Lecture 23 APPLICATIONS OF FINITE ELEMENT METHOD TO SCALAR TRANSPORT PROBLEMS

Transcription

1 COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms ctur 3 PPICTIONS OF FINITE EEMENT METHOD TO SCR TRNSPORT PROBEMS 3. PPICTION OF FEM TO -D DIFFUSION PROBEM Consdr th stady stat dffuson of a proprty n a on-dmnsonal doman dfnd n Fgur 3.. Th procss s govrnd y th dffrntal quaton d d S 0 (3.) whr s th dffuson coffcnt and S s th sourc trm. Boundary valus of at ndponts 0 and ar spcfd as (0), ( ), (3.) B t us dvd th doman n two nod lnar fnt lmnts. In any lmnt, th unknown functon can approxmatd as ( x) N( x) N( x) (3.3) whr N and N ar lnar shap functons, and, ar valus of at local nods and of th lmnt. Strong form of th wghtd rsdual statmnt s gvn y d d w S 0 0 (3.4) Prformng ntgraton y parts, and rqurng that th wght functon vanshs at th ndponts, w gt th followng wak form: dw d ws 0 0 (3.5) For Galrkn formulaton, w N. Susttuton of approxmaton (3.3) n Eq. (3.5) ylds th dscrt algrac systm Ku = f (3.6) whr K s th stffnss matrx, u s th vctor of nodal unknowns ϕ and f s calld th load vctor. Elmnts of th matrcs n th prcdng quaton ar gvn y whr (3.7) K K, and f f mn mn m m (3.8) Kmn Nm, k Nn, kd x, and f m NmS d x. Systm of lnar quatons s solvd usng sutal lnar solvr (.g. TDM would th st solvr for th prsnt xampl as K would tr-dagonal for lnar fnt lmnts). Nxt xampl llustrats th prcdng procss xplctly for a sampl hat conducton prolm. Dr K M Sngh, Indan Insttut of Tchnology Roork NPTE 3.

2 COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms Exampl 3. Consdr th stady stat hat conducton n a sla of wdth l = 0.5 m wth hat gnraton. Th lft nd of th sla (x = 0) s mantand at T = 373 K. Th rght nd of th sla (x = 0.5 m) s ng hatd y a hatr for whch th hat flux s kw/m. Th hat gnraton n th sla s tmpratur dpndnt and s gvn y Q = (73 T) W/m 3. Thrmal conductvty s constant at k = W/(m-K).Wrt down th govrnng quaton and oundary condtons for th prolm. Us th fnt dffrnc mthod (cntral dffrnc schm) to otan an approxmat numrcal soluton of th prolm. For th frst ordr drvatv, us forward or ackward dffrnc approxmaton of frst ordr. Choos lmnt sz h = 0., and us th TDM. (W hav chosn th sam as th on solvd prvously n Exampl 5. usng FDM to llustrat comparson twn FEM, FVM and FDM.) Soluton t us rcall that govrnng quaton for th stady stat hat conducton wth constant hat gnraton s th sla s k Q 0 () Gvn: Q = 73 T. Thus, Eq. () coms k ( f T) 0 () whr f = 73. ft nd of th sla s mantand at constant tmpratur; hnc oundary condton at ths nd s gvn y T (0) 373 () t th rght nd, hat nflux s spcfd. Thus, oundary condton at ths nd s k ( ) 000 (v) For dscrtzaton, lt us us a fnt lmnt msh of lnar lmnts. Gloal nods ar (x=0), (x=0.), 3(x=0.), 4(x=0.3), 5(x=0.4) and 6 (x=0.5). Each fnt lmnt s of wdth h = 0., whras fnt volums around oundary nods and 6 ar of wdth h Galrkn wghtd rsdual statmnt (3.4) for ths prolm taks th form dt w k ( f T) 0 (v) 0 Intgraton y parts ylds th followng wak form: dt dw dt k w k w( f T) (v) Th frst trm n th prcdng quaton would non-zro only for th lmnts at th oundary of th doman. Now, lt us us lnar shap functons for ntrpolaton n ach lmnt,.. T ( x) N( x) T N( x) T (v) For an lmnt of lngth h wth nod (locatd at x ) and nod (at x ), lnar shap functons ar Dr K M Sngh, Indan Insttut of Tchnology Roork NPTE 3.

3 COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms x x xx dn dn N( x), N( x), h h h h (v) Insrtng th approxmaton (v), wghtd rsdual formulaton for an lmnt can wrttn as dw dn dn dt k T T w( N( x) T N( x) T ) k w fw 0 (x) For Galrkn formulaton, wght functon w wll takn as on of th shap functons, N. Hnc, th prcdng quaton can wrttn as n matrx form as K K T K K T (x) whr dnm dnn Kmn NmNn d x and m Nm f d x, (x) wth an addtonal contruton to for th frst and last lmnt comng from spcfd flux. Thus, dn dn h K N N d x,, d d N N j h Kj N N j d x,, j, h f (for ntror lmnts) (x) h dt h f k, f, x0 N h N h dt f, f k d x x ssmlng all lmntal quatons, w gt KT =, whr K K, and f f (x) mn mn m m h h h h h h dt f k T d x x0 h h h T hf T3 hf h h T hf (xv) h 3 T 5 hf h h h T h dt f k d x x h h Dr K M Sngh, Indan Insttut of Tchnology Roork NPTE 3.3

4 COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms t us not that tmpratur s spcfd at x = 0 (flux kdt/ s unknown). Hnc, w modfy th frst quaton n Eq. (xv) y ncorporatng known tmpratur valu. Furthr, susttut h = 0., f = 73, valu of flux at rght-nd of th doman, and dvd last fv quatons y 0 (to mak all th dagonal lmnts to of sam ordr of magntud). Th rsultng dscrt systm s T 373 a d a T.73 0 a d a 0 0 T d a 0 T a d a T a d / T whr a = , d = Th prcdng systm s vry smlar to th dscrt systm (x) otand usng fnt dffrnc dscrtzaton n Exampl 5. (only coffcnts n last row dffr). Numrcal calculatons usng TDM ar gvn n th followng tal: (xv) W P E W E P P P * * W P T * ET P T x a d -a a d -a a d -a a d -a a d/ Comparson of rsults otand usng FEM, FVM and FDM usng th sam grd sz s gvn low: Exact FEM FVM FDM %Error FEM %Error FVM %Error FDM W can clarly osrv that th fnt lmnt and fnt volum rsults usng dntcal grd spacng ar mor accurat than thos otand usng FDM. Th prmary rason s us of frst ordr ackward dffrnc mthod usd n fnt dffrnc soluton for ncorporaton of flux oundary condton. Dr K M Sngh, Indan Insttut of Tchnology Roork NPTE 3.4

5 COMPUTTION FUID DYNMICS: FVM: pplcatons to Scalar Transport Prolms REFERENCES/FURTHER REDING Chung, T. J. (00). Computatonal Flud Dynamcs. nd Ed., Camrdg Unvrsty Prss, Camrdg, UK. Muraldhar, K. and Sundararajan, T. (003). Computatonal Flud Dynamcs and Hat Transfr, Narosa Pulshng Hous. Rddy, J. N. (005). n Introducton to th Fnt Elmnt Mthod. 3 rd Ed., McGraw Hll, Nw York. Znkwcz, O. C., Taylor, R.., Zhu, J. Z. (005). Th Fnt Elmnt Mthod: Its Bass and Fundamntals, 6 th Ed., Buttrworth-Hnmann (Elsvr). Dr K M Sngh, Indan Insttut of Tchnology Roork NPTE 3.5