Material Point Method Investigations of Trauma to Fluids and Elastic Solids Due to Finite Barriers

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1 Maeral Pon Mehod Inesgaons of Trama o Flds and Elasc Solds De o Fne Barrers J.L. Dean Dearmen of Aeronacal Engneerng Iowa Sae Uners Ames, Iowa 54 USA M.W. Roh Dearmen of Phscs Uners of Norhern Iowa Cedar Falls, Iowa USA Pal A. Gra Dearmen of Comer Scence Uners of Norhern Iowa Cedar Falls, Iowa USA Receed: Ocober 4, 8 Acceed: Janar 3, 9 ABSTRACT A Maeral Pon Mehod (MPM) algorhm s deeloed and led o nesgae how he dnamcs of (Langrangan) Naer-Sokes flds as well as ha of elasc solds s affeced b rama de o fne barrers. For he fld smlaons, maeral on arcles are laced n a wo dmensonal e wh aros nal and bondar condons and saonar errbaons o fld flow. Resls show ha edd crrens are resen no onl n he wake of he errbng obec b are also resonsble for dsron of lamnar flow sream from he barrer. An nfornael relean alcaon for sdden fne rama o an elasc sold noles smlaons of an arcraf srkng a large bldng nder arng ssem condons. The work resened here s nrodcor n nare; he oenal ramfcaons and morance of conned sd s dscssed and emhased. I. BACKGROUND AND IMPORTANCE Maeral Pon Mehod (MPM) smlaons hae a wde are of alcaons and can be sed n comaonal modelng of ssems hang lengh scales sannng seeral orders of magnde. Recen research on bologcal and geologcal ssems has shown ha fld flow n dcs and es hang small, abr obsrcons can dramacall affec he behaor of he ssem s a s he characer of fld flow. In fac s hogh ha he onse of ceran es of hear aacks noles he ranson from lamnar blood flow o chaoc, whch can be rggered b he resence of laqe or oher sorces of roghness on he essel neror []. Becase of he rad change n behaor of fld flow gen een small obsrcons, seems ha mch cold be ganed b condcng comer smlaons of fldc flow arond fne barrers. The MPM mehod conseres mass and momenm and s ncel sed o smlae a secal case of he Naer-Sokes eqaon, whch ms be modfed n order o deal wh a 7

2 reasonable model of fld flow. Howeer, he fld-fld scos neracon as well as ha for he fld-bondar resen n he Naer- Sokes eqaons s absen n MPM and so we hae deeloed an algorhm o smlae flds whch s a weddng of he wo mehods. The neracon of a fld wh a fne obsrcon can be hogh of as a rama o he fld. Snce a fld has elasc b does no sor shear forces ece hose ncrred b scos (for eamle, hose a a saonar bondar for no sl condons) s also of neres o eamne he effec of fne rama o an elasc bod. A anfll obos alcaon s an arcraf macng a large bldng. Snce Seember, man sch sdes hae been done and we wold lke o agmen he crren bod of sdes b comarng smlaons of arcraf bldng collsons. Ths aer resens relmnar resls for smlaons of collsons whch nesgae srcral desrcon de o beam ressre wae roagaon, and ses he sage for a more dealed sd where we ma beer ndersand how he os collson ressre wae roagaes hrogho he bldng. Ulmael, wh sohscaed enogh smlaons we ma be able o gan raccal nsgh no he consrcon of bldngs ha can whsand ncredbl ramac eens. II. COMPUTATIONAL DETAILS The smlaon of choce for or D fld flow as well as 3D collson smlaons s he Maeral Pon Mehod (MPM) [] wh an algorhm deeloed and ealaed b Z. Chen and R. Brannon.[] MPM s adanageos becase doesn' reqre a comle algorhm o smlae collsons or maeral falre. Insead, MPM mas he mass and momenm of gros of arcles n he smlaon o a backgrond grd and les he conseraon eqaons of mass as well as ha of momenm o lmael adance he ssem hrogh me. The se of he backgrond grd also aods enanglemen assocaed wh oher conacbased algorhms. To begn wh, he nal condons for he smlaon nclde a collecon of arcles wh nal osons { }, eloces { } and bondar condons chosen so as o reflec he moran hscs of he acal ssem as mch as ossble. Ne, he backgrond grd s defned, where comaonal sace s searaed no a defned nmber of grd cells n whch nodes are laced n each corner of each cell. The mass M of each arcle a me s maed o he nodes of he grd cells based on he shae fncon N ( reslng n he mass m of node a me : m N M ), N ( ) () The same s done for he momenm (M) of he arcles: N ( m ) ( M ) N ( ) () The shae fncons N ( ) are calclaed n he followng wa. The coordnaes of he 3D bo conanng a arcle are normaled so ha he ake on ereme ales of or a erces of he bo. So f a arclar bo has a ere a (,, ) and anoher ere a ( +, +, + ) across s dagonal, he normaled coordnaes are ( ( ( ) ) ) (3) and he egh shae fncons sed n he arcle-o-node mangs are calclaed from he normaled coordnaes as shown n Table so ha he are eqal o n for a ere (node) where he arcle s a and s ero for a gen node f he arcle s a an oher node. Deals for he D se are easl obanable from he 3D nformaon aboe and are gen elsewhere []. Once he mangs n eqaons () and () are comleed hen he forces a each node ms be calclaed. The nernal force (f ) 8

3 Node Coordnaes Shae Fncon (,,) (- )(- )(- ) (,,) (- )(- ) (,,) (- ) (,,) (- ) (- ) (,,) (- )(- ) (,,) (- ) (,,) (,,) (- ) Table. Eressons for he egh shae fncons sed o ma arcle roeres o each of he egh erces of he grd cbe resdes n. and sbseqenl he knemac arables assocaed wh each arcle n he smlaon are daed sng node o arcle mangs: a Nn Nn f m N ( ) ( m) N ( m ) (7) (8) (9) The new arcle knemac arables n eqaons (7)-(9) are maed back ono he nodes o oban daed nodal eloces. Now becase he smlaed maeral(s) ma hae deformed n beng daed, he elasc roeres of he ssem ms be aken no accon. The rae of sran ensor e s now calclaed from he ' ' eloc gradens: e ( ). Here he rmed arables reresen eloc comonens and, elcl n 3D we hae arses from elasc deformaon of he obec and s calclaed as N n ( f ) G ( ) s M (4) The eernal force s hen calclaed a each node e f c M gm (5) and hen he nodal acceleraons are obaned f sng a where he oal force s gen m b n f ( f ) ( f ) e momena are hen daed n me. The node ( m) ( m f (6) ) e w w w w () Sbseqenl, he 3d elasc sress-sran relaonsh s led o calclae he daed sress ensor s s s, where s kl C kl kl. Becase he sran ensor n eqn. () s smmerc, n general elds s nonserflos elemens and he sffness ensor C kl as well as he dslacemen ensor ε kl are cas n a 66 forma: 9

4 ( E )( ) () The algorhm hen rerns o mang arcle roeres o he grd (eqn. ) and s reeaed nl a desred me lm s reached. To smlae a fld, all he nondagonal elemens are remoed from eqaon () and fld-fld (arclearcle) neracon erms are added n he smlaon. The bondar laer behaor ma be modeled b eher addng anoher neracon or b enforcng arcle eloc bondar condons. III. RESULTS AND DISCUSSION a. Case : D Fld Smlaon For fld smlaons fld arcles are arbrarl laced whn a wo dmensonal e wh a nform nal eloc. A secfc amon of arcles are mananed hrogho he smlaons b lng erodc bondar condons (reseng arcles who hae reached he end of he e). A sngle saonar bondar of arbrar shae s also laced n he wake of he fld medm as a errbaon o he fld flow. For all es cases, fld arcles are confned whn he e wall defned b lgh gra lnes (Fgre ). Seeral smlaons are necessar o check arclar fld behaor (.e., scos neracons among fld elemens, localed orc, and fld mng ). The frs es nesgaes scos neracon beween fld elemens as well as oerall fld mng o ad n erfcaon of fld-lke behaor among elemens. Resls shown n Fgre are ndcae of redced scos behaor, and oer-all arcle mng s drecl de o scos shear force neracons beween fld elemens. In anel (a) nal lacemen of arcles are sch ha he saonar bondar wll mmedael case dsrbance n he fld medm. Vscos neracons become aaren n deformaons aearng whn he fld s nal condon. In anel (b) scos force neracons become ncreasngl aaren n he deformaon of he maeral. Panel (c) shows Parcle grong s aaren n fron of bondar wh a shae smlar o ha of nal lacemen deformaon shown n (b) and fld mng s aaren elsewhere. In sbseqen anels scos neracons case comlee mng of he fld. To roerl smlae deal flds and nesgae fld orces, an assmon ms be made ha he fld flls he enre regon beng nesgaed. Ths es s an aem o sasf ha assmon, and nesgae fld wake behnd he bondar. To emhase roaon, arcles dslang clockwse roaon aear red whle conerclockwse roaon s ndcaed wh ble. Inal condons are smlar o ha of he reos es, b gen an nal ercal eloc o frher emhase roaon near one end of he saonar bondar. Vorc s ndeed resen along he saonar bondar and n he fld wake, b onl n localed locaons de o a falre o manan he assmon ha he enre regon s flled wh fld. In fac, a flaw regardng MPM and fld smlaons s realed here n ha s dffcl o achee an enrel flled regon wh elemenal arcles of arbrar se and shae. Fgre shows he neracon of a fld hang wo nonero eloc comonens wh a saonar recanglar bondar. Panel (a) shows nal lacemen of fld arcles hrogho he enre regon; roaon s mmedael resen for arcles

5 (a) (d) (b) (e) (c) (f) Fgre. Smlaed horonal flow as a recanglar barrer. ha hae conaced he saonar bondar. In anel (b) here s falre o manan a comleel flled regon. Howeer, roaon s also realed n localed regons n he fld wake, arclarl behnd he boom edge of he bondar. The ne anel (c) shows ha localed orc s sll resen among fld arcles ha hae neraced wh he saonar bondar along he boom srface. For frher nesgaon on roaon and fld orc and general fld behaor, a crclar bondar s swaed wh he recanglar bondar, whle he fld arcles nal condons are smlar o ha of reos smlaons wh an nal nform ercal eloc [3]. Inesgaon of fld behaor beond a crclar dsrbance redcs erodc grongs of localed orc along he wake of he fld beond he bondar. Panel (a) shows he nal condon a he sar of he smlaon, and sace s no comleel flled wh arcles de o smlar sses rased n he reosl dscssed recanglar bondar smlaons. The ne anel (b) shows er slgh localed roaon s resen beond he bondar, smlar o ha n Fgre. Vscos neracons n fron of he bondar are also resen, and nflence arcles before he ass or nerac wh he bondar. Sbseqenl, n anel (c) erodc grongs of localed orc aears o be resen along he wake of he fld. b. Case : Bldng srcral desrcon de o ressre-wae roagaon The ne smlaon nesgaes srcral desrcon and ressre wae

6 (a) (b) (c) Fgre. Smlaed fld flow as a recanglar bondar wh a non-ero nal ercal eloc. roagaon hrogh a srcre de o rama from a nqe shae, smlar o ha of he Seember h aack. Maeral arcles whn he bldng are laced sch ha he combne o form nddal ercal beams sannng he lengh of he srcre. Each beam conans ff beam arcles sacked on o of each oher wh beams oal. Graaonal forces are no resen de o sses regardng srcral negr ror o collson, and o solae forces o hose onl de o collson. Ths

7 (a) (b) Fgre 3. Smlaed fld flow as a crclar bondar wh nal ercal eloc. (c) smlaon s led o romoe nesgaon of srcral falre de o rreglar shaed bod collsons. Case nesgaes he desrcon on he bldng cased b he nal lane collson, as well as ressre roagaon o creae a general srcral falre. Panel (a) shows he nal smlaon se, and anel (b) llsraes he nal mac of he arcraf no he bldng srcre. The arcraf s fselage generaes 3

8 Fgre 4. Smlaed arcraf-bldng collson. Noe he ressre wae roagaon hrogho srcre and oer all desrcon. relael lle mac o he srcre as a whole. Panel (c) shows secondar mac de o arcraf wngs and al. Pressre wae s generaed de o hs collson and begns roagang and down he srface of he srcre a he on of mac. In anel (d) a longdnal forwardl roagang ressre wae bsecs he srcre ha roagaes from he wngs. Ths ressre wae rodces he hghes leel of general desrcon whn he bldng. Noe: some arcles dear from he srcre de o reerberan forces de o mac. In sbseqen anels he lade ressre wae reaches he o of he bldng and begns o roagae n he oose drecon. The longdnal ressre wae begns o sread ercall casng seros srcral sses hrogho he base of he srcre, effecel slcng he bldng. As shown, he fselage generaes relael lle oerall desrcon of he bldng, de o he srcre beng comrsed of onl ercall sacked beams. The wng secons of he arcraf creae he hghes leel of desrcon hrogh a generaon of ressre waes ha roagae ercall and horonall hrogh he bldng. Frher nesgaon regardng 4

9 beam srengh, arcraf negr, and odoor condons s necessar o mroe hs smlaon. REFERENCES. h:// HearAack/wha_s_a_hear_aack.as. Z Chen, RM Brannon, Sanda Naonal Laboraores (SAND-48), 3. h:// Comaonal Smlaons Research Gro Dr. Mke Roh Phscs Dearmen Uners of Norhern Iowa Cedar Falls Iowa 5

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