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

Size: px
Start display at page:

Download "ICES REPORT 15-01. January 2015. The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712"

Transcription

1 ICES REPORT January 2015 A locking-fr modl for Rissnr-Mindlin 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 locking-fr modl for Rissnr-Mindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS," ICES REPORT 15-01, Th Institut for Computational Enginring and Scincs, Th Univrsity of Txas at Austin, January 2015.

2 A locking-fr modl for Rissnr-Mindlin 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 Rissnr-Mindlin 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 Poisson-Kirchhoff thory, bgan with th MS thsis of Papnfuss at th Univrsity of Washington in 1959 [24]. This four-nod 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 Rissnr-Mindlin. 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 dsign-through-analysis 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 T-splins, 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 dgrs-of-frdom, 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 Poisson-Kirchhoff thory, as anticipatd in [18]. With convnint, smooth basis functions providd by Isogomtric Analysis, th rol of Poisson-Kirchhoff thory is bing rconsidrd. A prim advantag is that no rotational dgrs-of-frdom 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 product-group 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 Rissnr-Mindlin 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 Rissnr-Mindlin 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 dgrs-of-frdom 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 Rissnr-Mindlin 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, non-uniform rational Powll-Sabin splins. W not in passing that Isogomtric Analysis has prviously bn applid to th standard displacmnt-rotation forms of th Rissnr-Mindlin plat and shll thoris [6, 8]. An outlin of th rmaindr of th papr follows: In Sction 2 w prsnt th Rissnr-Mindlin 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 Rissnr-Mindlin 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 Rissnr-Mindlin 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 so-calld 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 Rissnr-Mindlin 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 Rissnr-Mindlin 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 mathmatical-orintd 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 B-splins 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 lft-hand 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 non-vanishing 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 quasi-uniform. 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 locking-fr 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 lft-hand 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 tnsor-product NURBS-basd isogomtric analysis (Sction 3; triangular NURPS-basd 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 tnsor-product 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 tnsor-product 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 right-hand 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 quasi-uniform. 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, NURBS-basd isogomtric analysis is usd to prform numrical validations of th prsntd thory. W bgin with a brif summary of B-splins and NURBS (Non-Uniform Rational B-Splins. 3.1 B-splins and NURBS B-splins 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 B-splins ar C p 1 - continuous, and at a multipl knot of multiplicity k th continuity is rducd to C p k. Th B-splin 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 tnsor-product of th B-splin 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 locking-fr 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 h-rfinmnt 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 tnsor-product B-splins 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 tnsor-product B-splins 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 tnsor-product B-splins 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 tnsor-product 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 tnsor-product 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 NURPS-basd isogomtric discrtization to prform numrical validations. W first brifly summariz th construction and som proprtis of quadratic B-splins ovr a Powll-Sabin (PS rfinmnt of a triangulation and thir rational gnralization, th so-calld NURPS B-splins. Thn, th discrtizd modl is tstd on th sam xampls as dscribd in th prvious sction. 4.1 Quadratic PS and NURPS B-splins 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 Powll-Sabin 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 B-splin 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 Powll-Sabin (PS B-splins. Th PS B-splins 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 non-ngativ 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 B-splin lik basis, PS splins admit a straightforward rational xtnsion. A NURPS (Non-Uniform 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 B-splins 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 tnsor-product B-splin 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 B-splins. 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 tnsor-product 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 B-splins. 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 sub-optimal 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 Rissnr-Mindlin 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 vctor-valud and tnsor-valud 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 positiv-dfinitnss 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 positiv-dfinit, 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

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

More information

ME 612 Metal Forming and Theory of Plasticity. 6. Strain

ME 612 Metal Forming and Theory of Plasticity. 6. Strain Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

More information

QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

More information

(Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for the European Normal Black Scholes Formula) (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book. Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

More information

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS 25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

More information

The Normal Distribution: A derivation from basic principles

The Normal Distribution: A derivation from basic principles Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

More information

Adverse Selection and Moral Hazard in a Model With 2 States of the World

Adverse Selection and Moral Hazard in a Model With 2 States of the World Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

A Note on Approximating. the Normal Distribution Function

A Note on Approximating. the Normal Distribution Function Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

Architecture of the proposed standard

Architecture of the proposed standard Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th

More information

Introduction to Finite Element Modeling

Introduction to Finite Element Modeling Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

More information

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

More information

Statistical Machine Translation

Statistical Machine Translation Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

More information

Lecture 3: Diffusion: Fick s first law

Lecture 3: Diffusion: Fick s first law Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

More information

Question 3: How do you find the relative extrema of a function?

Question 3: How do you find the relative extrema of a function? ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

More information

Constraint-Based Analysis of Gene Deletion in a Metabolic Network

Constraint-Based Analysis of Gene Deletion in a Metabolic Network Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany

More information

Traffic Flow Analysis (2)

Traffic Flow Analysis (2) Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List. Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplete 2-Port Vector Network Analyzer Calibration Methods Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

More information

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

More information

Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing

Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif

More information

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000 hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

More information

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a high-lvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End

More information

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13) con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

More information

New Basis Functions. Section 8. Complex Fourier Series

New Basis Functions. Section 8. Complex Fourier Series Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

More information

Policies for Simultaneous Estimation and Optimization

Policies for Simultaneous Estimation and Optimization Policis for Simultanous Estimation and Optimization Migul Sousa Lobo Stphn Boyd Abstract Policis for th joint idntification and control of uncrtain systms ar prsntd h discussion focuss on th cas of a multipl

More information

AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 2008 Scoring Guidelines AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

More information

Lecture 20: Emitter Follower and Differential Amplifiers

Lecture 20: Emitter Follower and Differential Amplifiers Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

Entity-Relationship Model

Entity-Relationship Model Entity-Rlationship Modl Kuang-hua Chn Dpartmnt of Library and Information Scinc National Taiwan Univrsity A Company Databas Kps track of a company s mploys, dpartmnts and projcts Aftr th rquirmnts collction

More information

Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk. When speaking about forward or futures contracts, basis risk is the market Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

More information

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs

More information

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

More information

C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Reports with SAS C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

More information

Development of Financial Management Reporting in MPLS

Development of Financial Management Reporting in MPLS 1 Dvlopmnt of Financial Managmnt Rporting in MPLS 1. Aim Our currnt financial rports ar structurd to dlivr an ovrall financial pictur of th dpartmnt in it s ntirty, and thr is no attmpt to provid ithr

More information

MAXIMAL CHAINS IN THE TURING DEGREES

MAXIMAL CHAINS IN THE TURING DEGREES MAXIMAL CHAINS IN THE TURING DEGREES C. T. CHONG AND LIANG YU Abstract. W study th problm of xistnc of maximal chains in th Turing dgrs. W show that:. ZF + DC+ Thr xists no maximal chain in th Turing dgrs

More information

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among

More information

7 Timetable test 1 The Combing Chart

7 Timetable test 1 The Combing Chart 7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

More information

Planning and Managing Copper Cable Maintenance through Cost- Benefit Modeling

Planning and Managing Copper Cable Maintenance through Cost- Benefit Modeling Planning and Managing Coppr Cabl Maintnanc through Cost- Bnfit Modling Jason W. Rup U S WEST Advancd Tchnologis Bouldr Ky Words: Maintnanc, Managmnt Stratgy, Rhabilitation, Cost-bnfit Analysis, Rliability

More information

Performance Evaluation

Performance Evaluation Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism

More information

Parallel and Distributed Programming. Performance Metrics

Parallel and Distributed Programming. Performance Metrics Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

Deer: Predation or Starvation

Deer: Predation or Starvation : Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

More information

Production Costing (Chapter 8 of W&W)

Production Costing (Chapter 8 of W&W) Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary

More information

Sci.Int.(Lahore),26(1),131-138,2014 ISSN 1013-5316; CODEN: SINTE 8 131

Sci.Int.(Lahore),26(1),131-138,2014 ISSN 1013-5316; CODEN: SINTE 8 131 Sci.Int.(Lahor),26(1),131-138,214 ISSN 113-5316; CODEN: SINTE 8 131 REQUIREMENT CHANGE MANAGEMENT IN AGILE OFFSHORE DEVELOPMENT (RCMAOD) 1 Suhail Kazi, 2 Muhammad Salman Bashir, 3 Muhammad Munwar Iqbal,

More information

A Theoretical Model of Public Response to the Homeland Security Advisory System

A Theoretical Model of Public Response to the Homeland Security Advisory System A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial

More information

Budget Optimization in Search-Based Advertising Auctions

Budget Optimization in Search-Based Advertising Auctions Budgt Optimization in Sarch-Basd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY jonfld@googl.com Martin Pál Googl, Inc. Nw York, NY mpal@googl.com Intrnt sarch companis sll advrtismnt

More information

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009 Volum 3, Issu 1, 29 Statistical Approach for Analyzing Cll Phon Handoff Bhavior Shalini Saxna, Florida Atlantic Univrsity, Boca Raton, FL, shalinisaxna1@gmail.com Sad A. Rajput, Farquhar Collg of Arts

More information

Far Field Estimations and Simulation Model Creation from Cable Bundle Scans

Far Field Estimations and Simulation Model Creation from Cable Bundle Scans Far Fild Estimations and Simulation Modl Cration from Cabl Bundl Scans D. Rinas, S. Nidzwidz, S. Fri Dortmund Univrsity of Tchnology Dortmund, Grmany dnis.rinas@tu-dortmund.d stphan.fri@tu-dortmund.d Abstract

More information

Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost

Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 50-54 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin

More information

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST: .4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

More information

Electronic Commerce. and. Competitive First-Degree Price Discrimination

Electronic Commerce. and. Competitive First-Degree Price Discrimination Elctronic Commrc and Comptitiv First-Dgr Pric Discrimination David Ulph* and Nir Vulkan ** Fbruary 000 * ESRC Cntr for Economic arning and Social Evolution (ESE), Dpartmnt of Economics, Univrsity Collg

More information

On the moments of the aggregate discounted claims with dependence introduced by a FGM copula

On the moments of the aggregate discounted claims with dependence introduced by a FGM copula On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula - Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval - Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,

More information

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY 1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically

More information

An Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images

An Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images An Adaptiv Clustring MAP Algorithm to Filtr Spckl in Multilook SAR Imags FÁTIMA N. S. MEDEIROS 1,3 NELSON D. A. MASCARENHAS LUCIANO DA F. COSTA 1 1 Cybrntic Vision Group IFSC -Univrsity of São Paulo Caia

More information

SOFTWARE ENGINEERING AND APPLIED CRYPTOGRAPHY IN CLOUD COMPUTING AND BIG DATA

SOFTWARE ENGINEERING AND APPLIED CRYPTOGRAPHY IN CLOUD COMPUTING AND BIG DATA Intrnational Journal on Tchnical and Physical Problms of Enginring (IJTPE) Publishd by Intrnational Organization of IOTPE ISSN 077-358 IJTPE Journal www.iotp.com ijtp@iotp.com Sptmbr 015 Issu 4 Volum 7

More information

MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM. Hans G. Jonasson

MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM. Hans G. Jonasson MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM Hans G. Jonasson SP Tchnical Rsarch Institut of Swdn Box 857, SE-501 15 Borås, Swdn hans.jonasson@sp.s ABSTRACT Drum sound, that is th walking

More information

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final. Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

A Multi-Heuristic GA for Schedule Repair in Precast Plant Production

A Multi-Heuristic GA for Schedule Repair in Precast Plant Production From: ICAPS-03 Procdings. Copyright 2003, AAAI (www.aaai.org). All rights rsrvd. A Multi-Huristic GA for Schdul Rpair in Prcast Plant Production Wng-Tat Chan* and Tan Hng W** *Associat Profssor, Dpartmnt

More information

GOAL SETTING AND PERSONAL MISSION STATEMENT

GOAL SETTING AND PERSONAL MISSION STATEMENT Prsonal Dvlopmnt Track Sction 4 GOAL SETTING AND PERSONAL MISSION STATEMENT Ky Points 1 Dfining a Vision 2 Writing a Prsonal Mission Statmnt 3 Writing SMART Goals to Support a Vision and Mission If you

More information

STATEMENT OF INSOLVENCY PRACTICE 3.2

STATEMENT OF INSOLVENCY PRACTICE 3.2 STATEMENT OF INSOLVENCY PRACTICE 3.2 COMPANY VOLUNTARY ARRANGEMENTS INTRODUCTION 1 A Company Voluntary Arrangmnt (CVA) is a statutory contract twn a company and its crditors undr which an insolvncy practitionr

More information

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D 24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

More information

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives.

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives. Volum 3, Issu 6, Jun 2013 ISSN: 2277 128X Intrnational Journal of Advancd Rsarch in Computr Scinc and Softwar Enginring Rsarch Papr Availabl onlin at: wwwijarcsscom Dynamic Ranking and Slction of Cloud

More information

Factorials! Stirling s formula

Factorials! Stirling s formula Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

More information

User-Perceived Quality of Service in Hybrid Broadcast and Telecommunication Networks

User-Perceived Quality of Service in Hybrid Broadcast and Telecommunication Networks Usr-Prcivd Quality of Srvic in Hybrid Broadcast and Tlcommunication Ntworks Michal Galtzka Fraunhofr Institut for Intgratd Circuits Branch Lab Dsign Automation, Drsdn, Grmany Michal.Galtzka@as.iis.fhg.d

More information

A Project Management framework for Software Implementation Planning and Management

A Project Management framework for Software Implementation Planning and Management PPM02 A Projct Managmnt framwork for Softwar Implmntation Planning and Managmnt Kith Lancastr Lancastr Stratgis Kith.Lancastr@LancastrStratgis.com Th goal of introducing nw tchnologis into your company

More information

Quantum Graphs I. Some Basic Structures

Quantum Graphs I. Some Basic Structures Quantum Graphs I. Som Basic Structurs Ptr Kuchmnt Dpartmnt of Mathmatics Txas A& M Univrsity Collg Station, TX, USA 1 Introduction W us th nam quantum graph for a graph considrd as a on-dimnsional singular

More information

Current and Resistance

Current and Resistance Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9

More information

81-1-ISD Economic Considerations of Heat Transfer on Sheet Metal Duct

81-1-ISD Economic Considerations of Heat Transfer on Sheet Metal Duct Air Handling Systms Enginring & chnical Bulltin 81-1-ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv

More information

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 08-16-85 WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 Summary of Dutis : Dtrmins City accptanc of workrs' compnsation cass for injurd mploys; authorizs appropriat tratmnt

More information

Foreign Exchange Markets and Exchange Rates

Foreign Exchange Markets and Exchange Rates Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

Section 7.4: Exponential Growth and Decay

Section 7.4: Exponential Growth and Decay 1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

Efficiency Losses from Overlapping Economic Instruments in European Carbon Emissions Regulation

Efficiency Losses from Overlapping Economic Instruments in European Carbon Emissions Regulation iscussion Papr No. 06-018 Efficincy Losss from Ovrlapping Economic Instrumnts in Europan Carbon Emissions Rgulation Christoph Böhringr, Hnrik Koschl and Ulf Moslnr iscussion Papr No. 06-018 Efficincy Losss

More information

Free ACA SOLUTION (IRS 1094&1095 Reporting)

Free ACA SOLUTION (IRS 1094&1095 Reporting) Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 279-1062 ACA Srvics Transmit IRS Form 1094 -C for mployrs Print & mail IRS Form 1095-C to mploys HR Assist 360 will gnrat th 1095 s for

More information

Optimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress

Optimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress Shock and Vibration 9 (202) 445 46 445 DOI 0.3233/SAV-202-0685 IOS Prss Optimization dsign of structurs subjctd to transint loads using first and scond drivativs of dynamic displacmnt and strss Qimao Liu

More information

Gold versus stock investment: An econometric analysis

Gold versus stock investment: An econometric analysis Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 268-8662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag -7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin

More information

Category 7: Employee Commuting

Category 7: Employee Commuting 7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

Data warehouse on Manpower Employment for Decision Support System

Data warehouse on Manpower Employment for Decision Support System Data warhous on Manpowr Employmnt for Dcision Support Systm Amro F. ALASTA, and Muftah A. Enaba Abstract Sinc th us of computrs in businss world, data collction has bcom on of th most important issus du

More information

In the previous two chapters, we clarified what it means for a problem to be decidable or undecidable.

In the previous two chapters, we clarified what it means for a problem to be decidable or undecidable. Chaptr 7 Computational Complxity 7.1 Th Class P In th prvious two chaptrs, w clarifid what it mans for a problm to b dcidabl or undcidabl. In principl, if a problm is dcidabl, thn thr is an algorithm (i..,

More information

An International Journal of the Polish Statistical Association

An International Journal of the Polish Statistical Association STATISTICS IN TRANSITION nw sris An Intrnational Journal of th Polish Statistical Association CONTENTS From th Editor... Submission information for authors... 5 Sampling mthods and stimation CIEPIELA P.,

More information

Expert-Mediated Search

Expert-Mediated Search Exprt-Mdiatd Sarch Mnal Chhabra Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA chhabm@cs.rpi.du Sanmay Das Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA sanmay@cs.rpi.du David

More information

Meerkats: A Power-Aware, Self-Managing Wireless Camera Network for Wide Area Monitoring

Meerkats: A Power-Aware, Self-Managing Wireless Camera Network for Wide Area Monitoring Mrkats: A Powr-Awar, Slf-Managing Wirlss Camra Ntwork for Wid Ara Monitoring C. B. Margi 1, X. Lu 1, G. Zhang 1, G. Stank 2, R. Manduchi 1, K. Obraczka 1 1 Dpartmnt of Computr Enginring, Univrsity of California,

More information

Finite Elements from the early beginning to the very end

Finite Elements from the early beginning to the very end Finit Elmnts from th arly bginning to th vry nd A(x), E(x) g b(x) h x =. x = L An Introduction to Elasticity and Hat Transfr Applications x Prliminary dition LiU-IEI-S--8/535--SE Bo Torstnflt Contnts

More information

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

More information

the so-called KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through

the so-called KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through Liquidity and Information-Basd Trading on th Ordr Drivn Capital Markt: Th Cas of th Pragu tock Exchang Libor 1ÀPH³HN Cntr for Economic Rsarch and Graduat Education, Charls Univrsity and Th Economic Institut

More information

Examples. Epipoles. Epipolar geometry and the fundamental matrix

Examples. Epipoles. Epipolar geometry and the fundamental matrix Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon

More information

Key Management System Framework for Cloud Storage Singa Suparman, Eng Pin Kwang Temasek Polytechnic {singas,engpk}@tp.edu.sg

Key Management System Framework for Cloud Storage Singa Suparman, Eng Pin Kwang Temasek Polytechnic {singas,engpk}@tp.edu.sg Ky Managmnt Systm Framwork for Cloud Storag Singa Suparman, Eng Pin Kwang Tmask Polytchnic {singas,ngpk}@tp.du.sg Abstract In cloud storag, data ar oftn movd from on cloud storag srvic to anothr. Mor frquntly

More information

The Constrained Ski-Rental Problem and its Application to Online Cloud Cost Optimization

The Constrained Ski-Rental Problem and its Application to Online Cloud Cost Optimization 3 Procdings IEEE INFOCOM Th Constraind Ski-Rntal Problm and its Application to Onlin Cloud Cost Optimization Ali Khanafr, Murali Kodialam, and Krishna P. N. Puttaswam Coordinatd Scinc Laborator, Univrsit

More information

METHODS FOR HANDLING TIED EVENTS IN THE COX PROPORTIONAL HAZARD MODEL

METHODS FOR HANDLING TIED EVENTS IN THE COX PROPORTIONAL HAZARD MODEL STUDIA OECONOMICA POSNANIENSIA 204, vol. 2, no. 2 (263 Jadwiga Borucka Warsaw School of Economics, Institut of Statistics and Dmography, Evnt History and Multilvl Analysis Unit jadwiga.borucka@gmail.com

More information

A Graph-based Proactive Fault Identification Approach in Computer Networks

A Graph-based Proactive Fault Identification Approach in Computer Networks A Graph-basd Proacti Fault Idntification Approach in Computr Ntworks Yijiao Yu, Qin Liu and Lianshng Tan * Dpartmnt of Computr Scinc, Cntral China Normal Unirsity, Wuhan 4379 PR China E-mail: yjyu, liuqin,

More information

Important Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2

Important Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2 Installation and Opration Intrnt Tlphony Adaptr Aurswald Box Indx C I R 884264 03 02/05 Call Duration, maximum...10 Call Through...7 Call Transportation...7 Calls Call Through...7 Intrnt Tlphony...3 two

More information

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore Intrnational Journal of mrging Tchnology and dvancd nginring Wbsit: www.ijta.com (ISSN 2250-2459, Volum 2, Issu 4, pril 2012) n road outlin of Rdundant rray of Inxpnsiv isks Shaifali Shrivastava 1 partmnt

More information

CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER 4c. ROOTS OF EQUATIONS CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

More information

Epipolar Geometry and the Fundamental Matrix

Epipolar Geometry and the Fundamental Matrix 9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos.

More information

IHE IT Infrastructure (ITI) Technical Framework Supplement. Cross-Enterprise Document Workflow (XDW) Trial Implementation

IHE IT Infrastructure (ITI) Technical Framework Supplement. Cross-Enterprise Document Workflow (XDW) Trial Implementation Intgrating th Halthcar Entrpris 5 IHE IT Infrastructur (ITI) Tchnical Framwork Supplmnt 10 Cross-Entrpris Documnt Workflow (XDW) 15 Trial Implmntation 20 Dat: Octobr 13, 2014 Author: IHE ITI Tchnical Committ

More information

Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il

More information

SPECIFIC HEAT AND HEAT OF FUSION

SPECIFIC HEAT AND HEAT OF FUSION PURPOSE This laboratory consists of to sparat xprimnts. Th purpos of th first xprimnt is to masur th spcific hat of to solids, coppr and rock, ith a tchniqu knon as th mthod of mixturs. Th purpos of th

More information

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog

More information