Finite Element Analysis

Transcription

1 Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis

2 Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis

3 Wak Form Wak Form Fin u among th smooth functions that satisfy u l u such that l l w u AE l wat wb w with w EGR 54 Finit Elmnt Analysis 3

4 Continuity Smoothnss of wight functions an trial solutions A function is call a C n function if its rivativs of orr j for j n ist an ar continuous functions in th ntir omain. A rivativ of a C n function is C n- If th isplacmnt is a C function, strain is C - C continuity is sir C First rivativ is continuous cpt for a crtain point FEA mploys C functions 4

5 U Amissibl A trial solution that is smooth an satisfis th ssntial bounary conitions A wight function that is smooth an vanishs on th ssntial bounary conitions u u H, u u on t u Γ u U() is smooth nough an satisfis th ssntial bounary conition H : a st of functions that ar smooth, thy ar C continuous H C t u h traction bounary an isplacmnt bounaris Complmntary: t t u u, t u atural an ssntial BCs cannot b appli at th sam bounary points Any bounary is ithr an ssntial or a natural bounary an thir union is ntir BC EGR 59 Finit Elmnt Analysis 5

6 Construction of rial Solutions an Wight Functions You hav wak form ray, still you n to construct propr trial solutions an wight functions for ach lmnts on th omain in orr to fin solutions to your problm For th wak form, it is oftn ifficult to o th intgration, thus numrical intgration mtho such as Gauss Quaratur will b appli Important to unrstan th abov EGR 59 Finit Elmnt Analysis 6

7 In oay s Lctur Lagrang intrpolants Shap fns for iniviual lmnt rial solutions (shap/wt fns) for no lmnt rial solutions (shap/wt fns) for 3 no lmnt Dirct construction of shap functions (bas on th proprtis of shap fns- ronckr lta w may conclu whn constructing Lagrang intrpolants Global shap fns construction Chaptr 3 Wak Form Gauss Quaratur EGR 54 Finit Elmnt Analysis 7

8 Som Dfinitions Approimations: Wight functions rial solutions w() u() θ A continuous approimation (.g. isplacmnt, tmpratur tc.) for lmnt at θ θ h = α + α + α + A continuous global approimation θ ( )=θ h noal valu of approimation function for lmnt at EGR 59 Finit Elmnt Analysis 8

9 wo o Linar Elmnt Choos a linar polynomial to approimat th approimations oal valu In matri form p Goal: Eprss approimation (i.. isplacmnt) in trms of noal valus EGR 54 Finit Elmnt Analysis 9 p

10 Eprss th cofficints in trms of wo o Linar Elmnt M In matri form oal valus M oal valus EGR 54 Finit Elmnt Analysis

11 wo o Linar Elmnt Substituting M into p p yils p M Shap Function Etrmly important to FEA Approimat ion Shap function oal valus EGR 54 Finit Elmnt Analysis

12 h shap function now can b prss as Eprssions for Shap Fn l l M p M l M l l EGR 54 Finit Elmnt Analysis Shap functions for two-no lmnt

13 3 What Shap Functions Look Lik? l l l l J I if J I if IJ J I Important proprty of shap fns: For any shap fn I, it must b vanish (to b zro) at all nos othr than no I. Eg. at no I=, At no I= j I Shap Function # o # EGR 54 Finit Elmnt Analysis

14 Shap Functions Approimation such as isp. fil umbr of lmnt nos n I I n I oal valus such as noal isp Shap fns # of shap fn=# of lmnt nos Whn n =3, th shap functions ar S sc 4. on P8 3 3 l EGR 54 Finit Elmnt Analysis 4

15 Drivativ of Approimation B Do not forgt that wak form w vlop bfor ns hlp from approimat functions But in wak form, w hav first rivativ of u w th trial solutions an wight function First rivativ of approimation B l B EGR 54 Finit Elmnt Analysis st rivativ of shap function is B 5

16 Quaratic D Elmnt ow Lt s gt a littl mor compl by stuying Quaratic D lmnt with HREE nos EGR 54 Finit Elmnt Analysis 6

17 7 W will n a scon-orr polynomial approimation Quaratic D Elmnt α p In matri form p α α α l n n l l M p α p In matri form oal valus M M EGR 54 Finit Elmnt Analysis

18 8 Quaratic D Elmnt l n n l l M p α p Whn n =3, th shap functions ar l 3 3 EGR 54 Finit Elmnt Analysis

19 h shap fns vlop by th prvious mtho is call Lagrang intrpolants Any way that w may construct th shap functions irctly? EGR 54 Finit Elmnt Analysis 9

20 Dirct Construction of Shap Fns Bas on th proprtis of shap fns: For any shap fn I, it must b vanish (to b zro) at all nos othr than no I. I J IJ h most gnral form of quaratic polynomial can b prss as a b c ow our task bcoms to fin right a, b an c such that satisfis th ronckr lta Proprty I I if if I I J J EGR 54 Finit Elmnt Analysis

21 Dirct Construction of Shap Fns For th shap fn has to mt th rquirmnts: If w can asily conclu that a an b 3 a an b look goo!!, 3 c 3 t w n to trmin a goo c, that is to mak 3 3 c For th othr shap fns, w can o th sam thing 3 EGR 54 Finit Elmnt Analysis

22 Approimation of Wight Fn Common practic: o us th sam approimation for th wight functions an trial solutions n n I I w w I B w B w EGR 54 Finit Elmnt Analysis

23 Global Approimation ill now, w ar aling with LOCAL, but vntually w will n GLOBAL w notations h w h Global approimation of Global approimation of By gathring th iniviuals th trial solutions th wight functions L w L w nl n l nl n h l h L w w L w EGR 54 Finit Elmnt Analysis 3

24 Global Shap Functions n l æ n q h = å l ö n = å L l n l h ç w w L w = è = ø Global Shap Fn h n l l I I w h n l l I w I In Vctor notation Global noal isp EGR 54 Finit Elmnt Analysis 4

25 5 o bttr unrstan, lt s amin on ampl Eampl of Global Approimation L L L L EGR 54 Finit Elmnt Analysis

26 Eampl of Global Approimation Global shap function 6

27 Gauss Quaratur Sinc w now hav approimations for trial solutions an wight functions, w can plug into wak form an solv for unknowns, but th fact is: In gnral, th wak form cannot b intgrat in clos form Solution: numrical mtho-gauss Quaratur EGR 54 Finit Elmnt Analysis 7

28 Gauss Quaratur Lagrang intrpolants Shap fns for iniviual lmnt rial solutions (shap fns) for no lmnt rial solutions (shap fns) for 3 no lmnt?? Dirct construction of shap functions (bas on th proprtis of shap fns- ronckr lta w may conclu whn constructing Lagrang intrpolants Global shap fns construction Chaptr 3 Wak Form Gauss Quaratur EGR 54 Finit Elmnt Analysis 8

29 h Rol of Gauss Quaratur in FEA Finit lmnt procss will vntually hav such an intgral for stiffnss quations (structural mchanics prob. as an ampl) D b a I f D 3D I f,, I f, h intgrals is gnrally valuat numrically by Gauss Quaratur EGR 54 Finit Elmnt Analysis 9

30 Gauss Quaratur D Consir a gnral intgral blow: As w all know th Gauss quaratur formulas ar givn ovr a parnt omain [-,] h solution is to map th parnt omain [-,] to th physical omain [a, b] b I f a b Mapping I f I f a EGR 54 Finit Elmnt Analysis 3

31 Gauss Quaratur D Mapping Whn a b b a is a function of a b b a a a b b a a b b a b a b b a Rwrit th prssion a b Shap functions? a b b a a b EGR 54 Finit Elmnt Analysis 3

32 Gauss Quaratur D l a b b a b a J ow An b a iffrntial I f f J g J whr g f n n n i i i... f W g W g W g W g Jacobian Gaussian formula whr W ar th wights an ar th points at which th intgran is to b valuat i i hrfor I g J JWig n i i EGR 54 Finit Elmnt Analysis 3

33 Gauss Quaratur n gp n h numbr of intgration points n to intgrat a polynomial of orr p actly is: n gp p + EGR 54 Finit Elmnt Analysis 33

34 Gauss Quaratur Position an Wight EGR 54 Finit Elmnt Analysis 34

35 Eampl 4. on tbook Pag 88 So Eampl a b b a f g b 5 3 I f a which inicats a=, b=5, f ( )= n I g J J Wig i J W g W g i 3 n gp p+ =3+ = Us tabl to fin out ths valus EGR 54 Finit Elmnt Analysis 35

36 Eampl n I g J J Wig i J W g W g i 3 3 J{ J{ S mor about this ampl in ttbook P89 EGR 54 Finit Elmnt Analysis 36

37 h followings ar contnts in Chaptr 5 in Fish s ttbook EGR 54 Finit Elmnt Analysis 37

38 l w AE u D FEA Formulation Wak form for th D lastic bar l w otic that w is a scalar w b = w w ta w with wl h finit lmnt functions w, u an thir rivativs hav kinks an jumps at th lmnt intrfacs, rplac th intgral ovr th omain, l by th sum of th intgrals ovr iniviual lmnt omain,. Just lik iscrtization n l w A E u w b w ta EGR 54 Finit Elmnt Analysis 38

39 D FEA Formulation Submitting th approimations w u w,, u w B into th rorganiz wak function w B n l w B A E B b A t f Ω f Γ Etrnal boy forc matri EGR 54 Finit Elmnt Analysis Etrnal bounary forc matri 39

40 4 h two most important Eqs in FEA Elmnt Stiffnss an Etrnal Forcs Matrics B E A B B E A B t A t b A t b f EGR 54 Finit Elmnt Analysis Elmnt stiffnss matri Elmnt forc matri

41 Elmnt Matrics for -no Elmnt Dtrmin lmnt stiffnss matri using Eq in prvious sli EGR 54 Finit Elmnt Analysis 4

42 Elmnt Matrics for -no Elmnt EGR 54 Finit Elmnt Analysis 4

43 Elmnt Matrics for -no Elmnt t fin forc matri EGR 54 Finit Elmnt Analysis 43

44 D FEA Formulation Global stiffnss an forc n l L L n l f L f EGR 54 Finit Elmnt Analysis 44

45 45 Global systm of quations D FEA Formulation Global Systm Eqs f f r f u u u u f u f u u 3 3 u u u f r EGR 54 Finit Elmnt Analysis

46 rminology for FE Matrics EGR 54 Finit Elmnt Analysis 46

47 Discrt Eqs for Arbitrary BCs Gnralization By introucing arbitrary BCs EGR 54 Finit Elmnt Analysis 47

48 Discrt Eqs for Arbitrary BCs n n n l n n : lmnt omains, n n numbr of nos in lmnt numbr of lmnts valu of for lmnt at no valu of for lmnt at no n n C3D8 n n n l 3 () () EGR 54 Finit Elmnt Analysis 48

49 Discrt Eqs for Arbitrary BCs l w w AE AE u u l w w b b w ta w with wl w ta wu t u w w,, u w B w Ω: any limits of intgration such as [a,b] [,l] Γ t :prscrib traction bounary B EGR 54 Finit Elmnt Analysis 49

50 5 Discrt Eqs for Arbitrary BCs F w w t l A t b B A E B n F F E F E w w w, Partition th global solution an wight function as Introucing into th abov quation L L w w F n w f f L - L L w l n l EGR 54 Finit Elmnt Analysis

51 5 Discrt Eqs for Arbitrary BCs F w r w r f w r w r w r r w w F F E E F E F E W F is arbitrary ( natur of wight fn) r f f r r F F E F E F EF EF E E EGR 54 Finit Elmnt Analysis

52 5 Discrt Eqs for Arbitrary BCs F E EF F F F f E F EF E E E F E F E F EF EF E E f r f f r r B E E u E u EGR 54 Finit Elmnt Analysis Solv for F : Unknown raction

53 Eampl 5. on P99 EGR 54 Finit Elmnt Analysis 53

54 Eampl 5. on P7 EGR 54 Finit Elmnt Analysis 54

55 HW Problm 5.4 on Pag 4 in Fish s ttbook EGR 54 Finit Elmnt Analysis 55

56 Rfrncs A First Cours in Finit Elmnts Authors: Jacob Fish, Blytschko Publishr: Wily 7 h Finit Elmnt Mtho: Its Basis an Funamntals, Sith Eition Authors: Olk C Zinkiwicz, Robrt L aylor, J.Z. Zhu Publishr: Buttrworth-Hinmann 5 Finit Elmnt Procurs Author:.J. Bath Publishr: laus-jurgn Bath 7 Robrt Cook, Concpts an Applications of finit lmnt analysis,wily EGR 54 Finit Elmnt Analysis 56