9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 Dyic codestio d selective ss sclig i RADIOSS Explicit L.MORANCAY, G.WINKELMULLER. ALTAIR DEVELOPMENT Frce, rue de l reissce, 984 ANTONY Cédex (Frce) Abstrct : RADIOSS Explicit is o top of the idustril FE codes which re used for crsh siultio sice severl yers, s well s i other pplictio fields such s fluid structure iterctio or stpig siultio. Oe upo the i cocers of RADIOSS developet is to iprove its perforces while providig high qulity of results. Dyic codestio d selective ss sclig both llow icresig the criticl tie step without odifyig the low respose frequecies, either the rigid body oetu. Thus they re wellsuited for the siultio of processes for which the eergy cotet i the high frequecy doi is sll. The ipleettio of these ethods is discussed. Both the perforces d liits of these ethods re illustrted through soe use cses, issued fro crsh field. Keywords: explicit ethod, tie step, dyic codestio, selective ss sclig RADIOSS is registered trderk of Altir Egieerig Ic. Icresig the tie step of the explicit ethod. Stbility of the explicit ethod Let the discretized equtio of otio M u& &+ Cu& + Ku () The ss trix M results fro sseblig the eleetry ss trices M e E Ω F ext t ρ < Φ > < Φ > dv () Usig the cetrl differece schee which correspods to Newrk schee with γ/ d β0 d ssuig there is o dpig - the curret ccelertio is clculted t ech tie step + by solvig ext it it u& + F + F +, with + Ku+ M & F (5) The use of luped ss trix voids solvig siulteous equtios for ccelertios i (5). I the odl bsis, the recurret cotiutio becoes Let + + + k t + L, A f ω + t A with k Where ω is the eige frequecy ssocited to ode. 0 The stbility is isured if the spectrl rdius of A is less th for ll odes, let [,4] (7) (8)
9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 t < (9) ω x Coercil explicit codes geerlly provide wy of risig the tie step by ddig (digol) ss to the odes. Severl drwbcks re iheret to this ethod, s the o coservtio of rigid body trsltiol oetu, d the odifictio of the low frequecies of the syste.. Selective ss sclig Olovsso d l. [,3] proposed to dd dditiol ters to the luped ss trix, iig t decresig the highest frequecies of the syste, while ffectig the lower frequecies s little s possible. Strtig fro For which the eige frequecies re odified such s so tht low frequecies re less ffected. M M + Λ, with Λ αk (0) ω ω () + αω Sice o lierity would eed to copute M t ech tie step, they proposed to ssebly Λ fro costt eleetry trices such s the rigid trsltiol ccelertio of eleet is ot odified, let for 4-ode shell eleet 3 3 λ e where e is the totl ss of the eleet. () 3 3 Thus there is o chge i ierti effects o trsltiol rigid body ccelertio. i, Λ i x ext it ( F Fi ) i 0, x M u& M u& (3) i, i i, The iversio of trix would ot be coveiet if it is ot costt s i cse of o lier stiffess, d would still eed to ke lot of opertios t ech tie step for clcultig i i u& M F (4) Cosiderig tht M is strictly digolly doit, equtio (4) is pproxited usig itertive resolutio ethod up to soe tolerce. Cougte grdiet itertios re de up to where the iitil solutio is coputed usig the luped ss trix.3 Dyic codestio F Mu& ε F Mu& 0 (5) u& M F (6) 0.3. Adptive eshig Mesh refieet hs bee ipleeted for shell eleets, which better llows ppig of stte vribles th free re-eshig. It is ostly dedicted to stpig siultio d severl refieet criteri re vilble, tkig ito ccout the geoetry of cotctig tools, specilly. A o coforig esh is itied d kieticl coditios re iposed t o coforig odes.
9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 θ θ + θ θ θ FIG. No coforig esh X X + X It is possible to refie the esh up to soe levels, fro tie 0..3. Dyic codestio Let the reduced set of odes () of the corse esh. The dditiol odes () coig fro the ctul esh will be codesed oto odes (). Luped ss M t the reduced set of odes is coputed by cosiderig the corse esh. Forces coputed t codesed odes re pped to odes () usig the shpe fuctios φ of the corse esh so tht the equtio of otio t the reduced set of odes becoes M u& F + Φ ( ) F (8) Forces coputed t the codesed odes re still cosidered s ierti forces M ( u& u&& ) F with u&& Φ &, ( ) u& (8) But the luped ss M t codesed odes is set big eough, so tht the tie step c be coputed by oly cosiderig the odes of the corse esh. M is ot bsolute ss but it is oly reltive to the e displceet u& &. Therefore, icresig it does ot ffect globl oetu. O the cotrry to selective ss sclig, o trix iversio is eeded with dyic codestio. But dyic codestio is vilble oly for 4-ode d 3-ode shell eleets. Use cses. Box be The proble is described t figure : box be is clped o oe side d the other side is rigid body upo which costt force of 00 N is pplied. The sectio of the be is squre. 38 00 Force 00 N FIG. Proble descriptio Johso-Cook Elsto-Plstic teril lw E0000 MP, ρ0.0078 g -3 The be is eshed with 4-ode shell eleets. Due to syetry, oly hlf of the be is odelized. The corse esh which is cosidered for dyic codestio is show t figure 3 (defortio t 5 s is preseted). The tie step usig the corse esh is bout 0.57 µs, with dyic codestio s well s without. The fie esh is obtied fter levels of refieet (ech eleet of the corse esh is split ito 4x4 eleets). The fie esh is built this wy i order to ke possible the use of dyic codestio. But o rel dptive eshig is used sice the esh is refied fro tie 0 up to the fie esh. 3
9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 The defortio of the be fter 5 s whe usig the fie esh d defult tie step is show i Figure 3. The results deostrte to coverge to the fie esh. FIG. 3 Defortios of the corse esh d the fie esh t 5 s Figure 4 copres the results which re obtied with the fie esh depedig o the settigs which re for the tie step. () usig corse esh d defult tie step (roud 0.57 µs) (b) usig fie esh d defult tie step (roud 0.5 µs) (c) usig fie esh d ddig ss i order to eforce the tie step to 0.57 µs (d) usig fie esh d dyic codestio : tie step is the se s usig the corse esh (roud 0.57 µs) (e) usig fie esh d selective ss sclig i order to eforce the tie step to 0.57 µs (f) usig fie esh d selective ss sclig i order to eforce the tie step to µs FIG. 4 Displceet of the cetre of ss of the rigid body log the directio of the pplied lod The corse esh is clerly ot coverged, d the results of ddig ss o the fie esh re ot better. The results usig dyic codestio d selective ss sclig with tie step iposed to the tie step of the corse esh re closed to the results of the fie esh d defult tie step. There is little ore discrepcy with selective ss sclig d higher iposed tie step ( µs) but the results re still cceptble. The coputtio tie d the reltive cost of the specific tretets to dyic codestio d selective ss sclig versus the totl coputtio tie re preseted t figure 5: dyic codestio is fster th selective ss sclig, sice there is o eed to iverse trix. The reltive cost of itertios of cougte grdiet icreses with the iposed tie step, sice covergece eeds ore itertio to be chieved. 4
9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 () Defult tie step (d) Dyic codestio (e) Selective ss sclig, dt0.57 µs (f) Selective ss sclig, dt µs Nuber of cycles 74335 779 7576 0030 Elpsed tie 45 s 385 s 650 s 49 s Reltive cost 6,5% 45% 57% FIG. 5 Coputtio tie. Fo drop test A Fo block usig visco-elstic lw is supported t the botto by fixed shell. It is ipcted by rigid sphere with steel teril, which gets iitil velocity d is subitted to grvity. Rigid sphere (steel) Iitil velocity Fo block Iterfce to fixed shell FIG. 9 Proble descriptio plus defortio t xiu copressio (sectio view) Figure 0 shows the cotct force gist the support. Oe c otice the force pek correspodig to the desifictio phse of the fo. Results of ddig ss d selective ss sclig usig iposed tie step equl to 0.5 µs re close to those obtied with defult tie step. For iposed tie step equl to 5 µs, the behviour is better represeted with selective ss sclig th ddig ss. () Defult tie step - tiestep decreses fro 7 µs to 0. µs due to lrge out of copressio. (b) Addig ss, dt0.5 µs (c) Selective ss sclig dt0.5 µs (d) Addig ss, dt5 µs (e) Selective ss sclig, dt5 µs FIG. 0 Cotct force gist the support 3 Coclusio Dyic codestio d selective ss sclig both llow to icrese the tie step while gettig results closed to those obtied with defult tie step, t the opposite to ddig ss. Despite of the over-cost due to their ipleettio, they llow to reduce the coputtio tie. Selective ss sclig is ore expesive i soe cses, but its ipleettio kes it ore geerl d it is vilble for ll types of eleets. 5
9 èe Cogrès Frçis de Mécique Mrseille, 4-8 oût 009 Refereces [] Hugues T.J.R., The fiite eleet ethod: lier sttic d dyic fiite eleet lysis, Pretice Hll Egelwood Cliffs New Jersey 0763, 987. [] Olovsso L., Siosso K., Uosso M., Selective ss sclig for explicit fiite eleet lyses, Itertiol Jourl for Nuericl Methods i Egieerig, 63, 436-445, 005. [3] Olovsso L., Siosso K., Itertive solutio techique i selective ss sclig, Couictios i Nuericl Methods i Egieerig,, 77-8, 006. [4] RADIOSS Theory Mul, 9.0 versio, 008. 6