An Improved Isogeometrc Analyss Usng the Lagrange Mltpler Method N. Valzadeh 1, S. Sh. Ghorash 2, S. Mohammad 3, S. Shojaee 1, H. Ghasemzadeh 2 1 Department of Cvl Engneerng, Unversty of Kerman, Kerman, Iran, navd.valzadeh@gmal.com, saeed.shojaee@mal.k.ac.r 2 Department of Cvl Engneerng, KNToos Unversty of Technology, Tehran, Iran, s.sh.ghorash@gmal.com, ghasemzadeh@knt.ac.r 3 School of Cvl Engneerng, Unversty of Tehran, Tehran, Iran, smoham@t.ac.r 1Introdcton Isogeometrc analyss (IGA) s a novel comptatonal approach recently developed by Hghes et al. [1]wth the am of ntegratng compter aded desgn (CAD) nto strctral analyss. It ses non nform ratonal B-splnes (NURBS) for both descrpton of the geometry and approxmaton of thesolton feld. NURBS are the most common bass fnctons n the CAD systems. Usng CAD bass fnctons drectly n the fnte element analyss (FEA), leads to elmnate the exstng tme-consmng data converson process between CAD systems and fnte element packages n engneerng problems. Ths s one of the man deas behnd developng sogeometrc analyss and vastly smplfes mesh refnement of complex ndstral geometres [1].Althogh IGA has attracted many research nterests and sccessflly has been appled n dverse engneerng problems, bt t stll sffers from a nmber of drawbackswhch reqre frther nvestgaton. One of the key challenges wth IGA s the mposton of essental bondary condtons; hence, some specfc technqes have to be sed. As the NURBS bass fnctons n sogeometrc analyss,smlar to many meshfreebass fnctons, are not nterpolatory; they do not satsfy the kronecker-delta property. Therefore, mposton of nhomogeneos essental bondary condtons s not a straghtforward task, as t s the case n a homogeneos one.to mpose the essental bondary condtons n nmercal methods wth non-nterpolatory bass fncton, several strateges have been proposed [2]. Snce the Lagrange mltpler method [3] has been wdely appled n varos engneerng smlatons, t s now selected for mprovng the mposton of essental bondary condtons n IGA. 2 NURBS-based sogeometrc analyss A 2-D lnear elastcty problem wth the presence of body forces b and tracton forces t s consdered. The strong form eqaton and the bondary condtons are as follows:. σ b 0 n σ. n t on t (1-a) (1-b) on (1-c) where 0 and 0 represent the homogenos and nhomogeneos bondary condtons respectvely.
Implementng the vrtal dsplacement method, the followng weak form eqaton s then obtaned: T T T εσd bd td0 t where σ s the stress tensor and ε s the stran tensor. In sogeometrc approach, the dscretzaton s based on NURBS. Hence, the geometry and solton feld are approxmated as, np x(, ) R(, ) P, 1 np 1 patch h (, ) R(, ) d, patch where R (, ) s the NURBS bass fncton, P s the coordnate poston of the th control pont and d s the dsplacement vector. Ths yelds the element patch stffness matrx T K (, ) (, ) (5) B DB J dd e e 3Imposng essental bondary condtons It s well worth notng that drect mposton of homogeneos essental bondary condtons to the control varables, as dscssed n [1], has no dffclty, bt mposng nhomogeneos essental bondary condtons needs new developments [1]. For ths reason, the Lagrange mltpler method s employed n ths stdy as a scheme for treatment of essental bondary condtons. In ths method, sng the Lagrange mltpler λ, the essental bondary condton (Eq. 1-c) s converted nto an ntegral form: T λ ( ) d (6) Therefore, the standard weak form (2) s changed to nclde two new terms, T T T T T εσd bd td λ ( ) d λd0 t To obtan the dscretzed eqaton from ths weak form, the Lagrange mltplers mst be dscretzed. As dscssed n [2], among several possbltes for the choce of the nterpolaton space for the Lagrange mltpler, the NURBS and Lagrange shape fnctons can be chosen. In ths stdy, Lagrange mltplers are nterpolated on essental bondary sng the Lagrange shape fnctons. The Lagrange mltplers are defned at several essental bondary ponts. An approprate choce for these desred essental bondary ponts s the Grevlle abscssas [4], defned as, p 1... p g, 1,..., n (8) p where p s the NURBS order and n s the nmber of control ponts. Interpolatng the Lagrange mltplers sng 1D Lagrange shape fnctons on physcal bondary ponts correspondng to Grevlle abscssas, leads to n ( ) N ( ) (9) 1 where N( ) s the 1D Lagrange bass fncton and n s the nmber of the essental bondary ponts appled for ths nterpolaton. After some mathematcal manplaton, the dscretzed form of Eq. (7) can be descrbed as (2) (3) (4) (7) 2
K G f T (10) G 0 λ q where K and f are stffness matrx and force vector, respectvely, and the nodal matrx G j and the vector q are defned as: G nt j N ( ) R j (, ) dn ( k ) R j ( k, k )det( J( k )) w k k 1 q N ( ) d where R j(, ) s the 2D NURBS bass fncton, nt s the nmber of Gass ntegraton ponts, k denotes the poston of Gass pont n the parent element obtaned sng the Gass qadratre rle for ntegraton on a crve bondary and ( k, k ) s the poston of Gass pont n the parametrc space. 4Nmercal example Consder the potental problem on a nt sqare doman. The governng eqaton and bondary condtons are gven as follows: s 0 n n. t on t (13) on where s s the sorce term. The sorce term for ths problem s gven as: (11) (12) sx ( ) ysn( x) (14) The analytcal solton s also obtaned as, ( x) ( xsn( x)) y x[0,1] 2 (15) Prescrbed Drchlet bondary condtons are calclated from the exact solton and mposed on all sdes of the sqare. In ths problem, no tracton s consdered. The ntal geometry s constrcted by tensor prodct of qadratc NURBS bass fnctons. As all the weghts are assmed to be nty for ths problem, NURBS degenerates to B-splnes. The ntal parameterc space s gven by two knot vectors of the same form:0,0,0,0.5,1,1,1. Usng h- refnement strategy, meshes wth 16, 64, 256 and 1024 elements are consdered for the convergence stdy. The sogeometrc solton and the absolte error for the Lagrange mltpler method on a model wth 256 elements are plotted n Fg. 1. The L 2 and H 1 error norms are plotted n Fg. 2. As expected, cbc and qadratc convergence rates for the L 2 and H 1 error norms are obtaned for the Lagrange mltpler method (see Fg. 2), whereas only a qadratc convergence s obtaned wth the drect mposton of essental bondary condtons.it can be clearly observed that the Lagrange mltpler method presents speror accracy and hgher rate of convergence n comparson wth the drect method. 5 Conclson In ths paper, an effcent technqe based on the Lagrange mltpler method s sggested for mposton of essental bondary condtons n NURBS-based sogeometrc analyss. In ths approach, the Lagrange mltplers are defned on a set of physcal bondary ponts. In general, these nterpolaton ponts are Grvlle abscssas on the essental bondary. Among the 3
varos possble choces of the nterpolaton space for the Lagrange mltplers, the Lagrange shape fnctons have been adopted. The reslts of several 2D smlatons have llstrated that excellent solton accracy and rate of convergence are obtaned by ths approach. It also offers a far hgher rate of convergence n comparson wth the drect mposton of essental bondary condton. Fg. 1. Isogeometrc solton wth 256 elements (Left) and absolte error (Rght) based on the Lagrange mltpler method. References Fg. 2.Comparson of L 2 (Left) and H 1 error norm (Rght). [1] T.J.R. Hghes, J.A.Cottrell,Y.Bazlevs,Isogeometrc analyss: CAD, fnte elements, NURBS, exact geometry and mesh refnement, Compter Methods n Appled Mechancs and Engneerng, 194 (39-41): 4135 4195, 2005. [2] S.Fernández-Méndez, A. Herta,Imposng essental bondary condtons n mesh-free methods, Compter Methods n Appled Mechancs and Engneerng, 193(12-14): 1257 1275, 2004. [3] T.Belytschko, Y.Y. L, L.G, Element free Galerkn methods, Internatonal Jornal for Nmercal Methods n Engneerng, 37 (2): 229 256, 1994. [4] J.Hoschek, D.Lasser, Fndamentals of Compter Aded Geometrc Desgn, A.K. Peters, Ldt, Wellesley, Massachsetts,1993. 4
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