MESHLESS METHODS: ALTERNATIVES FOR SOLVING 2D ELASTICITY PROBLEMS

Transcription

1 nrnatonal Cvl Engneerng Conference "Towards Sustanable Cvl Engneerng Practce" Surabaa, August 25-26, 2006 MESHLESS METHODS: ALTERNATVES FOR SOLVNG 2D ELASTCTY PROBLEMS Effend TANOJO 1, Pamuda PUDJSURYAD 1 ABSTRACT: n recent ears mesless metods ave been developed as alrnatves to wellknown and wdel used fn element metod. Te man objectve n developng mesless metods s to overcome dffcult of mesng and remesng procedure of complex structural elements, wc s one of drawbacks of fn element metod. n mesless metods, element defnton s no longer needed to dscretze problem doman, onl nodal ponts defnton and boundar condtons of doman are necessar. Ts paper presents concepts bend some promnent mesless metods: element-free Galerkn, mesless local Petrov-Galerkn, and fn pont metod. Te dervatons of se metods for solvng 2D elastct problems wll also be presend and appled to a case problem. Te relts wll n be compared wt ones obtaned b fn element metod and exact solutons. KEYWORDS: Fn Element Metod, mesless, Element-Free Galerkn, Mesless Local Petrov- Galerkn, Fn Pont. 1. NTRODUCTON Fn element metod as been wdel-known and well-accepd to be used n man engneerng applcatons. However complex engneerng problems, uall ones nvolvng a contnuous cange n geometr of problem doman caused b crack propagaton or large deformaton, cannot be solved easl usng fn element metod, as requre a lot of remesng procedure to enre tat problem dscretzaton stll concdes wt geometr of real structure as t contnuousl canges. Mesless metods provde alrnatves to overcome ts problem as, unlke fn element metod, do not requre defnton of elements to dscretze problem doman. To dscretze problem doman, mesless metods requre onl defnton of nodal ponts. Te redscretzaton procedure of problem doman as t contnuousl canges s done b addng and/or movng nodal ponts onl, wtout need of redefnng elements n eac redscretzaton. Tus complex mesng and remesng procedure s no longer needed, or n some cases sgnfcantl reduced. Durng recent ears, man mesless metods ave been proposed and developed. Among m, element-free galerkn [1], mesless local petrov-galerkn [2], and fn pont metod [3] can be consdered as promnent ones. Ts paper wll present concept and dervaton of tree aforementoned metods for solvng 2D elastct problems. Te relts obtaned from se metods, as are appled to a st problem, wll be compared to ones obtaned b usng fn element metod. 1 Lecturer, Dept. of Cvl Engneerng, Petra Crstan Unverst, ndonesa 65

2 TANOJO and PUDJSURYAD 2. THEORY OF ELASTCTY Te governng dfferental equaton requred to be satsfed at all ponts n doman,, for two dmensonal problem, omogeneous and sotropc maral, s defned as σ j,j + F = 0, (1) were and j at ndex represent numbers 1 and 2 (or drecton x and n two dmensonal doman), σ j,j are stress components correspondng wt dsplacement feld u, and F are bod force components actng drecton at. Te boundar condtons of doman, Γ = Γ u U Γ t, are as follows: u = u at Γ u, t = t at Γ t, (2a) (2b) were rface tractons are defned b t = σ j. n j, were n j are unts normal to boundar Γ, u and t are prescrbed values of dsplacements and tractons at boundar Γ u and Γ t, respectvel. Γ u smbolzes essental boundar, Γ t smbolzes non-essental boundar. 3. MESHLESS METHODS Generall mesless metods, to keep ts local caracr, use a local approxmaton to represent values of unknown varables at some random nodal ponts wt tral functon. Next, most wdel used local approxmaton for mesless metods, movng least-squares approxmaton, wll be dscussed. t wll be followed n b concepts and dervatons of tree aforementoned mesless metods for solvng 2D elastct problems MOVNG LEAST-SQUARES (MLS) APPROXMATON n mesless metods, spatal dscretzaton can be obtaned b usng MLS approxmaton wc does not requre defnton of elements, but uses nodal selecton procedure nsad. Ts approxmaton s ntroduced b Lancasr and Salkauskas []. Tus, MLS approxmaton s ver table to form sape functon used n mesless metods. At doman, MLS approxmaton functon (tral functon), u (x), for dsplacement functon u(x) s expressed b vector of polnomal bass functon, p T (x), and vector of coeffcents, a(x), as follows m u (x) = p j (x) a j (x) = p T (x) a(x), (3) j= 1 were p j (x) s monomal at spatal coordna x T = x,, and m s number of monomal n bass functon. Te general form of bass functon wt degree s on two dmensonal can be expressed b p T (x) = 1,., x s, x s-1,.., x s-1, s. Te rm movng s used due to dependablt of coeffcent a to varable x, wereas wegt functon w (x) = w(x - x ) s defned for ever dfferent evaluaton pont x at doman. Consderng tat coeffcent a n Equaton 3 does not ave a pscal meanng, approxmaton of dsplacement feld values needs to be modfed b statng coeffcent a as d.o.f. at nodal u, so tat an equaton smlar to tat of nrpolaton functon used n fn element metod can be obtaned as 66

3 nrnatonal Cvl Engneerng Conference "Towards Sustanable Cvl Engneerng Practce" n u (x) = φ T (x). û = φ (x) û ; u (x ) u û () = 1 were φ (x) denos sape functon from MLS approxmaton, were φ (x) = 0 f wegt functon w (x) = 0, and n s number of nodes around evaluaton pont x, onl n areas were w (x) > 0 (also known as doman of defnton of pont x). Te pport of a wegt functon (also known as doman of nfluence of a node) refers to bdoman were value of wegt functon s not equal to zero. Te llustraton of MLS approxmaton s sape functon can be seen n Fgure 1. t s mportant to understand tat û s not real nodal values of unknown tral functon u (x), nsad t represents fcttous nodal values. Te dfference between u dan û on one dmensonal problems can be descrbed n Fgure 2. u (x) u û u Sape Doman of nfluence Fgure 1. llustraton of sape functon used n mesless metods[5] 3.2. ELEMENT-FREE GALERKN (EFG) x 1 x 2 Boundar node x Fgure 2. Te dfference between u dan û dsplacement[2] x EFG s frst ntroduced b Beltscko et al. [1]. Altoug defnton of elements s no longer needed n EFG metod, ts metod stll cannot be consdered as trul mesless because t requres use of a background cell n problem doman to execu numercal ngraton procedure (Fgure 3). Te weak form of governng equatons are satsfed globall n ts metod. Tus, numercal ngraton s done n wole doman. Ts procedure requres a lot of quadrature ponts to be defned and placed n eac of background cells. n EFG, dsplacement feld u acts as a tral functon, and vrtual dsplacement feld δv serves as a st functon. Multplng st functon to bot sdes of governng equaton, followed b ngratng on doman wll gve (σ j, j + F ) δv d = 0. (5) B applng dfferentaton b parts, Gauss dvergence orem, and notng tat t = σ j.n j (were t = 0 at Γ, except at boundar Γ t were t = t ), followng s obtaned Γ t t δv dγ + F δv d σ j δv, j d = 0 (6) 67

4 TANOJO and PUDJSURYAD Te sape functon of MLS approxmaton does not ave Kronecker delta propert, tus φ (x J ) δ J, were φ (x J ) s sape functon of nodal evaluad on nodal pont x J, and δ J s Kronecker delta (were δ J = 1 f = J, and δ J = 0 f J). Due to ts propert, tral functon does not satsf essental boundar condtons, u = u at Γ u. n EFG Lagrange multpler s ntroduced to overcome ts problem, gvng equaton σ j δv, j d F δv d Γ t t δv dγ (u u )δλ dγ λ δv dγ = 0, (7) Γ u Γ u were λ and δλ are Lagrange multpler and varaton of Lagrange multpler, respectvel. Mappng = ngraton Area (a) = Problem Doman = ngraton Area (Local bdoman) (b) Fgure 3. Numercal ngraton area (a) usng background cell over entre problem doman n EFG[6], and (b) over local bdoman n MLPG[6] 3.3. MESHLESS LOCAL PETROV-GALERKN (MLPG) Te weak form on MLPG are not satsfed globall on entre problem doman, but t s done n local bdoman (leadng to rm Local Smmetrc Weak Form) wc resdes entrel nsde global doman, tus MLPG can be consdered as trul mesless (Fgure 3). Moreover, essental boundar condtons (u = u pada Γ u ) also cannot be satsfed drectl n ts metod usng MLS approxmaton. MLPG solves ts problem b ntroducng use of a penalt functon. MLPG s frst ntroduced b Atlur et al. [2]. B multplng st functon to bot sdes of local weak form of governng equatons gven n Equaton 1 and boundar condtons gven n Equaton 2 and ntroducng penalt metod, followng s obtaned (σ j,j + b ) v d α Γ (u u ) v dγ = 0, (8) were Γ s nrsecton of Γ u wt boundar, u and v are tral functon and st functon respectvel, and α s penalt functon. Usng dfferentaton b parts and Gauss dvergence orem, and takng nto account non-essental boundar condtons, followng local smmetrc weak form s obtaned : 68

5 nrnatonal Cvl Engneerng Conference "Towards Sustanable Cvl Engneerng Practce" σ j v, j d + α Γ u v dγ Γ t v dγ = Γ st t v dγ + α Γ u v dγ + F v d, (9) were t = σ j.n j, and 3.. FNTE PONT (FP) st Γ s nrsecton of Γ t and boundar. Fn pont metod, proposed b Ona et al. [3], uses pont collocaton sceme to elmna ngral appearng n general wegd resdual form. = Collocaton Ponts Fgure. Collocaton ponts were governng dfferental equatons are satsfed n Fn Pont Metod [6] Wegd resdual metod s a common procedure to solve governng dfferental equaton numercall, were dsplacement functon u s approxmad b a tral functon u gvng W ( ) ( ) 0 A u + F d+ W B u t dγ+ W u u dγ=, (10) Γt Γu were W, W,danW deno wegt functons correspondng wt Equaton 1 and boundar condtons n Equaton 2 respectvel. Pont collocaton s n used to elmnas ngral appearng n Equaton 10 and wegt functon s set to W W W delta component as follows = = δ ( x x ) =, wt Drac δ 0, = 1, ( x x) x x x = x (11) Te dscre equatons n fn pont metod s obtaned b bsttutng approxmaton functon of dsplacement, u, to governng dfferental equaton stad n Equaton 1 and boundar condtons n Equaton 2, combned wt applcaton of pont collocaton procedure. Te dscre equatons are gven as u + F ] p = 0 at, p = 1, 2,... N r, (12a) [ A ( ) [ ( ) [ B ( ) u ] s = u at Γ u, s = 1, 2,... N u, (12b) u - t ] r = 0 at Γ t, r = 1, 2,... N t, (12c) 69

6 TANOJO and PUDJSURYAD were N r s number of nodal ponts on doman, except ones at boundar Γ u and Γ t, wle N u and N t are numbers of nodal ponts at boundar Γ u and Γ t respectvel. A and B smbolzes dfferental operator to defne governng dfferental equatons need to be satsfed u = σ j n j [7]. at dan Γ t respectvel, were A ( u ) = σ j, j and B ( ). TEST PROBLEM AND DSCUSSON Consder an nfn pla wt a central ole. Te ole s a crcle defned b x a 2, were a s radus of crcle. Te pla s bjecd to a unform nson, σ = 1, n x drecton, as sown n Fgure 5a. Te exact solutons for stresses, gven b Tmosenko et al. [8], are a ( x ) { } 2 3 3a σ, = 1 cos 2θ + cos θ + cos θ, (13a) x 2 r 2 2r 2 1 3a σ, = cos 2θ cos θ + cos θ, (13b) 2 r 2 2r a ( x ) { } 2 1 3a σ, = cos 2θ + sn θ + cos θ, (13c) x 2 r 2 2r a ( x ) { } were (r, θ) are polar coordnas and θ s meared from postve x axs counrclockwse. x σ σ x xx xx Fgure 5. (a) nfn pla wt a cenr ole, and (b) ts upper rgt part, wt tracton at our boundar [9] x Due to smmetr, onl a part of upper rgt quadrant of pla s modeled, as sown n Fgure 5b. Smmetr condtons are mposed on left (u x = 0, t = 0) and bottom (u = 0, t x = 0) edges, and nner boundar at a = 1 s tracton free. Te non-essental boundar condtons gven b exact soluton n Equaton 13 are mposed on rgt and top edges. Ts st problem as been evaluad prevousl b usng eac of tree aforementoned mesless metods. Beltscko et al. [1], Atlur et al. [9], Ona et al. [7] evaluad st problem usng EFG metod, MLPG metod, and FP metod, respectvel. Here, relts are presend and compared. Te varable to be compared s normal stress n orzontal drecton along lne x=0. Te relts from tree mesless metods wll n be compared b ones obtaned b usng fn element metod. Te fn element metod analss s done b SAP2000 [10] program. Te plotd stress relts are presend n Fgure 6. 70

7 nrnatonal Cvl Engneerng Conference "Towards Sustanable Cvl Engneerng Practce" σ x at x = 0 (a) (b) exact FEM (SAP2000) σ x at x = (c) Fgure 6. Te plotd stress relts from pla wt a cenr ole obtaned b usng (a) EFG metod [1] (b) MLPG metod [9] (c) FP metod [7], and (d) Fn Element Metod [5] As seen n Fgure 6, all metods used eld relatvel good relts compared to exact solutons gven n Equaton 13. Relts from EFG and MLPG metods also sow convergence propert. n relts obtaned b usng fn element metod, t can be seen tat plot s not as smoot as ones obtaned b usng mesless metod. Ts s due to fact tat sape functon obtaned from MLS approxmaton tself s alread smoot n geometr as sown n Fgure 1, and ts sape functon dervatves also ave smoot and contnuous caracr altoug bass functon used ma onl be a lnear one. Te approxmaton functon used n mesless metods also allows an ponts n doman to be approxmad b usng more number of nodes tan unknown varables due to ts least squares propert. Te fn element metod used n SAP2000 uses b-lnear bass functon n ts nrpolaton sceme. Due to ts lnear nrpolaton feature, smootness of plotd relts reles strongl on number of nodes used. Table 1 provdes a closer look at numercal values obtaned from plotd relts sown n Fgure 6. Table 1. σ x at x = 0 for pla wt a cenr ole [5] Metod used = 1 = 2 = 3 = = 5 Fn Element Metod (5 nodes) Element-free Galerkn (5 nodes) Mesless Local Petrov-Galerkn (5 nodes) Fn pont (60 nodes) Exact soluton (d) 71

8 TANOJO and PUDJSURYAD 5. CONCLUSONS Eac mesless metod as ts own specfc caracr and feature, but all of m empaszes on unnecesst to defne elements n dscretzaton of problem doman. EFG metod, beng frst well publsed mesless metod, s not trul mesless n a sense tat t stll requres a background cell to evalua weak form. MLPG does not requre a background cell and evaluas weak form over nrsectng local b-domans, refore t s trul mesless. FP metod as advantage over or metods snce absolul no ngraton sceme s needed n weak form evaluaton due to use of pont collocaton sceme, but t as been consdered tat b usng pont collocaton sceme, more nodes are requred to aceve same degree of correctness. Te st problem sows tat mesless metods ave advantage over fn element metod due to smootness of sape functon and sape functon dervatves used, asde from ts obvous advantage tat absolul no elements need to be defned n problem doman dscretzaton. Wt advantages and room for mprovement, mesless metods deserve more researc and atnton as an alrnatve to wdel-known fn element metod. 6. REFERENCES 1. Beltscko, T., Lu, Y.Y. and Gu, L., Element-Free Galerkn Metods. nrnatonal Journal for Numercal Metods n Engneerng; Vol. 37, 199, pp Atlur, S.N. and Zu, T., A New Mesless Local Petrov-Galerkn (MLPG) Approac n Computatonal Mecancs. Computatonal Mecancs; Vol. 22, 1998, pp Oña, E., delson, S., Zenkewcz, O.C., and Talor, R.L., A Fn Pont Metod n Computatonal Mecancs. Applcatons for Convectve Transport and Flud Flow. nrnatonal Journal for Numercal Metods n Engneerng; Vol. 39, 1996, pp Lancasr, P. and Salkauskas, K., Surfaces Generad b Movng Least-Squares Metods. Mamatcs of Computaton; Vol. 37, 1981, pp Sutanto, J.C., Wbowo, F.A., Pudjrad, P., and Tanojo, E., Pengenalan Metode Mesless Sebaga Alrnatf dar Metode Elemen Hngga untuk Menelesakan Problem Elaststas, PETRA Crstan Unverst Tess, Surabaa, Tanojo, E., An ratve Movng Least-Squares Formulaton For Problems n 2D Elastostatcs, Asan nsttu of Tecnolog Tess, Taland, Oña, E., Perazzo, F., and Mquel, J., A Fn Pont Metod for Elastct Problems. Compurs and Structures; vol. 79, 2001, pp Tmosenko and Gooder, Teor of Elastct. McGraw-Hll Book Compan, New York, Atlur, S.N. and Zu, T., Te Mesless Local Petrov-Galerkn (MLPG) Approac for Solvng Problems n Elasto-statcs. Computatonal Mecancs; Vol. 25, 2000, pp Compurs and Structures, nc.. CS Analss Reference Manual For SAP2000, ETABS, and SAFE., Compur and Structures, nc., Calforna,