CIVL 8/ D Boundary Value Problems - Triangular Elements (T6) 1/8

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1 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () A quadracall nrpolad rangular lmn dfnd b nod, hr a h vrc and hr a h mddl a ach d. h mddl nod, dpndng on locaon, ma dfn a ragh ln or a quadrac ln. SI-ODE RIAGULAR ELEMES () h lmn ma b vr hlpful n modlng om gomr hr h u of a quadrlaral lmn ma rul n an undrabl dformaon of h lmn and cau problm n h Jacoban mappng. SI-ODE RIAGULAR ELEMES () ranformaon and Shap Funcon - hr ar o approach o dvlop h nrpolaon or hap funcon for h quadrac rangular lmn. h fr approach bad on rprnng h gomr and h dpndn varabl a a funcon of h global coordna and. SI-ODE RIAGULAR ELEMES () ranformaon and Shap Funcon - hr ar o approach o dvlop h nrpolaon or hap funcon for h quadrac rangular lmn. h cond approach bgn h h parn lmn h h nrpolaon and hap funcon prd n rm of h local ara coordna. SI-ODE RIAGULAR ELEMES () For h fr approach, condr a ragh-dd rangular lmn hon blo. h varaon of h dpndn varabl u ovr h lmn ma b prd a:, u a b c d f SI-ODE RIAGULAR ELEMES () Fng h pron for u o h dfnon of h -nod rangl gvn abov rqur:, u a b c d f,,

2 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () h abov quaon rn for ach nodal valu of and rulng n quaon n h unknon a, b, c, d,, and f. Solvng h of quaon gv h follong nrpolaon. SI-ODE RIAGULAR ELEMES () Whr u =[u, u, u, u, u, u ] and h nrpolaon funcon ar: u, u u hr = - and = -. SI-ODE RIAGULAR ELEMES () Whr u =[u, u, u, u, u, u ] and h nrpolaon funcon ar: hr = - and = -. SI-ODE RIAGULAR ELEMES () h gomr of h rangular lmn ma alo b dcrbd ung h abov nrpolaon a: = An oparamrc lmn ma b formd b ung a valu of = hch u h nrpolaon funcon gvn abov. Hovr, a ubparamrc lmn ma alo b dfnd b ng =. In h ca, h nrpolaon funcon dfnd for a hrnod rangular ar ud. SI-ODE RIAGULAR ELEMES () h gomr of h rangular lmn ma alo b dcrbd ung h abov nrpolaon a: h nrpolaon or hap funcon n global coordna and ar mahmacall clum and rarl ud n FEM anal. An quvaln form of h hap funcon ma b drvd n rm of h local parnal lmn coordna. h funcon hav a rlavl mpl mahmacal form and ar mor ffcn n compung h lmnal marc. = SI-ODE RIAGULAR ELEMES () A dcrb abov, h cond approach o dvlopng a of nrpolaon or hap funcon for a -nod quadrac rangular lmn bgn h h parn lmn n local coordna. Condr h follong -nod rangl n local coordna and.,,

3 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () In h parn lmn h nrpolaon funcon ar gvn a:,,,, SI-ODE RIAGULAR ELEMES () h parn lmn nrpolaon funcon (, ) hav o bac hap. h bhavor of h funcon,, and mlar cp rfrnc a dffrn nod. h hap funcon hon blo:,,,, SI-ODE RIAGULAR ELEMES () h parn lmn nrpolaon funcon (, ) hav o bac hap. h cond p of hap funcon vald for funcon,, and. h funcon hon blo: SI-ODE RIAGULAR ELEMES () In ordr o undrand h naur of h ranformaon from h parn lmn o h global lmn, h chan rul ud o form h dffrnal rlaonhp: In mar noaon, h drvav ma b rn a: J SI-ODE RIAGULAR ELEMES () h drmna of h Jacoban mar J : J J J SI-ODE RIAGULAR ELEMES () Whn J pov vrhr n a rgon, h ranformaon ma b nvrd o drmn = (, ) and = (, ). h man ha for a gvn pon (, ) n h rangl hr a unqu corrpondng pon (, ) n h parn lmn. h J(, ) a maur of h panon or conracon of a dffrnal ara: d d J, d d h drmnan of h Jacoban mar, J, a of h nvrbl of h ranformaon = (, ) and = (, ).

4 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () Eampl Condr h quadrac rangular lmn gvn blo. (,) (,) ( aa, ) a8 a, 8 h rulng Jacoban : J a (,) (,) (,) h coordna ranformaon gvn a: a a8 SI-ODE RIAGULAR ELEMES () Eampl L condr vral ca hr h J ma bcom ngav. h mappng a nod and hav h valu + = n common. Subung h corrpondng and coordna for nod and no h pron for J gv: J, 8a hch clarl ngav a a /. SI-ODE RIAGULAR ELEMES () Eampl For valu of and l han /, h drmnan of h Jacoban ngav. (, ) (, ) (,) (.,.) (,) nror angl = 0 (,).. EVALUAIO OF MARICES - RIAGULAR ELEMES h lmnal marc for h Poon problm ar: k a A da d f h A fda hd o ha for a = / h nror angl a nod and qual o zro. EVALUAIO OF MARICES - RIAGULAR ELEMES h dvlopmn of boh h k and f rm dncal o ha prnd for ohr p of lmn: k JJ hr JJ = (J J + J J ) J. d d f fda d d J f A A EVALUAIO OF MARICES - RIAGULAR ELEMES h rm J and J ar h fr and cond ro of h nvr of h Jacoban mar. J J hr J h drmnan of J.

5 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 EVALUAIO OF MARICES - RIAGULAR ELEMES h valu of h mar ma b compud a: For a gnral quadrac rangular lmn, h mar JJ a funconal of and h h Jacoban J n h dnomnaor. h rulng pron for JJ vr dffcul o valua acl and h ngraon ar uuall don numrcall. EVALUAIO OF MARICES - RIAGULAR ELEMES hrfor, h rm k and f ma b ca n h follong form: G, dd hr G(,) a funcon of h varabl and. In prncpl, ma b pobl o valua h f rm, hovr, numrcal ngraon pcall mor praccal. For h k rm, h apparanc of h Jacoban J n h ngrand gnrall ndca h u of numrcal quadraur. EVALUAIO OF MARICES - RIAGULAR ELEMES hrfor, h gnral pron for k and f ar: k, JJ,, dd,, f J d d EVALUAIO OF MARICES - RIAGULAR ELEMES h gnral form for an -rm Gauan quadraur : G, dd G, hr, ar h rangular Gau pon and h gh. h dvlopmn of Gauan quadraur for a rangl dncal o ha prnd n prvou con. = = = EVALUAIO OF MARICES - RIAGULAR ELEMES h gnral form for an -rm Gauan quadraur : G, dd G, hr, ar h rangular Gau pon and h gh. hrfor k and f ma b valuad b: k JJ d d JJ f J d d f J f EVALUAIO OF MARICES - RIAGULAR ELEMES h follong abl conan h Gau pon and gh for =,, and. Accurac

6 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 EVALUAIO OF MARICES - RIAGULAR ELEMES h follong abl conan h Gau pon and gh for = and 7. Accurac EVALUAIO OF MARICES - RIAGULAR ELEMES Condr h ngral a and h : a d hr h ngraon along a boundar gmn of h lmn. Snc, h ngraon compud along a ngl d of h quadrac lmn, h nrpolaon funcon ar quadrac. k n h m l h l l h h h h h h l l ll hd l EVALUAIO OF MARICES - RIAGULAR ELEMES h varaon of and a funcon of along h boundar gvn a: h dffrnal arc lngh dl : I J l l I J l l dl d d dl d l dl d l l EVALUAIO OF MARICES - RIAGULAR ELEMES h ngral dfnng a and h ar: a l d I J l l h rulng lmnal ffn mar conrbu o h global m quaon f h lmn ha a d a par of h boundar. h hd ldh h rulng lmnal load vcor conrbu o h global m quaon f h lmn ha a d a par of h boundar. EVALUAIO OF MARICES - RIAGULAR ELEMES h global m quaon ar compod from h follong ummaon: K k a G G G h rulng m quaon ar, n mar form, gvn a: Ku F G G G F f h G G G SI-ODE RIAGULAR ELEMES () Eampl Condr h quadrac rangular lmn gvn blo. (, ). (, ) (,) (,) (.,.) (,) Calcula h valu of h ffn mar k for h abov lmn hn =. and =... k JJ d d JJ = (J J + J J ) J.

7 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () 7/8 SI-ODE RIAGULAR ELEMES () Eampl k JJ d d h valu of h mar ma b compud a: SI-ODE RIAGULAR ELEMES () Eampl k JJ d d J and J ar h fr and cond ro of h nvr of h Jacoban mar. J J J J J J J SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ k mmrc k mmrc.077 SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : k JJ d d JJ k mmrc.9787 k mmrc.079

8 CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () 8/8 SI-ODE RIAGULAR ELEMES () Eampl hrfor, h gnral pron for k : 7 k JJ d d JJ k mmrc.0889 SI-ODE RIAGULAR ELEMES () Eampl h rul for = ar no vr accura. Snc onl on Gau pon ud n h quadraur, h rul no unpcd. h valu for k ung Gauan quadraur h = and ndca ha h accurac of h valuaon ncra a h numbr of Gau pon ncra. = = = SI-ODE RIAGULAR ELEMES () PROBLEM # - Wr a compu ubroun calld QUAD ha calcula h componn of h k mar for a gnral quadrac rangular lmn ung Gauan quadraur. Whl ou ll no r a compl fn lmn program, h ubroun a crcal componn of h compuaonal procdur. A mpl drvr program ll b rqurd o dbug and our ubroun and pa gomrc nformaon. our program hould allo h ur o pcf h numbr of Gauan quadraur pon and h coordna nformaon. Chck our ork h h problm n our book on pag 9 and h ampl prnd n h no. End of -od rangular Elmn ()

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