Isogeometric Analysis of Soil Plasticity

Transcription

1 Gomatrials, 207, 7, 96-6 htt:// ISSN Onlin: ISSN Print: Isogomtric Analysis of Soil Plasticity Alx Stz *, Erika Tudisco, Ralf Dnzr 2, Ola Dahlblom Gotchnical Enginring, Dartmnt of Construction Scincs, Faculty of Enginring, Lund Univrsity, Lund, Swdn 2 Solid Mchanics, Dartmnt of Construction Scincs, Faculty of Enginring, Lund Univrsity, Lund, Swdn How to cit this ar: Stz, A., Tudisco, E., Dnzr, R. and Dahlblom, O. (207) Isogomtric Analysis of Soil Plasticity. Gomatrials, 7, htts://doi.org/0.4236/gm Rcivd: Fbruary 2, 207 Acctd: July 25, 207 Publishd: July 28, 207 Coyright 207 by authors and Scintific Rsarch Publishing Inc. This work is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY 4.0). htt://crativcommons.org/licnss/by/4.0/ On Accss Abstract In this ar w rsnt numrical simulations of soil lasticity using isogomtric analysis comaring th rsults to th solutions from convntional finit lmnt mthod. Isogomtric analysis is a numrical mthod that uss nonuniform rational B-slins (NURBS) as basis functions instad of th Lagrangian olynomials oftn usd in th finit lmnt mthod. Ths functions hav a highr-ordr of continuity, making it ossibl to rrsnt comlx gomtris xactly. Aftr a brif outlin of th thory bhind th isogomtric conct, w giv a rsntation of th constitutiv quations, usd to simulat th soil bhavior in this work. Th ar concluds with numrical xamls in two- and thr-dimnsions, which assss th accuracy of isogomtric analysis for simulations of soil bhavior. Th numrical xamls rsntd show, that for draind soils, th rsults from isogomtric analysis ar ovrall in good agrmnt with th convntional finit lmnt mthod in two- and thr-dimnsions. Thus isogomtric analysis is a good altrnativ to convntional finit lmnt analysis for simulations of soil bhavior. Kywords Isogomtric Analysis, NURBS, Numrical Mthods, Soil Plasticity. Introduction In th dsign of foundations and gotchnical structurs it is ssntial to rdict soil bhavior undr diffrnt loading conditions. In th last dcad finit lmnt analysis (FEA) has bcom a widly srad tool for rdicting soil bhavior. Much rsarch has bn carrid out to imrov th ability of simulating th bhavior of diffrnt soils and a numbr of nw constitutiv modls hav bn dvlod. Anothr imortant asct whn modling gotchnical roblms is th intraction btwn soil and structur, which can hav a larg influnc on th structural dsign []. DOI: /gm July 28, 207

2 A. Stz t al. Sinc first introducd by Hughs t al. in [2], isogomtric analysis (IGA) has bn usd for a broad numbr of nginring roblms, ranging from analysis of fluid-structur intraction [3] to analysis of shlls [4]. Som of th aras whr isogomtric analysis has shown to b advantagous comard to th standard finit lmnt analysis ar also of intrst in gotchnics. On xaml is th simulation of fluid flow through orous mdia, whr th highr-ordr basis functions of IGA lad to continuous rssur gradints ovr lmnt boundaris nsuring local mass consrvation, [5] and [6]. Th basic ida bhind isogomtric analysis is to us slins as basis functions for comutational analysis [2]. Originally th uros was to us th gomtry from th Comutr Aidd Gomtric Dsign (CAGD) softwar s also for analysis and consquntly rduc th tim rquird to rcrat modls for analysis. Howvr, th us of slin basis functions rovd to hav numrical bnfits comard to th standard Lagrangian basis-functions usd in FEA [7] [8]. Initially th IGA framwork suffrd from th lack of a mthod for local rfinmnt, frquntly sought during analysis. To ovrcom this initial drawback, a numbr of mthods hav bn dvlod to introduc local rfinmnts within th IGA framwork, whr T-slins [9] or Hirarchical NURBS [7] [0] ar th most widly usd mthods. Th highrordr smoothnss of th basis functions in IGA also ons u th ossibilitis of using quadratur ruls valuatd on lmnt boundaris rducing th total numbr of quadratur oints [2] [7]. Anothr intrsting ossibility within th IGA framwork is th us of isogomtric collocation mthods, a on-oint quadratur rul that rducs th comutational cost of analysis []. Sinc collocation mthods ar basd on th discrtization of th strong form of th artial diffrntial quations, it would b suitabl for th isogomtric conct. Th us of isogomtric collocation has shown otntial to incras th comutational fficincy of th isogomtric framwork, outrforming both th standard Galrkin form of isogomtric analysis and standard C 0 finit lmnt analysis with rgard to comutational costs [7]. As ointd out by Nguyn t al. [2] for nonlinar analysis th numrical rror strongly dnds on th intgration schm and th ordr of th Non-Uniform Rational B-slins (NURBS) basis functions. To valuat th isogomtric framwork for gotchnical alications, it is intrsting to assss how th isogomtric framwork rforms for soil lasticity. For simlicity and for stting u th numrical framwork w rstrict our slf to quadratic NURBS basis functions in this work. As soil tyically show strong lastic flow undr hydrostatic rssur, scial B-bar mthods ar not ncssary in our work, s [3] for a discussion of ths mthods in th cas of narly incomrssibl matrial bhavior. In this work w hav valuatd how isogomtric analysis rforms comard to th convntional finit lmnt mthod for soil lasticity in two and thr dimnsions. Th focus of th work is to valuat th convrgnc bhavior and msh siz dndncis for isogomtric analysis and to comar thm with rsults from convntional FEA. To rovid a background, w start with outlining th basic concts of isogomtric analysis, introducing th dfinitions of B- 97

3 A. Stz t al. slins and NURBS that ar usd as basis or sha functions in this work. W latr continu with th basic govrning quations for lasto-lasticity followd by a brif rcaitulation of th Druckr-Pragr constitutiv quations. Th ar is concludd with two- and thr-dimnsional numrical xamls, comaring th rsults to convntional finit lmnt analysis. 2. Isogomtric Analysis Th fundamntal ida bhind IGA is to mloy th sam basis functions for both gomtrical discrtization and analysis. Hrin lays also th most rofound diffrnc btwn IGA and standard FEA, whr isogomtric analysis utilizs th basis functions from CAGD, caabl of rrsnting th xact gomtry also for analysis. Whras, in convntional finit lmnt analysis, th icwis olynomials chosn to aroximat th solution filds ar also usd to aroximat th gomtry [2]. Th original rason to us th isogomtric analysis was to dcras th ovrall comutational cost by rmoving th nd to rcrat th gomtry and constructing a msh for analysis. In this work w hav analyzd th rformanc of IGA for soil lasticity using a NURBS-basd isogomtric formulation. To giv a brif introduction to th conct of IGA and to lucidat som of th diffrncs btwn convntional FEA and IGA this sction will rviw th basic concts of isogomtric analysis. For a mor xtnsiv dscrition th radr is rfrrd to [2]. 2.. Basic Conct of B-Slins To gt an ovrviw of th conct of NURBS-basd isogomtric framwork usd in this work w start by dfining a B-slin curv. A B-slin curv is a linar combination of B-slin basis functions, Ni,, i =, 2,, n and a st of d corrsonding control oints P i, i =, 2,, n. n ( ξ) N ( ξ) C = i, P i. () i= Th B-slin basis functions N i, ar constructd from a non-dcrasing st of coordinats in th aramtr sac, writtn as Ξ= { ξ, ξ2,, ξ n + + }, whr ξi dnots th ith knot of th knot vctor Ξ, indicats th ordr of th olynomial function and n rrsnts th numbr of basis functions ndd to construct a scific B-slin curv. With a givn knot vctor, ξ i, and a known olynomial ordr,, it is ossibl to construct a B-slin basis function. For th cas of ( = 0), th basis function tak a icwis constant sha givn from N i,0 ( ξ ) if ξ ξ < ξ, 0 othrwis. i i+ = (2) For =, 2, 3, th B-slin basis function can b constructd by using th xrssion, ξ ξ ξ ξ N N N i i+ + ( ξ) = ( ξ) + ( ξ) i, i, i+, ξi+ ξi ξi+ + ξi+ (3) 98

4 A. Stz t al. which is rfrrd to as th Cox-d Boor formula, first rsntd in [4] and [5]. On xaml of a scond ordr B-slin with thr control oints is illustratd in Figur. For olynomial dgrs < 2 th basis functions of IGA and convntional FEA tak th sam sha, howvr for 2 thy dviat in sha and suort. By lotting th B-slin basis functions and Lagrangian basis functions ovr th aramtr sac som of th diffrncs btwn IGA and convntional FEA ar mad visibl, s Figur 2 and Figur 3. Figur 3 shows that for 2, Lagrangian basis functions, th continuity varis btwn intrnal and cornr- or nd-nods, whras B-slin basis functions show a continuous and homognous attrn only shiftd rlativ to ach othr and with N i, (ξ) 0. Anothr imortant asct of slin basis functions, that distinguishs thm from convntional FEA basis functions, is that ach th ordr function has continuous drivativs across lmnt boundaris. Figur. A scond ordr B-slin curv, C(ξ). Figur 2. Rrsntation of quadratic B-slin basis function with knot vctor Ξ = {0, 0, 0,, 2, 3, 4, 5, 5, 5}. 99

5 A. Stz t al. Figur 3. Rrsntation of quadratic Lagrangian basis function NURBS Rrsntation On issu that ariss whn using B-slin, is that not all tys of gomtris can b rrsntd xactly using olynomial functions. To ovrcom this roblm rational B-slins wr introducd by Vrsrill in [6]. This gnralization of B- slins is constructd by introducing a non-ngativ wight, w i, to ach control oint and making us of th dfinition of rational functions as th ratio of two olynomials [7]. A NURBS body in Rd can b obtaind by rojctiv transformation of a B-slin body in Rd +, whr th wights, w, i ar th d + comonnts of rojctiv control oints. Non-Uniform Rational B-slins (NURBS) ar today standard in many CAGD softwar s and th fast and stabl algorithms mak thm a good choic also for analysis. To construct NURBS basis functions on can mak us of th basis functions for B-slins, R i ( ξ ) ( ξ ) ( ξ ) whr W(ξ) is calld a wighting function, dfind as Ni, wi = (4) W n ( ξ) ( ξ) W = N w. (5) iˆ = iˆ, If w i = for all i, thn Ri ( ξ) Ni ( ξ) for all i thn R ( ξ) = N ( ξ) i.., th N ( ξ ) ar scial cass of R ( ξ ) i i iˆ = for all i. In fact, if th valu of w i = a whr all th wights tak th sam valu. Using th NURBS basis functions Ri ( ξ ), i=, 2,, n and thir corrsonding control oints P, i a icwis NURBS curv, is constructd from n ( ξ) R ( ξ) i= i C = i P i. (6) To b abl to rform two- and thr-dimnsional analysis, NURBS surfacs and bodis nds to b dfind. This is don in a similar mannr. A NURBS body i 00

6 A. Stz t al. is givn by n m l qr ( ξηζ,, ) = R,, i jk ( ξηζ,, ) S P (7) i= j= k=,, i, jk, R ξηζ, is constructd from thr sts W ξηζ. qr,, whr th NURBS basis function, i, jk, (,, ) of knot vctors, and thir rsctiv wighting functions (,, ) N,, i, ( ξ) M jq, ( η) Lkr, ( ζ ) w qr i, jk, Ri, jk, ( ξηζ,, ) W ( ξηζ,, ) whr n m l = (8) ( ξηζ,, ) ( ξ) ( η) ( ζ) W = N M L w. (9) iˆ = ˆj= kˆ = iˆ, ˆjq, kr ˆ, iˆ, ˆjk, ˆ In th isogomtric framwork th conct of lmnts is rrsntd by th non-zro valud knot sans. This is illustratd in Figur 4, whr a two dimnsional lat is constructd from two knot vctors, a st of control oints P i,j and thir corrsonding wights w i,j. Th gomtry is constructd of on knot vctor, Ξ = {0, 0, 0, 0.5,,, } in th dirction of th arc of th hol and on knot vctor, = {0, 0, 0,,, }, in th radial dirction, thus crating a lat consisting of two lmnts. Th lmnt scific basis functions ar constructd from th nonzro valud basis functions in th activ knot san. Th numbr of activ functions in a knot san is dtrmind as n = ( + ) ( q+ ) ( r+ ). For analysis th basis functions ar valuatd at th chosn intgration oints of th arnt lmnt, s Figur 4(d). Ths lmnt scific NURBS basis functions will b dnotd Na ( ξηζ,, ) in th rmindr of this work. Figur 4. A NURBS surfac of th symmtric art of a lat with a circular hol constructd from two knot vctors, Ξ and, a st of control oints, P i,j and thir corrsonding wights, w i,j. (a) shows th lmnt msh (b) th control nt (c) th aramtr sac and basis functions (d) th arnt lmnt. 0

7 A. Stz t al. 3. Equilibrium Conditions (Hading 3) 3.. Strong and Wak Form In this sction w giv a brif rcaitulation of th strong and wak form of th quilibrium quations of th quasi-static balanc of linar momntum. Lt u i dnot th dislacmnt vctor, thn th infinitsimal strain tnsor, ε ij, is dfind as th symmtric art of th dislacmnt gradint u u i j εij = + in Ω (0) 2 xj x i Th govrning strong form of th quilibrium quation is givn as σ ij + fi = 0 in Ω x j whr f i is th body forc. Th strong form of th quilibrium quation is comlmntd by th ssntial and natural boundary conditions. u = g on Ω (2) i i g () σ n = h on Ω (3) ij j i h whr g i and h i ar known quantitis on Ω g and Ω h. Th variational or wak form of Equation () is formd by multilying th govrning quation with a tst function v i and rforming an intgration by arts ovr th domain Ω vi ijd vi fid vh i id A. x σ Ω= Ω+ Ω Ω (4) Ωh j Th wak form of th roblm is comlmntd by a constitutiv rlation ij ij kl (,{ }) σ = σ ε κ (5) with { κ } bing a st of intrnal variabls to dscrib th lastic bhavior of th matrial. For a mor dtaild drivation of th wak form, th radr is tntativly rfrrd to any of [8] or [9] Discrtization with Isogomtric Analysis Th main diffrnc btwn convntional FEA and IGA ar th basis functions usd for discrtization. In isogomtric analysis th sam basis functions that ar usd to discrtiz th gomtry ar also usd to solv for th aroximat dislacmnt fild u. Th only diffrnc to convntional FEA is that th basis functions in IGA ar lmnt-scific. Aftr solving th lmnt-scific basis functions and thir drivativs th rocdur of stablishing th stiffnss matrix and intrnal forc vctor is idntical to convntional FEA. Th lmnt scific basis function for ach lmnt is dtrmind as th non-zro functions in th knot san. Th dislacmnt fild for any givn lmnt can b solvd using th lmnt scific basis function N ξηζ,,, a ( ) n a= ( ξηζ,, ) u= Na a (6) a 02

8 A. Stz t al. with a a th lmnt dislacmnts. Th satial drivativs of th dislacmnt fild can b aroximatd by taking th drivativ of th lmnt-scific basis functions with rsct to th hysical coordinats x u= n a= Na a x To obtain th drivativs of th basis functions with rsct to th hysical coordinats on must us th chain rul, with ξ = { ξηζ,, } T Na Na ξ = x ξ x a. (7) (8). For a mor straightforward imlmntation w rwrit th dislacmnt fild using a vctor-matrix notation, i.., u= Na. Whr th matrix N contains th basis functions Na ( ξηζ,, ) for ach control oint in suort of an lmnt N 0 0 N 0 0 n N = 0 N 0 0 Nn 0 (9) 0 0 N 0 0 N n In a similar mannr w rwrit Equation (7) in matrix form, ε = Ba (20) whr th matrix B is an orator maing th lmnt discrt dislacmnts to th local strains. N N n x x N N n x2 x 2 N N n x3 x3 B = N N N n N n 0 0 x2 x x2 x N N N n N n 0 0 x3 x x3 x N N N n N n 0 0 x3 x2 x3 x2 (2) Th tst function v i can b discrtizd in th sam mannr as th dislacmnt fild. Introducing th rlations abov it is ossibl to rwrit Equation (4) in a rsidual format as T T T ( ) ra = Bσ dv NtdS NfdV = 0 (22) V S V with a th global unknown dislacmnt vctor. In this work w us a Nwton-Rahson mthod to solv this systm of nonlinar quations. Th Jaco- 03

9 A. Stz t al. bian matrix for th Nwton-Rahson mthod rads r = = a T J BD V atsb dv (23) with D ats th Voigt rrsntation of th algorithmic tangnt stiffnss. In ach Nwton itration w comut a dislacmnt incrmnt a by solving th linar quation J a= r( a). Aftrwards, w udat th dislacmnt vctor by a a + a and sto th itration if th rsidual r(a) < TOL with TOL a usr givn tolranc,.g. TOL = Constitutiv Modl (Hading 4) In this sction w will giv a brif rcaitulation of th Druckr-Pragr critrion and th thory of lasticity usd to valuat th influnc of th NURBS-basis functions on th lastic strains for granular matrials in this work. 4.. Druckr-Pragr Critrion Th Druckr-Pragr critrion can b sn as a smooth aroximation to th Mohr-Coulomb law and stats that lastic yilding bgins whn th J 2 and I invariants rach a critical combination. Although th Druckr-Pragr formulation is a rathr crud aroximation of ral soil bhavior it has th bnfit of bing straightforward to imlmnt and also lacks th singularitis that xist in th yild function of th Mohr-Coulomb critrion. Th yild function of th Druckr-Pragr can b xrssd using th first invariant of th strss tnsor, I, and th scond invariant of th dviatoric strss, J 2, I ( I, J2) = J2 + α k (24) 3 which forms a circular con in th rincial strss sac. Th matrial aramtrs k and αα, can b xrssd in trms of th matrials cohsion, c and intrnal friction, φ, by matching th Druckr-Pragr critrion to th Mohr- Coulomb critrion, sinφ c α α = and k = sinφ sinθ tanφ cosθ ± 3 whr θ in rrsnts th Lod angl. A ositiv sign in Equation (25) matchs th Druckr-Pragr con to th innr dgs of th Mohr-Coulomb surfac. Th matching to th Mohr-Coulomb critrion is illustratd in Figur Elasto-Plastic Constitutiv Modl W assum a small strain stting and thus additivly slit th strain tnsor ε into an lastic art ε and a lastic art ε as (25) ε = ε + ε. (26) For go-matrials in gnral, associativ flow ruls for th lastic strain ε dislays an xcssiv dilatant bhavior. This can b avoidd by using a non- 04

10 A. Stz t al. Figur 5. Projction of th Druckr-Pragr critrion matchd to th Mohr-Coulomb critrion on th dviator lan. associativ flow rul, that is, th otntial function is not qual to th yild function. In this work th otntial function is dfind by rlacing th angl of intrnal friction, φ, with th angl of dilation, ψ, in Equation (25) forming sinψ c α α = and k =. (27) sinψ sinθ tanψ cosθ ± 3 Th otntial function is thn writtn as I ( I, J2) = J2 + α k. (28) 3 At th surfac of th Druckr-Pragr yild function th volution of th lastic strain is dtrmind as, εij = λ (29) σ with λ th lastic multilir and th drivativ of th otntial function is givn from = sij + δα ij (30) σ 2 J 3 ij with S ij th dviatoric art of th strss tnsor. Furthrmor w assum a bi-linar hardning modl for th matrial cohsion c and th angl of intrnal friction φ, as dictd in Figur 6. Both hardning aramtrs ar drivn by th accumulatd dviatoric lastic strain E d givn by Ed = ij ij with ij = εij εkkδij (3) 2 3 Th rsntd modl is comlmntd by th loading/unloading conditions 0, λ 0 and λ = 0. With th volution quation for th lastic strain and th just dscribd hardning modl at hand w gt th st of intrnal variabls 2 ij 05

11 A. Stz t al. Figur 6. Bi-linar strain dndnt hardning. for our lasticity modl as { κ} { ε, c, φ } =, s also Equation (5). For th numrical intgration of th constitutiv volution quations w us a standard Eulr backward schm. Th radr is rfrrd to ([20], Sc. 7.2) which givs a dtaild dscrition of th numrical imlmntation rocdur and also th format of th algorithmic tangnt stiffnss D ats. W lik to mntion that at th ax of th yild surfac con, th rturn vctor is containd by a comlmntary con, illustratd in Figur 7, s [20] for furthr dtails. Th IGA and FEA as wll as th lasto-lastic Druckr-Pragr modl discussd abov hav bn imlmntd in our in-hous Fortran cod. 5. Numrical Studis To validat th rformanc of isogomtric analysis for soil-lasticity, thr numrical bnchmark modls hav bn stablishd. Th modls hav all bn simulatd for soils in saturatd conditions using th Druckr-Pragr critrion. Th modls valuatd consist of on two-dimnsional modl of a stri footing and two thr-dimnsional cylindrical soil rofils subjctd to a rscribd forc and rscribd dislacmnt rsctivly. 06

12 A. Stz t al. 5.. Two-Dimnsional Study A two-dimnsional modl of a stri footing on sandy silt has bn analyzd. Du to th symmtric rortis of th roblm th analysis contains only half of th footing. Th gomtry and boundary conditions ar shown in Figur 8. To modl th load from a flxibl footing a vrtical rssur of 80 kpa is alid in 50 qual sts. Th roblm is solvd for lan strain conditions using quadratic NURBSbasd IGA and convntional FEA with 5 diffrnt mshs. In ordr to comar th two mthods th lmnt mshs hav bn constructd using quadratic isoaramtric lmnts for both IGA and convntional FEA. Th lmnt siz rangs from a coars msh with lmnts of 2 2 mtrs down to a msh with lmnts of mtrs. Th msh data and dgrs of frdom for ach msh ar shown in Tabl. Th matrial aramtrs of th soil in th two dimnsional modl ar shown in Tabl 2. Figur 9 shows th ground surfac dislacmnts undr th flxibl footing. To mak th comarison clarr, w comar th rsults at two oints along th ground surfac. Th first oint, A, is lacd at th cntr of th footing and th scond oint, B, is lacd at th dg of th footing, s also Figur 8. It can b sn that th dislacmnts ar in good agrmnt at cntr of th footing but Figur 7. Illustration of rturn to th ax of th yild surfac for th Druckr-Pragr critrion. Figur 8. Gomtry and boundary conditions for th flxibl footing modl. 07

13 A. Stz t al. Tabl. Msh data for th two-dimnsional simulations. min. l. siz [m] nl ndofs (FEA) ndofs (IGA) Tabl 2. Matrial aramtrs for th soil. Matrial Paramtr Valu [Unit] Young s modulus, E 00 [Ma] Poisson s ratio, ν Cohsio, Cohsio, c in 20 [kpa] c 23 [kpa] Angl of intrnal friction, Angl of intrnal friction, φ in 20 [dg] φ 22 [dg] angl of dilatation, ψ 5 [dg] ) Accumulatd lastic strai, ( E d Figur 9. Ground surfac dislacmnts undrnath along th loading surfac. 08

14 A. Stz t al. that thr is a minor diffrnc btwn th dislacmnts from th isogomtric- and convntional finit lmnt analysis at th dg of th footing (x = 2). Figur 0 dislays th load/dislacmnt rsons in oint A and B for th finst msh. By studying th load/dislacmnt rsons in Figur 0, th diffrncs btwn th IGA and FEA rsults in oint B bcom mor aarnt. Th rsults for oint B showd in Figur 0 ar of scial intrst. Th NURBS basis functions usd in th isogomtric analysis rovids continuous strss filds at th dg of th footing, whr discontinuous strsss can b xctd. Th continuous strss will in turn affct th dvlomnt of lastic strains at th dg of th footing. Th ffcts of th continuous strss filds that rsult from th NURBS basis functions can b sn in Figur, whr th volution of th lastic zon undr th footing is dislayd. From load sts in Figur it can b sn that th initial lastic strains ar slightly diffrnt at th dg of th footing in th convntional finit lmnt analysis comard to th isogomtric analysis. Th diffrnc in lastic strains at th dg of th footing corrsonds to th divrging dislacmnts sn in Figur 9 and Figur 0. For th isogomtric analysis th lastic strains originat undr th cntr of th footing and rachs th dg of th footing during incrasd loading. In th finit lmnt analysis, howvr, th lastic strain originats both undr th cntr of th footing and at th dg of th footing simultanously. Studying th dislacmnts in oint A and B for ach lmnt msh, w comar th msh siz dndncis in Figur 2. Th figur show th dviation of th dislacmnt for ach msh comard to th Figur 0. Load/dislacmnt rsons at oint A and B for th finst msh (4800 lmnts). 09

15 A. Stz t al. Figur. Evolution of th ffctiv lastic strains. Th ffctiv lastic strains from th init lmnt analysis and isogomtric analysis at six diffrnt load sts. Figur 2. Comarison of th convrgnc of th dislacmnt btwn FEA and IGA, for oint A (a) and for oint B (b). rsults from th finst msh usd. Th grahs show that th convrgnc for th isogomtric analysis and th convntional finit lmnt analysis ar comarabl Thr-Dimnsional Studis Th two thr-dimnsional studis in this work ar comosd of a cylindrical soil rofil with a 2: hight/diamtr roortion. In th first xaml, th cylindrical soil rofil is subjctd to a rscribd dislacmnt at th to of th cylindr, in th scond study th soil rofil is subjctd to a confining rssur and an incrasd vrtical load at th to of th cylindr. Th gomtry and boundary conditions of th thr-dimnsional xamls ar rsntd in Figur 3. In both xamls, a rfct lasto-lastic Druckr-Pragr critrion, matchd to th comrssiv mridian of th Mohr-Coulomb critrion has bn usd to modl soil lasticity. Th matrial rortis for th soil in both studis ar givn in Tabl 3. For both IGA and convntional FEA 27 nod isoaramtric lmnts hav bn usd to modl th cylindr. Th msh data usd ar rsntd in Figur 4. Using NURBS aramtrization an xact cylindr can b modld. Whras, for convntional FEA th gomtry dnds on th dnsity of th msh. 0

16 A. Stz t al. Tabl 3. Matrial aramtrs for th soil in thr-dimnsional studis. Matrial Paramtr Valu [Unit] Young s modulus, E 000 [ka] Poisson s ratio, ν Cohsio, c 5.5 [kpa] Angl of intrnal friction, ϕ 0 [dg] angl of dilatation, ψ 5 [dg] Figur 3. Boundary conditions for th thr-dimnsional modls. Th dislacmnt-controlld analysis is fixd in th bottom of th cylindr and has no horizontal constraints at th to (a). In th load controlld analysis th bottom of th cylindr is assumd to b fixd (b). Figur 4. Th lft hand sid shows a horizontal viw of th finit lmnt msh and on th right hand sid shows th convntional FEA (to) and IGA (bottom) mshs viwd from th to. Th figur also includs msh data usd in th simulations Dislacmnt Controlld Analysis In th dislacmnt controlld analysis a total strain of 7% is alid ovr 00 qual tim by rscribing a dislacmnt at th to surfac. To comar th rsults from th isogomtric analysis with th finit lmnt analysis, Figur 5 shows th alid rssur normalizd to th cohsion against th axial strains. To study th ffcts of th msh dnsity, Figur 6 shows th avrag vrtical

17 A. Stz t al. Figur 5. Load/dislacmnt rsons for th dislacmnt controlld analysis of th cylindrical soil rofil. Figur 6. Dviation of th avrag dislacmnts on th to surfac of th saml. strss comonnts ovr th to surfac of th soil rofil for ach msh normalizd to th vrtical strss from th finst msh. From Figur 6 it is clar that th dviation of th vrtical strsss is slightly lss ronouncd in th rsults from th isogomtric analysis comard to th convntional finit lmnt m- 2

18 A. Stz t al. thod. Comaring th rsults from th two dimnsional xaml, th authors judg that th diffrnc ariss from th bnficial gomtrical aramtrization using NURBS in th isogomtric analysis Forc Controlld Analysis In th forc-controlld xaml th soil rofil is subjctd to a confining rssur, σ c, and an axial load, σ v, acting on th to surfac as illustratd in Figur 3(b). Th analysis is run ovr 00 load sts. Th load is alid by first incrasing th confinmnt rssur to 3 kpa aftr which th vrtical load is incrasd to 4 kpa. To comar th rsults from th isogomtric analysis to th rsults from th convntional finit lmnt analysis, th alid forc, normalizd to th cohsion, is lottd against th axial strains in Figur 7. Th rsults show that th solutions ar in good agrmnt but that thr is a diffrnc btwn th dislacmnts from th isogomtric analysis comard to th finit lmnt rsults. Comaring th dviation of th dislacmnt at th to surfac for th load controlld xaml, th influnc of using th xact gomtry in th isogomtric analysis bcoms mor vidnt than for th dislacmnt-controlld analysis, s Figur 8 and Figur 6. This would b xctd as th confining rssur σ c will b mor accurat using a soil rofil mad out of a rfct cylindr, than on discrtizd with Lagrangian olynomials. This can b sn in Figur 8 whr th rsults from th coarsr mshs clarly ar in rror. 6. Summary and Conclusion Th aim of this work is to valuat th isogomtric framwork for numrical Figur 7. Dviation of th avrag dislacmnts on th to surfac of th saml. 3

19 A. Stz t al. Figur 8. Dviation of th avrag dislacmnts of th to surfac of th cylindr. analysis of soil bhavior. To comar IGA to convntional FEA, th Druckr- Pragr critrion has bn imlmntd togthr with a NURBS-basd isogomtric framwork. A numrical study has bn conductd using NURBS-basd isogomtric analysis comaring th rsults with rsults from analysis rformd using th finit lmnt mthod. Th numrical xamls rsntd show, that for draind soils; th rsults from isogomtric analysis ar ovrall in good agrmnt with th convntional finit lmnt mthod in two- and thr-dimnsions. Howvr, th rsults from th two-dimnsional xaml rsntd illustrat that th highr continuity of th basis functions usd in IGA can hav an ffct on th lastic strains whr abrut strss changs can b xctd. For th thrdimnsional xamls rsntd in this ar th isogomtric analysis has rformd as good as or bttr than th finit lmnt mthod, comaring th load/dislacmnt rsons. To comar th comutational fficincy of IGA and convntional FEA is not within th sco of this study. Furthr, gotchnical alications lik rtaining walls in wak soil or installations of friction ils oftn involv comlicatd contact roblms and fluid flow, hnc could bnfit from using th isogomtric framwork. Acknowldgmnts Th Dvlomnt Fund of th Swdish Construction Industry, SBUF, suortd this work. Th suort is gratfully acknowldgd. Rfrncs [] Potts, D.M. and Zdravkovi c, L. (999) Finit Elmnt Analysis in Gotchnical Enginring: Thory. Philadlhia Univrsity, Philadlhia. 4

20 A. Stz t al. htts://doi.org/0.680/faigt [2] Hughs, T.J.R., Cottrll, J.A. and Bazilvs, Y. (2005) Isogomtric Analysis: CAD, Finit Elmnts, NURBS, Exact Gomtry and Msh Rfinmnt. Comutr Mthods in Alid Mchanics and Enginring, 94, htts://doi.org/0.06/j.cma [3] Bazilvs, Y., Gohan, J.R. and Hughs, T.J.R. (2009) Patint-Scific Isogomtric Fluid-Structur Intraction Analysis of Thoracic Aortic Blood Flow Du to Imlantation of th Jarvik 2000 Lft Vntricular Assist Dvic. Comutr Mthods in Alid Mchanics and Enginring, 98, htts://doi.org/0.06/j.cma [4] Bnson, D.J., Bazilvs, Y., Hsu, M.C. and Hughs, T.J.R. (200) Isogomtric Shll Analysis: Th Rissnr-Mindlin Shll. Comutr Mthods in Alid Mchanics and Enginring, 99, htts://doi.org/0.06/j.cma [5] Irzal, F., Rmmrs, J.J., Vrhoosl, C.V. and Borst, R. (203) Isogomtric Finit Elmnt Analysis of Porolasticity. Intrnational Journal for Numrical and Analytical Mthods in Gomchanics, 37, htts://doi.org/0.002/nag.295 [6] Nguyn, M.N., Bui, T.Q., Yu, T. and Hiros, S. (204) Isogomtric Analysis for Unsaturatd Flow Problms. Comutrs and Gotchnics, 62, htts://doi.org/0.06/j.comgo [7] Schillingr, D., Evans, J.A., Rali, A., Scott, M.A. and Hughs, T.J. (203) Isogomtric Collocation: Cost Comarison with Galrkin Mthods and Extnsion to Adativ Hirarchical NURBS Discrtizations. Comutr Mthods in Alid Mchanics and Enginring, 267, htts://doi.org/0.06/j.cma [8] Evans, J.A., Bazilvs, Y., Babuska, I. and Hughs, T.J.R. (2009) N-Widths, Su-Infs, and Otimality Ratios for th K-Vrsion of th Isogomtric Finit Elmnt Mthod. Comutr Mthods in Alid Mchanics and Enginring, 98, htts://doi.org/0.06/j.cma [9] Bazilvs, Y., Calo, V.M., Cottrll, J.A., Evans, J.A., Hughs, T.J.R., Liton, S., Scott, M.A. and Sdrbrg, T.W. (200) Isogomtric Analysis Using T-Slin. Comutr Mthods in Alid Mchanics and Enginring, 99, htts://doi.org/0.06/j.cma [0] Rali, A. and Hughs, T.J.R. (205) An Introduction to Isogomtric Collocation Mthods. In: Br, G., Ed., Isogomtric Mthods for Numrical Simulation, Sringr, Brlin, htts://doi.org/0.007/ _4 [] Auricchio, F., Viga, L.B., Hughs, T.J.R., Rali, A. and Sangalli, G. (200) Isogomtric Collocation Mthods. Mathmatical Modls and Mthods in Alid Scincs, 20, htts://doi.org/0.42/s [2] Nguyn, K.D. and Nguyn, H.X. (207) Isogomtric Analysis of Linar Isotroic and Kinmatic Hardning Elastolasticity. Vitnam Journal of Mchanics, 38. [3] Elgudj, T., Bazilvs, Y., Calo, V.M. and Hughs, T.J. (2008) Projction Mthods for Narly Incomrss-Ibl Linar and Non-Linar Elasticity and Plasticity Using Highr-Ordr Nurbs Elmnts. Comutr Mthods in Alid Mchanics and Enginring, 97, htts://doi.org/0.06/j.cma [4] Cox, M.G. (972) Th Numrical Evaluation of B-Slins. IMA Journal of Alid Mathmatics, 0, htts://doi.org/0.093/imamat/ [5] Boor, C. (972) On Calculating with B-Slins. Journal of Aroximation Thory, 6, htts://doi.org/0.06/ (72) [6] Vrsrill, K.J. (975) Comutr-Aidd Dsign Alications of th Rational B-Sli Aroximation. Ph.D. Thsis, Syracus Univrsity, Syracus. 5

21 A. Stz t al. [7] Pigl, L.A. and Tillr, W. (995) Th NURBS Book, Monograhs in Visual Communication. Sringr, Brlin. [8] Bath, K.J. (996) Finit Elmnt Procdurs. Prntic Hall, Englwood Cliffs. [9] Hughs, T.J.R. (202) Th Finit Elmnt Mthod: Linar Static and Dynamic Finit Elmnt Analysis. Courir Cororation, North Chlmsford. [20] Nto, E.A.D.S., Prić, D. and Own, D.R.J. (2008) Comutational Mthods for Plasticity: Thory and Alications. Wily, Chichstr. htt://catdir.loc.gov/catdir/toc/ci0824/ html htts://doi.org/0.002/ Submit or rcommnd nxt manuscrit to SCIRP and w will rovid bst srvic for you: Accting r-submission inquiris through , Facbook, LinkdIn, Twittr, tc. A wid slction of journals (inclusiv of 9 subjcts, mor than 200 journals) Providing 24-hour high-quality srvic Usr-frindly onlin submission systm Fair and swift r-rviw systm Efficint tystting and roofrading rocdur Dislay of th rsult of downloads and visits, as wll as th numbr of citd articls Maximum dissmination of your rsarch work Submit your manuscrit at: htt://arsubmission.scir.org/ Or contact gm@scir.org 6