CIVL 7/ D Boundary Value Problems - Quadrilateral Elements (Q8) 1/9

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1 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES (Q) h nx n our lmn dvlopmn logcl xnon of h qudrlrl lmn o qudrclly nrpold qudrlrl lmn dfnd by gh nod, four h vrc nd four h mddl ch d. h mddl nod, dpndng on locon, my dfn rgh ln or qudrc ln. EIGH-ODE QUADRILAERRAL ELEMES (Q) rnformon nd Shp Funcon - hr r wo pproch o dvlop h nrpolon or hp funcon for lmn. h fr pproch bd on rprnng h gomry nd h dpndn vrbl funcon of h globl coordn x nd y. Q Q EIGH-ODE QUADRILAERRAL ELEMES (Q) rnformon nd Shp Funcon - hr r wo pproch o dvlop h nrpolon or hp funcon for lmn. h cond pproch bgn wh h prn lmn wh h nrpolon nd hp funcon xprd n rm of h locl r coordn. EIGH-ODE QUADRILAERRAL ELEMES (Q) For h fr pproch, condr rgh-dd rcngulr lmn hown blow: h vron of h dpndn vrbl u ovr h lmn my b xprd : Q, u x y bx cy dx xy fy gx y hxy EIGH-ODE QUADRILAERRAL ELEMES (Q) For h fr pproch, condr rgh-dd rcngulr lmn hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) For h fr pproch, condr rgh-dd rcngulr lmn hown blow: Fng h xpron for u o h dfnon of h gh-nod qudrlrl gvn bov rqur:, u x y bx cy dx x y fy gx y hx y h bov quon wrn for ch nodl vlu of x nd y rulng n gh quon n h gh unknown, b, c, d,, f, g, nd h.,,,

2 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES (Q) For h fr pproch, condr rgh-dd rcngulr lmn hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) For h fr pproch, condr rgh-dd rcngulr lmn hown blow: Solvng h of quon nd collcng rm n u rul n h nrpolon funcon. u x, y u u whr u = [ u, u, u, u, u, u, u, u ] h nrpolon funcon hv h propry h (x, y ) =, n ohr word, (x, y )= f =, nd zro lwhr. u x, y u u whr u = [ u, u, u, u, u, u, u, u ] EIGH-ODE QUADRILAERRAL ELEMES (Q) h gomry of h qudrlrl lmn my lo b dcrbd ung h bov nrpolon : x x x An oprmrc lmn my b formd by ung vlu of = whch u h nrpolon funcon gvn bov. Howvr, ubprmrc lmn my lo b dfnd by ng =. In h c, h nrpolon funcon dfnd for four qudrlrl r ud. y y y EIGH-ODE QUADRILAERRAL ELEMES (Q) h gomry of h qudrlrl lmn my lo b dcrbd ung h bov nrpolon : x x x A w found n h dvlopmn of h qudrc rngulr lmn, h globl coordn pproch no h mo ffcn mhod for dcrbng h nrpolon ovr h lmn. h gh nrpolon or hp funcon n globl coordn x nd y r mhmclly clumy nd rrly ud n FEM nly. y y y EIGH-ODE QUADRILAERRAL ELEMES (Q) h gomry of h qudrlrl lmn my lo b dcrbd ung h bov nrpolon : x x x An quvln form of h hp funcon my b drvd n rm of h locl prnl lmn coordn. h funcon hv rlvly mpl mhmcl form nd r mor ffcn n compung h lmnl mrc. y y y EIGH-ODE QUADRILAERRAL ELEMES (Q) A dcrbd bov, h cond pproch o dvlopng of nrpolon or hp funcon for n gh-nod qudrlrl lmn bgn wh h prn lmn n locl coordn. Condr h followng gh-nod qudrlrl n locl coordn nd. x x(,) y y (,) y x

3 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES (Q) In h prn lmn h nrpolon funcon r gvn :,,,,,,,, EIGH-ODE QUADRILAERRAL ELEMES (Q) h prn lmn nrpolon funcon (, ) hv wo bc hp. h bhvor of h funcon,,, nd r mlr xcp rfrnc dffrn nod. h hp funcon hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) h prn lmn nrpolon funcon (, ) hv wo bc hp. h bhvor of h funcon,,, nd r mlr xcp rfrnc dffrn nod. h hp funcon hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) h prn lmn nrpolon funcon (, ) hv wo bc hp. h cond yp of hp funcon vld for funcon,,, nd. h funcon hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) h prn lmn nrpolon funcon (, ) hv wo bc hp. h cond yp of hp funcon vld for funcon,,, nd. h funcon hown blow: EIGH-ODE QUADRILAERRAL ELEMES (Q) In mnnr dncl o h ud n vry lmn w hv dvlopd o fr, h nur of h rnformon from h prn lmn o h globl lmn, h chn rul ud o form h dffrnl rlonhp: x y x y x y x y In mrx noon, h drvv my b wrn : x y x x y y J x

4 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES (Q) h drmn of h Jcobn mrx J : x y x y x y y x J J x y x y x y y x EIGH-ODE QUADRILAERRAL ELEMES (Q) Whn J pov vrywhr n rgon, h rnformon my b nvrd o drmn = (x, y) nd = (x, y). h mn h for gvn pon (x, y) n h rngl hr unqu corrpondng pon (, ) n h prn lmn. h J(, ) mur of h xpnon or conrcon of dffrnl r: dx dy J, d d h drmnn of h Jcobn mrx, J, of h nvrbly of h rnformon x = x(, ) nd y = y(, ). EIGH-ODE QUADRILAERRAL ELEMES Exmpl Condr h gh-nod qudrlrl lmn gvn blow. (,) (,) (,) x EIGH-ODE QUADRILAERRAL ELEMES Exmpl Condr h gh-nod qudrlrl lmn gvn blow. (,) (,) (,) x (,) (,) y (,) (,) y (,) (, ) (,) (,) (, ) (,) h coordn rnformon gvn : x x y y h coordn rnformon gvn : x x y y EIGH-ODE QUADRILAERRAL ELEMES Exmpl Condr h gh-nod qudrlrl lmn gvn blow. (,) (,) (,) (,) (,) (,) (, ) (,) h rulng Jcobn : J, x y EIGH-ODE QUADRILAERRAL ELEMES Exmpl L condr vrl c whr h J my bcom ngv. h mppng nod h h vlu of = = -. Subung h corrpondng nd coordn for nod no h xpron for J gv: J, h rulng Jcobn : J,

5 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EIGH-ODE QUADRILAERRAL ELEMES Exmpl h J : J, whch zro = / nd /. For vlu of x nd y grr hn /, h drmnn of h Jcobn ngv. (,) (,) (.,) (,) (,) (,.) Inror ngl = 0 (,) (,) x y.. EIGH-ODE QUADRILAERRAL ELEMES Exmpl h J : J, o h for = / h nror ngl nod qul. (0.,) (,) (,) (,) (,) Inror ngl = (,0.) (,) (,) x y o h for = / h nror ngl nod qul 0. EIGH-ODE QUADRILAERRAL ELEMES Exmpl Drvd Vrbl - h drvd vrbl for h Poon or Lplc quon r h prl drvv u/x nd u/y. h rm r compud funcon of poon n n lmn ung h hp funcon nd h nodl vlu. h prl drvv my b combnd o gv h norml or drconl drvv: u nxj n J u n y EVALUAIO OF MARICES - Q ELEMES h lmnl mrc for h Poon problm r: k da f f da x x y y A A d h hd EVALUAIO OF MARICES - Q ELEMES h dvlopmn of boh h k nd f rm dncl o h prnd for ohr yp of lmn: k JJ whr JJ = (J J + J J ) J. d d f fda d d J f A A EVALUAIO OF MARICES - Q ELEMES h rm J nd J r h fr nd cond row of h nvr of h Jcobn mrx. J y y J x x whr J h drmnn of J. J y y x x J J J

6 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EVALUAIO OF MARICES - Q ELEMES h vlu of h mrx my b compud : EVALUAIO OF MARICES - Q ELEMES h k rm: k JJ For gnrl qudrc qudrlrl lmn, h x mrx JJ funconl of nd wh h Jcobn J n h dnomnor. h rulng xpron for JJ vry dffcul o vlu xcng nd h ngron r uully don numrclly. d d EVALUAIO OF MARICES - Q ELEMES hrfor, h rm k nd f my b c n h followng form: G, dd whr G(,) complcd funcon of h vrbl nd. In prncpl, my b pobl o vlu h f rm, howvr, numrcl ngron ypclly mor prccl. For h k rm, h pprnc of h Jcobn J n h ngrnd gnrlly ndc h u of numrcl qudrur. EVALUAIO OF MARICES - Q ELEMES hrfor, h gnrl xpron for k nd f r: k, JJ,, dd,, f J dd W cn u h Gun qudrur dvlopd for on dmnonl ngrl nd pply h pproxmon for boh nd. EVALUAIO OF MARICES - Q ELEMES h gnrl form for n -rm Gun qudrur :,, I G dd G w w whr, r h Gu pon nd w nd w r h corrpondng wgh. hrfor k nd f my b vlud by: d d w w k JJ JJ d d w w f J f J f EVALUAIO OF MARICES - Q ELEMES Condr h ngrl nd h : d h h d whr h ngron long boundry gmn of h lmn. Snc, h ngron compud long ngl d of h qudrc lmn, h nrpolon funcon r qudrc. m m l m

7 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 EVALUAIO OF MARICES - Q ELEMES Condr h ngrl nd h : d h h d whr h ngron long boundry gmn of h lmn. Snc, h ngron compud long ngl d of h qudrc lmn, h nrpolon funcon r qudrc. EVALUAIO OF MARICES - Q ELEMES h vron of x nd y funcon of long h boundry gvn : h dffrnl rc lngh dl : x x x x y y y y I J m m I J m m dl dx dy dl x y d h m h m l h mm h h h hmm l dl x y d x x y y x x xm x y y ym y EVALUAIO OF MARICES - Q ELEMES h ngrl dfnng nd h r: I J m m hd ld l d h h h rulng x lmnl lod vcor conrbu o h globl ym quon f h lmn h d pr of h boundry. EVALUAIO OF MARICES - Q ELEMES h globl ym quon r compod from h followng ummon: K k F f h G G G G G G h rulng ym quon r, n mrx form, gvn : Ku F G G G EVALUAIO OF MARICES - Q ELEMES PROBLEM # - Wr compur progrm ubroun clld QQUAD h clcul h componn of h k mrx for gnrl qudrc qudrlrl lmn ung Gun qudrur. h gnmn mlr o Problm # xcp n ghnod qudrlrl ud. Chck your work wh h problm n h your xbook on pg nd. Exmpl - Condr h m problm of oron of homognou oropc prmc br w olvd bfor, xcp ung ngl Q lmn. y Ln of Symmry (0,0) x (,0) cro-conl r x (,) y

8 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) /9 Exmpl - Rcll, h non-dmnonl Poon quon govrnng h problm., 0 n wh 0 on x y G h r nd orqu for h Prndl r funcon r: xz G yz G G dd Exmpl - Elmnl Formulon - Ung lnr rngulr lmn h lmnl ffn mrx componn r: For lmn : d d w w k JJ JJ k ymrc. Exmpl - Elmnl Formulon - h lodng funcon of f = gv r of lmnl lod vcor of: d d w w f J f J f For lmn : f Ambly - Snc hr on lmn mbly no dffcul: (,) (0,0) (,0) Exmpl - Conrn - For h modl, = 0 on h boundry, hrfor,,,,, nd = (,) Exmpl - Conrn - For h modl, = 0 on h boundry, hrfor,,,,, nd = Soluon - Solvng h bov quon gv: G (0,0) (,0)

9 CIVL / -D Boundry Vlu Problm - Qudrlrl Elmn (Q) 9/9 Exmpl - Conrn - For h modl, = 0 on h boundry, hrfor,,, nd = 0. Soluon - Solvng h bov quon gv: (,) (,) (0,0) (,0) Soluon - Solvng h bov quon gv: (0,0) (,0) Q lmn lmn (0,0) (,) (,0) (,) (0,0) (,0) Exmpl - Compuon of Drvd Vrbl - h ol orqu my b clculd : G dd G A G dd End of -od Qudrlrl Elmn (Q).0G.9G xc.g lmn