Three Point Bending Analysis of a Mobile Phone Using LS-DYNA Explicit Integration Method

Transcription

1 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) Three Poi Bedig Aalysis o a Mobile Phoe Usig LS-DYNA Explici Iegraio Mehod Feixia Pa, Jiase Zhu, Ai O. Helmie, Rami Vaaparas NOKIA Ic. Absrac I his aricle, he 3 poi bedig aalysis o a mobile phoe usig LS-DYNA explici iegraio mehod is discussed. Sice here are a large umber o coac pairs deied i he FEA model, ad he FEA model is very large i a 3 poi bedig aalysis o a phoe, i is much more coveie o use he explici mehod ha he implici mehod. However, usig explici procedure o a quasi-saic aalysis requires some special cosideraio. Sice a quasi-saic soluio, is by deiiio, a log-ime soluio. I oe requires a excessive umber o small ime icremes. I is compuaioally impracical o coduc he simulaio i is aural ime scale. I real aalysis, he quasi-saic eve is ariicially acceleraed by wo approaches o reduce he compuaio ime. Oe approach is o use mass scalig. Aoher approach is o icrease he loadig rae. These wo approaches are closely relaed ad should work ogeher. I hey are properly used, he speed o he aalysis could be icreased subsaially wihou severely degradig he qualiy o he quasi-saic soluio. We discuss i his aricle how he loadig rae ad mass scalig acor aec each oher, how o selec proper values o hese wo parameers, ad how o use hese wo approaches i he 3 poi bedig aalysis o a mobile phoe. 1. Iroducio The explici soluio mehod is origially developed o model high-speed impac eves i which ieria plays a domia role i he soluio. Ou-o-balace orces are propagaed as sress waves bewee eighborig elemes while solvig or a sae o dyamic equilibrium. Explici ime iegraio mehod has bee used exesively or phoe drop es simulaio. I ac, he explici mehod has prove o be valuable i solvig saic problems as well. For cerai ypes o saic problems, i is much more coveie o use explici mehod ha implici mehod. The explici mehod could more readily resolve complicaed coac problems ha he implici mehod. I addiio, as models become very large, he explici procedure requires less sysem resources ha he implici procedure. However, applyig he explici dyamic procedure o a quasi-saic problem requires some special cosideraios. Sice a saic soluio is, by deiiio, a log-ime soluio, i is oe compuaioally impracical o coduc he simulaio i is aural ime scale, which would require a excessive umber o small ime icremes. To obai a ecoomical soluio, he eve mus be acceleraed i some way. The goal is o model he process i he shores ime period i which ierial orces remai isigiica. There are wo approaches o accelerae he eve. Oe approach is o use mass scalig. Aoher approach is o icrease he loadig rae. I should be meioed ha hese wo approaches are closely relaed. They would aec each oher. I hese wo approaches are properly used, he speed o he aalysis could be icreased subsaially wihou severely degradig he qualiy o he quasi-saic soluio; he ed resul o he slow case ad a somewha acceleraed case would be early he same. Oherwise, he eve would be acceleraed o a poi a which ieria eecs domiae, he soluio would ed o localize, ad he resuls would be quie diere rom he quasi-saic soluio

2 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece I his aricle, we are goig o discuss he use o explici mehod o solve a 3 poi bedig problem o a phoe. Sice here are a large umber o coac pairs deied i he FEA model, ad he FEA model is very large i a 3 poi bedig aalysis o a phoe, i is much easier o use he explici mehod ha he implici mehod. We sar wih examiig how he loadig rae ad mass scalig acor aec each oher, ad how o selec proper values o hese wo parameers wih he aid o a simple mass sprig sysem. The we discuss i more deail he 3 poi bedig aalysis o he phoe.. Approaches o reduce he compuaio ime i a quasi-saic aalysis I usig explici mehod o solve a quasi-saic problem, he quasi-saic eve is usually acceleraed by mass scalig ad loadig rae scalig approaches. To eiciely use hese approaches, i is impora o udersad how he loadig rae scalig ad he mass scalig would aec he perormace o a explici aalysis. Accordig o [1], he ollowig ad hoc rules have bee used o deermie wheher a quasi-saic aalysis is successul: i) The kieic eergy o he deormed srucure shall o exceed a small racio (abou 5%) o is ieral eergy hroughou mos o he ime period o he explici aalysis. ii) The raio o he kieic eergy o he ieral eergy shall be less ha 0.1% a he seady sae. iii) The ime rae o chage o he ieral eergy shall be egligible a he seady sae. iv) The maximum ou-o-plae deormaio shall reach a cosa value a he seady sae. Esseially wheher a aalysis is quasi-saic or o depeds o wheher he dyamic vibraio erms are small eough or o i he respose o he sysem. I wha ollows, we use a simple mass-sprig sysem o discuss i ur he eec o loadig rae scalig ad mass scalig o he perormace o a quasi-saic aalysis..1 The eec o loadig rae Figure 1 shows a mass sprig sysem ad he loadig proile o ha sysem. The load is ramped o F 0 wihi, ad he keeps a he cosa value o F 0. m F k F F 0 Figure 1. Mass sprig sysem ad he loadig proile o ha sysem. 13-3

3 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) The exac soluio o he respose o he mass sprig sysem is as ollows: F 1 0 ( ) x = siω, k ω F 0 x( ) = 1 siω cos ω k ω, 0 (1), () where ω is he aural requecy o he sysem. The raio o he kieic eergy o he ieral eergy is give by: Ekieic mf0 (1 cosω) ( ) = Ei eral k 1 1 siω ω si ω si ( ) ω Ekieic mf0 ( ) = Ei eral k 1 siω cos ( ) ω ω From equaios (1), (), ad (4), we id a ecessary codiio or he vibraio erms i equaios (1) ad () be very small, ad he raio o kieic eergy o ieral eergy i equaio (4) is less ha a cerai value (0.1%) a he seady sae. This ecessary codiio is: he value o ω be big eough. A good rule o humb is o selec, 0, (3),. (4) such ha ω 0π. (5) This is equivale o he relaio o 10, where T is he aural period o he sysem []. This T rule o humb is also valid or a real applicaio o complicaed FEA model, excep ha ω should be replaced by he requecy o he 1 s aural mode o he sysem. Accordig o equaio (1), a sie wave is superimposed o he liear soluio o he displaceme respose o he sysem. The requecy or period o he 1 s aural mode o a real applicaio could be obaied by ruig requecy aalysis. I could also be esimaed by a ew rials o he explici aalysis, as will be discussed urher i Secio The eec o mass scalig For a quasi-saic aalysis, he chage i he mass o a objec wo aec he deormaio o ha objec i a soluio coverges. O he oher had, i a explici aalysis, he mass desiy o a maerial would aec he ime sep size o umerical iegraio. The ceral dierece scheme, which is he mos commoly used explici algorihm, is oly codiioally sable, he sabiliy limi beig approximaely equal o he smalles ime required or a soud wave o ravel L hrough ay o he eleme i he mesh. Tha is, Δ = mi mi, where L mi is he smalles S 13-33

4 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece dimesio i a eleme, ad S is he speed o soud ravelig hrough he eleme. I is well E kow ha he speed o soud ravelig hrough a eleme is proporioal o, where E ρ ad ρ are he Youg s modulus ad mass desiy o he maerial, respecively. Accordig o he above relaios, ariicially icreasig he maerial desiy ρ by a acor o decreases he wave speed by a acor o ad icreases he sable ime icreme by a acor o. Thereore, we could eiciely reduce he compuaio ime by icreasig he mass desiy o hose elemes wih relaively large E ad/or small L mi. Boh LS-DYNA ad ABAQUS/EXPLICIT have eaure o se a smalles ime sep size permied or umerical iegraio [3,4]. I ay elemes i he FEA model could o saisy his ime sep size limi, mass scalig would be doe o hese elemes. I LS-DYNA, he parameer dms i CONTROL_TIMESTEP card could be used o se he miimum ime sep size permied i he aalysis [3]. I simulaios ivolvig a rae-depede maerial or rae-depede dampig, such as dashpos, mass scalig is he oly opio or reducig he soluio ime. I such simulaios icreasig he loadig rae is o a opio because maerial srai raes icrease by he same acor as he loadig rae. Whe he properies o he model chage wih he srai rae, ariicially icreasig he loadig rae ariicially chages he process. However, as he icrease o mass causes he decrease o he requecy o he 1 s aural mode o he sysem, excessive mass scalig wihou icreasig he rampig ime o loadig could lead o erroeous soluio, sice he codiio i (5) would o be saisied. O he oher had, oo much mass scalig could dramaically chage he mass disribuio ad hus he dyamic behavior o he sysem. Someimes, maually scale he mass desiy o he pars would be beer ha usig he eaure o auomaic mass scalig. We could sar wih uiormly scalig he mass desiy o he whole FEA model, ad he scale cerai idividual elemes accordig o he miimum ime sep size limi. I is more secure o keep he perceage o he added mass hrough idividual eleme mass scalig low (less ha 5%). O he oher had, ogeher wih he approach o icreasig loadig rae, i we oly uiormly scale he mass desiy o he whole FEA model bu do o scale idividual eleme mass, he he mass scalig would o be helpul o reduce he compuaio ime. For isace, i we scale he mass o he whole FEA model by a acor o 10, he approximaely he sable ime sep size or umerical iegraio would be icreased by a acor o 10, bu he requecy o he 1 s aural mode o he sysem would be decreased by a acor o 10. I we wa o keep ω 0π, he he rampig ime eeds o be icreased by a acor o 10. As a resul, wih boh he sable ime sep size ad he rampig ime icreased by a acor o 10, he oal compuaio ime would be he same as i he case o o mass scalig. I addiio, i is see rom equaio (4) ha he raio o he kieic eergy o he ieral eergy whe is proporioal o m. To keep he m value o a cosa, he icrease o he global mass by a acor o 10 would require he icrease o he rampig ime by a acor o 10. This would lead o he same compuaio ime

5 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) 3. Three poi bedig aalysis o a phoe We use explici mehod o coduc a 3 poi bedig aalysis o a phoe wih commercially available FEA soware LS-DYNA. The loadig codiio is as ollows: The phoe is acig dowward. A load o 130N is applied a he ceer area o a circular shape wih diameer o 19mm. The wo eds o he phoe are suppored. Oe ed is ixed, ad he oher ed could move i he logiudial direcio (y axis). Figure shows a schemaic diagram o he laeral view o he 3 poi bedig es o he phoe. P=130N Figure. Schemaic diagram o a phoe uder 3 poi bedig es. 3.1 Selecio o mass scalig acor ad rampig ime or loadig Wihou acceleraig he eve, more ha a week o compuaio ime would be eeded o simulae he quasi-saic bedig process. This is because he ime sep size wih a explici aalysis o he phoe is as small as 5x10-5 ms, ad he ime period or he quasi-saic bedig is i he order o a ew secods. We cu he compuaio ime by mass scalig ad loadig rae scalig. We have ried various mass scalig o he 3 poi bedig aalysis o he phoe. Wihou mass scalig, he sable ime sep size is abou 5x10-5 ms. We se dms=-3x10-4 ms i LS-DYNA. Tha meas he miimum ime sep size permied i he aalysis is 3x10-4 ms. The ime sep size has bee elarged by a acor o 6. We ried he ollowig our mass scalig approaches: 1) No uiorm mass desiy scalig was doe o he whole FEA model. Through mass scalig o hose elemes ha could o saisy he 3x10-4 ms ime sep size limi, he mass added was 397% o he physical mass (M 0 ) o he phoe. So he oal mass o he FEA model was 4.97M 0. ) The mass desiy o he whole FEA model was scaled by a acor o 4, ad he 31% o (4 M 0 ) was added o he model so he oal mass o he model is 5.4 M 0. 3) The mass desiy o he whole FEA model was scaled by a acor o 8, ad he 4.03% o (8 M 0 ) was added o he model so he oal mass o he model is 8.3 M 0. 4) The mass desiy o he whole FEA model was scaled by a acor o 10, ad he 1.4% o (10 M 0 ) was added o he model so he oal mass o he model is M 0. Table 1 liss he mass scalig deails or hese our cases. I urs ou ha i he curre aalysis, all hese approaches could lead o coverge soluio wih load ramped rom 0 o 130N i 30ms. However, i some oher cases, he irs approaches may cause some rouble. I is recommeded ha eiher approach 3 or approach 4 be used

6 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece Table 1. Four cases wih diere deails o mass scalig Cases Physical mass Mass added by uiorm mass scalig Mass added by idividual mass scalig Toal mass aer mass scalig M M 0 3 M M M M M M M M M M 0 Nex we discuss quaiaively he reducio o he compuaio ime by approaches 3 ad 4. Usig approach 3, he mass o he model has bee icreased by a acor o 8.3, so he aural requecy o he sysem could be reduced by a acor o.9. As a resul, he rampig ime may eed o be icreased by a acor o.9 (.9 ). Sice he ime sep size has bee icreased 0 by a acor o 6, he compuaio ime could be reduced o abou 48% o he compuaio ime eeded wihou mass scalig. Similarly, usig approach 4, he mass o he model has bee icreased a acor o 10.14, so he aural requecy o he sysem could be reduced by a acor o 3.. As a resul, he rampig ime may eed o be icreased by a acor o 3., ad he compuaio ime could be reduced o abou 53% o he compuaio ime eeded wihou mass scalig. Figure 3 shows a ime sequece o he maximum delecio o he phoe rom a 3 poi bedig aalysis. I ha aalysis, he mass scalig deail o Case 4 i Table 1 has bee used. The rampig ime o loadig is se o be 50ms. I is see ha he aural period is abou 3 ms. Thereore, accordig o equaio (5), a rampig ime equal or greaer ha 30 ms would be eeded or a quasi-saic aalysis. Figure 3. Time sequece o he ceer delecio o a phoe rom a 3 poi bedig aalysis

7 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) We have ried o use a rampig ime o 1ms, 5ms, 0ms, 30ms, 40ms, ad 50ms, respecively. The simulaio resuls coverge or all hese rampig ime periods excep i he case o 1ms rampig ime. I he case o rampig he load wihi 1ms, he kieic eergy is oo big ad he resul could o coverge. Figure 4 shows he maximum delecio o he phoe versus he load applied o he ceer area o he phoe wih rampig ime periods o 5ms, 0ms, 30ms, 40ms, ad 50ms, respecively. I is see ha excep he case wih rampig ime o 5ms, he respose is very close o quasi-saic. As a resul, o obai accurae eough soluio wih miimum compuaio ime, a rampig ime o 30ms is recommeded or he 3 poi bedig aalysis o he phoe. Figure 4. The maximum delecio o he phoe versus he load applied o he ceer area o he phoe wih various rampig ime periods. 3. Simulaio resuls Figures 5 ad 6 show he coour plos o he verical displaceme o he phoe rom 3 poi bedig quasi-saic aalysis

8 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece Figure 5. Coour plo o he displaceme i z direcio o he back o he phoe rom quasisaic 3 poi bedig aalysis. Figure 6. Coour plo o he displaceme i z direcio o he ro o he phoe rom quasisaic 3 poi bedig aalysis

9 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) As meioed beore, a rampig ime o 30 ms is recommeded or he quasi-saic aalysis. I eeds abou 35 hour CPU ime o complee he aalysis usig oe HP UNIX worksaio J6700. Figure 7 shows he ime sequeces o he kieic eergy, ieral eergy, oal eergy, hourglass eergy, ad exeral work durig he 3 poi bedig aalysis o he phoe. The load is ramped wihi 30ms. I is see ha he kieic eergy is very small ad is egligible relaive o he ieral eergy. So is he hourglass eergy. The ieral eergy is very close o he oal eergy o he sysem ad is very close o he exeral work doe o he sysem. For urher clariicaio, Figure 8 shows he ime sequece o he kieic eergy. I is see rom Figures 7 ad 8 ha he kieic eergy is less ha 5% o he ieral eergy hroughou mos o he ime period o he aalysis. The raio o he kieic eergy o he ieral eergy is less ha 0.07% a he seady sae. The ieral eergy reaches a seady sae whe he load is held a a cosa value. The ieral eergy o he seady sae is abou 11 N.mm. To show ha 30ms rampig ime is eough log or he accuracy o he soluio, we show i Figure 9 he eergy ime sequeces durig he 3 poi bedig aalysis o he phoe whe usig 50ms as he rampig ime. Wih 50ms as he rampig ime, i eeds abou 58 hour CPU ime o iish he aalysis usig oe HP UNIX worksaio J6700. I is see orm Figure 9 ha he ieral eergy o he seady sae is abou 11 N.mm. This shows ha boh he simulaio resuls wih rampig ime o 30ms ad rampig ime o 50ms coverge o he same soluio. This reveals ha he rampig ime o 30ms is eough log or accurae soluio o he 3 poi bedig quasi-saic aalysis. Based o he above discussio, we see ha he quasi-saic aalysis wih a rampig ime o 30ms is successul. Figure 7. Eergy ime sequeces durig he 3 poi bedig aalysis o he phoe whe usig 30ms as he rampig ime

10 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece Figure 8. Time sequece o kieic eergy durig he 3 poi bedig aalysis o he phoe whe usig 30ms as he rampig ime. Figure 9. Eergy ime sequeces durig he 3 poi bedig aalysis o he phoe whe usig 50ms as he rampig ime

11 9 h Ieraioal LS-DYNA Users Coerece Simulaio Techology (3) Figure 10 shows he ime sequeces o he maximum delecio o he phoe durig 3 poi bedig aalysis whe usig 30ms ad 50ms as rampig ime, respecively. I is see ha he maximum delecio i he seady sae is abou 0.305mm wih eiher o he rampig ime. Figure 10. Time sequeces o he maximum delecio o he phoe durig 3 poi bedig aalysis. I addiio, we have checked he peelig sress o he solder iercoecs bewee he elecroic packages ad he PWB, he 1 s pricipal srai o LCD glass, he Vo Mise sress o he PWB ad he phoe cover, ec. 4. Coclusios The 3 poi bedig aalysis o a phoe has bee coduced usig explici iegraio mehod. I has bee show ha or a quasi-saic problem wih large umber o poeial closures/opeigs i coac, ad wih large size o FEA model, i is much more coveie o use he explici mehod ha he implici mehod. I usig explici mehod o solve a quasi-saic problem, he compuaio ime could be reduced by loadig rae scalig ad mass scalig. The loadig rae scalig ad mass scalig echiques should be boh cosidered sice hey aec each oher. I hey are properly used, he speed o he aalysis could be icreased dramaically wihou severely degradig he qualiy o he quasi-saic soluio. A good rule o humb is o selec he loadig rae such ha he rampig ime o loadig is abou 10 imes o he period o he 1 s aural mode o he sysem. The requecy or period o he 1 s aural mode o a real applicaio could be esimaed by ruig requecy aalysis or by a ew rials o he explici aalysis. Mass scalig could icrease he sable ime sep size o explici iegraio, bu a he same ime i would icrease he period o he 1 s aural mode o he sysem. Too much mass scalig could dramaically chage he mass disribuio ad hus he dyamic behavior o he sysem

12 Simulaio Techology (3) 9 h Ieraioal LS-DYNA Users Coerece Someimes, maually scale he mass desiy o he pars would be beer ha usig he eaure o auomaically mass scalig. I he 3 poi bedig aalysis o he phoe, we sar wih uiormly scalig he mass desiy o he whole FEA model, ad he scale hose idividual elemes ha sill could o saisy he miimum ime sep size limi. The sable ime sep size o he aalysis has bee icreased by a acor o 6 wih he aid o mass scalig. Based o he mass scalig acor, a 30ms o rampig ime o loadig is recommeded i compariso wih a ew secods i real 3 poi bedig es. The loadig rae has bee acceleraed by orders o magiude. Accordig o he ime sequeces o he eergy proile ad he maximum delecio o he phoe, he 3 poi bedig quasi-saic aalysis is successul. The oal CPU ime eeded o iish he aalysis is abou 35 hours usig oe HP UNIX worksaio J6700. The compuaio ime could be reduced by usig muliple CPU ad/or usig aser compuer. Ackowledgemes The auhors would like o hak Fujii Takaharu or maagig he research projec. Thaks ad appreciaios also exed o he members i he NRC Dallas produc iegraio group. Wihou he grea helps o hem, i would be impossible o iish his work. Reereces [1] J. T. Wag, T. Che, D. W. Sleigh, ad A. Tessler, Simulaig oliear deormaios or solar sail membraes usig explici ime iegraio, 45h AIAA/ASME/ASCE/AHS/ASC Srucures, Srucural Dyamics ad Maerials Coerece, Palm Sprigs, Calioria, AIAA , April 19-, 004, pp. 15, (837KB). [] Geig sared wih ABAQUS/EXPLICIT keywords versio, ABAQUS Versio 6.4 Documeaio. [3] LS-DYNA keyword user s maual, April 003, versio 970. [4] ABAQUS aalysis user s maual, ABAQUS Versio 6.4 Documeaio. 13-4