theappoxmatng quadatc pogammng poblem of equaton.(17 can be tanfeed nto olvng a dualpogammng poblem by ung the dual theoy.

8 6.Numecal example A beam lke D tuctue hown n Fg.1. The beam wth dmenon.3 m.1m mply uppoted at both end. The degn doman. m.1 m equally dvded nto 8 4 fou-node plane te element. The mateal aumed wth Young modulu E = 1GPa, Poon ato v =.3, A concentc foce wth auto-powe pectal appled at the mddle pont A along vetcal decton at the bottom of the plate a hown n Fg. 1 and all mode dampng ato n =. ae 17 pecfed, the pecbed dynamc te epone contant lmt.5 1. Hee, the paamete.46 and L 1. elected,fg. the optmal topology obtaned by the popoed method. The optmzaton htoe of the weght and the maxmum dynamc te epone fo the tuncated mode numbe 16 ae llutated n Fg. and 3, epectvely. The optmaltopology obtaned by the popoed method hown n Fg. 4. Fg. 1. The ntal tuctue model Fg.. the optmzaton htoy of the topology weght

9 Fg. 3. the optmzaton htoy of the maxmum dynamc te epone Fg. 4. The optmal topology of the plane te plate obtaned by the popoed method 7.Concluon (1A methodology fo the topology optmzaton of contnuum tuctue wth wth dynamc te epone contant popoed fo mnmzng the tuctual weght. Ramp ntepolaton cheme ued to pevent localzed mode,and a vay dynamc epone contant method adopted to make ue that the appoxmate expeon effectve and avod objectve ocllaton phenomenon n optmzaton teaton.the entvty of the dynamc te epone wth epect to the degn vaable deved. (The example ued to pove the valdty and effcency of the popoed method fo the dynamc te epone contant optmzaton poblem,the optmzed topology can tably convege to optmal oluton. The obtaned optmal degn wth pecbed dynamc te epone by the popoed method, wll have a lowe weght than the ntal, gue degn. The popoed contnuum tuctual topologcal optmzaton wth dynamc te epone contant ae effectve, and undeed localzed mode can be pevented.

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