CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid poor approimaion for an drivd varia. B uiizing highr-ordr mn ioparamric mn h dficinci of h inar mn ar ovrcom. ISOPARAMERIC ELEMENS h rm ioparamric man ha an qua numr of paramr ar ud o rprn h gomr and h dpndn varia. In ohr word, h am hap funcion ud o inrpoa h dpndn varia ar ud o rprn h gomr. A uparamric mn i on whr h gomr i dfind fwr paramr han ud o inrpoa h ouion. A uprparamric mn i on whr h gomr i dfind mor paramr han ud o inrpoa h ouion. ISOPARAMERIC ELEMENS h rm ioparamric man ha an qua numr of paramr ar ud o rprn h gomr and h dpndn varia. u un i i 6-nod rianguar mn (6) N N i i i i 8-nod quadriara mn (Q8) h four-nod quadriara mn i vr imiar o h four-nod rcanguar mn. Boh provid iinar inrpoaion ovr h mn for h gnra ouion and a inar approimaion of h fir drivaiv. Howvr, h quadriara mn i no rricd o rcanguar hap; in fac i ma rprn an numr of rciinar hap. h gnra approimaion ovr a quadriara mn i aumd o : u, un, un, un, un, u NuN,,,, h mna coordina ar and. h inrpoaion funcion N i (, ) ar idnica o h rcanguar mn inrpoaion funcion. N, N, 1 1 1 1 1 N, N, 1 1 1 1
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 h ranformaion from goa coordina (, ) o mna or parna coordina (, ) i accompihd uing h foowing coordina mapping:,,,,, N N N N,,,,, N N N N N N N N h inrpoaion funcion for a quadriara mn ar rfrrd a hap funcion inc h prion dfin h hap of h mn. L ook a how h in and poin in h parn mn (, ) ar mappd or ranformd o h goa m (, ).,, * (,) 1 1,, * (,) 1 1 1, 1, (,) 1 a 1 1 1, 1, (,) 1 a 1 1 In gnra raigh in in h - coordina m ar mappd ino raigh in in h quadriara mn. For amp, poin a and ar mappd o poin a* and * in h quadriara. h raigh in = 1 in h parn mn, conncing nod and, ma mappd ino h quadriara a:,, * (,) 1 1,, * (,) 1 1 1, 1, (,) 1 a 1 1 1, 1, (,) 1 a 1 1 In ordr o undrand h naur of h ranformaion, h chain ru i ud o form h diffrnia raionhip: In mari noaion h drivaiv ma wrin a: J whr J J h drminan of h Jacoian mari, J, i a of h invriii of h ranformaion = (, ) and = (, ). Whn J i poiiv vrwhr in a rgion, h ranformaion ma invrd o drmin = (,) and = (, ). hi man ha for a givn poin (, ) in h quadriara hr i a uniqu corrponding poin (, ) in h parn mn.
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 h mn of h Jacoian mari ma compud a: 11 1 11 1 1 11 1 11 hrfor, J ha h form: J j1 j j j whr h conan j i ar funcion of h coordina of h nod (funcion of h gomr of h mn). h conan j i acua zro o ha J i a inar funcion of and. h drminan J gomrica rprn h raionhip wn h diffrnia ara da(, ) = d d and h diffrnia ara da(, ). J da, da, J d d PROBLEM #1 - Show ha h conan j dfining h drminan of h Jacoian J i zro. Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 h mn of h Jacoian mari ma cacuad a: 1 1 1 1 18 Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 h mn of h Jacoian mari ma cacuad a: 1 1 1 19 1 1 1 1 Eamp - h drminan i: 10 8 10 9 178 188 170 J 8 Conidr h ocaion of nod hr a = 19 and =, hn h ranformaion i givn : 67 6 1 98 111 66 79 1 177 J 8
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 Eamp - Ovr h inrva -1 1 and -1 1 h drminan of h Jacoian, J, i poiiv, indicaing ha hr ar no prom wih h mapping from - o -. h ara of h quadriara ma compud a: 98 A A d d J d d 1 177 d d 8 66 79 177 Eamp - h parn mn in h - m and h hap of h quadriara mn ar hown in h figur ow: 1, 0 10,10 * 19, (,) (,) 0, 11 1 a 1 1 1 1 111 Eamp - Conidr h ocaion of nod hr a = 16 and = 1, hn h ranformaion i givn : 9 9 7 5 7 11 9 7 J 0 1 5 J Eamp - Ovr h inrva -1 1 and -1 1 h drminan of h Jacoian, J, i no awa poiiv. In fac, J i ngaiv for ( + ) 5/. For poin (, ) who vau aif ( + ) 5/, h mapping oca hm ouid of h quadriara. h prom i aociad wih h fac h rgion i conv aong h id dfind nod,, and. 9 7 J 0 8 1 8 1 Eamp - h parn mn in h - m and h hap of h quadriara mn ar hown in h figur ow: 1, 0 10,10 (,) 16, 1 (,) 0, 11 1 a 1 1 1 1 h mna maric for h Poion prom ar: NN NN k da f f da N A A a NN d h Nh d
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 5/8 h dvopmn of oh h k and f rm i idnica o ha prnd for rcanguar mn: k JJ d d f Nf J d d whr JJ = (J 1 J 1 + J J ) J. h vau of h mari ma compud a: Rca, J 1 and J ar h fir and cond row of h invr of h Jacoian mari. 1 1 J J1 J J J J h mn of h invr of h Jacoian ar: 11 1 11 1 1 11 1 11 J For a gnra quadriara mn, h mari JJ i a funcion of and wih h Jacoian J in h dnominaor. h ruing prion for JJ i vr difficu o vaua ac and h ingraion ar uua don numrica. hrfor, h rm k and f ma ca in h foowing form: G, dd whr G(,) i a compicad funcion of h varia and. In princip, i ma poi o vaua h f rm, howvr, numrica ingraion i pica mor pracica. For h k rm, h apparanc of h Jacoian J in h ingrand gnra indica h u of numrica quadraur. I i foruna ha h imi of h ingra conform o h forma ud in Gau-Lgndr quadraur. Sinc h ingraion ar in oh h and -dircion, on approach i o app Gauian quadraur para in ach dircion. hrfor, h gnra prion for k and f an N-rm Gauian quadraur i: k JJ d d ww JJ f Nf Jdd i n i n 1 j1 1 j1 whr h ucrip ij rfr o h vauaion of h quani a i and j. n n i j ij ij ij,,,, ww N f J i j i j i j i j ij Conidr h ingra a and h : a NN d h Nh d whr h ingraion i aong a oundar gmn of h mn. Sinc, h ingraion i compud aong a ing id of h quadriara mn, h quadriara hap funcion rduc o inar funcion.
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 6/8 h ru for hi ingraion ar idnica o prion oaind for inar rianguar and rcanguar mn. j a h k h k 0 i Nk N k k k Nk hn k k hn d N h N N N N d Nk 1 N Evauaing h ingra, h a and h rm rduc o: j a k h k 0 i k k 1 k k h Nk 1 N h h k h 6 hk h h ruing a mna iffn mari and h 1 h mna oad vcor conriu o h goa m quaion if h mn ha a id a par of h oundar. h goa m quaion ar compod from h foowing ummaion: K k a F f h G G G G G G h ruing m quaion ar, in mari form, givn a: Ku F G G G Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 Cacua h vau of h iffn mari k for h aov mn whn = 19 and = 1. k JJ d d f Nf J d d 1 1 Eamp - Rca JJ = (J 1 J 1 + J J ) J. h vau of h mari ma compud a: whr J 1 and J ar h fir and cond row of h invr of h Jacoian mari. 1 1 1 J J J 1 J J1 J Eamp - h mn of h invr of h Jacoian ar: 1 1 1 1 1 1 1 1 1 1 1 1
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 7/8 Eamp - h cacuaion ar prformd compuing h vau of ach componn of,, J 1, J, and J maric a ach Gau poin, prforming h mari muipicaion, and fina umming h ru for a h Gau poin. h u of numrica quadraur a hi poin in h formuaion ma gnra om rror. Howvr, if nough Gau poin ar ud h numrica ru ar qui accura whn compard o h ac vau for h ingraion. Eamp - A wo-dimniona quadraur uing N = i acua Gau poin and uing N = i 16 quadraur poin. 1 N = 1 Gau poin 1 N = 1 N = Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N 1 0.09605 0.101695 0.09605 0.101695 0.6559 0.101695 0.6559 k 0.09605 0.101695 mmric 0.6559 Eac k 0.5519 0.0106 0.781 0.0975 0.7519 0.05195 0.6917 0.6771 0.878 mmric 0.91507 Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k 0.55081 0.1989 0.86 0.0966 0.790 0.006 0.705 0.669 0.71 mmric 0.9171 0.5519 0.0106 0.781 0.0975 0.7519 0.05195 0.6917 0.6771 0.878 mmric 0.91507 Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k 0.551 0.0101 0.790 0.097 0.7515 0.05188 0.695 0.6770 0.876 mmric 0.9159 0.5519 0.0106 0.781 0.0975 0.7519 0.05195 0.6917 0.6771 0.878 mmric 0.91507 Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k 0.5519 0.0106 0.781 0.0975 0.7519 0.05195 0.6917 0.6771 0.878 mmric 0.91507 0.5519 0.0106 0.781 0.0975 0.7519 0.05195 0.6917 0.6771 0.878 mmric 0.91507
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 8/8 Eamp - h ru for N = 1 ar no vr accura. Sinc on on Gau poin i ud in h quadraur, hi ru i no unpcd. h vau for k uing Gauian quadraur wih N = and indica ha h accurac of h vauaion incra a h numr of Gau poin incra. Eamp - Drivd Varia - h drivd varia for h Poion or Lapac quaion ar h paria drivaiv u/ and u/. h rm ar compud a funcion of poiion in an mn uing h hap funcion and h noda vau. h ru uing N = ar ac o fiv or i dcima pac. Eamp - Drivd Varia - h paria drivaiv ma comind o giv h norma or dirciona drivaiv: nin jn u u n un u u n n u u J u u J1 u 1 u J u u J Eamp - Drivd Varia - Rca J 1 and J ar h fir and cond row of J -1, rpciv. Rca ha u / i: u [ Nu] u u u [ Nu] hrfor: u u J1 u J u h dirciona or norma drivaiv i: u nj n J u n 1 End of Quadriara mn (Q)