Some computational aspects for solving deep penetration problems in geomechanics

Transcription

1 Compu Mech (29) 44: DOI 1.17/s ORIGINAL PAPER Some compuaional aspecs for solving deep peneraion problems in geomechanics Daichao Sheng Majidreza Nazem John P. Carer Received: 8 December 28 / Acceped: 17 April 29 / Published online: 7 May 29 Springer-Verlag 29 Absrac Peneraion problems in geomechanics involve he inserion or inrusion of solid bodies ino he ground. Such problems are exremely difficul o model numerically, because hey usually involve severe mesh disorion caused by large deformaion and fricional conac. In his paper, an Arbirary Lagrangian Eulerian mehod is used o overcome he mesh disorion problem. Some specific issues associaed wih he ALE mehod, such as node relocaion and remapping of conac hisory variables, are discussed. The ALE mehod, incorporaed wih an auomaic load sepping scheme and a smooh conac discreisaion echnique, is hen used o analyse he peneraion of axial displacemen piles ino he ground. Keywords Peneraion problems Fricional conac Mesh disorion ALE mehod Mesh moion 1 Inroducion Peneraion problems in geomechanics involve he inserion or inrusion of solid bodies ino he ground. The solid body can be a displacemen pile used o suppor a srucural load or a esing device used o measure soil properies. Numerical modelling of such problems can improve our undersanding of he physical processes involved, which in urn can lead o beer inerpreaion of es resuls and more accurae esimaion of pile capaciy [9]. However, such a simulaion is exremely complex, mainly because he problem involves D. Sheng (B) M. Nazem J. P. Carer School of Engineering, The Universiy of Newcasle, Newcasle, NSW, Ausralia daichao.sheng@newcasle.edu.au large deformaions ha make i difficul o design an appropriae finie elemen mesh. In principle, he soil elemens beneah and around he peneraing body should be sufficienly small o achieve good accuracy in he numerical resuls. However, he use of fine elemens ineviably leads o severe mesh disorion and even negaive elemen Jacobians due o he very large deformaion in he soil around he peneraing body. One of he early aemps o solve peneraion problems numerically was made by van den Berg [29], who combined inerface elemens wih an Eulerian formulaion of large deformaion o model a fixed volume of soil mass sreaming pas a fixed cone. This mehod successfully avoided he mesh disorion problem, bu i is limied o known maerial flows a boundaries. Therefore, he mehod canno simulae he complee peneraion process righ from he ground surface. In addiion, as wih oher Eulerian mehods where he discreisaion does no move wih he maerial, he approach has difficulies in racking maerial hisory. I is hus difficul o model hisory-dependen maerial behaviour, such as he sress-pah dependence exhibied by many soils. Hu and Randolph [8] presened a novel mehod ha combines small srain analysis wih coordinae updaing and remeshing. They successfully applied heir mehod o model he peneraion of spudcan foundaions, and i has he marked benefi of being heoreically sraighforward and simple for implemenaion. However, he efficiency and accuracy of he mehod for solving problems involving significan roaion need furher research. Liyanapahirana e al. [13] presened a new mehod for modelling he large deformaions associaed wih pile driving. By successively acivaing predefined soil pile inerface and pile elemens, hey were able o model he large peneraion of open-ended piles wih negligible wall hickness. Because he inacive pile and inerface elemens mus be defined a priori, hey canno occupy space when hey

2 55 Compu Mech (29) 44: (a) (b) Fig. 1 Mesh for large deformaion analysis of pile insallaion, showing a he fines soil elemens beneah he pile ha are possible for a successful Updaed Lagrangian analysis (pile soil fricion coefficien µ =.1) and b ypical mesh disorion if he inerfacial fricional coefficien is larger han.1 are inacive. I is hus difficul o exend heir mehod o handle closed-ended piles or open-ended piles wih a significan wall hickness. In a differen approach, Sheng e al. [2,21, 24] used he conac mechanics faciliy in he commercial code ABAQUS o model cone peneraion ess in dry soils, bu encounered serious problems wih mesh and ime sep effecs. Very coarse elemens and exremely small ime seps had o be used, leading o inaccurae and ime-consuming analyses. For example, i was found ha he smalles soil elemen ha can be used in a Lagrangian formulaion mus have a dimension which is approximaely half he radius of he peneraing body. Figure 1a shows a ypical deformed mesh from such an analysis assuming axi-symmeric condiions. I can be noed ha mos radial deformaion in he soil occurs wihin he firs annular column of elemens around he peneromeer. This deformaion paern indicaes he imporance of he discreisaion of he soil underneah he peneromeer. Alhough he soil elemens should be sufficienly small o capure he localised deformaion, furher refinemen of he soil elemens shown in Fig. 1a can cause negaive elemen Jacobians and numerical breakdown. Susila and Hryciw [28] used he remeshing echnique in ABAQUS o model saic cone peneraion ess in a Drucker Prager maerial. However, he same problems occurred wih mesh fineness and convergence, and he analyses had o assume a rigid peneromeer. The fricional conac a he soil srucure inerface also makes i difficul o obain a sable, convergen soluion. Firs, he fricion a he inerface affecs he shear deformaion in he soil elemens around he peneraing body, and higher inerfacial fricion exaggeraes he disorion of hese elemens. The meshes shown in Fig. 1a are for an inerfacial fricion coefficien of.1. Furher increasing his quaniy will ineviably lead o unaccepable mesh disorion or negaive elemen Jacobians (Fig. 1b). Second, he sudden change of he normal direcions along he peneraing body (a corners or verices) can cause oscillaion of he compued conac forces. For example, i is observed ha he compued pile resisances are srongly oscillaory due o he cyclic release of he soil nodes from he pile ip and he shoulder [26,2,24]. A ypical example is shown in Fig. 2. The oscillaions occur whenever a soil node is released from verical compression beneah he conical end. Once he soil node moves pas he ransiion poin a he shoulder of he pile, he high verical compression changes largely o a shearing force. Because his shear force is resriced by he fricion law a he inerface, a sudden drop in he oal resisance is expeced. Aferwards he resisance increases again unil he nex soil node moves pas he verex. These sudden changes in he verical force can lead o divergence of he numerical analysis if hey are no handled properly. The magniude of hese sudden changes, of course, depends on he discreisaion of he soil beneah and around he pile. In heory, he observed oscillaions in he oal resisance would vanish if he soil beneah he peneraing cone could be discreised ino infiniely small elemens. Anoher compuaional challenge in he modelling of peneraion problems arises from he large variaion of he maerial siffness. For example, a soil modelled by he Modified Cam Clay Model has essenially zero siffness a zero mean sress, such as near he ground surface, bu a seel pile has a Young modulus which is of he order of 1 GPa. Very ofen, he peneraing body is simply reaed as a rigid body, o avoid he ill-condiioning of he siffness marix. Indeed, i seems ha mos commercial codes are limied o his approach when hey are used o simulae peneraion problems [9,21]. This is inconvenien when we are also ineresed in he sresses in he peneraing body, and hus have o consider is finie siffness as well. Very recenly, Sheng e al. [24] proposed a number of numerical algorihms o improve he soluions of pile peneraion problems. These include a smooh discreisaion of he pile surface o enhance he performance of node-o-segmen conac elemens, and an auomaic load sepping scheme for linearisaion of he discreised governing equaions. Alhough hese algorihms have led o significan improvemen of he numerical soluions in erms of robusness and efficiency, he fundamenal problem of mesh disorion remains o be solved.

3 Compu Mech (29) 44: Fig. 2 Oscillaion in prediced pile resisance and he sudden change of conac sresses a he verex (pile radius =.2 m) Toal Resisance (kn) µ =.1 µ =.2 µ = Peneraion (m) α F T = F N µ F V = F N sin α + F T cos α F V = F H µ F H = F N cos α - F T sinα F N An effecive mehod o ackle he mesh disorion in large deformaion problems is he so-called Arbirary Lagrangian Eulerian mehod. The ALE mehod is based on he separaion beween maerial and mesh displacemens [2,3,6,7,12,32]. Such an ALE mehod ypically involves wo subseps during every ime sep: a Lagrangian sep followed by an Eulerian sep. In he Lagrangian sep, he governing equaions are solved o obain he maerial displacemens. A he end of his sep, he mesh may be disored. In he Euler sep, a new and beer mesh is generaed for he deformed domain o obain he mesh displacemens. All kinemaic and saic variables are hen ransferred from he disored mesh o he new mesh. The key issues in his mehod hus include he mesh refinemen mehod and he remapping of variables beween he wo meshes. The mehod used in his paper closely follows ha recenly proposed by Nazem e al. [16] and Sheng [25], bu includes some improvemens o heir node relocaion scheme. The key challenge when applying his ALE mehod o fricional conac problems is he remapping of hisory variables for all he conac elemens. Such variables include he elasic and plasic slip funcions of each slave node. The ALE mehod is hen incorporaed ino an auomaic load sepping scheme wih he smooh conac discreisaion echnique presened by Sheng e al. [24] and Sheng [25]. The effeciveness of hese numerical enhancemens is demonsraed by simulaions of he insallaion of displacemen piles. 2 Arbirary Lagrangian Eulerian mehod 2.1 Updaed Lagrangian soluion The Arbirary Lagrangian Eulerian (ALE) mehod used here is based on he operaor-spli echnique [2,16] and is associaed wih he Updaed Lagrangian (UL) formulaion. We hus firs briefly review he UL formulaion of large deformaions. For simpliciy, we limi our discussion o ime-independen problems of a single phase soil ha does no involve pore pressure dissipaion. In order o accommodae he definiions of sress raes, we use index noaion in his par of he paper. The index noaion is hen changed o vecor and marix noaion in he following secion, o faciliae he expressions for ime sepping and ieraions when solving he global equaions. The principle of virual work saes ha if δu is a virual displacemen field saisfying he displacemen boundary condiions, hen equilibrium is saisfied provided: ( ) σ ij δε ij dv + δu i b i dv + δu i q i ds α V α V α S α + ( N δg N + T δg T )ds = (1) S c where δε denoes he variaion of he srain ensor derived from he virual displacemens, σ is he Cauchy sress ensor, b he body force vecor, q he disribued force acing on he boundary S α of he volume V α, N and T are respecively he normal and angenial forces a he conac surface S c,δg N and δg T are respecively he virual normal and angenial gap, and he summaion is over he number of bodies. For a nonlinear problem, Eq. 2 is ypically applied incremenally. We assume ha he analysis sars a ime and all sae variables ha saisfy equilibrium are known up o ime. Furher loading and deformaion will require he equilibrium o be saisfied a ime +. The principle of virual work becomes: α V + α σ + ij δε ij dv = R + + C + (2) where R denoes he virual work resuling from body forces and surface racions, C denoes he virual work resuling from he conac racions, and he superscrip denoes he ime when he quaniies are measured. The quaniy R

4 552 Compu Mech (29) 44: involves a volume inegraion over V + and a surface inegraion over S +. The quaniy C involves surface inegraions over he conac surface (Sc + ). For geomerically nonlinear problems, he configuraion (V + and S + ) in Eq. 2 is no known and has o be ransferred o a known configuraion, for example he one a he sar of he curren ime sep (Updaed Lagrangian) or he one a ime zero (Toal Lagrangian). In eiher case, he second Piola Kirchhoff sress and he Green Lagrangian srain are usually inroduced in place of σ and ε, respecively, o eliminae he effecs of rigid body moion on he Cauchy sress ensor. For problems involving large-slip conac, he conac consrains are always described using he curren configuraion and herefore he Updaed Lagrangian formulaion is preferable. Since consiuive laws for geomaerials can seldom be wrien convenienly in erms of he second Piola Kirchhoff sress and he Green Lagrangian srain, we decompose he Cauchy sress rae ino a frame-independen sress rae due o sraining and a sress rae due o rigid body moion. Using he Jaumann sress rae as he frame-independen sress rae, we have dσ J ij = dσ ij σ ik d kj σ jk d ki = C ijkl dε kl (3) where C ijkl is he sress srain ensor derived from he consiuive relaions in erms of he Cauchy sresses and he linear srains, and is he spin ensor given by ij = 1 ( ui u ) j (4) 2 x j x i Inroducing (3) ino he virual work equaion, we obain he following equilibrium equaion for he UL mehod based on he Jaumann sress rae according o V ( ) C ijkl dε kl δ dεij dv + V + V ( σ ik d kj + σ jk d ) ( ) ki δ dεij dv σij δ ( ) dη ij dv = R + + C + V σij δ ( ) dε ij dv where δ(dη) is he variaion of he nonlinear par of he incremenal Green Lagrange srain ensor. Linearisaion of he erms on he lef-hand side in he above equaion will generally lead o a siffness marix corresponding o maerial non-lineariy (he firs inegraion on he lef-hand-side) as well as a siffness marix corresponding o he geomeric nonlineariy (he second and hird inegraions on he lefhand-side). Linearisaion of he virual work due o conac (5) racions requires he discreisaion of he conac surfaces, which will be discussed laer. A Gauss poins, he sress incremens are found by inegraing dσ ij in Eq. 3 along given srain incremens and spin ensor incremens. Nazem e al. [16] have recenly discussed alernaive inegraion schemes for geomaerials ha experience srain hardening. 2.2 ALE soluion A he end of he Updaed-Lagrangian soluion oulined above, he mesh may be disored since i moves along wih he maerial. To avoid mesh disorion, he mesh and he maerial displacemens can be separaed from each oher, hus allowing he mesh o move independenly from he maerial. This assumpion leads o he formulaion of he Arbirary Lagrangian Eulerian (ALE) mehod, which adds an addiional Euler sep following each UL sep. In he Euler sep, a new mesh is generaed for he deformed domain. All kinemaic and saic variables are hen ransferred from he old mesh o he new mesh using he relaion beween he maerial derivaive and he mesh derivaive [16] f = f r + ( v i vi r ) f (6) x i where f is an arbirary funcion, v i is he maerial velociy, vi r he mesh velociy, f denoes he ime derivaive of f wih respec o maerial coordinaes, and f r represens he ime derivaive of f wih respec o mesh (grid poins) coordinaes. The erm v i vi r is called he convecive velociy. Because he mesh displacemens are decoupled from he maerial displacemens, we can see ha he new mesh o be esablished in he Euler sep can be arbirary, provided wo basic condiions are saisfied: 1. The new mesh should conform o he deformed boundaries of he domain and he maerials. 2. The opology and conneciviy of he new mesh should remain he same as he old mesh. These requiremens ensure ha he mapping of variables beween he new and old meshes can be carried ou maerial-by-maerial and elemen-by-elemen. The firs requiremen also implies ha he nodes on a maerial or domain boundary should remain on i, even hough he locaions of oher inernal nodes in he new mesh are arbirary. We also noice ha, for an iniially opimal mesh for a homogeneous domain, if he displacemens a all he boundary nodes are prescribed and such displacemens ensure an opimal division of he boundaries, hen an elasic analysis based on he prescribed displacemens should resul in an opimal disribuion of inernal nodes. Here he word opimal loosely means ha he mesh qualiy is as good as he undeformed

5 Compu Mech (29) 44: Mesh a Mesh a he end of UL sep. Gives maerial displacemens Relocae boundary nodes. Apply displacemens of hese nodes o he original mesh and assume linear elasic maerial. Mesh a +. Gives he mesh displacemens Fig. 3 General procedure of he ALE node relocaion mehod mesh. Because he displacemens a all boundary nodes are prescribed, he elasic properies used in such an analysis may be arbirary as long as he maerial is assumed o be homogeneous and isoropic. Wih hese resuls in mind, our firs goal in he Euler sep is hen o obain he opimal division of he maerial and domain boundaries obained from he UL sep, or in oher words, o relocae he nodes on he boundaries so ha hey are opimally locaed. We also assume ha he iniial mesh before he UL sep is opimal. Comparing he relocaed nodes wih he nodes in he iniial mesh gives us he mesh displacemens of he nodes on all he boundaries. Wih he boundary displacemens now known, we carry ou an elasic analysis on he iniial mesh o obain he opimal mesh displacemens of he inernal nodes. Because we prescribe he displacemens along all boundaries and maerial inerfaces, he acual values of he elasic parameers used in his analysis are no imporan, as previously indicaed, and one se of elasic parameers can be used for he enire domain, regardless of he presence of real maerial inerfaces. The new mesh so obained also shares he same conneciviy and opology as he old mesh. The procedure oulined above is illusraed in Fig. 3. Once he mesh displacemens are obained, he sae variables such as sresses and hardening parameers are hen ransferred o he new mesh. The general procedure for he ALE mehod consiss of he following seps. remapped o he new mesh can be sae variables, such as sresses and hardening parameers, or nodal variables such as displacemens. If he sresses and hardening parameers are remapped, he consisency condiion may no be saisfied and he objeciviy of he soluion is no guaraneed. If he displacemens are remapped, he sresses and hardening parameers have o be re-inegraed over he accumulaed srains and hence i is very difficul o deal wih iniial sresses and sress-pah dependence [17,18]. Neiher mehod will guaranee he equilibrium a he end of Sep 4. In addiion, when fricional conac is involved, hisory variables such as elasic and plasic slip have o be remapped o he new mesh, giving anoher poenial source of equilibrium imbalance. If equilibrium is indeed no saisfied, addiional ieraions may be carried ou a he end of Sep 4 o reduce he unbalanced forces. These ieraions can be done wih he fixed mesh, i.e., under he assumpion of small deformaion. The resuling displacemens from hese ieraions are hen added o he oal nodal displacemens. Furher deails on node relocaion and he mapping of hisory variables may refer o [16]. 2.3 Remapping of conac hisory variables The procedure for remapping sae variables, such as sresses and hardening parameers, follows ha suggesed in Nazem e al. [16] and Nazem [15]. Since he siffness of a peneraing srucure is usually much larger han ha of he soil involved, we can reasonably assume ha he disorion of he srucure elemens is negligible. When he surface of he srucure is aken as he maser surface and he soil nodes on i are aken as slave nodes [24], he maser segmens remain unchanged when he slave is relocaed during he mesh moion sep of he ALE mehod. This feaure significanly simplifies he remapping of conac hisory variables, and he only one ha has o be remapped is he slip funcion a each slave node. This slip funcion records he oal angenial disance ha he slave node has slid along he maser surface. Consider a slave (soil) surface coming ino conac wih a maser (srucure) surface, as shown in Fig. 4. Theslave surface consiss of he slave nodes S1 S5, while he maser surface is defined by he nodes M1 M4. The curren locaions of slave nodes S1 S5 represen he UL soluion and 1. Perform an UL sep o find he maerial displacemens. 2. Relocae he nodes on all boundaries of he domain and beween maerials. 3. Opimise he mesh by performing an elasic analysis. 4. Remap hisory variables from he old mesh ono he new mesh. The oal nodal displacemens are hen he summaion of he maerial and mesh displacemens. The hisory variables M1 NS1 S1 S2 NS2 S3 M2 NS3 S4 NS4 M3 M4 NS5 Fig. 4 Relocaion of slave nodes (arrows indicae he relaive movemen of maser and slave surfaces) S5

6 554 Compu Mech (29) 44: he new slave nodes NS1 NS5 represen heir new locaions afer he node relocaion algorihm. Wih he slip funcions a nodes S1 S5 known, he slip funcions a nodes NS1 NS5 can be found by inerpolaion. Before he inerpolaion is done, a conac search has o be performed o idenify he new slave nodes ha come ino conac wih a maser segmen. In Fig. 4, he slip funcion a NS2 can be inerpolaed from he slip funcions of S2 and S3. If NS1 comes ino conac wih he segmen M1 M2, is slip funcion is hen se o zero. On he oher hand, he slip funcion a NS4 is preferably exrapolaed from hose a S3 and S4, insead of inerpolaing beween S4 and S5. In eiher case, remapping he slip funcions a slave nodes is likely o cause some drif from equilibrium, jus as he remapping of oher sae variables may do so. Therefore, an addiional equilibrium check afer remapping of all sae and hisory variables is recommended. 3 Discreisaion of conac surfaces and governing equaions Linearisaion of he virual work equaion (1) or(5) requires he linearisaion of he virual work due o conac forces, which in urn requires he discreisaion of he conac surfaces [31,11]. In his paper, he so-called node-o-segmen conac elemen is used, where conac consrain is enforced for each slave node while he maser surface is discreised ino curved segmens. The smooh discreisaion of he maser surface uses Bézier polynomials [24]. The discreised virual work due o normal and angenial conac forces respecively akes he form n c C N = N δg N ds δu T s [ε N A s g Ns B N (ξ)] S c n c s=1 = δus T Fc Ns (7) s=1 n c C T = T δg T ds δus T B ξ (ξ) x s ξ S s=1 c n c γ A s = δus T Fc Ts (8) s=1 where n c is he oal number of slave nodes, δu s he nodal virual displacemens of he conac elemen consising of one slave node and wo maser nodes, A s he conac area associaed wih he slave node, ε N he penaly parameer for normal conac, g Ns he normal gap, γ he angenial sress which will depend on he consiuive law for slip conac and on he angenial penaly parameer for sick conac, x s are he coordinaes of he projecion of he slave node on he maser segmen, ξ he disance beween x s and he maser node ha is closes o he slave node, B N and B ξ are Bézier polynomials, and F c Ns and Fc Ts are he nodal forces of he conac elemen. The linearisaion of equaions (7) and (8) gives he elemen angen marices for normal and angenial conac. Combining equaions (1), (7) and (8) leads o a se of global equaions of he form δε T σ dv + δu T b dv α V α V α + δu T ds + ( N δg N + T δg T ) ds S α σ S c = δu T ( G(U) + F c N (U) + Fc T (U)) = δu T ( G(U) + G c (U) ) = δu T R(U) = (9) where U is used in place of u o indicae he discreised global displacemen field, G(U) denoes he domain conribuions o he residual vecor, G c (U) denoes he conac conribuions given by (7) and (8), and R(U) is he global residual vecor. The global angen marix is obained by linearising (9)a agivenu according o K(U) = R(U) U = (G(U) + Gc (U)) U = ( K ep (U) + K nl (U) + K Ns (U) + K Ts (U) ) (1) where K ep is he siffness marix due o he maerial siffness, K nl he siffness marix due o geomeric nonlineariy, and K Ns and K Ts are he angen marices due o normal and angenial conac. Wih he maerial, geomery and conac siffness marices derived, he auomaic load sepping scheme presened in [24] can hen be modified o solve equaion (9). The modified algorihm is presened below. Noe ha he Euler sep in he ALE mehod includes he seps in he algorihm below. If he sub-incremen is large enough o cause significan mesh disorion, hese seps can also be included in each sub-incremen wihin each coarse incremen. In addiion, an equilibrium check can be included a he end of sep If equilibrium is no saisfied, addiional ieraions may be carried ou before Sep 12, o reduce he unbalanced forces. In he algorihm below, seps represen he Lagrangian sep, and seps represen he Eulerian sep in he ALE mehod. The auomaic sepping scheme is carried ou in sep 4.5, where wo user-defined olerances are used o conrol he sep size error: DTOL and ITOL.

7 Compu Mech (29) 44: Algorihm auomaic load sepping incorporaed ino he ALE mehod 1. Iniialise algorihm: se U=. 2. Check for conac: g acive node. Ns 3. Se all acive nodes o sae of sick. 4. LOOP over coarse load seps: n = 1, 1 ex ex 4.1. Solve: 1 = [ ( n 1) ] ( n n 1) U K U F F and updae U = Un 1+ U 1. n 4.2. Check for conac: g acive node. 1 ex ex 4.3. Solve: 2 = [ ( n 1+ 1) ] ( n n 1) Ns U K U U F F. n U U R = Un 4.5. IF R DTOL, ieraion i = 1,, convergence. 5. END LOOP. ex Check for convergence R( U) ITOL F exi ieraion wih a new sep size for nex subsep. i i 1 i 1 ( n ) ( n ) Solve: = ( ) 1 δ U K U R U. i i 1 i i i Updae U = U + δ U and Un = U 1+ U n Check for conac: g acive node. Ns 4.6. ELSE, reduce sep size and resar sep ENDIF Updae maerial coordinaes X M = X M +U Compue nodal sresses using a sress recovery procedure Check he boundaries and relocae he nodes on hem wherever necessary Compue he new mesh coordinaes by performing an elasic analysis and sore hem in X G Remap sae variables and conac hisory variables ono he new mesh Updae he oal displacemens vecor by U U XM + X G. n 4 Numerical examples The firs example is a purely academic problem where he effecs of he mesh fineness and he ALE mehod on he soluion are sudied. A relaively shor pile is chosen o reduce he compuaional work. The finie elemen meshes and maerial properies adoped in he analyses are shown in Fig. 5. Linear riangular elemens are used boh for he soil and he pile. Higher order elemens could be used here, bu in such cases special echniques have o be used o ensure he conac forces are ransferred correcly [31]. Four ypes of mesh are used, wih he raio beween he widh of he soil elemens beneah he pile and he pile radius varying beween.125 o 1. An elasic pile of radius.4 m is pushed ino he soil o a deph of 2. m. The peneraion is achieved by imposing a oal verical displacemen a he wo op nodes of he pile shaf. The soil is modelled as a non-associaed Mohr Coulomb maerial, wih he properies given in Fig. 5. The pile is modelled as an elasic maerial wih a Young s modulus E = kpa, which is imes he modulus of he soil. Oher maerial properies defined in Fig. 5 are: ν = Poisson s raio, c = cohesion, φ = fricional angle, ψ =

8 556 Compu Mech (29) 44: (a) Mesh A (b) Mesh B (c) Mesh C (d) Mesh D (e) UL: Mesh A (f) UL: Mesh B (g) ALE: Mesh B (h) ALE: Mesh C (i) ALE: Mesh D Fig. 5 Original and deformed meshes (Mohr Coulomb soil: E = 1 4 kpa, ν =.3, c = 1kPa,φ = 3 o,ψ = 2 o,γ = 2 kn/m 3. Elasic pile: E = kpa, ν =.3. Dimension of iniial soil domain: 2.4 m 4.8 m. Dimension of pile shaf:.4 m 3 m. Cone angle: 6 o. Pile soil inerfacial fricion coefficien:.1. Peneraion: 2.5 m) dilaion angle, and γ = uni weigh of he soil. The pilesoil inerfacial fricion coefficien is se o.1. The penaly parameers in he normal and angenial direcions are se o 1 6 kn/m 3. I was found ha his value can be increased or decreased by an order of one, wihou causing significan change in he numerical resuls.

9 Compu Mech (29) 44: Deph/Radius ALE: Mesh C Pile Resisance (kpa) UL: Mesh A ALE: Mesh D Fig. 6 Mesh effecs on he pile resisances ALE: Mesh B UL: Mesh B The Updaed Lagrangian mehod was employed o analyse mesh A and mesh B shown in Fig. 5a and b, respecively. The deformed meshes from he UL analyses are illusraed in Fig. 5e and f, respecively. In addiion, he Arbirary Lagrangian Eulerian mehod was also used o analyse he same meshes. Figure 5g shows he deformed mesh from he ALE analysis for mesh B. This is no much differen from he mesh in Fig. 5f, excep ha he nodes along he boundaries as well as he inernal nodes are relocaed. The meshes shown in Fig. 5h and i were also obained from ALE analyses. Figure 6 shows he prediced pile resisances as a funcion of he deph of peneraion for he four meshes. As expeced, he UL and ALE analyses give almos he same curve for Mesh B. A any peneraion deph he prediced pile resisance decreases as he mesh becomes finer. The prediced resisances for mesh A differ from hose for mesh D by up o a facor of 2. However, he differences beween Mesh C and Mesh D are no as pronounced as hose beween mesh A and mesh B, indicaing ha he numerical soluion is converging as he mesh is refined. The difference ( 15%) beween he resuls for mesh C and D indicaes ha very fine elemens have o be used in such analyses. I is believed ha he convergence rae will improve as he soil becomes less dilaan (e.g., a soil modelled by he modified Cam clay model). I is also observed ha he numerical oscillaions in he prediced pile resisances become less pronounced for he finer meshes such as C and D. In he second example, a seel pile of radius.2 m is pushed ino a Mohr Coulomb soil o a deph of 2.5 m. The finie elemen mesh shown in Fig. 7, where he widh of he fines soil elemens is roughly one eighh of he pile radius, was used for he ALE analysis. Figure 7 indicaes ha he ALE analysis can simulae he peneraion of he pile o a deph of 12.5 pile radii wihou generaing any significan or unwaned mesh disorion. Four-noded linear elemens are used in he analysis. Due o he dilaan behaviour prediced by he Mohr Coulomb model under shear, he oal volume of he soil increases as he pile is insered. The volume increase can be confirmed by esimaing he oal volume occupied by all soil elemens in Fig. 7. In Fig. 8, he prediced oal resisances, i.e. he reacions a he wo op nodes where displacemens are prescribed, are ploed agains he normalised peneraion deph. All UL soluions in he figure are obained using a coarser mesh wih he widh of he fines soil elemens equal o one quarer of he pile radius. Wih such a coarse mesh, he UL analysis can also furnish wih some resuls, provided ha seps wih unbalanced forces larger han he prescribed olerance are allowed o coninue. The prediced oal resisance from he UL analysis shows srong oscillaions, similar o hose shown in Fig. 2. The UL analysis using unsmoohed pile segmens canno finish wih a complee soluion, due o severe mesh disorion. Comparing he curves denoed by UL (µ =.1, unsmoohed pile segmens) and UL (µ =.1), we see ha he smooh discreisaion of he conac surfaces reduces he oscillaions significanly. Comparing he curves denoed by UL (µ =.1) and UL (µ =.), we see ha he degree of oscillaion depends also on he amoun of pile soil inerfacial fricion. The ALE soluions illusraed in Fig. 8 were obained using he smooh discreisaion of he conac surfaces. These soluions, even hough sill somewha oscillaory, are much smooher han he UL soluions. In heory, o compleely remove he oscillaions, we need o use very fine soil elemens. A key advanage of he proposed ALE mehod is ha i can effecively solve he oscillaion problem by using a fine mesh. In Fig. 9 he compued oal resisance is compared wih he analyical soluion for cone peneraion es by Yu [33]. The soil is now represened by he Tresca model wih he undrained shear srengh of 1 kpa. The soil cone inerface is assumed o be fricionless. The analyical soluion of Yu [33] is based on cylindrical caviy expansion heory and is for seady sae cone peneraion only, i.e., i does no consider he peneraion near he ground surface. Figure 9 shows ha he compued cone resisance is somewha larger han he analyical soluion for a smooh cone. This overesimaion is likely due o he linear elemens used for in he analysis. This ype of elemens is known o be locking in axisymmeric problems and higher order elemens should be used o avoid

10 558 Compu Mech (29) 44: Fig. 7 Deformed meshes (Mohr Coulomb soil: E = 1 4 kpa, ν =.3, c = 1 kpa, φ = 35 o, ψ = 1 o,γ = 2 kn/m 3. Elasic pile: E = kpa, ν =.3. Dimension of iniial soil domain: 2 m 5 m. Dimension of pile shaf:.2 m 3 m. Cone angle: 6 o. Pile soil fricion coefficien:.1. Peneraion: 2.5 m) Toal Resisance (kpa) Toal Resisance (kpa) UL ( µ =.1, non-smooh pile segmen) ALE ( µ =.1) -2-4 ALE predicion of cone resisance wih smooh surface Deph / Pile Radius -6-8 ALE ( µ =.) UL ( µ =.1) Deph/Radius UL ( µ =.) Yu (21) for smooh cone Yu (21) for rough cone -14 Fig. 8 Prediced load displacemen curves (µ: inerfacial fricion coefficien. The UL soluions were obained wih a coarser mesh) Fig. 9 Prediced load displacemen curve (Tresca soil: E = 1 4 kpa, ν =.4, c = 1 kpa, γ = 2 kn/m 3. Elasic pile: E = kpa, ν =.3. Dimension of pile shaf:.2 m 3 m. Cone angle: 6 o. Pile soil fricion coefficien: )

11 Compu Mech (29) 44: Szz σrr σzz σ θθ Sxy σr θ Fig. 1 Sress disribuions obained from ALE analysis (σ rr : radial sress; σ zz : axial sress; σ θθ : circumferenial sress; σ rθ : shear sress; legends in kpa and ension posiive; Soil: Mohr Coulomb; Pile: elasic; Pile soil fricion coefficien:.1) he locking [27]However, higher order elemens require more complex conac formulaion [4,5]. Figure 1 shows some sress conours in he soil and he pile. The sress bulbs beneah he pile are clearly shown in he figures for radial, verical and circumferenial sresses. The shear sress also shows a concenraion area beneah he pile end. In he pile, he maximum radial and circumferenial sresses occur in he pile end, while he larges verical sress occurs a he pile shoulder. The key advanage here is ha he pile is reaed as a deformable body, compared o he rigid piles assumpion in he analyses of [26]or[2], Sheng 24). Some elemen overlapping is observed near he ip. The overlapping is caused by he penaly mehod and he smooh discreisaion of he pile surface.

12 56 Compu Mech (29) 44: σ σ σ rr, kpa Sxx rr, kpa Sxx rr, kpa Sxx Fig. 11 Radial sress disribuions around a pile pushed ino Modified Cam Clay (Elasic pile: E = 1 8 kpa, ν =.3, shaf radius =.2 m, lengh = 3.5 m. Mohr Coulomb soil:.5 m hick, E =1 4 kpa, ν =.3, c = 1kPa,φ = 3 o, ψ = 2 o,γ = 2 kn/m 3. Modified Cam Clay soil: 4.95 m hick, ν =.3,φ = 3 o,γ = 2 kn/m 3, λ =.2,κ =.5, N = 3., OCR = 1. Pile soil fricion coefficien: ) All he soluions were obained wih he auomaic scheme described previously using 1, coarse load seps. The ALE analyses for he fine mesh require approximaely 6 8 h CPU imeonanibmt41lapopwiha1.6ghzpeniumprocessor and 1.5 GB RAM. The UL analyses for a coarse mesh, wih one quarer of he elemens in he fine mesh, require approximaely 2 3 h of CPU ime on he same machine. Allowing for he difference in he degrees of freedom in hese meshes, we see he ALE mehod is almos as efficien as he UL mehod. A sandard Newon Raphson scheme (see [24]) was also ried wih 1, load seps, bu failed o provide a reasonable soluion. The failures of he Newon Raphson scheme were caused eiher by lack of convergence, or by numerical breakdown if he non-converged seps were allowed o coninue. Figure 11 shows he ALE mesh for a pile peneraing ino a Modified Cam Clay (MCC) soil. The soil is fully drained and is properies are given in he figure: λ is he slope of he normal compression line (NCL), κ he slope of he unloading-reloading line, e N he void raio on he NCL when he mean sress is one uni (kpa), and OCR he overconsolidaion raio. The descripion and implemenaion of he specific MCC consiuive model in he finie elemen procedure can be found in Sheng e al. [22]. Because he MCC soil has a zero elasic bulk modulus a zero mean sress, he op layer of soil elemens (.5 m hick) was modelled using he Mohr Coulomb model. The deformed meshes in Fig. 11 show ha a par of he iniial ground surface is in conac wih he pile, and ha he ALE meshes susain he opimal form of he iniial undeformed mesh. We can also noice ha he ground Deph / Pile Radius ALE ( µ =., OCR=1.) Toal Resisance (kpa) UL ( µ =.1, OCR=1.) ALE ( µ =., OCR=3.) ALE ( µ =.1, OCR=3.) Fig. 12 Prediced load displacemen curves in Modified Cam Clay soil (µ: inerfacial fricion coefficien) surface remains more or less a he same level as he pile peneraes, indicaing ha he oal soil volume decreases. The load displacemen curves shown in Fig. 12 furher demonsrae he effeciveness of he new algorihms in reducing

13 Compu Mech (29) 44: oscillaions. I is noed ha he UL analysis for a fricional pile does no finish wih a complee soluion, due o severe mesh disorion. 5 Conclusions The ALE mehod presened in his paper is effecive for handling he mesh disorion in peneraion problems and does no cause much increase in he CPU ime. The smooh discreisaion of conac surfaces can reduce he oscillaion in he load displacemen curves significanly, while he auomaic load sepping scheme is more robus in solving he sysems of nonlinear equaions. These algorihms advance he simulaion of peneraion problems in geomechanics considerably. Addiional enhancemens may include he incorporaion of morar-based fricional conac formulaion for higher order elemens in he soil domain [4], he incorporaion of elemen separaion modelling and efficien conac search algorihms for group piles and 3D peneraion problems, as well as he developmen of coupled displacemen and pore pressure mehods wih dynamic effecs. References 1. Abbo AJ, Sloan SW (1996) An auomaic load sepping algorihm wih error conrol. In J Numer Mehods Eng 39: Benson DJ (1989) An efficien, accurae and simple ALE mehod for nonlinear finie elemen programs. Comp Mehods Appl Mech Eng 72: Belyschko T, Kennedy JM (1978) Compuer models for subassembly simulaion. Nucl Eng Des 49: Fischer KA, Wriggers P (26) Moar based fricional conac formulaion for higher order inerpolaion using he moving fricion cone. Comp Mehods Appl Mech Eng 195(37 4): Fischer KA, Sheng D, Abbo AJ (27) Modeling of pile insallaion using conac mechanics and quadraic elemens. Comp Geoech 34(6): Gadala MS, Wang J (1998) ALE formulaion and is applicaion in solid mechanics. Comp Mehods Appl Mech Eng 167: Ghosh S, Kikuchi N (1991) An arbirary Lagrangian Eulerian finie elemen mehod for large deformaion analysis of elasicviscoplasic solids. Comp Mehods Appl Mech Eng 86: Hu Y, Randolph MF (1998) A pracical numerical approach for large deformaion problems in soils. In J Numer Anal Mehods Geomech 22: Huang W, Sheng D, Sloan SW, Yu HS (24) Finie elemen analysis of cone peneraion in cohesionless soil. Comp Geoech 31: Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian Eulerian finie elemen formulaion for incompressible viscous flow. Comp Mehods Appl Mech Eng 58: Laursen TA (22) Compuaional conac and impac mechanics. Springer, Berlin 12. Liu WK, Belyschko T, Chang H (1986) An arbirary Lagrangian Eulerian finie elemen mehod for pah-dependan maerials. Comp Mehods Appl Mech Eng 58: Liyanapahirana DS, Deeks AJ, Randolph MF (2) Numerical modelling of large deformaions associaed wih driving of openended piles. In J Numer Anal Meh Geomech 24: Lopez RJ (21) Advanced engineering mahemaics. Addison Wesley, New York 15. Nazem M (26) Numerical algorihms for large deformaion problems in geomechanics. Ph.D. hesis, School of Engineering, The Universiy of Newcasle, Ausralia 16. Nazem M, Sheng D, Carer JP (26) Sress inegraion and meshing refinemen for large deformaion in geomechanics. In J Numer Mehods Eng 65: Peric D, Hochard C, Duko M, Owen DRJ (1996) Transfer operaors for solving meshes in small srain elaso-plasiciy. Comp Mehods Appl Mech Eng 137: Peric D, Vaz MJr, Owen DRJ (1999) On adapive sraegies for large deformaions of elaso-plasic solids a finie srains: compuaional issues and indusrial applicaions. Comp Mehods Appl Mech Eng 176: Pos DM, Zdravkovic L (21) Finie elemen analysis in geoechnical engineering, vol 1, heory. Thomas Telford, London 2. Sheng D, Axelsson K, Magnusson O (1997) Sress and srain fields around a peneraing cone. In: Pieruszczak S, Pande GN (eds) Numerical models in geomechanics. Balkema, Roerdam, pp Sheng D, Eigenbrod KD, Wriggers P (25) Finie elemen analysis of pile insallaion using large-slip fricional conac. Comp Geoech 32(1): Sheng D, Sloan SW, Yu HS (2) Aspecs of finie elemen implemenaion of criical sae models. Compu Mech 26: Sheng D, Sloan SW (21) Load sepping mehods for criical sae models. In J Numer Mehods Eng 5: Sheng D, Wriggers P, Sloan SW (26) Improved numerical algorihms for fricion conac in pile peneraion analysis. Comp Geoech 33: Sheng D (27) Fricional conac for pile insallaion. In: Wriggers P, Nackenhors U (eds) IUTAM symposium on compuaional mehods in conac mechanics. Springer, Heidelberg, pp Simo JC, Meschke G (1993) A new class of algorihms for classical plasiciy exended o finie srains. Appl Geomaer Compua Mech 11: Sloan SW, Randolph MF (1984) Numerical predicion of collapse loads using finie elemen mehods. In J Numer Anal Mehods Geomechan 6: Susila E, Hryciw RD (23) Large displacemen FEM modelling of he cone peneraion es (CPT) in normally consolidaed sand. In J Numer Anal Mehods Geomech 27: Van den Berg P (1994) Analysis of soil peneraion. Ph.D. hesis, Technische Universiei Delf 3. Wang CX, Carer JP (22) Deep peneraion of srip and circular fooings ino layered clays. In J Geomech 2(2): Wriggers P (22) Compuaional conac mechanics. Wiley, Chicheser 32. Yamada T, Kikuchi F (1993) An arbirary Lagrangian Eulerian finie elemen mehod for incompressible hyperelasiciy. Comp Mehods Appl Mech Eng 12: Yu HS (24) Caviy expansion mehods in geomechanics. Kluwer, Dordrech 34. Zhou H, Randolph MF (27) Compuaional echniques and shear band developmen for cylindrical and spherical peneromeers in srain-sofening clay. In J Geomech ASCE 7(4):