ICES REPORT January The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712


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1 ICES REPORT January 2015 A lockingfr modl for RissnrMindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS by L. Birao da Viga, T.J.R. Hughs, J. Kindl, C. Lovadina, J. Niirann, A. Rali, and H. Splrs Th Institut for Computational Enginring and Scincs Th Univrsity of Txas at Austin Austin, Txas Rfrnc: L. Birao da Viga, T.J.R. Hughs, J. Kindl, C. Lovadina, J. Niirann, A. Rali, and H. Splrs, "A lockingfr modl for RissnrMindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS," ICES REPORT 1501, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, January 2015.
2 A lockingfr modl for RissnrMindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS L. Birão da Viga T.J.R. Hughs J. Kindl C. Lovadina J. Niirann A. Rali H. Splrs January 13, 2015 Abstract W study a rformulatd vrsion of RissnrMindlin plat thory in which rotation variabls ar liminatd in favor of transvrs shar strains. Upon discrtization, this thory has th advantag that th shar locking phnomnon is compltly prcludd, indpndnt of th basis functions usd for displacmnt and shar strains. Any combination works, but du to th apparanc of scond drivativs in th strain nrgy xprssion, smooth basis functions ar rquird. Ths ar providd by Isogomtric Analysis, in particular, NURBS of various dgrs and quadratic triangular NURPS. W prsnt a mathmatical analysis of th formulation proving convrgnc and rror stimats for all physically intrsting quantitis, and provid numrical rsults that corroborat th thory. Dpartmnt of Mathmatics, Univrsity of Milan, Via Saldini 50, Milan, Italy. Institut for Computational Enginring and Scincs, Univrsity of Txas at Austin, 201East 24th Strt, Stop C0200 Austin, TX , USA. Dpartmnt of Civil Enginring and Architctur, Univrsity of Pavia, Via Frrata 3, 27100, Pavia, Italy. Dpartmnt of Mathmatics, Univrsity of Pavia, Via Frrata 1, Pavia, Italy. Dpartmnt of Civil and Structural Enginring, Aalto Univrsity, PO Box 12100, AALTO, Finland. Dpartmnt of Civil Enginring and Architctur, Univrsity of Pavia, Via Frrata 3, 27100, Pavia, Italy. Dpartmnt of Computr Scinc, Univrsity of Luvn, Clstijnnlaan 200A, 3001 Hvrl (Luvn, Blgium. Dpartmnt of Mathmatics, Univrsity of Rom Tor Vrgata, Via dlla Ricrca Scintifica, Rom, Italy. 1
3 1 Introduction Finit lmnt thin plat bnding analysis, basd on th PoissonKirchhoff thory, bgan with th MS thsis of Papnfuss at th Univrsity of Washington in 1959 [24]. This fournod rctangular lmnt mployd C 1 continuous intrpolation functions, but was dficint in th sns that its basis functions wr not complt through quadratic polynomials. Throughout th 1960s th dvlopmnt of quadrilatral and triangular thin plat lmnts was a focus of finit lmnt rsarch. Th problm was surprisingly difficult. Evntually thr wr tchnical succsss but th lmnts wr complicatd, hard to implmnt, and difficult to us in practic. For an arly history, w rfr to Flippa [16]. To circumvnt th difficultis, in th 1970s attntion was rdirctd to th thick plat bnding thory of RissnrMindlin. In this cas only C 0 basis functions wr rquird for th displacmnt and rotations, so th continuity and compltnss rquirmnts wr asily satisfid with standard isoparamtric basis functions, but nw difficultis aros associatd with shar locking in th thin plat limit in which th transvrs shar strains must vanish. Nvrthlss, through a numbr of clvr idas, som tricks and som fundamntal, succssful lmnts for many applications bcam availabl. Th simplicity and fficincy of ths lmnts ld to immdiat incorporation in industrial and commrcial structural analysis softwar programs, and that has bn th situation vr sinc. Rsarch in th subjct thn bcam somwhat stagnant for a numbr of yars, but rcntly things changd. Isogomtric Analysis was proposd by Hughs, Cottrll & Bazilvs in 2005 [18]. Th motivation for its dvlopmnt was to simplify and rndr mor fficint th dsignthroughanalysis procss. It is oftn said that th dvlopmnt of suitabl Finit Elmnt Analysis (FEA modls from Computr Aidd Dsign (CAD fils occupis ovr 80% of ovrall analysis tim [12]. Som dsign/analysis nginrs claim that this is an undrstimat and th situation is actually wors prhaps 90%, or vn mor. Whatvr th prcis prcntag, it is clar that th intrfac btwn dsign and analysis is brokn, and it is th statd aim of Isogomtric Analysis to rpair it. Th way this has bn approachd in Isogomtric Analysis is to rconstitut analysis within th functions utilizd in nginring CAD, such as, for xampl, NURBS and Tsplins, making it possibl, at th vry last, to prform analysis within th rprsntations providd by dsign, liminating rdundant data structurs and unncssary gomtric approximations. This has bn th primary focus of Isogomtric Analysis, but in th procss of its dvlopmnt nw analysis opportunitis hav also prsntd thmslvs. On manats from a basic proprty of th functions utilizd in CAD thy ar smooth, usually at last C 1 continuous, mor oftn C 2 continuous, and thy do not rquir drivativ dgrsoffrdom, on of th paradigmatic dficincis of th arly thin plat bnding lmnts. CAD functions also hav othr advantagous proprtis, but w will not go into ths hr. It turns out that smoothnss alon has cratd nw opportunitis in plat bnding lmnt rsarch. Th first and most obvious to b xploitd by Isogomtric Analysis is r 2
4 nwd intrst in th long abandond PoissonKirchhoff thory, as anticipatd in [18]. With convnint, smooth basis functions providd by Isogomtric Analysis, th rol of PoissonKirchhoff thory is bing rconsidrd. A prim advantag is that no rotational dgrsoffrdom ar rquird. This immdiatly rducs th siz of quation systms and th computational ffort ncssary to solv problms. In finit dformation analysis thr is a furthr advantag. This stms from th fact that finit rotations involv a productgroup structur, SO(3, a significant complication. It is compltly obviatd by rotationlss thin bnding lmnts. Anothr innovation that occurrd, du to Long, Bornmann & Cirak [21] and Echtr, Ostrl & Bischoff [15] in th contxt of shlls, was again motivatd by th xistnc of smooth basis functions. It was obsrvd that if on could dal with scond drivativs apparing in strain nrgy xprssions, thn a chang of variabls from rotations in RissnrMindlin plat thory to transvrs shar strains would liminat transvrs shar locking, indpndnt of th basis functions mployd. Why wasn t this amazing formulation utilizd prviously for th dvlopmnt of plat and shll lmnts? Th answr sms to b thr simply wr not convnint smooth basis functions availabl within th finit lmnt paradigm. Shar locking has bn th fundamntal obstacl to th dsign of ffctiv plat lmnts basd on RissnrMindlin thory. It is rmarkabl that th discovry of a complt and gnral solution has occurrd narly 50 yars aftr th widsprad adoption of th thory as a framwork for th drivation of plat and shll lmnts. Givn th abov, th qustion ariss as to what othr opportunitis might b providd by clvr changs of variabls? On answr has bn prsntd in th work of Kindl t al. [19] who hav shown that smooth basis functions with only translational displacmnt dgrsoffrdom can also b mployd succssfully for thick bnding lmnts. Th pric to pay in th formulation of [19] is that squars of third drivativs appar in th strain nrgy xprssion, but ths ar no problm for C 2 continuous Isogomtric Analysis basis functions. All ths nw idas hav cratd a rnaissanc in th dvlopmnt of mthods to solv problms involving thin and thick bnding lmnts. It is clar that ths and othr rlatd concpts will gnrat considrabl intrst in th coming yars. In this papr w undrtak th mathmatical analysis of a class of mthods for RissnrMindlin plat thory basd on th chang of variabls introducd in [15, 21]. Th dpndnt variabls ar thn th transvrs displacmnt of th plat and th transvrs shar strain vctor. Th chang of variabls rsults in squars of scond drivativs of th displacmnt in th strain nrgy, which yild to splin discrtizations of C 1, or highr, continuity. It is apparnt that th shar strain vctor is not as physically appaling or implmntationally convnint as th rotation vctor. Howvr, by utilizing wakly nforcd rotation boundary conditions, by way of Nitsch s mthod [23], ths issus ar circumvntd. Nitsch s mthod also provids othr analytical bnfits in that it allviats boundary locking, a potntial problm ncountrd in th prsnt thory for clampd boundary conditions. In addition to C p 1 NURBS dis 3
5 crtizations, w considr quadratic C 1 triangular NURPS, that is, nonuniform rational PowllSabin splins. W not in passing that Isogomtric Analysis has prviously bn applid to th standard displacmntrotation forms of th RissnrMindlin plat and shll thoris [6, 8]. An outlin of th rmaindr of th papr follows: In Sction 2 w prsnt th RissnrMindlin plat modl, first in trms of displacmnt and rotation variabls and thn in an quivalnt form in trms of displacmnt and transvrs shar strain variabls. W introduc th discrt form of th thory and summariz stability and convrgnc rsults that apply to displacmnts, shar strains, rotations, bnding momnts, and transvrs shar forc rsultants. In Sction 3 w dscrib and prform numrical tsts with NURBS spacs. In Sction 4 w do likwis with quadratic NURPS, and in Sction 5 w prsnt concluding rmarks. Th tchnical dtails of proofs ar postpond until Appndix A so as not to intrrupt th flow of th main idas and rsults in th body of th papr. 2 Th modl and discrtization W start by considring th classical RissnrMindlin modl for plats. Lt Ω b a boundd and picwis rgular domain in R 2 rprsnting th midsurfac of th plat. W subdivid th boundary Γ of Ω in thr disjoint parts (such that ach is ithr void or th union of a finit sum of connctd componnts of positiv lngth, Γ = Γ c Γ s Γ f. Th plat is assumd to b clampd on Γ c, simply supportd on Γ s and fr on Γ f, with Γ c, Γ s dfind to prclud rigid body motions. W considr for simplicity only homognous boundary conditions. Th variational spac of admissibl solutions is givn by X = { (w, θ H 1 (Ω [H 1 (Ω] 2 : w = 0 on Γ c Γ s, θ = 0 on Γ c }. Following th RissnrMindlin modl, s for instanc [4] and [17], th plat bnding problm rads as { Find (w, θ X, such that a(θ, η + µkt 2 (θ w, η v = (f, v, (v, η X, (1 whr µ is th shar modulus and k is th socalld shar corrction factor. In th abov modl, t rprsnts th plat thicknss, w th dflction, θ th rotation of th normal fibrs and f th applid scald normal load. Morovr, (, stands for th standard scalar product in L 2 (Ω and th bilinar form a(, is dfind by a(θ, η = (Cε(θ, ε(η, (2 with C th positiv dfinit tnsor of bnding moduli and ε( th symmtric gradint oprator. 4
6 Providd th boundary conditions satisfy th conditions abov, th bilinar form apparing in problm (1 is corciv, in th following sns (s Proposition A.1 in Appndix A. Thr xists a positiv constant α dpnding only on th matrial constants and th domain Ω such that a(η, η + µkt 2 (η v, η v ( α η 2 H 1 (Ω + t 2 η v 2 L 2 (Ω + v 2 H 1 (Ω, (v, η X. (3 Th abov rsult stats th corcivity of th bilinar form on th product spac X. It is wll known that th discrtization of th RissnrMindlin modl poss difficultis, du to th possibility of th locking phnomnon whn th thicknss t << diam(ω. W will hr considr an altrnativ modl that dos not suffr from such a drawback. To simplify notation, and without any loss of gnrality, w will assum µk = 1 in th following. 2.1 An quivalnt formulation Th modl prsntd hr, and originally introducd in th contxt of shlls in [21, 15], is drivd by simply considring th following chang of variabls: (w, θ (w, γ with θ = w + γ, (4 and changing th RissnrMindlin formulation accordingly. Th physical intrprtation of γ is th transvrs shar strain. Rmark 2.1. In th nginring practic, th shar strain is most frquntly dfind as γ = θ + w, instad of γ = θ w (s (4, which is mor common in mathmaticalorintd paprs. For all sufficintly rgular functions τ : Ω R 2 and v : Ω R, lt th nrgy norm b givn by v, τ 2 = τ + v 2 H 1 (Ω + t 2 τ 2 L 2 (Ω + v 2 H 1 (Ω. (5 W thn dfin th variational spacs X = { closur of C (Ω [C (Ω] 2 with rspct to th norm, }, X = {(v, τ X : v = 0 on Γ c Γ s, v + τ = 0 on Γ c }. It is immdiatly vrifid that H 2 (Ω [H 1 (Ω] 2 X H 1 (Ω [L 2 (Ω] 2. Morovr, not that X xactly corrsponds to X up to th chang of variabls (4. Lt th variational problm for th hirarchic modl b givn by { Find (w, γ X, such that a( w + γ, v + τ + t 2 (6 (γ, τ = (f, v, (v, τ X, 5
7 whr w rcall that w now hav µk = 1 for notational simplicity. Problm (6 is clarly quivalnt to (1 up to th abov chang of variabls. In particular, th corcivity proprty (3 translats immdiatly into a( v + τ, v + τ + t 2 (τ, τ α τ, v 2, (v, τ X, (7 for th sam positiv constant α. 2.2 Discrtization of th modl W introduc a pair of finit dimnsional spacs for dflctions and shar strains: W h H 2 (Ω, Ξ h [H 1 (Ω] 2. W assum that th two spacs abov ar gnratd ithr by som finit lmnt or isogomtric tchnology, and ar thrfor associatd with a (physical msh Ω h. In th following, w will indicat by K Ω h a typical lmnt of th msh, and dnot by h K its diamtr. W dnot by h th maximum msh siz. Morovr, w will indicat by E h th st of all (possibly curvd dgs of th msh, by E h th gnric dg, and by h its lngth. As usual w assum that th boundary parts Γ c, Γ s, Γ f ar unions of msh dgs. Furthrmor, w will assum that th st of lmnts K in th family {Ω h } h is uniformly shap rgular in th classical sns. W thn considr th discrt spac with (partial boundary conditions X h = { (v h, τ h W h Ξ h : v h = 0 on Γ c Γ s } ; s also Rmark 2.3 blow. Th rotation boundary condition on Γ c will b nforcd with a pnalizd formulation in th spirit of Nitsch s mthod [23], through th introduction of th following modifid bilinar form. Lt a h ( w h + γ h, v h + τ h = a( w h + γ h, v h + τ h ( Cε( wh + γ h n ( vh + τ h Γ c ( Cε( vh + τ h n ( wh + γ h Γ c + β tr(c ( w h + γ h ( v h + τ h, c( 1 (8 for all (w h, γ h, (v h, τ h in X h and for β > 0 a stabilization paramtr. Abov, c( is a charactristic quantity dpnding on th sid. Rmark 2.2. For a shap rgular family of mshs, on can choos c( = h or c( = (Ara K 1/2, whr K is th lmnt containing. This lattr choic has bn mployd in th numrical tsts of Sction 3, whil th formr has bn usd in Sction 4. Howvr, whn a sid blongs to an lmnt with a larg aspct ratio, a diffrnt choic could b prfrabl. For xampl, significantly 6
8 thin lmnts may b usd in th prsnc of boundary layrs, and on may choos c( = h, whr h is th lmnt siz in th dirction prpndicular to th boundary. W ar now abl to prsnt th proposd discrtization of th modl (6: { Find (wh, γ h X h, such that a h ( w h + γ h, v h + τ h + t 2 (9 (γ h, τ h = (f, v h, (v h, τ h X h. Rmark 2.3. It is not wis to us a dirct discrtization of th spac X by nforcing all th boundary conditions on W h Ξ h. Indd, th clampd condition on rotations w h + γ h = 0 on Γ c, w h W h, γ h Ξ h, is vry difficult to implmnt, and can b a sourc of boundary locking unlss th two spacs W h, Ξ h ar chosn in a vry carful way. To illustrat why a poor approximation might occur on Γ c, lt us considr th following xampl. Lt Ω = (0, 1 (0, 1 and Γ c = [0, 1] {1}. Slct W h = S p,p p 1,p 1 (Ω and Ξ h = S p 1,p 1 p 2,p 2 (Ω Sp 1,p 1 p 2,p 2 (Ω, whr Sp,q r,s (Ω dnots th spac of Bsplins of dgr p and rgularity C r with rspct to th x dirction, and of dgr q and rgularity C s with rspct to th y dirction. Imposing w h + γ h = 0 on Γ c implis in particular w h y (x, 1 = γ 2,h(x, 1 x [0, 1], whr γ 2,h is th scond componnt of th vctor fild γ h. Sinc w h y (x, 1 S p p 1 (0, 1 and γ 2,h(x, 1 S p 1 p 2 (0, 1, it follows that γ 2,h (x, 1 S p 1 p 2 (0, 1 Sp p 1 (0, 1 = Sp 1(0, 1. Hnc γ 2,h (x, 1 is ncssarily a global polynomial of dgr at most p 1 on Γ c, rgardlss of th msh. This mans that on Γ c γ 2,h cannot convrg to γ 2, scond componnt of γ, as th msh siz tnds to zro, in gnral. Instad, as w will prov in th nxt sction, th mthod proposd abov is fr of locking for any choic of th discrt spacs W h, Ξ h. 2.3 Stability and convrgnc rsults In th prsnt sction w show th stability and convrgnc proprtis of th proposd mthod. All th proofs can b found in Appndix A. In what follows, w st c( = h in (8, which is a suitabl choic for shap rgular mshs, s Rmark 2.2. W start by introducing th following discrt norm v h, τ h 2 h = v h, τ h 2 + h 1 v h + τ h 2 L 2 (, (10 for all (v h, τ h X h. In th thortical analysis of th mthod w will mak us of th following assumptions on th solution rgularity and spac approximation proprtis. p 1 7
9 A1 W assum that th solution w H 2+s (Ω and γ H 1+s (Ω for som s > 1/2. A2 W assum that th following standard invrs stimats hold v h H 1 (K Ch 1 K v h 2 (K, τ h H 1 (K Ch 1 K τ h 2 (K, for all v h W h and τ h Ξ h with C indpndnt of h. A3 W assum th following approximation proprtis for X h. Lt s = 2, 3. For all (v, τ (H s (Ω [H 1 (Ω] 2 X thr xists (v h, τ h X h such that τ τ h Hj (Ω Ch 1 j τ H 1 (Ω, j = 0, 1, with C indpndnt of h. v v h Hj (Ω Ch s j v Hs (Ω, j = 0,..., s, Morovr, w will mak us of th following natural assumption, in ordr to avoid rigid body motions: A4 W assum that Γ c Γ s has positiv lngth and that ithr i Γ c has positiv lngth, or ii Γ s is not containd in a straight lin. W now introduc a corcivity lmma stating in particular th invrtibility of th linar systm associatd with (9. Lmma 2.1. Lt hypothss A2 and A4 hold. Thr xist two positiv constants β 0, α such that, for all β β 0, w hav a h ( v h +τ h, v h +τ h +t 2 (τ h, τ h α τ h, v h 2 h, (v h, τ h X h. (11 Th constant α only dpnds on th matrial paramtrs and th domain Ω, whil th constant β 0 dpnds only on th shap rgularity constant of Ω h. Lt A1 hold. By an intgration by parts, it is immdiatly vrifid that th schm (9 is consistnt, in th sns that a h ( w + γ, v h + τ h + t 2 (γ, τ h = (f, v h, (v h, τ h X h, (12 whr (w, γ is th solution of problm (6 and th lfthand sid maks sns du to th rgularity assumption A1. Not that condition A1 could b significantly rlaxd by intrprting th intgrals on Γ c apparing in (8 in th sns of dualitis. By combining th corcivity in Lmma 2.1 with th consistncy proprty (12, th following convrgnc rsult follows. 8
10 Proposition 2.1. Lt A1 and A2 hold. Lt (w, γ b th solution of problm (6 and (w h, γ h X h b th solution of problm (9. Thn, if β β 0, w hav w w h, γ γ h 2 h 2 ( C inf + (v h,τ h X h j=0 h 2(j 1 K K Ω h h 2(j 1 K K Ω h w v h 2 H j+1 (K γ τ h 2 H j (K + t 2 γ τ h 2 L 2 (Ω, (13 with C dpnding only on th matrial paramtrs and th domain Ω. W now stat a convrgnc rsult concrning bnding momnts and shar forcs, th quantitis of intrst in nginring applications. To this aim, w first dfin th scald bnding momnts and th scald shar forcs as follows: M = Cε( w + γ, Q = t 2 µκγ = t 2 γ, (14 whr (w, γ is th solution to problm (6. Th abov quantitis ar known to convrg to nonvanishing limits, as t 0 (s,.g., [3] and [9]. Onc problm (9 has bn solvd, w can dfin th scald discrt bnding momnts and th scald discrt shar momnts as M h = Cε( w h + γ h, Q h = t 2 γ h. (15 W hav th following stimats. Proposition 2.2. Lt A1 and A2 hold. Lt (w, γ b th solution of problm (6 and (w h, γ h X h b th solution of problm (9. Thn, if β β 0, w hav M M h 2 (Ω + t Q Q h 2 (Ω C w w h, γ γ h h. (16 Morovr, lt th msh family {Ω h } h b quasiuniform. Thn, w hav ( h Q Q h 2 (Ω C w w h, γ γ h h + h inf Q s h s h Ξ 2 (Ω, (17 h ( Q Q h H 1 (Ω C w w h, γ γ h h + h inf Q s h 2 (Ω. (18 s h Ξ h In addition, w can formulat th following improvd rsult rgarding th rror for th rotations in th L 2 norm and th rror for th dflctions in th H 1 norm. An analogous rsult (possibly with a smallr improvmnt in trms of h, t also holds if th additional hypothss in th proposition ar not satisfid; w do not dtail hr this mor gnral cas. Proposition 2.3. Lt th sam assumptions and notation of Proposition 2.1 hold. Morovr, lt assumption A3 hold, Γ c = Γ and lt th domain Ω b 9
11 ithr rgular, or picwis rgular and convx. Thn, th following improvd approximation rsult holds θ θ h 2 (Ω C(h + t w w h, γ γ h h, (19 w w h H 1 (Ω C(h + t w w h, γ γ h h + γ γ h 2 (Ω, (20 whr θ h = w h + γ h and th constant C dpnds only on th matrial paramtrs and th domain Ω. Not that th last trm apparing in (13 is not a sourc of locking sinc t 1 γ = tq is a quantity that is known to b uniformly boundd in th corrct Sobolv norms (s,.g., [3] and [9]. Th constants apparing in Propositions 2.1 and 2.3 ar indpndnt of t, and so th rsults shown stat that th proposd mthod is lockingfr rgardlss of th discrt spacs adoptd. This is a vry intrsting proprty that is missing in th standard mthods for th Rissnr Mindlin problm. Th accuracy of th discrt solution (9 will only dpnd on th approximation proprtis of th adoptd discrt spacs, and will not b hindrd by small valus of th plat thicknss. W also rmark that th norms for th scald shar forcs apparing in th lfthand sid of (17 and (18, ar indd th usual norms for which a convrgnc rsult can b stablishd (s,.g., [9] and [10]. In th following two sctions w will prsnt a pair of particular choics for W h, Ξ h within th framwork of standard tnsorproduct NURBSbasd isogomtric analysis (Sction 3; triangular NURPSbasd isogomtric analysis (Sction 4. For such choics w can apply Propositions 2.1, 2.3 in ordr to obtain th xpctd convrgnc rats in trms of h. For xampl, in th cas of standard tnsorproduct NURBS, combining Proposition 2.1 with th approximation stimats in [5, 7] w obtain th following convrgnc rsult. Corollary 2.1. Lt th sam assumptions and notation of Proposition 2.1 hold. Lt standard isoparamtric tnsorproduct NURBS of polynomial dgr p for all variabls b usd for th spac X h. Thn, providd that th solution of problm (6 is sufficintly rgular for th righthand sid to mak sns, th following rror stimats holds for all 2 s p: w w h, γ γ h h Ch s 1( w H s+1 (Ω + θ H s (Ω + t γ H s 1 (Ω, (21 M M h 2 (Ω+t Q Q h 2 (Ω Ch s 1( (22 w H s+1 (Ω + θ H s (Ω + t γ H s 1 (Ω. Morovr, lt th msh family {Ω h } h b quasiuniform. Thn, h Q Q h 2 (Ω + Q Q h H 1 (Ω Ch s 1( (23 w H s+1 (Ω + θ H s (Ω + t γ H s 1 (Ω + γ H s 2 (Ω. Th constant C dpnds only on p, th matrial paramtrs and th domain Ω. 10
12 Th abov corollary can also b combind with Proposition 2.3 to obtain improvd rror stimats for th L 2 norm of th rotations and th H 1 norm of th dflctions. Also rcalling dfinition (5, on can asily obtain a convrgnc rat of (h + t h s 1 for such quantitis. Finally, not that, lik for all high ordr mthods, th rgularity rquirmnt in Corollary 2.1 may b too dmanding du to th prsnc of layrs in th solution. This situation is typically dalt with by making us of rfind mshs, an xampl of which is shown in th numrical tsts. 3 Isogomtric discrtization with NURBS In this sction, NURBSbasd isogomtric analysis is usd to prform numrical validations of th prsntd thory. W bgin with a brif summary of Bsplins and NURBS (NonUniform Rational BSplins. 3.1 Bsplins and NURBS Bsplins ar picwis polynomials dfind by th polynomial dgr p and a knot vctor [ξ 1, ξ 2,..., ξ n+p+1 ], whr n is th numbr of basis functions. Th knot vctor is a st of paramtric coordinats ξ i, calld knots, which divid th paramtric spac into intrvals calld knot spans. A knot can also b rpatd, in this cas it is calld a multipl knot. At a singl knot th Bsplins ar C p 1  continuous, and at a multipl knot of multiplicity k th continuity is rducd to C p k. Th Bsplin basis functions of dgr p ar dfind by th following rcursion formula. For p = 0, { 1, ξ i x < ξ i+1, N i,0 (x = 0, othrwis. For p 1, N i,p (x = x ξ i N i,p 1 (x + ξ i+p+1 x N i+1,p 1 (x. ξ i+p ξ i ξ i+p+1 ξ i+1 A bivariat NURBS function R p,q i,j is dfind as th wightd tnsorproduct of th Bsplin functions N i,p and M j,q with polynomial dgrs p and q, R p,q N i,j (x, y = i,p (xm j,q (yω i,j n m, N l,p (xm r,q (yω l,r l=1 r=1 whr ω i,j ar calld control wights. Following th isogomtric concpt, NURBS ar mployd to both rprsnt th gomtry and to approximat th solution, i.. th isoparamtric concpt is invokd. Accordingly, th unknown variabls 11
13 w and γ ar approximatd by w h (x, y = n w m w i=1 j=1 R pw,qw i,j (x, yŵ i,j, γ h (x, y = n γ m γ i=1 j=1 R pγ,qγ i,j (x, yˆγ i,j, with n w, m w th numbrs of basis functions in th two paramtric dirctions for w h, and n γ, m γ th numbrs for γ h. Th tst functions v and τ ar discrtizd accordingly. Sinc th rotations θ ar not discrtizd in this approach, rotational boundary conditions cannot b imposd in a standard way by applying thm dirctly on th rspctiv dgrs of frdom at th boundary. Instad, such boundary conditions ar nforcd by th modifid bilinar form introducd in quation (8. Displacmnt boundary conditions ar nforcd in a standard way through th displacmnt dgrs of frdom on th boundary. 3.2 Numrical tsts In this sction, th proposd mthod is tstd on diffrnt numrical xampls. W always mploy th sam polynomial ordr and th highst rgularity for all th unknowns, but diffrnt choics can b mad. In ordr to dmonstrat th lockingfr bhavior of this mthod w considr both thick and thin plats. Furthrmor, w invstigat an xampl which xhibits boundary layrs. As an rror masur, an approximatd L 2 norm rror for a variabl u is computd as u x u h 2 u x 2 (x = (u i,y i G x u h 2, (x i,y i G u2 x whr G is a uniform grid in th paramtr domain [0, 1] 2 mappd onto th physical domain Squar plat with clampd boundary conditions Th first xampl consists of a unit squar plat [0, 1] 2 with an analytical solution as dscribd in [11]. Th plat is clampd on all four sids, and subjct to a load givn by f(x, y = E [ 12y(y 1(5x 2 12(1 ν 2 5x + 1(2y 2 (y x(x 1(5y 2 5y x(x 1(5y 2 5y + 1(2x 2 (x y(y 1(5x 2 5x + 1 ]. 12
14 Th analytical solution for th displacmnt w is givn by w(x, y = 1 3 x3 (x 1 3 y 3 (y 1 3 2t2 [ y 3 (y 1 3 x(x 1(5x 2 5x + 1 5(1 ν + x 3 (x 1 3 y(y 1(5y 2 5y + 1 ]. W prform an hrfinmnt study using qual polynomial dgrs p = 2, 3, 4, 5 for w h and γ h, for th cas of a thick plat with t = 10 1 and a thin plat with t = Th matrial paramtrs ar takn to b E = 10 6 and ν = 0.3. Figur 1(a shows th convrgnc plots for th thick plat, whras Figur 1(b thos for th thin plat. Dashd lins indicat th rfrnc ordr of convrgnc. As can b sn, th convrgnc rats for all polynomial ordrs ar at last of ordr p. In addition, w study th convrgnc for bnding momnts and shar forcs sinc ths ar of prim intrst in th nginring dsign of plats. Bnding momnts m and shar forcs q ar obtaind as m = t 3 Cε( w + γ, q = kµtγ. Th xact solution for bnding momnts and shar forcs is ( m xx = K b 2 y 3 (y 1 3 (x x 2 (5x 2 5x ν(x 3 (x 1 3 (y y 2 (5y 2 5y + 1, ( m yy = K b 2 ν(y 3 (y 1 3 (x x 2 (5x 2 5x x 3 (x 1 3 (y y 2 (5y 2 5y + 1, m xy = m yx = K b (1 ν3y 2 (y 1 2 (2y 1x 2 (x 1 2 (2x 1, ( q x = K b 2 y 3 (y 1 3 (20x 3 30x x 1 + 3y(y 1(5y 2 5y + 1x 2 (x 1 2 (2x 1, ( q y = K b 2 x 3 (x 1 3 (20y 3 30y y 1 + 3x(x 1(5x 2 5x + 1y 2 (y 1 2 (2y 1, Et 3 whr K b = 12(1 ν 2 is th plat bnding stiffnss. For th rror masur, th 2 2 Euclidan norms of m and q ar usd, m = m 2 ij and q = 2 qi 2. Th i=1 j=1 i=1 13
15 ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#dof 1 c2*(#dof 3/2 c3*(#dof 2 c4*(#dof 5/2 ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#dof 1 c2*(#dof 3/2 c3*(#dof 2 c4*(#dof 5/ (#dof 1/ (#dof 1/2 (a (b Figur 1: Squar plat with clampd boundary conditions. L 2 norm approximation rror of displacmnts with tnsorproduct Bsplins for (a t = 10 1 and (b t = ( m x m h 2 / m x p=2 p=3 p=4 p=5 c1*(#cp 1/2 c2*(#cp 1 c3*(#cp 3/2 c4*(#cp 2 ( m x m h 2 / m x p=2 p=3 p=4 p=5 c1*(#cp 1/2 c2*(#cp 1 c3*(#cp 3/2 c4*(#cp (#dof 1/ (#dof 1/2 (a (b Figur 2: Squar plat with clampd boundary conditions. L 2 norm approximation rror of bnding momnts with tnsorproduct Bsplins for (a t = 10 1 and (b t = convrgnc plots for bnding momnts ar prsntd in Figur 2, and thos for shar forcs in Figur 3. Rcalling that m = t 3 M (rsp. m h = t 3 M h and q = t 3 Q (rsp. q h = t 3 Q h, w notic that th convrgnc rats for th rlativ rrors displayd in Figurs 2 and 3 ar in accordanc with th thortical rsults of Corollary 2.1 (s stimats (22 and (23. In particular, w rmark that Figur 3(b displays an O(1 convrgnc rat for th L 2 norm of th shar forc rrors, whn p = 2, in agrmnt with stimat (23, for s = 2. In othr words, thr is no convrgnc. 14
16 ( q x q h 2 / q x p=2 p=3 p=4 p=5 c1*(#cp 1/2 c2*(#cp 1 c3*(#cp 3/2 c4*(#cp 2 ( q x q h 2 / q x p=2 p=3 p=4 p=5 c2*(#cp 1/2 c3*(#cp 1 c4*(#cp log (#dof 1/ log (#dof 1/2 10 (a (b Figur 3: Squar plat with clampd boundary conditions. L 2 norm approximation rror of shar forcs with tnsorproduct Bsplins for (a t = 10 1 and (b t = y x Figur 4: Quartr of annulus plat. Gomtry stup Quartr of an annulus with clampd and simply supportd boundary conditions Th scond tst consists of a quartr of an annulus with an innr diamtr of 1.0 and outr diamtr of 2.5, as shown in Figur 4. Th plat thicknss is t = 0.01 and th matrial paramtrs ar E = 10 6 and ν = 0.3. Th plat is loadd with a uniform load f(x, y = 1 and two boundary conditions ar considrd: (a all dgs ar clampd, (b all dgs ar simply supportd. For both cass, this xampl xhibits boundary layrs. Thrfor, w adopt a rfinmnt stratgy in ordr to bttr captur th boundary layrs. Givn that th knot vctors rang from 0 to 1, w introduc as a first stp additional knots at 0.1 and 0.9 in 15
17 y y x x Figur 5: Quartr of annulus with boundary rfinmnt. (a Initial modl, (b boundary rfind msh ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#dof 1 c2*(#dof 3/2 c3*(#dof 2 c4*(#dof 5/2 ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#cp 1 c2*(#cp 3/2 c3*(#cp 2 c4*(#cp 5/ (#dof 1/2 (a log (#dof 1/2 10 (b Figur 6: Quartr of annulus with clampd boundary conditions. L 2 norm approximation rror of displacmnts with tnsorproduct NURBS and (a boundary rfinmnt, (b uniform rfinmnt. both dirctions, s Figur 5. Thn, w prform uniform rfinmnt of th givn knot spans. In th following, w prform convrgnc studis with and without th boundary rfinmnt stratgy. Th rfinmnt is prformd such that th total numbr of dgrs of frdom is comparabl in both cass. Sinc analytical solutions ar not availabl for ths problms, w us as rfrnc th solutions obtaind on a vry fin msh (an ovrkill solution with quintic lmnts and comput th L 2 norm approximation rrors for th displacmnt. Figur 6 shows th rsults for th clampd cas, (a with boundary rfinmnt and (b with uniform rfinmnt, whil in Figur 7 th rsults for th 16
18 ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#dof 1 c2*(#dof 3/2 c3*(#dof 2 c4*(#dof 5/2 ( w x w h 2 / w x p=2 p=3 p=4 p=5 c1*(#dof (#dof 1/2 (a (#dof 1/2 (b Figur 7: Quartr of annulus with simply supportd boundary conditions. L 2  norm approximation rror of displacmnts with tnsorproduct NURBS and (a boundary rfinmnt, (b uniform rfinmnt. simply supportd cas ar plottd, (a with boundary rfinmnt and (b with uniform rfinmnt. As xpctd, du to th prsnc of boundary layrs, in both th clampd and th simply supportd cas, th tsts confirm that a suitabl boundary rfind msh is ndd to achiv optimal ordrs of convrgnc. With a uniform rfinmnt, suboptimal rsults ar instad obtaind; this is mor pronouncd in th simply supportd cas. 4 Isogomtric discrtization with NURPS In this sction, w us a triangular NURPSbasd isogomtric discrtization to prform numrical validations. W first brifly summariz th construction and som proprtis of quadratic Bsplins ovr a PowllSabin (PS rfinmnt of a triangulation and thir rational gnralization, th socalld NURPS Bsplins. Thn, th discrtizd modl is tstd on th sam xampls as dscribd in th prvious sction. 4.1 Quadratic PS and NURPS Bsplins Lt T b a triangulation of a polygonal (paramtric domain Ω in R 2, and lt V i = (Vi x, V y i, i = 1,..., N V, b th vrtics of T. A PS rfinmnt T of T is th rfind triangulation obtaind by subdividing ach triangl of T into six subtriangls as follows (s also Figur 8. Slct a split point C i insid ach triangl τ i of T and connct it to th thr vrtics of τ i with straight lins. For ach pair of triangls τ i and τ j with a common dg, connct th two points C i and C j. If τ i is a boundary triangl, thn also connct C i to 17
19 Figur 8: A triangulation T and a PowllSabin rfinmnt T of T. an arbitrary point on ach of th boundary dgs. Ths split points must b chosn so that any constructd lin sgmnt [C i, C j ] intrscts th common dg of τ i and τ j. Such a choic is always possibl: for instanc, on can tak C i as th incntr of τ i, i.. th cntr of th circl inscribd in τ i. Usually, in practic, th barycntr of τ i is also a valid choic, but not always. Th spac of C 1 picwis quadratic polynomials on T is calld th Powll Sabin splin spac [26] and is dnotd by S 1 2(T. It is wll known that th dimnsion of S 1 2(T is qual to 3N V. Morovr, any lmnt of S 1 2(T is uniquly spcifid by its valu and its gradint at th vrtics of T, and can b locally constructd on ach triangl of T onc ths valus and gradints ar givn. Dirckx [13] has dvlopd a Bsplin lik basis {B i,j, j = 1, 2, 3, i = 1,..., N V } of th spac S 1 2(T such that B i,j (x, y 0, N V i=1 j=1 3 B i,j (x, y = 1, (x, y Ω. (24 Th functions B i,j will b rfrrd to as PowllSabin (PS Bsplins. Th PS Bsplins B i,j, j = 1, 2, 3, ar constructd to hav thir support locally in th molcul Ω i of vrtx V i, which is th union of all triangls of T containing V i. It suffics to spcify thir valus and gradints at any vrtx of T. Du to th structur of th support Ω i, w hav B i,j (V k = 0, x B i,j(v k = 0, for any vrtx V k V i. Morovr, w st B i,j (V i = α i,j, x B i,j(v i = β i,j, 18 y B i,j(v k = 0, y B i,j(v i = γ i,j.
20 Figur 9: Location of th PS points (black bullts, and a possibl PS triangl associatd with th cntral vrtx (shadd. Th triplts (α i,j, β i,j, γ i,j can b spcifid in a gomtric way in ordr to satisfy (24. To this aim, for ach vrtx V i, i = 1,..., N V, w dfin thr points such that {Q i,j = (Q x i,j, Q y i,j, j = 1, 2, 3}, α i,1 α i,2 α i,3 Q x i,1 Q y i,1 1 β i,1 β i,2 β i,3 Q x i,2 Q y i,2 1 = γ i,1 γ i,2 γ i,3 Q x i,3 Q y i,3 1 Vi x V y i Th triangl with vrtics {Q i,j, j = 1, 2, 3} will b rfrrd to as th PS triangl associatd with th vrtx V i and will b dnotd by T i. Finally, for ach vrtx V i w dfin its PS points as th vrtx itslf and th midpoints of all th dgs of th PS rfinmnt T containing V i, s Figur 9. It has bn provd in [13] that th functions B i,j, j = 1, 2, 3, ar nonngativ if and only if th PS triangl T i contains all th PS points associatd with th vrtx V i. From a stability point of viw, it is prfrabl to choos PS triangls with a small ara. Bing quippd with a Bsplin lik basis, PS splins admit a straightforward rational xtnsion. A NURPS (NonUniform Rational PS basis function is dfind as B i,j (x, yω i,j R i,j (x, y = N V l=1 r=1, 3 B l,r (x, yω l,r whr ω i,j ar positiv control wights. In our plat contxt, similar to th discrtization with NURBS, th unknown variabls w and γ ar approximatd by w h = N V w i=1 j=1 3 R i,j (x, yŵ i,j, γ h = N V γ i=1 j=1. 3 R i,j (x, yˆγ i,j, 19
21 whr N V w is th numbr of vrtics for w h and N V γ is th numbr of vrtics for γ h. Th tst functions v and τ ar discrtizd accordingly. Displacmnt boundary conditions ar nforcd in a standard way through th displacmnt dgrs of frdom on th boundary whil rotation boundary conditions ar nforcd by th modifid bilinar form introducd in quation (8. PS and NURPS Bsplins hav alrady bn succssfully mployd to solv partial diffrntial problms [30], in particular in th isogomtric nvironmnt [32, 31]. Crtain splin spacs of highr dgr and smoothnss (rgularity hav also bn dfind on triangulations ndowd with a PS rfinmnt, and thy can b rprsntd in a similar way as in th quadratic cas. W rfr to [27] for C 2 quintics and to [28] for a family of splins with arbitrary smoothnss. Morovr, th quadratic cas has bn xtndd to th multivariat stting in [29]. Unfortunatly, thy ar lacking th sam flxibility of any combination of polynomial dgr and smoothnss in contrast with th tnsorproduct Bsplin cas. On th on hand, it is known how to construct stabl splin spacs on triangulations with a sufficintly high polynomial dgr with rspct to th global smoothnss (s,.g., [20]. In particular, on can quit asily do dgrlvation for th abov mntiond xisting spacs (i.., raising th polynomial dgr and kping th original smoothnss. On th othr hand, it is xtrmly challnging to construct splin spacs on triangulations with a vry high smoothnss rlativly to th dgr (lik th highst continuity C p 1 for a dgr p 2. Anothr intrsting point of furthr invstigation is th construction of splin spacs with mixd smoothnss. 4.2 Numrical tsts In this sction, w solv th sam xampls as illustratd in Sction 3.2, using quadratic PS or NURPS Bsplins. In particular, th sam rror masur as dscribd bfor is adoptd. Dspit th fact that PS/NURPS splins can b dfind on arbitrary triangulations, w will only considr rgular mshs in our xampls, in ordr to b abl to mak a fair comparison with tnsorproduct splins. Of cours, in ral applications on should xploit this fatur and us triangulations gnratd by an adaptiv rfinmnt stratgy. For rsults with adaptiv PS/NURPS approximations in isogomtric analysis, w rfr to [32, 31] Squar plat with clampd boundary conditions W prform th sam tst dscribd in Sction using quadratic PS B splins both for dflctions and rotations dfind on uniform triangulations. Th coarsst triangulation is dpictd in Figur 10 (lft, and th approximation rror for th displacmnt is shown in Figur 11. Th dashd lin indicats th rfrnc ordr of convrgnc. As can b sn, th convrgnc rat is of ordr 2. Figur 12 rprsnts th approximation rror for bnding momnts and shar forcs. W rmark that shar forcs sm to convrg lik O(h in th L 2 norm for this cas. Howvr, othr numrical tsts (not rportd hr xhibit an 20
22 Figur 10: Uniform triangulation and its mapping to a quartr of an annulus. ( w x w h 2 / w x t=10 1 t=10 3 c1*(#dof (#dof 1/2 Figur 11: Squar plat with clampd boundary conditions. L 2 norm approximation rror of displacmnts for t = 10 1 and t = 10 3 with triangular PS splins. O(1 convrgnc rat, in agrmnt with th thortical stimat (23, with s = Quartr of an annulus with clampd and simply supportd boundary conditions W prform th sam tst dscribd in Sction using NURPS Bsplins. As bfor w considr a clampd and a simply supportd cas and in both cass w prform boundary rfinmnt and uniform rfinmnt. Th coarsst uniform msh and its imag ar shown in Figur 10. Th imags of som of th boundary rfind mshs ar shown in Figur 13. Sinc thr ar no analytical solutions availabl, w hav takn as rfrnc solutions th NURPS approximations on a fin msh (an ovrkill solution: w hav usd a triangulation consisting of 21
23 ( m x m h 2 / m x m, t=10 1 m, t=10 3 c*(#dof 1/2 ( q x q h 2 / q x q, t=10 1 q, t=10 3 c*(#dof 1/ log (#dof 1/ log (#dof 1/2 10 Figur 12: Squar plat with clampd boundary conditions. L 2 norm approximation rror of bnding momnts (lft and shar forcs (right for t = 10 1 and t = 10 3 with triangular PS splins Figur 13: Som boundary rfind mshs triangls according to th two rfinmnt schms. Figur 14 shows th approximation rror for th displacmnt. Th dashd lin indicats th rfrnc ordr. As can b sn, boundary rfinmnt yilds improvd rsults for both cass. In particular, th following rmark holds for th invstigatd rang of dgrs of frdom. For th simply supportd cas, th boundary rfinmnt schm achivs th corrct convrgnc rat, whras uniform rfinmnt producs a suboptimal convrgnc rat. For th clampd cas, both th boundary rfinmnt schm and th uniform rfinmnt schm giv optimal convrgnc rats, but th formr procdur xhibits a numrical bttr constant in th rror plots. 22
24 1 1.5 uniform boundary c1*(#dof uniform boundary c1*(#dof 1 ( w x w h 2 / w x ( w x w h 2 / w x log (#dof 1/ log (#dof 1/2 10 Figur 14: Quartr of annulus with clampd (lft and simply supportd (right boundary conditions. Uniform and boundary rfind mshs ar considrd. L 2  norm approximation rror of displacmnts computd with triangular NURPS. 5 Conclusions In this papr w mathmatically and numrically invstigatd th rformulatd variational formulation of RissnrMindlin plat thory in which th rotation variabls ar liminatd in favor of th transvrs shar strains. Boundary conditions on th rotations wr nforcd wakly by way of Nitsch s mthod to mak th implmntation asir and to ovrcom possibl boundary locking phnomna (s Rmark 2.3. A distinct advantag of this thory is that shar locking is prcludd for any combination of trial functions for displacmnt and transvrs shar stains. Howvr, scond drivativs of th displacmnt appar in th strain nrgy xprssion and ths rquir basis functions of at last C 1 continuity. To dal with th smoothnss rquirmnts w mployd Isogomtric Analysis, spcifically various dgr NURBS of maximal continuity, and quadratic triangular NURPS. Th numrical rsults corroboratd th thortical rror stimats for displacmnt, bnding momnts and transvrs shar forc rsultants. Acknowldgmnts L. Birão da Viga, C. Lovadina and A. Rali wr supportd by th Europan Commission through th FP7 Factory of th Futur projct TERRIFIC (FoF ICT , Rfrnc: T. J. R. Hughs was supportd by grants from th Offic of Naval Rsarch (N , th National Scinc Foundation (CMMI and SINTEF (UTA , with th Univrsity of Txas at Austin. J. Kindl and A. Rali wr supportd by th Europan Rsarch Council through th FP7 Idas Starting Grant n ISOBIO. Jarkko Niirann was supportd by Acadmy of Finland (dcision numbr H. Splrs was supportd by th Rsarch Foundation Flandrs and by th 23
25 MIUR Futuro in Ricrca Programm through th projct DREAMS. A Proofs of th thortical rsults In th prsnt sction w prov all th thortical rsults prviously prsntd in th body of th papr. In th following w will assum th obvious condition that 0 < t < diam(ω, whr diam(ω dnots th diamtr of Ω. W will nd th following rsults. First Korn s inquality (s [14]. Thr xists a positiv constant C such that ε(η 2 L 2 (Ω + η 2 L 2 (Ω C η 2 H 1 (Ω, η H1 (Ω 2. (25 Scond Korn s inquality (s [14]. Suppos that Γ c > 0. Thn, thr xists a positiv constant C such that ε(η 2 L 2 (Ω C η 2 H 1 (Ω, η H1 (Ω 2, such that η Γc = 0. (26 Agmon s inquality (s [1, 2]. Lt b an dg of an lmnt K. Thn C a (K > 0 only dpnding on th shap of K such that ϕ 2 L 2 ( C a(k ( h 1 ϕ 2 L 2 (K + h ϕ 2 H 1 (K, ϕ H 1 (K. (27 Clarly, (27 also holds for vctorvalud and tnsorvalud functions. A.1 Corcivity of th continuous problm Proposition A.1. Lt assumption A4 hold. Thn thr xists a positiv constant α dpnding only on th matrial constants and th domain Ω such that a(η, η + µkt 2 (η v, η v ( α η 2 H 1 (Ω + t 2 η v 2 L 2 (Ω + v 2 H 1 (Ω, (v, η X. (28 Proof. It is asy to s that th hypothss on Γ c and Γ s ar sufficint to prvnt rigid body motions. W procd by considring th two diffrnt cass. i Lt Γ c hav positiv lngth. Thn, from th positivdfinitnss of C and th scond Korn s inquality, w gt a(η, η = (Cε(η, ε(η C 1 ε(η 2 L 2 (Ω C 2 η 2 H 1 (Ω. Thrfor, stimat (28 follows from a littl algbra and th Poincaré inquality for v. ii Lt Γ c hav zro lngth. Thn Γ s is not containd in a straight lin, and, sinc Γ c Γ s has positiv lngth, it follows that Γ s has positiv lngth. It is nough to prov that on has a(η, η + η v 2 L 2 (Ω ( η C 2 H 1 (Ω + v 2 H 1 (Ω, (v, η X. (29 24
26 By contradiction, suppos that stimat (29 dos not hold. Thn, thr xists a squnc {(v k, η k } X such that { a(ηk, η k + η k v k 2 L 2 (Ω 0, for k + ; η k 2 H 1 (Ω + v k 2 H 1 (Ω = 1. (30 Up to xtracting a subsqunc, th scond quation of (30 shows that η k η 0 wakly in H 1 (Ω 2 ; v k v 0 wakly in H 1 (Ω. (31 By Rllich s Thorm w infr that η k η 0 in L 2 (Ω 2 ; v k v 0 in L 2 (Ω. (32 Thrfor, rcalling that C is positivdfinit, from a(η k, η k 0 (cf. (30, (32 and (25, w gt that {η k } is a Cauchy squnc in H 1 (Ω 2. Thus, w hav η k η 0 in H 1 (Ω 2 and ε(η 0 = 0. (33 Morovr, sinc {η k } is a Cauchy squnc in L 2 (Ω 2, from η k v k 2 L 2 (Ω 0 (cf. (30, w hav that also { v k } is a Cauchy squnc in L 2 (Ω 2. Thrfor, from (31 and (32 w obtain that v k v 0 in H 1 (Ω and v 0 = η 0. Hnc, from (33 w gt ε( v 0 = 0, which implis that v 0 is an affin function. Sinc v 0 = 0 on Γ s and Γ s is not containd in a straight lin, it follows that v 0 = 0 in Ω. Thrfor, η 0 = 0 and w hav provd that (η k, v k (0, 0 in H 1 (Ω 2 H 1 (Ω, which contradicts th scond quation of (30. A.2 Stability and convrgnc analysis In th prsnt sction w giv th proofs of th rsults in Sction 2.3. W nd th following Korn s typ inquality. Lmma A.1. Suppos that Γ c has positiv lngth. Thn, thr xists a positiv constant C such that ε(v 2 L 2 (Ω + v 2 L 2 (Γ c C v 2 H 1 (Ω, v H1 (Ω 2. (34 Proof. By contradiction. If (34 dos not hold, thn thr xists a squnc {v k } in H 1 (Ω 2 such that { ε(vk 2 L 2 (Ω + v k 2 L 2 (Γ c 0, for k + ; v k 2 H 1 (Ω = 1. (35 Up to xtracting a subsqunc, th scond quation of (35 and Rllich s thorm show that thr xists v 0 H 1 (Ω 2 such that 25
27 { vk v 0 wakly in H 1 (Ω 2 ; v k v 0 strongly in L 2 (Ω 2. (36 From (35 and (36 w gt that {(v k, ε(v k } is a Cauchy squnc in L 2 (Ω 2 L 2 (Ω 4 s. Using th first Korn inquality (25 w dduc that {v k } is a Cauchy squnc also in H 1 (Ω 2, and thus v k v 0, strongly in H 1 (Ω 2. Thrfor, from th first quation of (35 w hav ε(v 0 = 0 in Ω; v 0 Γc = 0. (37 Equation (37 asily implis v 0 = 0. Thrfor, v k 0, which is in contradiction with v k H 1 (Ω = 1 (cf. (35. Proof of Lmma 2.1. W distinguish two cass. i Γ c has zro lngth. In this cas w hav a h ( v h + τ h, v h + τ h = a( v h + τ h, v h + τ h, v h, τ h h = v h, τ h, for vry (v h, τ h X h ; s (8 and (10. Thrfor, stimat (11 immdiatly follows from stimat (7, sinc Γ c with vanishing lngth implis X h X. ii Γ c has positiv lngth. First, for vry (v h, τ h X h w will show that (cf. (10: a h ( v h + τ h, v h + τ h ( C v h + τ h 2 H 1 (Ω + h 1 v h + τ h 2 L 2 (. (38 For notational simplicity, w st θ h := v h + τ h. Thn, rcalling (8, w hav ( a h (θ h, θ h = a(θ h, θ h 2 Cε(θh n θh + β tr(c h 1 θ h 2 Γ c E h Γ c = a(θ h, θ h + β 2 tr(c θ h 2 h 1 ( 2 Cε(θh n θh + β Γ c 2 tr(c h 1 Applying (34 with v = θ h, w obtain θ h 2. a h (θ h, θ h C K θ h 2 H 1 (Ω ( 2 Cε(θh n θh + β Γ c 2 tr(c h 1 θ h 2, (39 26
28 for a suitabl positiv constant C K. For ach dg Γ c E h, lt th symbol K dnot an lmnt of Ω h such that K. W now hav, by simpl algbra and using (27: ( 2 Cε(θh n θh = ( ( 2 Cε(θh n θh Γ c E h Γ c ( 2 Cε(θh n 2 ( θ h 2 ( C C ( 2CC ε(θ h 2 ( θ h 2 ( ( γh ε(θ h 2 L 2 ( + 1 θ h 2 L γ h 2 ( ( (C C C a (K γ ε(θ h 2 L 2 (K + h2 ε(θ h 2 H 1 (K + C C θ h 2 L γ 2 h 2 ( (40 for positiv constants {γ } Γc E h to b chosn. By using th invrs inquality and stting from (40 it follows that ( 2 Cε(θh n θh Γ c ε(θ h 2 H 1 (K C inv(k h 2 K ε(θ h 2 L 2 (K, ( C(K = C C C a (K 1 + C inv (K h2 h 2, K ( C(K γ ε(θ h 2 L 2 (K + C C θ h 2 L γ h 2 (. Thrfor, from (39 and (41 w gt a h (θ h, θ h C K θ h 2 H 1 (Ω C(K γ ε(θ h 2 L 2 (K + ( β 2 tr(c C C h 1 θ h 2 γ E h Γ c ( C K Choosing C(K γ θ h 2 H 1 (Ω + ( β 2 tr(c C C γ γ = C K 2 C(K 1 and β 0 = γc K + 2C C γtr(c from (42 w dduc that, for vry β β 0, w hav ( a h (θ h, θ h C K θ h 2 H 2 1 (Ω + 27 h 1 with γ = h 1 min γ, θ h 2. (41 θ h 2. (42,
29 Rcalling that θ h := v h + τ h, w gt that (38 holds. Thrfor, (11 follows from (38, (5, (10 and th Poincaré inquality applid to v h (rcall that v h Γc = 0 and Γ c > 0. Finally not that, du to th uniform shap rgularity of th lmnts K in {Ω h } h, it is asy to chck that th constant β 0 is uniformly boundd from abov indpndntly of th msh siz h. Proof of Proposition 2.1. In th following, C will dnot a gnric positiv constant indpndnt of h. Givn any pair (v h, τ h in X h, w dnot by w E = w h v h, γ E = γ h τ h and by w A = w v h, γ A = γ τ h. By applying first th corcivity Lmma 2.1 and thn using th linarity of th bilinar forms and th consistncy condition (12, w gt α γ E, w E 2 h a h ( w E + γ E, w E + γ E + t 2 (γ E, γ E = a h ( w A + γ A, w E + γ E + t 2 (γ A, γ E. (43 By dfinitions (8 and (2 and standard algbra w gt from (43 γ E, w E 2 h C T 1/2 A T 1/2 E + t 2 γ A 2 (Ω γ E 2 (Ω, (44 whr th scalar trms ar givn by T A = w A + γ A 2 H 1 (Ω + + h 1 w A + γ A 2 L 2 (, h Cε( w A + γ A n 2 L 2 ( and T E = w E + γ E 2 H 1 (Ω + h Cε( w E + γ E n 2 L 2 ( + w E + γ E 2 L 2 (. h 1 Trm T A can b boundd by using (27, as alrady don in (40. Without again showing th dtails, and following th sam notation for K introducd in (40, w gt T A C ( w A + γ A 2 H 1 (Ω + ( h 2 w A 2 H 3 (K + w A 2 H 2 (K + h 2 γ A 2 H 2 (K + γ A 2 H 1 (K + h 2 w A 2 H 1 (K + h 2 γ A 2 L 2 (K. From th abov bound, a triangl inquality, and th dfinition of w A, γ A, w gt T A C 2 ( j=0 h 2(j 1 K K Ω h w v h 2 H j+1 (K + 28 h 2(j 1 K K Ω h γ τ h 2 H j (K. (45
30 W now bound T E. Again, using th Agmon inquality (27 and invrs stimats as don in (40, w gt for all E h Γ c : h Cε( w E + γ E n 2 L 2 ( (h C 2 w E + γ E 2 H 2 (K + w E + γ E 2 H 1 (K C w E + γ E 2 H 1 (K. Combining th abov bound with th dfinition of T E and (10 yilds T E + t 2 γ E 2 L 2 (Ω C w E, γ E 2 h. (46 Now, rcalling (44 and using (46 w asily gt γ E, w E h C ( T 1/2 A + t 1 γ A 2 (Ω. Finally, rcalling that th abov inquality holds for all (v h, τ h X h, th bound in (45 concluds th proof. Proof of Proposition 2.2. W only sktch th proof. Estimat (16 immdiatly follows from (5, by rcalling (14 and (15. W prov (17 only for th cas of a simply supportd plat. In this cas, a h (, = a(,, bcaus Γ c is th mpty st. Howvr, w notic that diffrnt boundary conditions can b dalt with using a similar tchniqu. Lt s h and τ h b givn in Ξ h. Using (6 and (9, w gt (Q h s h, τ h = (Q h Q, τ h + (Q s h, τ h = a ( (w w h + (γ γ h, τ h + (Q sh, τ h. Choosing ψ h = h 2 (Q h s h, and using th invrs inquality w hav Q h s h H 1 (Ω Ch 1 Q h s h 2 (Ω, h 2 Q h s h 2 L 2 (Ω = h2 a ( (w w h + (γ γ h, (Q h s h Hnc, w obtain + h 2 (Q s h, Q h s h C w w h, γ γ h h h Q h s h 2 (Ω + h Q s h 2 (Ω h Q h s h 2 (Ω. h Q h s h 2 (Ω C w w h, γ γ h h + h Q s h 2 (Ω. Thrfor, th triangl inquality givs h Q Q h 2 (Ω C w w h, γ γ h h + 2h Q s h 2 (Ω, from which w infr h Q Q h 2 (Ω C w w h, γ γ h h + 2h inf Q s h 2 (Ω, s h Ξ h 29
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