The LS-TaSC Software
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1 The LS-TaSC Sofware TOPOLOGY AND SHAPE COMPUTATIONS USING THE LS-DYNA SOFTWARE THEORY MANUAL Jauary 2014 Verso 3.0 Copyrgh LIVERMORE SOFTWARE TECHNOLOGY CORPORATION
2 All Rghs Reserved Corporae Address Lvermore Sofware Techology Corporao P. O. Box 712 Lvermore, Calfora Suppor Addresses Lvermore Sofware Techology Corporao 7374 Las Posas Road Lvermore, Calfora Tel: Fax: Emal: Webse: Lvermore Sofware Techology Corporao 1740 Wes Bg Beaver Road Sue 100 Troy, Mchga Tel: Fax: Dsclamer Copyrgh Lvermore Sofware Techology Corporao. All Rghs Reserved. LS-DYNA, LS-OPT ad LS-PrePos are regsered rademarks of Lvermore Sofware Techology Corporao he Ued Saes. All oher rademarks, produc ames ad brad ames belog o her respecve owers. LSTC reserves he rgh o modfy he maeral coaed wh hs maual whou pror oce. The formao ad examples cluded here are for llusrave purposes oly ad are o eded o be exhausve or all-clusve. LSTC assumes o lably or resposbly whasoever for ay drec of drec damages or accuraces of ay ype or aure ha could be deemed o have resuled from he use of hs maual. Ay reproduco, whole or par, of hs maual s prohbed whou he pror wre approval of LSTC. All requess o reproduce he coes hereof should be se o sales@lsc.com. 22-Ja-14 2
3 I mus say looks a b lke scece fco o may people Ofr Shor, Jue 2009, whle evaluag he alpha verso. 3
4 1. TOPOLOGY THEORY 1.1. Backgroud The radoal approach for solvg opology opmzao problems s based o sesvy aalyss ha s expesve o oba for lear-sac problems. However, dervg aalycal sesves for dyamc aalyss s very dffcul due o he complex eracos amog maeral oleares, geomery ad mesh, ad rase aure of load ad boudary codos. Numercal compuao of sesves s also o praccal due o he hgh compuaoal expese. Hece he coveoal sesvy based approach of opology opmzao s o praccal for crashworhess problems. To overcome he aforemeoed dffcules opology opmzao, a dffere approach was proposed. Ths approach does o requre grades ad hece here s o eed o compue he sesves. I verso 1, he approach was refer o as Hybrd Cellular Algorhm [1,2], bu academcs dog a leraure revew should also cosul oher sadard vews of opology opmzao ad our pae porfolo o udersad wha s currely acually mplemeed. Wh here beg o cellular algorhm he curre verso, he mehodology s bes referred o as LS-TaSC Implemeao The algorhm for srucural opmzao s show pcorally Fgure 1-1. Afer defg he problem, he opology s evolved usg he smple rules defed o he varables. The cosras are accommodaed durg he sae updae procedure. Fgure 1-1: The opology opmzao algorhm 4
5 Defo The pu daa s used o defy he desg doma ad desg maeral model. The pu daa comprses of mehod daa e.g., umber of eraos, covergece olerace, ad he problem daa, e.g. load cases, desg par, ec Creag he varables The fe eleme model s mapped o desg varables. Each desg varables s assged o a sold eleme he desg doma. For exruso ad symmery cosras, he equaly cosras are defed bewee he varables. For casg cosras, equaly cosras are esablshed Flerg of resuls Pas work were based o he srucured grd arrageme of cells. Ths assumpo would breakdow for dusral applcaos where srucured grds are o always possble. Hece, a radus based sraegy s used o defy eghbors. I hs sraegy, a vrual sphere of user-defed radus s placed a he cerods of a eleme. All elemes ha are wh hs sphere are cosdered he eghbors of he correspodg eleme, ad he resuls are averaged over he elemes he eghborhood U w U w. (2) Maeral Parameerzao The maeral model s parameerzed usg a so-called desy approach. I hs approach, a desg varable s drecly lked o he dvdual maeral eleme such ha each varable has s ow maeral model. The maeral properes correspodg o he values of desg varables are obaed usg a approprae erpolao model. The sold soropc maeral wh pealzao (SIMP) model [6] s he mos popular erpolao mehod. Ths model s power law approach ha drves he ermedae maeral properes owards he boudares o oba a 0-1 opology. Accordg o SIMP model, he maeral properes are defed as, x) x, (3) ( 0 p E x) x E, (4) ( 0 q x) x, (5) ( 0 q E x) x E, (6) h( h0 where deoes he desy of he maeral, E represes he Youg s modulus, s he yeld sress, ad E h s he sra hardeg modulus. The las wo maeral properes represe maeral o-leares ad are requred for dyamc problems lke crash ha volve maeral yeldg. The subscrp 0 refers o he base maeral properes. The desg varable x, also kow as relave desy, vares from 0 o 1 where 0 dcaes vod ad 1 represes full maeral. A more dealed descrpo of he maeral model 5
6 parameerzao, oe should refer o Bedsøe ad Sgmud [7], ad Pael [8]. The elemes wh desg varable value less ha a user-defed mmum value are deleed o mprove umercal sably. To eable he use of very large FE models, hs approached was mplemeed usg a dscree maeral model approach [9] Desg Obecves ad Cosras The ypcal goal of opology opmzao s o oba a srucure wh he bes use of he maeral. Complace ad he sra eergy desy are he mos commoly used obecves for lear-sac problems. For dyamc problems lke crashworhess smulaos, he srucure eeds o absorb maxmum eergy whle maag he srucural egry ad keepg he peak loads rasmed o he occupas low. Followg he formulao proposed by Pael [8], he goal of obag uform eral eergy desy he srucure s defed as he obecve for opmzao. Ths cocep s smlar o he fully-sressed desg ad uform sra eergy desy approaches (Hafka ad Gurdal [10], Paak ad Hopks [11]) ha are well esablshed leraure for lear-sac problems. The use of he eral eergy desy opmzao, s relaoshp wh he desg sesvy formao for crash problems, ad s usefuless for rakg varables has bee exesvely suded by Öma [12,13]. The opmzao problem s formulaed as, N L * m w U ( x ) U, (7) x 1 subec o : N 1 C x m ( x ) V M l C x C u, 1.0. * 1,2,..., J where U represes he eral eergy desy of he h eleme, V s he volume of h eleme, U * represes eral eergy desy se po, ad C s he h cosra. There are L load cases wh a oal of J cosras. The superscrps l ad u represe lower ad upper bouds o he cosras, respecvely Desg Varable Ialzao The desg varables are alzed o sasfy he maeral cosra. All elemes are assged he same desg varable values. All assocaed feld varables are also alzed o zero Smulao o Oba Feld Varables The elemes he fe eleme model are modfed by chagg he maeral models, addg or deleg elemes, a each erao. So he pu deck s re-wre a each erao. Ths modfed pu deck s aalyzed usg LS-DYNA [11]. Oe ca ake advaage of mulple processors usg he MPP verso of LS-DYNA. The releva feld (8) 6
7 varables for all elemes are obaed from he oupu o compleely defe he sae of each varable. For mulple load case codos, he sae varable s based o he oupu from smulaos of dffere load cases. For dyamc problems, was observed ha accoug for he hsory of evoluo duces sably by reducg he eleme deleo rae. Hece, he feld varable (eral eergy desy) of h varable a erao s updaed by defg a weghed sum o he feld varable of hree prevous eraos as follows, U ( x ) U ( x ). (9) 0 where x s he desg varable assocaed wh he h varable a erao. If he load cases are a mxure of dyamc ad sac problems, he hs weghg s followed for all he load cases Global Cosra Hadlg I presece of cosras oher ha he mass cosras, he arge mass cosra s adused o sasfy he srucural cosras. The mass arge (M * ) s creased proporo o he cosra volao for all cosras excep force cosras for whch he mass arge s reduced. * * M M M, c M K (10) / J, c where J s he oal umber of cosras, K s he coeffce used o scale he cosra volao of he h cosra, ad ε s he volao of he h cosra. The oal chage mass arge (ΔM) s bouded o allow gradual chages he srucure Dyamc Load Case Weghg The desred behavor s k1 C1 offse 1 k2c2 offse wh C he cosra value, k a scale facor, ad a offse added as show. The wegh w of load case s adused o chage cosra C. The arge value s compued as ( kc offse ) C arg e from whch we compue w ( C arg k C offse )/ C / w wh he dervave approxmaed as ±1 ad a e maxmum boud s place o eraos. 0 w o esure covergece a reasoable umber of 7
8 Sae Updae Rules Ths s he hear of opology opmzao mehod. I hs sep, he sae of a varable s updaed based o he sae of s eghbors. The sae updae s carred ou wo seps: 1. Feld varable updae: The feld varable (eral eergy desy) of a varable s updaed as accoug for he feld varable values of s eghbors as, U 1. (10) U 0 2. Varable/Maeral Updae: Oce he feld-varable sae of each varable s defed, he desg varable s updaed o reflec he chages. Whle umerous rules are proposed leraure [6] o updae desg varables, a corol based rule used by Pael [8] s mplemeed here (Fgure 1-2). The chage he desg varable of h varable (Δx ) s compued as, * * x K U U / U (11) 0. * where K s a scalg facor ad U deoes he eral eergy desy se po. The desg varable s updaed as, 1 x x x. (12) The chage he varable s cosraed by he bouds o he value of he desg varable.e., 1 I. f x 1 LB, he x LB, 1 II. f x 1 UB, he x UB, ad oly cera dscree values are allowed. dx =0 dx =K(U /U * 1) Correc X +1 X +1 = x + dx dm =M +1 -M M= M +1 If M-M * < ε Yes No U * =U * (M/M * ) Fgure 1-2: Desg varable updae. dx= dx Sop The mass of each eleme s he calculaed by usg he approprae maeral model assocaed wh he desg varables. If he oal mass of he srucure mees he cosra, he oal chage desg varables hs erao s calculaed, ad he desg varable updae s cosdered compleed. If he mass cosra s o sasfed, he IED se po s updaed eravely o accommodae he maeral cosra as, * * * * U U U M / M. (13) 8
9 where M s he mass of he srucure Soppg Crera Two ermao codos are used o sop he opmzao process. 1. The umber of eraos has exceeded he maxmum umber of eraos, or 2. The chage he opology s smaller ha he olerace,.e., N dx x. (14) The umercal oscllaos covergece are lmed by averagg he oal chage opology over wo eraos. 1 9
10 2. SURFACE DESIGN THEORY 2.1. Backgroud The radoal approach for solvg shape desg problems s based o sesvy aalyss ha s expesve o oba for lear-sac problems. However, dervg aalycal sesves for dyamc aalyss s very dffcul due o he complex eracos amog maeral oleares, geomery ad mesh, ad rase aure of load ad boudary codos. Numercal compuao of sesves s also o praccal due o he hgh compuaoal expese. Hece hs approach s o praccal for crashworhess problems. To overcome he aforemeoed dffcules, a dffere approach was proposed. Ths approach does o requre grades ad hece here s o eed o compue he sesves. The mehodology s bes referred o as LS-TaSC Implemeao The algorhm s show pcorally Fgure 1-1. Afer defg he problem, he surface shape s evolved usg he smple rules defed o he varables. Fgure 2-1: The surface desg algorhm Defo The pu daa s used o defy he desg problem. The pu daa comprses of mehod daa e.g., umber of eraos, covergece olerace, ad he problem daa, e.g. load cases, desg surface, ec Creag he varables The dscree surface s mapped o desg varables. The ormal dsplaceme o each ode he desg surface assged o a desg varable. For exruso ad symmery cosras, he equaly cosras are defed bewee he varables. 10
11 Flerg of resuls A radus based sraegy s used o defy eghbors. I hs sraegy, a vrual sphere of user-defed radus s placed a he cerods of a eleme. All elemes ha are wh hs sphere are cosdered he eghbors of he correspodg eleme, ad he resuls are averaged over he elemes he eghborhood U w U w. (2) Desg Obecve The goal of shape desg s o oba surface wh a uform sress. The opmzao problem s formulaed as, 1 m [ U( x ) U (7) x arg e ] where U represes he desg feld (ypcally he vo Mses sress) a he ode assocaed wh desg varable x, ad U arg e represes he arge value of he desg feld. (8) Targe Sress The goal of shape desg s o oba surface wh a uform sress. I order o complee hs ask we eed o defe a arge sress. There are he followg possbles of selecg a arge sress: Average over he surface The maxmum value o he surface The mmum value o he surface A user-defed value Usg he above arge sresses should be oed ha he goal becomes more suble ha obag he a surface wh a uform sress: f selecg he maxmum s as he arge sress, he he wegh of he srucure wll be reduced; whle f he mmum s seleced, he he average sress s reduced Desg Varable Ialzao All desg varables are also alzed o zero Smulao o Oba Feld Varables The elemes he fe eleme model are modfed he odal locaos for all eraos. So he pu deck s re-wre for all eraos. The releva feld varables for all odes are obaed from he oupu o compleely defe he sae of each varable. For 11
12 mulple load case codos, he sae varable s based o he oupu from smulaos of dffere load cases. For dyamc problems, was observed ha accoug for he hsory of evoluo duces sably by reducg he eleme deleo rae. Hece, he feld varable (eral eergy desy) of h varable a erao s updaed by defg a weghed sum o he feld varable of hree prevous eraos as follows, U ( x ) U ( x ). (9) 0 where x s he desg varable assocaed wh he h varable a erao. If he load cases are a mxure of dyamc ad sac problems, he hs weghg s followed for all he load cases Varable Updae Ths s he hear of shape desg mehod. I hs sep, he sae of a varable s updaed based o he sae of s eghbors. The sae updae s carred ou wo seps: 1. Feld varable updae: The feld varable (eral eergy desy) of a varable s updaed as accoug for he feld varable values of s eghbors as, 0 U 1. (10) U 0 2. Varable updae: Oce he feld-varable sae of each varable s defed, he desg varable s updaed o reflec he chages.. 0 The chage feld value requred s U U arg e x he requred moveme of ode ormal o he surface.. Now compue U x / x wh Soppg Crera Two ermao codos are used o sop he opmzao process. 1. The umber of eraos has exceeded he maxmum umber of eraos, or 2. The chage he opology s smaller ha he olerace,.e., N dx x. (14) Refereces 1. A Tovar, Boe Remodelg as a Hybrd Cellular Auomao Opmzao Process, PhD hess, Uversy of Nore Dame,
13 2. NM Pael, B-S Kag, JE Reaud, Crashworhess Desg usg a Hybrd Cellular Auomaa Algorhm, I Proceedgs of he 2006 Ieraoal Desg Egeerg Techcal Coferece, DETC , Phladelpha PA, Sep 10-13, P Haela, B Km, O he Use of Eergy Mmzao of CA Based Aalyss Elascy, Srucural ad Muldscplary Opmzao, 23, 23-33, J Forsberg, L Nlsso, Topology Opmzao Crashworhess Desg, Srucural ad Muldscplary Opmzao, 33, 1-12, hp://mahworld.wolfram.com, Las accessed 23-March MP Bedsøe, O Sgmud, Maeral Ierpolao Schemes Topology Opmzao, Archves of Appled Mechacs, 69, , MP BedsØe, O Sgmud. Topology Opmzao: Theory, Mehods ad Applcaos, Sprger-Verlag, Berl, NM Pael, Crashworhess Desg Usg Topology Opmzao, PhD hess, Uversy of Nore Dame, Goel T, Roux WJ. Topology opmzao for desgg egeerg produc. US pae 8,126,684, fled Aprl 10, 2009, ad ssued February 8, RT Hafka, Z Gurdal, MP Kama, Elemes of Srucural Opmzao, Kluwer Academc Publshers, Dordrech, The Neherlads, 2 d ed., SN Paak, DA Hopks, Opmaly of Fully-Sressed Desg, Compuer Mehods Appled Mechacs ad Egeerg, 165, , M Öma, Opmzao ad Robusess of Srucural Produc Famles, PhD Thess, Lköpg Uversy, M Öma ad L Nlsso, Srucural Opmzao based o Ieral Eergy Dsrbuo, Egeerg Opmzao, 45(4), , JO Hallqus, LS-DYNA Maual verso 971, Lvermore Sofware Techology Corporao, Ocober RT Hafka, Z Gurdal, MP Kama, Elemes of Srucural Opmzao, Kluwer Academc Publshers, Dordrech, The Neherlads, 2 d ed., SN Paak, DA Hopks, Opmaly of Fully-Sressed Desg, Compuer Mehods Appled Mechacs ad Egeerg, 165, ,
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