An efficient TFETI based solver for elasto-plastic problems of mechanics

MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 57 AN EFFICIENT TFETI BASED SOLVER FOR ELASTO-PLASTIC PROBLEMS OF MECHANICS Ma in CERMAK1, Tomas KOZUBEK1 1Depa men o Applied Ma hema ics, Facul y o Elec ical Enginee ing and Compu e Science, VSB–Technical Uni e si y o Os a a, 17. lis opadu 15, 708 33 Os a a Po uba, Czech Republic ma in.ce mak@ sb.cz, omas.kozube[email p o ec ed] Abs ac . This pape illus a es how o implemen e ec i e sol e s o elas o-plas ic p oblems. We conside he ime s ep p oblems o mula ed by nonlinea a ia ional equa ions in e ms o displacemen s. To ea nonlinea i y and nonsmoo hness we use semismoo h New on me hod. In each New on i e a ion we ha e o sol e linea sys em o algeb aic equa ions and o i s nume ical solu ion we use TFETI algo i hm. In ou benchma k we demons a e ou app oach on on Mises plas ici y wi h iso opic ha dening and use e u n mapping concep . Keywo ds Domain decomposi ion, elas o-plas ici y, TFETI. 1. In oduc ion The pape is o ganized as ollows. We b ie ly e iew he TFETI me hodology ha ans o ms he la ge p imal p oblem o elas os a ics in e ms o displacemen s in o he smalle and be e condi ioned dual one in e ms o he Lag ange mul iplie s (p essu es) whose condi ioning is u he imp o ed by using he p ojec o s de ined by he na u al coa se g id. Fu he we b ie ly e iew he elas o- plas ici y me hodology o on Mises plas ici y wi h iso opic ha dening. We illus a e he e iciency o ou algo i hm on he solu ion o 3D elas o-plas ic model benchma k and gi e encou aging esul s o nume ical expe imen s. 2. P oblem o Elas os a ics Le us conside an iso opic elas ic body ep esen ed in a e e ence con igu a ion by a domain  in ,2,3 dd, wi h he su icien ly smoo h bounda y  as in Fig. 1. Suppose ha  consis s o wo disjoin pa s U  and F  , UF    , and ha he displacemen s :d U U and o ces :d F F a e gi en. The mechanical p ope ies o  a e de ined by he Young modulus E, he Poisson a io ν, and he densi y  . Fig. 1: Model p oblem. Le 2222 :sym   C, 22 ,, , s ym      CC , ijkl ijlk klij ccc  , whe e : ijkl c and :d g deno e he componen o he elas ici y enso C and a ec o o body o ces, espec i ely. Fo any su icien ly smoo h displacemen :d u, he o al po en ial ene gy is de ined by 1 () (,) 2F J add      uuugu Fu  , (1) whe e   ijkl a( , ) = c ij kl d     u u , (2) and kl 1 () = 2 kl lk uu x x        u. We suppose ha he elas ici y enso sa is ies na u al physical es ic ions so ha (,) (,)and(,) 0aa a  u u uu . (3) Now le us in oduce he Sobole space MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 58 1() d VH and le {: on} U KV   U . be i s non-emp y, i he e exis s a unc ion  1 0 d H    u such ha 0U uU , closed, and con ex subse . The displacemen K u o body in equilib ium sa is ies ( ) ( ) o any J JKu . (4) Condi ions ha gua an ee exis ence and uniqueness may be exp essed in e ms o coe ci i y o J. Mo e gene al bounda y condi ions, such as p esc ibed no mal displacemen s and pe iodici y, may be conside ed wi hou any concep ual di icul ies. 3. TFETI Domain Decomposi ion To apply he TFETI domain decomposi ion, we ea he body om he pa o he bounda y wi h he Di ichle bounda y condi ion, decompose he body in o subdomains, assign each subdomain a unique numbe , and in oduce new “gluing” condi ions on he a i icial in e subdomain bounda ies and on he bounda ies wi h imposed Di ichle da a. Mo e speci ically, he o iginal body  is decomposed in o a sys em o s homogeneous iso opic elas ic bodies, each o which occupies, in a e e ence con igu a ion, a subdomain p  in , 2,3 dd, 1, ,ps . A e decomposi ion each bounda y p  o p  consis s o h ee disjoin pa s p U , p F , and p G  , pp p p UFG , wi h he co esponding displacemen s p U and o ces p F inhe i ed om he o iginally imposed bounda y condi ions on . Fo he a i icial in e subdomain bounda ies, we use he ollowing no a ion: pq G  deno es he pa o p  ha is glued o q  and p G  deno es he pa o p  ha is glued o he o he subdomains. Ob iously pq qp GG  . An auxilia y decomposi ion o he p oblem wi h enumbe ed subdomains and a i icial in e subdomain bounda ies is in Fig. 2. Fig. 2: TFETI domain decomposi ion wi h subdomain enumbe ing and aces o disc e iza ion. The gluing condi ions equi e con inui y o he displacemen s and o hei no mal de i a i es ac oss he in e subdomain bounda ies. The mechanical p ope ies o p  a e de ined by he Young modulus p E, he Poisson a io p  , and he densi y p  . Le 2222 : pp sym   C, 22 ,, , pp s ym      CC , ppp ijkl ijlk klij ccc, whe e : p ijkl cand p g deno e again he en ies o he elas ici y enso and a ec o o body o ces, espec i ely. Fo any su icien ly smoo h displacemen 1 : s d  u, he o al po en ial ene gy is de ined by 1 1 () ( , ) ( ) 2p s ppp p p p J ad         uuugu  () p F pp d    Fu , (5) whe e       , ppp p ppp ijkl ij kl ac d      u u , (6) and  1 2 pp pp kl kl pp lk uu x x        u. Le us in oduce he p oduc Sobole space 11 1 () () dsd HHV, (7) and le     1,..., : on , on s pp ppq pq UG        V U be i s non-emp y, closed, and con ex subse . The displacemen   u o he sys em o subdomains in equilib ium sa is ies     o any JJ   u . (8) The ini e elemen disc e iza ion o 1 s    wi h a sui able numbe ing o nodes esul s in he quad a ic p og amming (QP) p oblem 1 min subjec o 2   uuKu u Bu c  , (9) whe e 1 diag( , , ) s  KKK deno es a symme ic posi i e semide ini e block-diagonal s i ness ma ix o o de n, B deno es an mn  ull ank cons ain ma ix, n   is a load ec o , and m c is a cons ain ec o . The algo i hm o sol ing he minimiza ion p oblem (9) can ound in [4]. The diagonal blocks p K ha co espond o he subdomains p  a e posi i e semide ini e spa se ma ices wi h known ke nels, he igid body modes. This is a g ea MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 59 ad an age because all blocks can be e ec i ely egula ized and hen decomposed using any s anda d spa se Cholesky ype ac o iza ion me hod o nonsingula ma ices [2], [4]. The ma ix B wi h he ows i b and he ec o c wi h he en ies i c en o ce he p esc ibed displacemen s on he pa o he bounda y wi h imposed Di ichle condi ion and he con inui y o he displacemen s ac oss he auxilia y in e aces. A pa allel and nume ically scalable algo i hm o he nume ical solu ion o (9) is in oduced in [4] wi h scalabili y demons a ed up o 315 millions o unknowns and 4800 co es. 4. Elas o-Plas ici y Elas o-plas ic p oblems a e he so-called quasi-s a ic p oblems whe e he his o y o loading is aken in o accoun . We conside he on Mises elas o-plas ici y wi h he s ain iso opic ha dening and inc emen al ini e elemen me hod wi h he e u n mapping concep [1]. The elas o-plas ic de o ma ion o a body  a e loading is desc ibed by he Cauchy s ess enso  , he small s ain enso  , he displacemen u, and he nonnega i e ha dening pa ame e ߢ. Symme ic enso is ep esen ed by he ec o and i s de ia o ic pa is deno ed by he symbol de . Le us deno e he space o con inuous and piecewise linea unc ions cons uc ed o e a egula iangula ion o  wi h he disc e iza ion no m h by h VV, whe e   1(): 0 on d U VH    . Le * 01 0kN  , (10) be a pa i ion o he ime in e al * [0, ] . Then he solu ion algo i hm a e ime and space disc e iza ions has he o m: Algo i hm 1: 1. Ini ial s ep: 000 0, 0, 0 hhh  u. 2. o 0, , 1kN  do (load s ep). 3. F om p e ious s ep we know: , , kkk hhh u   and compu e , , hhh   u   , hhhh V   uu , (11)   ,, kk hhhh T      , (12)   ,, kk hhhh T     . (13) 4. Solu ion     ,, kk hhh h  u is subs i u ed in o equa ion o equilib ium:     ,, , ,, kk hh h h k hh h h dx V         u . (14) This leads o a nonlinea sys em o equa ions wi h unknown h  u which is sol ed using he New on me hod. The linea ized p oblem a ising in each New on s ep is sol ed by he TFETI algo i hmic scheme [4], [8]. I is possible because he s i ness ma ix o linea ized sys em is symme ic and posi i e semide ini e. 5. Then we compu e alues o he nex s ep: 11 1 ,, kk k k kk hhhh h hhhh          uuu . 6. enddo. Fo e u n mapping concep we de ine ope a o s R M T  and R M T  . Thei o m a e RM h TT    and RM h TT     Now we can go om enso no a ion ,, hh h   o he algeb aic no a ion ,, hh h σεκ o s ess, s ain and ha dening a iables. Abo e we conside he ollowing no a ion. Le C deno e he Hook’s ma ix, E ep esen linea ope a o de , ,   be he Lame coe icien s, k h  be he inc emen o he igh hand side, and k hh h σσCε. Le s  i ( , ) 0, i ( , ) 0, k hhh h k hR hh P P         Cεσκ σ Cεnσκ    (15) 1 0i (,)0, i ( , ) 0, k hh h k Rhh P zzP           σκ κ Cp σκ  ‖‖ (16) whe e 32 (,) 33 k R hh m P H   σκ , (17)  () 3 ,2,1 2 () h h de z de    σ nCp σ ‖‖ ‖‖ , (18) and plas ici y unc ion 3 (,) () ( ) 2 k k hh h mh Pde YHσκ σ κ‖‖ , (19) whe e ,0. m YH  The unc ion  R  n is semismoo h and po en ial, ha ’s why we can sol e his linea ized sys em by algo i hm T-FETI. We wan o achie e quad a ic con e gence, and he e o e we compu e he de i a i e o . R M T  . The o m o R M T  is 0 2 + 3 3 ()'() 2 3() k RM mh k mh YH THde            κ εEC σCε ‖‖ MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 60 2 ()(()) () kkT hh k h de de de           σCεσCεE σCε‖‖. (20) I we ep esen a unc ion hh V by he ec o n   and omi index k hen (14) can be ew i en as he sys em o nonlinea equa ions ()F  u , (21) whe e ( ), ( ( )), ( ) , , R Mn hh FT dx      w w w  ,(), n hh    w w . (22) Simila ly we build he angen s i ness ma ix    1 ,' ,,, R Mn k Tu wdx        K w w whe e 1k u is displacemen om p e ious New on s ep. 5. Nume ical Expe imen s Desc ibed algo i hms we e implemen ed in Ma Sol lib a y [5] de eloped in Ma lab en i onmen and es ed on he solu ion o 3D p oblems. Le us conside a 3D pla e wi h a hole in he cen e (due o symme y only a qua e o he whole s uc u e is used). The geome y o he body wi h aces o decomposi ion and disc e iza ion is depic ed in Fig. 3. In Fig. 4 we see a zoom o Fig. 3 nea he hole. Symme y condi ions a e p esc ibed on he le and lowe sides o . The su ace load ( ) 450sin(2 ) g  [MPa], [0,1/ 4] [sec], is applied o he uppe side o  . The elas o-plas ic ma e ial pa ame e s a e E = 206900 [MPa], 0.29  ,   450 MPaY,   100 MPa m H and he ime in e al [0,1/ 4][sec] is di ided in o 50 s eps. We conside a mesh wi h 4450 nodes and 19008 e ahed ons. A simila nume ical example was also in es iga ed in [3]. In he n- h New on i e a ion we compu e an app oxima ion n u by sol ing he cons ained linea p oblem o he o m   1 min 2 n nnn nn  Bu o uKu u    , using he scalable TFETI algo i hmic scheme p oposed in [4]. We s op he New on me hod in e e y ime s ep i   11 / nn n n uu u u   ‖ ‖‖ ‖‖ ‖ is less han 9 10  . No ice ha he maximum numbe o he New on i e a ions is small o all ime s eps, he e o e he me hod is sui able o he p oblem. In he ollowing igu es, we depic plas ic and elas ic elemen s and on Mises s ess in he xy plane wi h he z coo dina e 0 [mm]. In Fig. 5, 6, 7 and 8, we can see which elemen s a e plas ic (g ay colo ) and which a e elas ic (whi e colo ) in chosen ime s eps. Pa icula ly, in ime s eps 1-12 we obse e only elas ic beha io , and in ime s eps 13-50 plas ic beha io o some elemen s. The maximum alue o ha dening a each ime s ep is depic ed in Fig. 9. The on Mises s ess dis ibu ion on de o med mesh is showed in Fig. 10. Fig. 3: Geome y in [mm] wi h aces o decomposi ion and disc e iza ion. Fig. 4: Zoom o Fig. 3 nea he hole. Fig. 5: Plas ic and elas ic elemen s a e 1 ime s ep. MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 61 Fig. 6: Plas ic and elas ic elemen s a e 20 ime s eps. Fig. 7: Plas ic and elas ic elemen s a e 35 ime s eps. Fig. 8: Plas ic and elas ic elemen s a e 50 ime s eps. Fig. 9: Maximum alues o ha dening in ime i e a ions. Fig. 10: Von Mises s ess dis ibu ion on he de o med mesh (scaled 10x). 6. Conclusion and Goals We ha e p esen ed an e icien algo i hm o he nume ical solu ion o elas o-plas ic p oblems. These p oblems lead o he quasi-s a ic p oblems, whe e each nonlinea and nonsmoo h ime s ep p oblem is sol ed by he semismoo h New on me hod. In each New on i e a ion we ha e o sol e an auxilia y (possibly o la ge size) linea sys em o algeb aic equa ions. We p oposed a new app oach how o sol e such sys em e icien ly using in a sense op imal algo i hm based on ou To al-FETI a ian o FETI domain decomposi ion me hod. We illus a ed he e iciency o ou algo i hm on he solu ion o 3D elas o-plas ic model benchma k and ga e esul s o nume ical expe imen s. The esul s indica e ha he algo i hm may be e icien . Nowadays we adap his app oach o he solu ion o con ac p oblems. Acknowledgemen s This wo k has been suppo ed by he g an GA CR 103/09/H078. Re e ences [1] BLAHETA, Radim. Nume ical me hods in elas o-plas ici y. P ague: PERES Publishe s, Documen a Geonica 1998, 1999. ISBN 80-902465-8-3. [2] BRZOBOHATY, Tomas, Zdenek DOSTAL, Pe KOVAR, Tomas KOZUBEK, and Alexand os MARKOPOULOS. Cholesky decomposi ion wi h ixing nodes o s able e alua ion o a gene alized in e se o he s i ness ma ix o a loa ing s uc u e. In e na ional jou nal o nume ical me hods in enginee ing [online]. 2011, ol. 88, iss. 5, p. 1384-1405. ISSN 1097-0207. A ailable a : h p://dx.doi.o g/10.1002/nme.3187. [3] GRUBER, P. G. and Jan VALDMAN. Solu ion o One-Time S ep P oblem in Elas oplas ici y. SIAM Jou nal on Scien i ic Compu ing. 2008, ol. 31, iss. 2., p. 1558-1580. ISSN 1064- 8275. A ailable a : h p://dx.doi.o g/10.1137/070690079>. MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 10 | NUMBER: 1 | 2012 | MARCH © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 62 [4] KOZUBEK, Tomas, Vi VONDRAK, Ma in MENSIK, Da id HORAK, Zdenek DOSTAL, Vacla HAPLA, Pa la KABELIKOVA and Ma in CERMAK. To al FETI domain decomposi ion me hod and i s massi ely pa allel implemen a ion. Compu e s and S uc u es. 2011, submi ed. ISSN 0045-7949. [5] KOZUBEK, Tomas, Alexand os MARKOPOULOS, Tomas BRZOBOHATY, Radek KUCERA, Vi VONDRAK and Zdenek DOSTAL. Ma Sol - MATLAB e icien sol e s o p oblems in enginee ing [online], 2012. A ailable a : h p://ma sol. sb.cz. [6] CERMAK, Ma in. New on me hod o elas o-plas ici y p oblems. In: P oceedings o CVUT, 2010. P ague, 2010. [7] CERMAK, Ma in, Tomas KOZUBEK and Alexand os MARKOPOULOS. An e icien solu ion o elas o-plas ic p oblems in mechanics. In: Semina Nume icka Analyza. Rozno pod Radhos em, 2011. p. 16-20. ISBN 978-80-86407- 19-7. [8] CERMAK, Ma in, Tomas KOZUBEK and Alexand os MARKOPOULOS. An e icien pa allel sol e o elas o- plas ic p oblems o mechanics. In: P oceedings o he Second In e na ional Con e ence on Pa allel, Dis ibu ed, G id and Cloud Compu ing o Enginee ing. F ance, 2011. ISBN 978-1- 905088-43-0. A ailable a : h p://dx.doi.o g/10.4203/ccp.95.5. Abou Au ho s Ma in CERMAK was bo n in 1983 in H anice, g adua ed om he Facul y o Elec ical Enginee ing and Compu e Science o he VSB-Technical Uni e si y o Os a a in 2008 in he ield o In e ace be ween COMSOL and OOSol o Sol ing Con ac P oblems. Cu en ly he is a Ph.D. s uden a he Depa men o Applied Ma hema ics o he VSB-Technical Uni e si y o Os a a, a esea ch assis an (Ph.D.) a he Cen e o Excellence IT4inno a ions. His cu en esea ch in e es s conce n mainly he TFETI Domain decomposi ion and elas o-plas ic p oblems o equali y and inequali y. Tomas KOZUBEK was bo n in 1975 in Ka ina, g adua ed om he Facul y o Elec ical Enginee ing and Compu e Science o he VSB-Technical Uni e si y o Os a a in 1998 in he ield o Fic i ious Domain Me hods o he Nume ical Solu ion o PDEs. He comple ed his Ph.D. s udies in he ield o Shape Op imiza ion in 2002 and habili a ed in he ield o Nume ical Solu ion o Pa ial Di e en ial Equa ions using Wa ele s and Fic i ious Domain App oach in 2007. Cu en ly he is an associa e p o esso a he Depa men o Applied Ma hema ics o he VSB-Technical Uni e si y o Os a a, a esea ch p og amme manage a he Cen e o Excellence IT4inno a ions, and a main coo dina o o he SPOMECH p ojec ocused on eliable solu ion o nonlinea p oblems in mechanics.