Variational Approach in FEM Part II

Transcription

1 COIUUM & FIIE ELEME MEHOD aratonal Approach n FEM Part II Prof. Song Jn Par Mchancal Engnrng, POSECH

2 Fnt Elmnt Mthod vs. Ralgh-Rtz Mthod On wants to obtan an appromat solton to mnmz a fnctonal. On of th hstorcall famos appromat mthods for ths nd of problm s Ralgh-Rtz Mthod, and th othr modrn mthod s th Fnt Elmnt Mthod. () Fnd () to mnmz. ()

3 Fnt Elmnt Mthod vs. Ralgh-Rtz Mthod ) Ralgh-Rtz Mthod: ~ ( ) whr n c ( ) : appromat solton satsfng th ssntal B.C. () : tral fnctons (dfnd ovr th whol doman) ~ ( ) ( c, c,, cn to b mnmzd w.r.t. c. ~ ( ),,, n : n qatons for n nnown c s. c hn, ) hrfor, hs mthod s vr smpl and as to ndrstand. Howvr, t s not as to fnd a faml of tral fnctons for th ntr doman satsfng th ssntal bondar condtons whn gomtr s complcatd. h solton to ths troblsom pont can b fond n th Fnt Elmnt Mthod.

4 Fnt Elmnt Mthod vs. Ralgh-Rtz Mthod ) Fnt Elmnt Mthod ~ ( ) ( ) : nodalvals () : shapfnctons hn, ( ~ ) (,,, n ) to b mnmzd w.r.t.,,, n : n qatons for n nnown s. In ths cas, th shap fnctons can b fond mor asl than th tral fnctons wthot havng to worr abot satsfng th ssntal bondar condtons, whch mas FEM mch mor sfl than Rallgh-Rtz Mthod. In ths rgard, th Fnt Elmnt Mthod s as modrnzd appromaton mthod stabl for comptr nvronmnt.

5 Eampl: ght strng problm va two mthods w() () d d l ( ) w d to bmnmzd. * Spcal cas: w()=w (constant)

6 Eampl: ght strng problm va two mthods ) Ralgh-Rtz Mthod ~ A sn A sn l l (ot: ral fnctons satsf ssntal B.C.) A A l -w A l A l 4l w A, A 3 ~ 4l w sn 3 l ~ wl wl (ot:.9 vs..5 )

7 Eampl: ght strng problm va two mthods ) Fnt Elmnt Mthod d ( to b mnmzd s qvalnt to d l ) w d l d d w d d d for an δ. Introdc lmnts to th sstm as dpctd blow l,,3 : lmnt nmbr,, 3, 4 : nod nmbr l : local coordnat l

8 Eampl: ght strng problm va two mthods Introdc th appromat solton va ntrpolaton fnctons (or shap fnctons) for ach lmnt. ( ) ( ) hn, d d d d ( ) d d d d d d and th varaton of fnctonal ovr ach lmnt s smmd to rslt n th varaton of th whol sstm,.., () () (3)

9 Eampl: ght strng problm va two mthods Lt s consdr an lmnt for l d d w dl d d (hncforth l for convnnc) l d d K l f d d d d f f w d d (n an ndcal form) K (n a matr form) d d d l w

10 Eampl: ght strng problm va two mthods whr K l d d d d d : lmnt stffnss matr f l w d : wor - qvalnt nodal forc smmaton K F for an K F

11 Eampl: ght strng problm va two mthods Lnar lmnt for smplct ( ) ( ) ( ) ( ) ( ) ( l l ) ( ) ( ) d d d d d d l d d d d, d l d l

12 l K l wl d w f / / wl l Global matr qaton B.C. = 4 = wl 3 9 Eampl: ght strng problm va two mthods

13 Eampl: ght strng problm va two mthods ots:. st and 4 th qatons ar not to b sd (or obtand mor prcsl) snc and 4 ar not arbtrar, bt zro. As a mattr of fact, howvr, ntrodcton of bondar condtons rplacs thos qatons. h racton forc F and can b obtand from th st and 4 th F4 qatons, rspctvl.. Advantags of varatonal approach ovr th drct on: ) Us of scalar qantt (nrg) vrss vctors, ) Eas n tratmnt of dstrbtd load 3. ratmnt of concntratd loads: F c c

14 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton * Stad stat hat condcton, flow thogh poros mda, torson, tc. S S z nˆ n ),, ( z f z z z B.C. on ),, ( ),, ( on ),, ( S z h z g n z n n S z z z

15 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton h abov partal dffrntal qaton wth bondar condtons s qvalnt to th followng va ratonal prncpl: Mnmz.. J for an.

16 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton Wth man lmnts ntrodcd, on can sm th contrbtons of ach lmnt to th fnctonals as dscrbd blow: J J J J Introdc th appromat solton n trms of shap fnctons and nodal vals of ovr ach lmnt: S

17 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton

18 S S z ds h ds f z z J g S f S C R R K K J FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton

19 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton whr K C z z z (stffnss matr d to condcton) K S h S ds (forcng matr d to convcton) R f f (forcng matr d to dstrbtd hat sn) R S g ds S (forcng matr d to dstrbtd hat otfl)

20 FEM for Scond-Ordr Ellptc Partal Dffrntal Eqaton Aftr th assmbl procdr, on can obtan J J K K R R C S f S for an arbtra, wth for nods on S. K K R R C S f S K F K K C K S F R f R S forcng matr d to hat sorc and hat fl

21 aratonal Prncpl for Dformaton A. Prncpl of rtal Dsplacmnt (Wor) For a gvn statcall admssbl strss fld, consdr an nmatcall admssbl vrtal dsplacmnt and trnal wor d to th vrtal dsplacmnt. t n qlbrm stat, f : vrtal dsplacmnt

22 f f ds n f ds t W t whr,, aratonal Prncpl for Dformaton

23 aratonal Prncpl for Dformaton Prncpl of vrtal dsplacmnt W t t ds f Phscal manng t ds f Fnd An vrtal dsplacmnt f, t, f ar n qlbrm Prncpl of vrtal vloct (powr) W t t t t ds f t t

24 aratonal Prncpl for Dformaton ots:. Knmatcall admssbl vrtal dsplacmnt whr prscrbd.. Statcall admssbl strss fld satsfs not onl th qlbrm qaton bt also th prscrbd tracton bondar condton (.., natral bondar condton),.. t n t 3. Prncpl of vrtal vloct (powr) W t t 4. If nrtal trm s ncldd n th bod forc trm, th prncpl of vrtal dsplacmnt can b tndd to fcttos qlbrm stat. t t v f v ds f t v t ds δd t ( can b rplacd wth rspctvl.), v, d

25 aratonal Prncpl for Dformaton B. Prncpl of Mnmm Potntal Enrg Prncpl of vrtal dsplacmnt W t t ds f For an lastc bod, thr sts stran nrg dnst, U o, sch that U o thn U o U o U o U

26 aratonal Prncpl for Dformaton whr U U o : stran nrg W t U Dfn th potntal nrg as hn W t hrfor t ds f wth fd U t and ρf

27 aratonal Prncpl for Dformaton Dfnng th total potntal nrg p as U lds p p In smmar, th dformaton of an lastc bod (lnar or nonlnar) s govrnd b mnmzng th fnctonal, th total potntal nrg, p U o tds f Wth U o and,, ot: Prncpl of rtal Wor s vald for an matral, whras Prncpl of Mnmm Potntal Enrg s vald onl for lastc matrals.

28 aratonal Prncpl for Dformaton C. Prncpl of Complmntar rtal Wor Consdr a varaton of statcall admssbl strss fld and trnal forcs f, t whl png nmatcall admssbl dsplacmnt. f, n W t t ds, n on S Dfn a complmntar vrtal wor * ( ) n ds f, f f as * W d W * d W t

29 aratonal Prncpl for Dformaton ow as a contrpart to th Prncpl of Mnmm Potntal Enrg for an lastc matral, consdr th cas for an lastc matral for whch a complmntar stran nrg dnst sts as blow: U * o (a constttv qaton) hn w can rwrt th rght hand sd of th prsson for * W as W * U * o U * o U * o U and th lft hand sd can b rwrttn n trms of th complmntar potntal nrg as follows * * t ds f wth * t ds f (png fd)

30 aratonal Prncpl for Dformaton hrfor, on can hav * * *, t, f U <otal Complmntar potntal nrg> * For an statcall admssbl strss, forc sstm ot: * p,t, f : lads to dsplacmnt-basd FEM ldng a stffnss matr : lads to qlbrm-basd FEM ldng a flblt matr

31 Dsplacmnt-basd FEM for Elastct Lt s appl th prncpl of vrtal dsplacmnt (or prncpl of mnmm potntal nrg) to th two-dmnsonal lastc dformaton problm. Prncpl of rtal Dsplacmnt (Wor): t ds f p t ds f (wth, t, f p fd) Prncpl of Mnmm Potntal Enrg: p U o tds whr U aratonal prsson: o,, p t ds f ( ) ( ) f F

32 Dsplacmnt-basd FEM for Elastct In ths scton, lt s ncld th nrta forc and concntratd forcs as th most gnral FEM formlaton for lastct. So, consdr th followng varatonal prsson as th startng form. p t ds ( ) ( ) f F

33 v ), ( ), ( v v v v v Dsplacmnt appromaton va shap fnctons Acclraton 3 Dsplacmnt-basd FEM for Elastct

34 , C Stran matr:.g. Strss-Stran rlaton (constttv law): Dsplacmnt-basd FEM for Elastct

35 strss plan for E C stran plan for ) )( ( E C C U ) ( ) ( p F f ds t C ) ( ) ( p F f ds t C and th total potntal nrg bcoms Dsplacmnt-basd FEM for Elastct ot: Wth ths notaton, th stran nrg can b rprsntd as

36 Dsplacmnt-basd FEM for Elastct ot: In cas of ntal stran nt, s to b rplacd wth nt. In cas of ntal strss, nt Dfn th forc matrcs as follows: C t t t : tracton forc f F ( ) f f F F ( ) ( ) : bod forc : concntratd forc appld at -th poston

37 v ), ( ), ( p p p F F v ) ( ) ( ) ( ) ( ) ( f ds t C f ds t C S S p rtal dsplacmnt and corrspondng stran: araton of total potntal nrg: For an lmnt: Dsplacmnt-basd FEM for Elastct

38 b d S p m F F K f ds t C M F F F K c b d p an for Introdcng th matr notatons for appromatd dsplacmnt fld and constttv law and so on nto th abov qaton gvs: Aftr assmbl, on fnall obtans: Dsplacmnt-basd FEM for Elastct

39 Dsplacmnt-basd FEM for Elastct M K F F F F d b c whr m K C F t d S ds F f b ( ) F C F c : lmnt mass matr : lmnt stffnss matr : wor qvalnt nodal forc : wor qvalnt nodal forc : concntratd forc p K F s th fnctonal to b mnmzd. rslts n K F.) p