ICES REPORT Discontinuous and enriched Galerkin methods for phase-field fracture propagation in elasticity

Transcription

1 ICES REPORT Fbruary 2016 Discontinuous and nrichd Galrkin mthods for phas-fild fractur propagation in lasticity by Prashant Mital, Thomas Wick, Mary F. Whlr, Grgina Pnchva Th Institut for Computational Enginring and Scincs Th Univrsity of Txas at Austin Austin, Txas Rfrnc: Prashant Mital, Thomas Wick, Mary F. Whlr, Grgina Pnchva, "Discontinuous and nrichd Galrkin mthods for phas-fild fractur propagation in lasticity," ICES REPORT 16-05, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, Fbruary 2016.

2 Accptd for publication in Numrical Mathmatics and Advancd Applications, ENUMATH-2015 Procdings, Springr, 2016 Discontinuous and nrichd Galrkin mthods for phas-fild fractur propagation in lasticity Prashant Mital Thomas Wick Mary F. Whlr Grgina Pnchva In this work, w introduc discontinuous Galrkin and nrichd Galrkin formulations for th spatial discrtization of phas-fild fractur propagation. Th nonlinar coupld systm is formulatd in trms of th Eulr-Lagrang quations, which ar subjct to a crack irrvrsibility condition. Th rsulting variational inquality is solvd in a quasi-monolithic way in which th irrvrsibility condition is incorporatd with th hlp of an augmntd Lagrangian tchniqu. Th rlaxd nonlinar systm is tratd with Nwton s mthod. Numrical rsults complt th prsnt study. Kywords: Phas fild fractur; Discontinuous Galrkin; SIPG; NIPG; IIPG; augmntd Lagrangian mthod 1 Introduction Fractur propagation in lasticity, plasticity, and porous mdia is currntly on of th major rsarch topics in mchanical, nrgy, and nvironmntal nginring. In this papr, w concntrat spcifically on fractur propagation in lasticity. W considr a variational approach for brittl fractur introducd in [6], which has bn latr formulatd in trms of a thrmodynamically-consistnt phas fild tchniqu [8]. In fact, variational and phas fild formulations for fractur ar activ rsarch aras as attstd in rcnt yars,.g., [3, 9, 2, 4, 1, 10]. Our motivations for mploying a phas fild modl ar that fractur nuclation, propagation, kinking, and curvilinar paths ar automatically includd in th modl; post-procssing of strss intnsity factors and rmshing rsolving th crack path ar avoidd. Furthrmor, th undrlying quations ar basd on continuum mchanics principls that can b tratd with adaptiv Galrkin finit lmnts. In this work, w xtnd xisting Galrkin formulations for phas-fild fractur with rgard to two major aspcts: Spatial discrtization of th displacmnt fild with discontinuous Galrkin (DG) finit lmnts rsulting in NIPG [12] and IIPG mthods [5] and an nrichd Galrkin (EG) formulation [13]; Formulation of a quasi-monolithic augmntd Lagrangian itration for th nonlinar coupld displacmntphas-fild systm. Ths framworks ar formulatd in Sctions 2-4 and ar substantiatd with numrical tsts in Sction 5. Cntr for Subsurfac Modling, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, Austin, Txas 78712, USA RICAM, Austrian Acadmy of Scincs Altnbrgr Str Linz, Austria Cntr for Subsurfac Modling, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, Austin, Txas 78712, USA Cntr for Subsurfac Modling, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, Austin, Txas 78712, USA 1

3 Mital, Wick, Whlr, Pnchva 2 Th phas-fild fractur modl W limit our attntion to 2-dimnsional problms and lt Ω R 2, b a smooth, opn, connctd and boundd st. W dnot th L 2 scalar product with (, ), and assum that th crack C is a 1-dimnsional st, not ncssarily connctd, containd in Ω. Using th variational/phas-fild approach to fractur [6, 3], th crack C is rprsntd using a continuous phas fild variabl ϕ : Ω [0, 1]. This valu of th phas fild variabl intrpolats btwn th brokn (ϕ = 0) and unbrokn (ϕ = 1) stats of th matrial. Th diffusiv transition zon btwn ths two stats is controlld by a rgularization paramtr ε > 0. Imposing a crack irrvrsibility condition ϕ ϕ n 1 (whr ϕ n 1 := ϕ(t n 1 ) dnots th prvious tim stp solution), and furthr ingrdints for a thrmodynamically consistnt phas-fild framwork [8] rsult in th following Eulr-Lagrang formulation: Formulation 1: Find vctor-valud displacmnts and a scalar-valud phas-fild variabl (u, ϕ) {ū+v } W such that ( ((1 κ)ϕ 2 + κ ) σ(u), (w)) = 0 w V, (1) as wll as, (1 κ)(ϕσ(u) : (u), ψ ϕ) + G c ( 1 ε (1 ϕ, ψ ϕ) + ε( ϕ, ψ ϕ) ) 0 ψ W in L (Ω), (2) whr V := H 1 0 (Ω), W in := {w H 1 (Ω) w ϕ n 1 1 a.. on Ω} and W := H 1 (Ω). Furthrmor, σ = σ(u) = 2µ s +λ s tr()i is th strss tnsor with µ s, λ s > 0, and (u) = 0.5( u+ u T ) is th linarizd strain tnsor. Th critical nrgy rlas rat is G c > 0. Th domain is subjct to boundary conditions, and w assum Γ D, with th possibly non-homognous and tim-dpndnt Dirichlt boundary conditions ū. Morovr, κ is a rgularization paramtr for th lastic nrgy boundd blow by 0, such that κ ε, s.g., [3]. To trat crack irrvrsibility, w us th augmntd-lagrangian formulation dscribd in [14]. To apply this mthod, w bgin by approximating th tim drivativ t ϕ using th backward diffrnc t ϕ t ϕ = ϕ ϕn 1 t 1 t ((λ + γ(ϕ ϕ n 1 )) + ), t = t n t n 1. Hr, λ and γ ar a pnalization function and paramtr, rspctivly, and ϕ n 1 is th phas fild solution at th prvious tim stp. Morovr, (x) + := max{0, x}. 3 A quasi-monolithic incrmntal formulation W choos a quasi-monolithic approach [7] as this rducs algorithmic complxity and has bn dmonstratd to b numrically robust and fficint whn ϕ is rplacd by th xtrapolation ϕ in th first trm of Formulation 2. Th rason for choosing ϕ is th nd to circumvnt th non-convxity of th undrlying nrgy functional. 2

4 DG and EG for phas-fild fractur Algorithm 1 Solution algorithm For ach tim t n : Lt m = 0; choos initial λ m L 2 (Ω), γ > 0. rpat Lt k = 0; choos initial Ũk V h W h. rpat Find δu k solving A (U k )(δu k, Ψ) = A(U k )(Ψ) Updat Ũk+1 Ũk + δu k Updat k k + 1 until Stopping critrion U k U k 1 TOL 2 is satisfid. St U m+1 = (u m+1, ϕ m+1 ) = Ũk Updat λ m+1 = min(0, λ m + γϕ m+1 ) + (λ m + γ(ϕ m+1 ϕ n 1 )) + Updat m m + 1 until Stopping critrion max( u m 1 u m, λ m 1 λ m ) TOL i is satisfid. (u n, ϕ n ) = (u m+1, ϕ m+1 ) Formulation 2: Find U := {u, ϕ} {ū + V } W, whr V := H 1 0 (Ω) and W := H1 (Ω), such that A(U)(Ψ) = 0 Ψ := {w, ψ} V W, (3) whr A( )( ) is th following smi-linar form ( ((1 A(U)(Ψ) = κ) ϕ 2 + κ ) ) σ(u), (w) + (1 κ)(ϕσ(u) : (u), ψ) ( + G c 1 ) ε (1 ϕ, ψ) + ε( ϕ, ψ) + 1 ( ) (λ + γ(ϕ ϕ n 1 )) +, ψ. (4) t Hr ϕ is a linar xtrapolation of tim-laggd ϕ, i.. ϕ ϕ := ϕ(ϕ n 1, ϕ n 2 ), with ϕ n 1, ϕ n 2 dnoting solutions to prvious tim stps. Solving th nonlinar variational problm (4) is prformd with Nwton s mthod and lin sarch backtracking. Th rsulting solution algorithm is outlind in Algorithm 1. 4 Spatial discrtization with DG and EG In this sction w stablish ky notations for DG and EG followd by th mathmatical statmnt of th discrt variational forms. On a conforming subdivision E h of a polygonal domain Ω subdividd into lmnts E w dfin th discontinuous finit lmnt subspac to b D k (E h ) = {v L 2 (Ω) : E E h, v E P k (E)}, (5) whr P k (E) dnots th spac of picwis polynomials of total dgr lss than or qual to k on E. W also dfin th spac of CG approximating polynomials nrichd with discontinuous picwis constants Hr D C k (E h) is th CG approximating spac dfind as D C0 k (E h) := D C k (E h) D 0 (E h ). (6) D C k (E h) = {v C(Ω) : E E h, v E P C k (E), v Γ D = 0}, (7) 3

5 Mital, Wick, Whlr, Pnchva whr P C k (E) dnots th spac of continuous picwis polynomials of total dgr lss than or qual to k on E. (a) D 1 C (E h ) (b) D 1 (E h ) (c) D C0 1 (E h ) D 2 C (E h ) D 2 (E h ) D C0 2 (E h ) Figur 1: Support points for bilinar and biquadratic basis functions. (a) CG: all support points in rd. (b) DG: support points for lft lmnt in rd, for right lmnt in blu. Th common dg has two sts of support points - on from ach lmnt. (c) EG: support points from CG approximating spac in rd, picwis constants in blu. Only th picwis constant dgr of frdom is discontinuous across th common dg. In ordr to dscrib th vctor-valud displacmnts, w considr th spacs of vctor functions that gnraliz (5) and (6): D k (E h ) = (D k (E h )) d, D C0 k (E h) = (Dk C0(E h)) d, whr d is th numbr of spatial dimnsions. W not that th functions in D k (E h ) and D C0 k (E h) ar discontinuous along th dgs (or facs) of th msh. Now, considr two nighboring lmnts E1 and E 2 that shar a common sid. Naturally thn, thr ar two tracs of w D k (E h ) along. W considr n to b th normal vctor associatd with to b orintd from E1 to E2 and dfin: {w} = 1 2 (w E1 ) (w E2 ), [w] = (w E1 ) (w E2 ) = E 1 E 2. W xtnd this dfinition to lmnts on th boundary Ω as: {w} = [w] = (w E 1 ) = E1 Ω. Furthr, w dnot by th lngth of an dg in d = 2. W now stat th quations corrsponding to a discontinuous spatial discrtization dirctly from inspction of th monolithic formulation (4) and th DG-schm for pur linar lasticity,.g., [11]. W pursu a discontinuous rprsntation of th displacmnt variabl u only, rcognizing that th rgularization in th cas of th phas fild variabl ϕ nforcs its continuity across th crack. W augmnt (4) with th jump and pnalization trms to dfin th discrt incrmntal smi-linar form A(U h )(Ψ h ) = ( (1 κ) ϕ 2 + κ ) ) σ(u), (w) Γ h Γ D +η Γ D E E h E { ((1 κ) ϕ 2 + κ ) } { σ(u) n [w] +η ((1 κ) ϕ 2 +κ ) } σ(w) n [u] Γ h (((1 κ) ϕ 2 + κ ) σ(w) n )(u g D ) + Γ h δ β [u][w] + +(1 κ)(ϕσ(u) : (u), ψ) +G c ( 1 ε (1 ϕ, ψ) +ε( ϕ, ψ) ) + 1 t Γ D δ β (u g D )w ) ( (λ + γ(ϕ ϕ n 1 )) +, ψ whr η = 1 (NIPG) or η = 0 (IIPG); and δ > 0 and β > 0 (hr β = 1) ar th DG-pnalization and, (8) 4

6 DG and EG for phas-fild fractur suprpnalization paramtrs, rspctivly. Th DG-CG variational problm rads: Find U h := {u, ϕ} {ū h + Vh DG } Wh CG such that A(U h )(Ψ h ) = 0 Ψ h := {w, ψ} V DG h W CG h. (9) Th EG-CG variational problm rads: Find U h := {u, ϕ} {ū h + Vh EG } Wh CG such that A(U h )(Ψ h ) = 0 Ψ h := {w, ψ} V EG h W CG h. (10) Th tst and trial spacs ar Vh DG := [D k (E h )] 2, Vh EG := [Dk C0(E h)] 2, Wh CG := Dk C(E h). Our formulation nforcs both Dirichlt and Numann boundary conditions wakly. Th us of homognous Numann boundary conditions for u and ϕ rsults in a formulation xclusivly dpndnt on Γ D. Th dirctional drivativ of (8) ndd for th Nwton itrations is computd analytically. 5 A numrical tst: singl dg notchd tnsion Th singl dg notchd tnsion tst is a widly usd xprimntal mthodology usd to charactriz th fractur toughnss of various matrials in plan-strain. ū = (0, ū y(t)) Figur 2: Schmatic of th singl-dg-notchd tnsion tst (lft), th final phas-fild crack pattrn at T = s (middl), and comparison of load-displacmnt curvs from our monolithic schm with CG, DG-IIPG and EG-IIPG against rsults rportd by Mih t al. [9] and Histr t al. [7]. W considr a squar plat with a horizontal notch placd at half-hight, running from th right outr surfac to th cntr of th spcimn. Th plat is subjct to zro displacmnt boundary conditions on th bottom surfac, and tim-dpndnt displacmnt on th top surfac. Th lft and right surfacs ar considrd to b tractionfr. Th problm stup is shown in Figur 2. Th matrial paramtrs ar chosn as λ = kN/mm 2, µ = kN/mm 2 and G c = kn/mm. Th displacmnt boundary condition on th top surfac is takn to b ū y (t) = tᾱ with ᾱ = 1 mm/s. Th xpctd rspons of this tst is th build-up of th strss concntration in th vicinity of th crack-tip, followd by unstabl, catastrophic crack growth. Our first objctiv is to study h-convrgnc for fixd ε. W choos t = s for th first 50 loading stps, aftr which t = s. This adaptivity in th tim stp is ncssary to captur th rapid movmnt of th crack tip. W choos ε = mm, κ = , and run our cod for h 1 = mm, h 2 = mm, and h 3 = mm. W valuat th surfac load vctor on th top surfac of Ω as τ = (F x, F y ) = Ω top σ(u)nds. In this xampl, w ar particularly intrstd in F y. Our findings for th surfac load volution with varying h ar shown in Figur 3 for th IIPG flavors of EG and DG. It is obsrvd 5

7 Mital, Wick, Whlr, Pnchva that our approach is stabl with spatial msh rfinmnt, and that our solution convrgs as w us finr mshs. Comparison to litratur valus ar displayd in Figur 2 at right. Figur 3: W fix κ and ε on th coarsst msh lvl and vary h. Lft: CG. Middl: DG-IIPG. Right: EG-IIPG. Spatial convrgnc is obsrvd for all schms. Rsults obtaind from DG-NIPG and EG-NIPG ar vry similar and ar thrfor not prsntd hr. Th SIPG mthod (η = 1) yilds unsatisfactory findings, which ar not shown in this work. With th rsults of our schm duly validatd, w procd to study th rlativ fficincy of th schms by comparing th numbr of Nwton itrations takn by ach of thm to convrg. W first invstigat th variation in th numbr of Nwton stps takn with th pnalization paramtr δ. Not that whn w multiply Equations (8) throughout by t, our ffctiv pnalization of th jump bcoms δ t. This is an important dtail that cannot b ovrlookd whil using DG/EG for th phas fild quations bcaus for instanc, using δ = 10 5 with t = s givs an ffctiv pnalization of δ t = 1 which is not sufficintly larg and producs spurious rsults. In th cas of an adaptiv tim stp siz (w usually tak t = s for th first 50 stps, and a smallr tim stp thraftr), th product δ t is rportd for th smallr tim stp. W vary th valus of th ffctiv pnalization and plot th cumulativ numbr of Nwton stps as a function of tim for h = mm, ε = 2h[mm], and κ = Th rsults of this study with IIPG and NIPG ar shown in Figur 4. In Figur 5, w obsrv that DG and EG schms tak much fwr Nwton itrations to convrg than CG spcially aftr th onst of crack growth (approximatly t = s). Figur 4: Nwton convrgnc prformanc with h = mm, t = {10 4, t < 0.005; 10 5, t 0.005}, ε = 2h[mm], and κ = and varying pnalty δ. Lft: DG-NIPG. Right: EG-NIPG. Lft: DG-IIPG. Right: EG-IIPG. Convrgnc is fastr for highr valus of pnalization. For a bttr comparison of th fficincy, w run a tst with th sam physical paramtrs as abov, but with 6

8 DG and EG for phas-fild fractur Figur 5: Singl dg notchd tnsion tst rsults using CG, DG-IIPG and EG-IIPG. Lft: load vs. displacmnt curv. Right: Nwton convrgnc prformanc for constant t = 10 5 s. a uniform tim stp of t = 10 5 s throughout. Th motivation is to supprss th ffct of adaptiv tim stpping on th Nwton prformanc and to giv an unbiasd comparison. Sinc th computational burdn with such a simulation is significant, w only considr th IIPG cas with δ t = Ths rsults ar shown in Figur 5. As w can s, DG and EG tak roughly th sam numbr of Nwton itrations (1800) whil CG taks significantly mor (2520). W also obsrv that th load-displacmnt curvs for all thr mthods ar in rasonabl agrmnt. Hnc, w can conclusivly stat that th Nwton mthod convrgs in fwr itrations for th DG and EG schms than for th CG schm. Furthrmor by inspcting Figur 1, w s that EG has significantly fwr dgrs of frdom than DG. Rfrncs [1] M. AMBATI, T. GERASIMOV, AND L. DE LORENZIS, A rviw on phas-fild modls of brittl fractur and a nw fast hybrid formulation, Computational Mchanics 55:2 (2015), [2] M. J. BORDEN, C. V. VERHOOSEL, M. A. SCOTT, T. J. HUGHES, AND C. M. LANDIS, A phas-fild dscription of dynamic brittl fractur, Computr Mthods in Applid Mchanics and Enginring 217 (2012), [3] B. BOURDIN, G. A. FRANCFORT, AND J. J. MARIGO, Th variational approach to fractur, Journal of lasticity 91:1-3 (2008), [4] S. BURKE, C. ORTNER, AND E. SÜLI, An adaptiv finit lmnt approximation of a variational modl of brittl fractur, SIAM Journal on Numrical Analysis 48:3 (2010), [5] C. DAWSON, S. SUN, AND M. F. WHEELER, Compatibl algorithms for coupld flow and transport, Computr Mthods in Applid Mchanics and Enginring 193:23 (2004), [6] G. A. FRANCFORT AND J.-J. MARIGO, Rvisiting brittl fractur as an nrgy minimization problm, Journal of th Mchanics and Physics of Solids 46:8 (1998), [7] T. HEISTER, M. F. WHEELER, AND T. WICK, A primal-dual activ st mthod and prdictor-corrctor msh adaptivity for computing fractur propagation using a phas-fild approach, Computr Mthods in Applid Mchanics and Enginring 290 (2015),

9 Mital, Wick, Whlr, Pnchva [8] C. MIEHE, F. WELSCHINGER, AND M. HOFACKER, Thrmodynamically consistnt phas-fild modls of fractur: Variational principls and multi-fild f implmntations, Intrnational Journal for Numrical Mthods in Enginring 83:10 (2010), [9] C. MIEHE, M. HOFACKER, AND F. WELSCHINGER, A phas fild modl for rat-indpndnt crack propagation: Robust algorithmic implmntation basd on oprator splits, Computr Mthods in Applid Mchanics and Enginring 199:45 (2010), [10] A. MIKELIĆ, M. F. WHEELER, AND T. WICK, A quasi-static phas-fild approach to prssurizd fracturs, Nonlinarity 28:5 (2015), [11] B. RIVIÈRE, Discontinuous galrkin mthods for solving lliptic and parabolic quations: thory and implmntation, SIAM, [12] B. RIVIÈRE, M. F. WHEELER, K. BANAŚ, ET AL., Discontinuous galrkin mthod applid to a singl phas flow in porous mdia, Computational Goscincs 4:4 (2000), [13] S. SUN AND J. LIU, A locally consrvativ finit lmnt mthod basd on picwis constant nrichmnt of th continuous galrkin mthod, SIAM Journal on Scintific Computing 31:4 (2009), [14] M. F. WHEELER, T. WICK, AND W. WOLLNER, An augmntd-lagrangian mthod for th phasfild approach for prssurizd fracturs, Computr Mthods in Applid Mchanics and Enginring 271 (2014),