NONCONFORMING FINITE ELEMENTS FOR REISSNER-MINDLIN PLATES

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1 NONCONFORMING FINITE ELEMENTS FOR REISSNER-MINDLIN PLATES C. CHINOSI Dipartimnto di Scinz Tcnologi Avanzat, Univrsità dl Pimont Orintal, Via Bllini 5/G, 5 Alssandria, Italy chinosi@mfn.unipmn.it C. LOVADINA AND L.D. MARINI Dipartimnto di Matmatica, Univrsità di Pavia, and IMATI CNR, Via Frrata, 7 Pavia, Italy s: carlo.lovadina@unipv.it, marini@imati.cnr.it W rport on rcnt rsults about som nonconforming finit lmnts for plats, introducd and analyzd in [6], [] and [9]. All th lmnts ar locking-fr and xhibit optimal convrgnc rats. Introduction As it is wll-known (cf. [], for instanc), th numrical tratmnt of th Rissnr-Mindlin modl rquirs spcial car, in ordr to avoid th so-calld shar locking phnomnon and th occurrnc of spurious mods. Th shar locking has its roots from th shar nrgy trm, which for small thicknss nforcs th Kirchhoff constraint. It turns out that for simpl low-ordr lmnts this constraint is gnrally too svr, thus compromising th quality of th obtaind discrt solution. A gnral and commonly adoptd stratgy to ovrcom th problm consists in modifying, at th discrt lvl, th shar nrgy trm, with th aim of rducing its influnc. In most cass th modification can b intrprtd as th rsult of (or dirctly ariss from) a mixd approach to th problm. Howvr, a carlss choic of th shar rduction procdur may caus a loss of stability, typically nlightd by th prsnc of undsirabl oscillating componnts in th discrt solution (th spurious mods). W point out that nowadays a wid choic of good lmnts ar avalaibl

2 in th litratur (s, for instanc, [] [3], [5] [7], [] [5], and th rfrncs thrin). Th aim of this Not is to discuss som finit lmnt schms rcntly proposd and studid in [6], [9], and []. Ths lmnts ar all basd on a mixd nonconforming approach, and thy hav som faturs which sm to b favorabl for a possibl xtnsion to th mor complx (and mor intrsting) cas of shll problms. Indd, th mthods w will considr tak advantag of low-ordr polynomial approximations; thy ar locking-fr and optimally convrgnt; finally, onc th mixd variabls (i.. th shar strsss) hav bn liminatd, all th rmaining unknowns (i.. rotations and dflctions) shar th sam nods and dgrs of frdom. Th papr is organizd as follows. In Sction w rcall th Rissnr- Mindlin quations, and w brifly prsnt th lmnts of [6] and []. Sction is dvotd to numrical rsults.. Th Rissnr-Mindlin problm and th nonconforming lmnts Th Rissnr-Mindlin quations for a clampd plat rquir to find (θ, w, γ) such that div C ε(θ) γ = in Ω, () div γ = g in Ω, () γ = λt ( w θ) in Ω, (3) θ =, w = on Ω. () In ()-(3), Ω is th midplan, rgular and boundd, t is th thicknss, C is th tnsor of bnding moduli, and λ is th shar modulus (incorporating also th shar corrction factor). Morovr, θ rprsnts th rotations, w th transvrsal displacmnt, γ th scald shar strsss and g a givn transvrsal load. Finally, ε is th usual symmtric gradint oprator. Th classical variational formulation of problm () () is Find (θ, w, γ) H (Ω) H (Ω) L (Ω) : a(θ, η) + ( v η, γ) = (g, v) (η, v) H (Ω) H (Ω), ( w θ, τ ) λ t (γ, τ ) = τ L (Ω), (5)

3 3 whr (, ) is th innr-product in L (Ω) (or in L (Ω)), and a(θ, η) := C ε(θ) : ε(η) dx. (6) Ω W now introduc nonconforming finit lmnt approximations of problm () () using th approach dtaild in [6]. Lt thn T h b a rgular dcomposition of Ω into triangular lmnts T (s [], for xampl), and lt us st H (T h ) := T T h H (T ). Following [], w dfin suitabl jump oprators. Lt E h dnot th st of all th dgs in T h, and Eh in th st of intrnal dgs. Lt b an intrnal dg of T h, shard by two lmnts T + and T, and lt ϕ dnot a function in H (T h ), or a vctor in H (T h ). For a scalar function ϕ H (T h ) w dfin its jump as [ϕ] = ϕ + n + + ϕ n whil th jump of a vctor ϕ H (T h ) is givn by [ϕ] = (ϕ + n + ) S + (ϕ n ) S E in h, (7) E in h, (8) whr (ϕ n) S dnots th symmtric part of th tnsor product, and n + (rsp. n ) is th outward unit normal to T + (rsp. to T ). On th boundary dgs w dfin jumps of scalars as [ϕ] = ϕn, and jumps of vctors as [ϕ] = (ϕ n) S, whr n is th outward unit normal to Ω. Following th idas of [6], w now slct finit lmnt spacs Θ h H (T h ), W h H (T h ), and Γ h L (Ω), with th proprty: h W h Γ h, whr h dnots th gradint oprator lmnt by lmnt. Th discrt problm is thn Find (θ h, w h, γ h ) Θ h W h Γ h : a h (θ h, η h ) + (γ h, h v h R h η h ) = (g, v h ) (η h, v h ) Θ h W h, ( h w h R h θ h, τ h ) λ t (γ h, τ h ) = τ h Γ h. (9) Abov, th bilinar form a h (, ) is dfind, for picwis rgular functions, by a h (θ, η) := C ε(θ) : ε(η) dx + p Θ (θ, η), () T T h T whr p Θ is a pnalty trm givn by p Θ (θ, η) := κ [θ] : [η] ds E h ( := lngth of th dg ), ()

4 and κ is a positiv constant having th sam physical dimnsion as C (for instanc, for smooth C on could tak κ as C valuatd at th midpoint of ). Furthrmor, R h : H (T h ) Γ h is a suitabl rduction oprator, to b dfind cas by cas. Rmark.. W point out that liminating γ h from systm (9), our schm is quivalnt to th following problm involving only th rotations and th vrtical displacmnts: Find (θ h, w h ) Θ h W h : a h (θ h, η h ) + λt ( h w h R h θ h, h v h R h η h ) = (g, v h ) (η h, v h ) Θ h W h. () Rmark.. Th pnalty trm p Θ dfind in () maks th bilinar form a h (, ) of () to b corciv on th spac Θ h. Sinc Θ h will b mad by nonconforming vctorial functions, dropping th trm p Θ, or taking κ xcssivly small, may caus a loss of stability for th schm at hand. This occurnc is brifly highlightd in Sction... Th nonconforming bubbl lmnt This is th lmnt prsntd and analyzd in [6], dfind as follows. First, on a gnric triangl T T h w dfin: B NC (T ) := Span {χ }, (3) whr χ dnots th nonconforming bubbl of P, i.., th polynomial of dgr vanishing at th two Gauss points of ach dg. In barycntric coordinats this bubbl has th xprssion (for instanc), χ = 3(λ + λ + λ 3). () Th schm is thn givn by th following choics. Th finit lmnt spacs ar Θ h = { η : η T ( P (T ) B NC (T ) ) }, [η] ds = E h, (5) W h = { v : v T P (T ) B NC } (T ), [v] ds = E h, (6) Γ h = { τ : τ T P (T ) B NC (T ) }. (7)

5 5 Th rduction oprator R h : H (T h ) Γ h is dfind locally by: (η R h η) dx = T T h, η H (T h ), (8) T div(η R h η) dx = T T h, η H (T h ). (9) T.. Th conforming bubbl lmnt This lmnt is also mntiond in [6], and it diffrs from th prvious on only for th choic of th typ of bubbl. Dfin, on a gnric triangl T T h : B 3 (T ) := Span {b 3 }, () whr b 3 dnots th standard cubic bubbl. In barycntric coordinats its xprssion is, for instanc, b 3 = 7λ λ λ 3. () This schm is charactrizd by th following choics. Th finit lmnt spacs ar Θ h = { η : η T ( P (T ) B 3 (T ) ) }, [η] ds = E h, () W h = { v : v T P (T ) B 3 (T ), } [v] ds = E h, (3) Γ h = { τ : τ T P (T ) B 3 (T ) }. () Th rduction oprator R h : H (T h ) Γ h is th sam dfind in (8)-(9)..3. Th P NC P NC P lmnt This lmnt has bn introducd and analyzd in [], and it is charactrizd by th following choics. Th finit lmnt spacs ar Θ h = { η : η T (P (T )) }, [η] ds = E h, (5) W h = { v : v T P (T ), } [v] ds = E h, (6) Γ h = { τ : τ T (P (T )) }. (7)

6 6 R h is simply th L -projction oprator onto th picwis constant functions (s (8)). All th lmnts prsntd in Sctions..3 ar first ordr convrgnt for th kinmatic variabls. Mor prcisly, for ach mthod it holds (s [6] and []) θ θ h,h + w w h,h Ch, (8) whr,h dnots th H -brokn norm. Furthrmor, it has bn provd in [9] that for th lmnt of Sction.3 on has th L bounds: θ θ h + w w h Ch. (9). Numrical rsults In this sction w prsnt som numrical rsults showing th bhavior of th nonconforming lmnts of Sctions..3. For comparison purposs, w considr also th stabl and first ordr convrgnt Arnold-Falk lmnt, proposd and analyzd in []. As a tst problm w tak an isotropic and homognous plat Ω = (, ) (, ), clampd on th whol boundary, for which th analytical solution is xplicitly known (s [8] or [9] for all th dtails). W analyz th convrgnc proprtis of th lmnts by considring diffrnt uniform dcompositions as shown in Figur (lft) with h = /, /8, /6, /3, and kping th thicknss sufficintly small (t =.). Morovr, in th pnalty trm () w st κ = C uclidan norm of C. In Figurs and 3 w rport th rlativ rrors, in th L -norm, for rotations and dflction rspctivly. Instad, Figurs and 5 show th rlativ rrors for rotations and dflction, rspctivly, in th nrgy norm. W conclud this Not by brifly commnting on th ffct of choosing a too small pnalty paramtrs κ in () (cf. Rmark.). To this aim, w considr th sam tst problm as bfor, but w us th msh shown in Figur (right). Figurs 6 7 display th profils of th analytical solution for th first componnt of th rotation vctor θ, and for th dflction w, rspctivly. Figurs 8 9 show th profils of th corrsponding discrt solutions, obtaind by th lmnt of Sction.3 with th rasonabl choic κ = C ; w may obsrv that th profils ar quit accuratly capturd. Figurs show th profils of th discrt solutions, this tim with th wrong choic κ = C 5 ; undsirabl oscillations clarly appar on th approximation of both th rotations and th vrtical displacmnts.

7 7 (,) (,) (,) (,) (,) (,) (,) (,) Figur. Adoptd mshs. Rlativ rotation rrors P NC P NC P lmnt conforming bubbl lmnt nonconforming bubbl lmnt Arnold Falk lmnt E θ 3 msh paramtr:/h Figur. Rlativ rotation rrors in L -norm vrsus /h. Rfrncs. D.N. Arnold, F. Brzzi and L.D. Marini, A family of discontinuous Galrkin finit lmnts for th Rissnr-Mindlin plat, J. Sci. Comp. (to appar).. D.N. Arnold and R.S. Falk, A uniformly accurat finit lmnt mthod for

8 8 Rlativ dflction rrors P NC P NC P lmnt conforming bubbl lmnt nonconforming bubbl lmnt Arnold Falk lmnt E w 3 msh paramtr:/h Figur 3. Rlativ dflction rrors in L -norm vrsus /h. Rlativ rotation rrors P NC P NC P lmnt conforming bubbl lmnt nonconforming bubbl lmnt Arnold Falk lmnt E θ msh paramtr:/h Figur. Rlativ rotation rrors in th nrgy norm vrsus /h. th Rissnr-Mindlin plat, SIAM J. Numr. Anal. 6, 76 9 (989). 3. F. Auricchio and C. Lovadina, Analysis of kinmatic linkd intrpolation mth-

9 9 Rlativ dflction rrors P NC P NC P lmnt conforming bubbl lmnt nonconforming bubbl lmnt Arnold Falk lmnt E w msh paramtr:/h Figur 5. Rlativ dflction rrors in th nrgy norm vrsus /h. x analytical θ x Figur 6. Profil of th analytical solution θ. ods for Rissnr-Mindlin plat problms, Comput. Mthods Appl. Mch. Engrg. 9, 65 8 ().. F. Brzzi and M. Fortin, Mixd and Hybrid Finit Elmnt Mthods, Springr

10 analytical w x 5 8 x Figur 7. Profil of th analytical solution w. θ κ = C x x Figur 8. Profil of th approximation to θ with κ = C. Vrlag, F. Brzzi, M. Fortin and R. Stnbrg, Error analysis of mixd-intrpolatd lmnts for Rissnr-Mindlin plats, Math. Modls Mthods Appl. Sci., 5 5 (99).

11 w κ = C x 5 8 x Figur 9. Profil of th approximation to w with κ = C. θ κ = C Figur. Profil of th approximation to θ with κ = C F. Brzzi and L.D. Marini, A nonconforming lmnt for th Rissnr-Mindlin plat, Computrs & Structurs 8, 55 5 (3). 7. D. Chapll and R. Stnbrg, An optimal low-ordr locking-fr finit lmnt mthod for Rissnr-Mindlin plats, Math. Modls and Mthods in Appl. Sci.

12 w κ = C 5 x x Figur. Profil of th approximation to w with κ = C 5. 8, 7 3 (998). 8. C. Chinosi and C. Lovadina, Numrical analysis of som mixd finit lmnt mthods for Rissnr-Mindlin plats, Comput. Mchanics 6, 36 (995). 9. C. Chinosi, C. Lovadina and L.D. Marini, Nonconforming locking-fr finit lmnts for Rissnr-Mindlin plats, submittd to Comput. Mthods Appl. Mch. Engrg... P.G. Ciarlt, Th Finit Elmnt Mthod for Elliptic Problms, North- Holland, R.S. Falk and T. Tu, Locking-fr finit lmnts for th Rissnr-Mindlin plat, Math. Comp. 69, 9 98 ().. C. Lovadina, A low-ordr nonconforming finit lmnt for Rissnr-Mindlin plats, SIAM J. Numr. Anal. (to appar). 3. R. Stnbrg, A nw finit lmnt formulation for th plat bnding problm, in Asymptotic Mthods for Elastic Structurs, ds. P.G. Ciarlt, L. Trabucho and J. Viaño, Waltr d Gruytr & Co., 9 (995).. R.L. Taylor and F. Auricchio, Linkd intrpolation for Rissnr-Mindlin plat lmnts: Part II A simpl triangl, Int. J. Numr. Mthods Eng. 36, (993). 5. A. Tsslr and T.J.R. Hughs, A thr-nod Mindlin plat lmnt with improvd transvrs shar, Comput. Mthods Appl. Mch. Engrg. 5, 7 (985).