1 Isoparametric Concept

Transcription

1 UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric concpt in on-dimnsion is a mthod of standardizing th ncssary computations to build th stiffnss matrix and th righthand sid forcing vctor: kij = dni Ω dx AE dn j dx dx () fi = Ni b dx. Ω () Hr Ni (x) is th shap function. In th basic st up, on uss Lagrang intrpolation polynomials for ths functions and th unknown u h (x) = i uh i Ni (x) for x Ω. In th isoparamtric stting w first dfin shap functions ovr th fixd parnt domain [, ]. Ths functions ar dnotd as N i (ξ) and ar dtrmind by way of th Lagrang intrpolation formula applid to th parnt lmnt. In th parnt lmnt th nods ar always qually spacd. To b abl to valuat () and () w nd, howvr, Ni (x). In th isoparamtric stting ths ar dfind by postulating a connction btwn th parnt lmnt and th physical lmnt. Th connction is known as th isoparamtric map, x (ξ); w plac th suprscript on th x to rmind us that th mapping is lmnt-bylmnt. Graphically th map is simply a point mapping from [, ] to Ω ; s Fig. for th quadratic isoparamtric mapping cas. Mathmatically, th isoparamtric map for an ξ( x ) x ξ x x x - 0 x (ξ) Figur : Isoparamtric mapping, x(ξ), for th quadratic lmnt.

2 lmnt is dfind by th rlation x (ξ) := i x i N i (ξ). () Th rquirmnt on th mapping is that it b on-to-on and hnc invrtibl. For us this can b rprsntd by th rquirmnt j := dx /dξ > 0 for ξ (, ). With () dfind w can now dfin th shap functions ovr th physical domain as Rmarks N i (x ) = N i (ξ(x )) = N i (ξ) ξ(x ). (4). Not that (4) mploys th invrs of th isoparamtric map, hnc th arlir statmnt that th map nds to b invrtibl.. If th isoparamtic map turns out to b linar for a particular lmnt, thn shap functions (in (4)) ovr th physical lmnt turn out to just b our original Lagrang polynomial shap functions. Howvr is th mapping is not linar, thn th shap functions in (4) ar slightly diffrnt.. In th cas whr th shap functions ovr th physical lmnt ar no longr Lagrang polynomials, th primary rquirmnt for our rror stimats to still hold is that thy b abl to xactly rprsnt arbitrary linar functions ovr a singl lmnt (whn daling with problms with at most first drivativs in th wak form). This is oftn call th compltnss rquirmnt (for problms with singl drivativs in th wak form). 4. Obsrv that th unknown fild ovr a singl lmnt is u h (x ) = i uh i N i (x ). Whn paramtrizd ovr th parnt lmnt this xprssion bcoms u h (ξ) = i uh i N i (ξ), which has th xact sam form as (). This is th origin of th trminology: isoparamtric. Th paramtrization of th gomtry and unknown fild is th sam.. Consqunc If w us th rlations outlind abov and chas through th chain rul w com to th final xprssions kij = dn i [,] dξ (ξ)ae dn j dξ (ξ) dξ j (5) fi = N i (ξ)b j dξ. (6) [,] All th ntris ar standard/uniform for all lmnts. Th lmnt gomtry is compltly containd in th Jacobian: j.

3 . Exampl: Linar Isoparamtric Elmnt Considr a gnric linar lmnt with nods at x and x. Th isoparamtric shap functions ovr th parnt domain ar givn by N (ξ) = ξ and th shap function drivativs ar givn by N (ξ) = + ξ (7) Th lmnt Jacobian is givn by dn dξ (ξ) = dn dξ (ξ) =. (8) j = dx dξ = dn x dξ (ξ) + dn x dξ (ξ) = x x = h, (9) whr h is th physical lngth of th lmnt. Assuming that b and AE ar constants, this dlivrs th rsult that for all lmnts kij = AE [ ] (0) h ( ) fi = bh. (). Exampl: Quadratic Isoparamtric Elmnt Considr a quadratic isoparamtric lmnt with nods at x, x = (x + x )/ and x. Not th uniform spacing of th physical nods should giv us a linar isoparamtric map. (In th nxt xampl w will trat th cas whr th physical nods ar not uniformly spacd.) In this cas th nods in th parnt domain ar locatd at -, 0, and. Th rsulting shap functions ovr th parnt domain ar N (ξ) = ξ(ξ ) N (ξ) = ( ξ)( + ξ) N (ξ) = ξ(ξ + ). () Th drivativs of th shap functions ar dn dξ = ξ / dn dξ = ξ dn dξ Using th givn (uniform) nodal spacings, th isoparamtric map is linar: = ξ + /. () x (ξ) = x i N i (ξ) = x ξ(ξ ) + x + x ( ξ ) + x ξ(ξ + ) i= ( ) = x x + x + x ξ + x x ξ + x + x = h ξ + x + x (4)

4 Plugging in w find kij = AE h f i = bh / / (5). (6) Rmarks. In ths two xampls th isoparamtric map is linar and thus th rsulting (physical domain) shap functions ar just th Lagrang shap functions..4 Exampl: Quadratic Isoprimtric Shap Functions To apprciat th last rmark, considr th cas of x = 0.0, x = 0.6, and x =.0. Th spacing of th physical nods is no longr uniform. In this cas, and th invrs function is givn by x (ξ) = ξ( + ξ)/ + 0.6( ξ ) (7) ξ(x) = 0.5 ( x ). (8) If on plots th standard Lagrang shap functions ovr this lmnt and compars thm to th isoparamtric ons ovr th sam physical lmnt (i.. plot (4)), on ss that thy diffr from ach othr. Both sts possss th Kronckr proprty but thy ar slightly diffrnt from ach othr. Shap Function x N (x) N (ξ(x)) N (x) N (ξ(x)) N (x) N (ξ(x)) Figur : Isoparamtric shap functions vrsus Lagrang shap functions whn x (x + x )/. Cas shown is for x = 0.0, x = 0.6, and x =.0 4

5 Th diffrnc in th shap functions also rsults in diffrnt lmnt forc vctor and stiffnss matrix. For th Lagrang cas applid to this lmnt on has th rsults shown in (5) and (6) with h =. If howvr w us th isoparamtric shap functions ovr this lmnt on finds kij = AE fi = b which ar clarly diffrnt from (5) and (6) with h =. Rmarks (9), (0). Dspit th diffrncs, th isoparamtric concpt still lads to a valid and convrgnt finit lmnt solution. Furthr, in multi-dimnsional problms it prmits on to gnrat shap functions that can b usd on gnral gomtris. Not that th Lagrang shap functions hav critical failurs in two- and thr-dimnsions for gnral shapd lmnts th isoparamtric concpt is ndd. 5