FEM FOR HEAT TRANSFER PROBLEMS دانشگاه صنعتي اصفهان- دانشكده مكانيك

Transcription

1 FEM FOR HE RNSFER PROBLEMS 1

2 Fild problms Gnral orm o systm quations o D linar stady stat ild problms: For 1D problms: D D g Q y y (Hlmholtz quation) d D g Q d

3 Fild problms Hat transr in D in h h ( D D ) ( ) y q y t t Hat conduction Hat convctction Hat supply y t t Not: h h, g Q q t t z Hat convction on th surac 3

4 Fild problms Hat transr in long D body q y y Not: Hat supply Hat conduction t=1 y z D =, D y = y, g = and Q = q Rprsntativ plan 4

5 Fild problms Hat transr in 1D in d hp hp q d Hat conduction Hat convctoin Hat supply Not: D, g hp, Q q hp P 5

6 Fild problms Hat transr across composit wall d d Hat conduction q Hat supply Hat convction y Hat convction Not: D, g, Q q 6

7 Fild problms orsional dormation o bar 1 1 G G y ( - strss unction) Not: D =1/G, D y =1/G, g=, Q= Idal irrotational luid low y y Not: D = D y = 1, g = Q = ( - stramlin unction and - potntial unction) 7

8 Fild problms ccoustic problms P P w P y c Not: w g, D c = D y = 1, Q = P - th prssur abov th ambint prssur ; w - wav rquncy ; c - wav vlocity in th mdium 8

9 Wightd Rsidual pproach For FEM Establishing FE quations basd on govrning quations without nowing th unctional. D D g Q y y pproimat solution: ( (, y)) (Strong orm) w ( (, y ))d d y (Wa orm) Wight unction 9

10 Wightd Rsidual pproach For FEM Discrtiz into smallr lmnts to nsur bttr approimation In ach lmnt, (, y) N(, y) Φ whr N N1 N1 N n d 3 Using N as th wight unctions 1 Galrin mthod R N ( (, y))ddy Rsiduals ar thn assmbld or all lmnts and norcd to zro. 1

11 1D Hat ransr Problms 1D in d hp hp q d Hat conduction Hat convctoin Hat supply (Spciid boundary condition) : thrmal conductivity h : convction coicint : cross-sctional ara o th in P : primtr o th in : tmpratur, and : ambint tmpratur in th luid d h b d (Convctiv hat loss at r nd) 11

12 1D in Using Galrin approach, R j d N D g Q d i d d 1D Hat ransr Problms j j N D Q d g d N i i d whr; D =, g = hp, and Q = hp + q (1) 1 i (i) j (n) n+1 (i) 1 i j

13 1D in 1D Hat ransr Problms Intgration by parts o irst trm on right-hand sid, j j j d dn d j R N D D d Q d g d d i d d N N i i i Using ( ) N( ) j d j dn dn R N D ( D d ) d i d d i D j j ( QN d ) ( gn Nd ) i b i Q g 13

14 1D Hat ransr Problms 1D in R b [ ] whr D g Q ( ) j d d N N j d D D B DB d i d d i B d N d (Strain matri) (hrmal conduction) g i j gn Nd (hrmal convction) Q i j QN d (Etrnal hat supplid) b N D d d i j (mpratur gradint at two nds o lmnt) 14

15 1D Hat ransr Problms 1D in For linar lmnts, j i N ( ) Ni N j l l (Rcall 1D truss lmnt) B dn d j i 1 1 d d l l l l E hror, or truss lmnt l 1 j l (Rcall stinss D D d i 1 l l l 1 1 matri o truss lmnt) l 15

16 1D in 1D Hat ransr Problms j j l j 1 i hpl g g d i l l 6 1 i l lor truss lmnt (Rcall mass matri o truss lmnt) Q j j l Ql 1 l 1 Q d ( q hp ) i 1 1 i Hat supply Hat convction l 16

17 1D in b 1D Hat ransr Problms d d j d d i N D d d i d d i d j d j b L br or b b b L R (Lt nd) (Right nd) t th intrnal nods o th in, b L () and b R () vanish upon assmbly. t boundaris, whr tmpratur is prscribd, no nd to calculat b L () or b R () irst. 17 (n1) i1 i1 (n1) i d D d i 1 d D d i (n) i i+1 i (n) i+1 d D d i d D d i 1

18 1D Hat ransr Problms 1D in Whn thr is hat convction at boundary,.g. b R d hb d h h b j h Sinc b is th tmpratur o th in at th boundary point, b = j hror, b i R h j h M s 18

19 1D Hat ransr Problms 1D in b R M s whr M h, s h For convction on lt sid, b L M s whr h M, s h 19

20 1D Hat ransr Problms 1D in hror, [ ] D g M Q S R R Rsiduals ar assmbld or all lmnts and norcd to zro: Ku = F sam orm or static mchanics problm

21 1D Hat ransr Problms 1D in Dirct assmbly procdur R or R R Elmnt 1: R (1) (1) (1) (1) R (1) (1) (1) (1) 1 1 (1) () 1 3 1

22 1D in Dirct assmbly procdur (Cont d) Elmnt : R () () () () R () () () () 1 3 R : (1) (1) (1) (1) D Hat ransr Problms Considring all contributions to a nod, and norcing to zro (Nod 1) R R : ( ) ( ) (Nod ) (1) (1) (1) (1) () () (1) () R : (Nod 3) () () () () 1 3

23 1D Hat ransr Problms 1D in Dirct assmbly procdur (Cont d) In matri orm: (1) (1) (1) (1) (1) () () (1) () () () () 1 3 (Not: sam as assmbly introducd bor) 3

24 W 3 cm C 1D Hat ransr Problms 1D in ampl: Hat transr in 1D in Calculat tmpratur distribution using FEM. W W 3, h. 1, C cm C cm C 1 cm 8 C 8 cm.4 cm 4 linar lmnts, 5 nods (1) () (3) (4)

25 1D in Elmnt 1,, 3:, ( ) M Elmnt 4:, ( ) M S S not rquird rquird 1D Hat ransr Problms W 3 cm 8 C C W.1 C1 cm cm C h 8 cm (1) () (3) (4).4 cm l 3.4 hpl W.6 C W.93 C h W C hpl h 5.6W.1.4.8W 5

26 1D in For lmnt 1,, 3 1D Hat ransr Problms D 1 1 hpl 1 1,, l g, Q hpl 1 1 1,, For lmnt hpl 1 L h 1 hpl 1 h 6 D g M, Q S

27 1D Hat ransr Problms 1D in * Q Hat sourc (Still unnown) 1 = 8, our unnowns liminat Q * * Q Solving: { } 7

28 Composit wall d d Hat conduction q Hat supply Convctiv boundary: d hb d d hb d 1D Hat ransr Problms at = at = H y ll quations or 1D in still applis cpt and G Q vanish. (1) () (3) Rcall: Only or hat convction 1 1 hror, M l 1 1, 8 S

29 Composit wall 1D Hat ransr Problms ampl: Hat transr through composit wall Calculat th tmpratur distribution across th wall using th FEM. 5 W.1 cm h C C W. cm C C W.6 cm C 5 cm linar lmnts, 3 nods (1) () 1 3 9

30 Composit wall For lmnt 1, L. 1.1 W C W h.11.1 C h W 1 1 L 1 1 h 1D Hat ransr Problms 5 C W.1 cm h C W. cm 5 cm (1) () 1 3 C C W.6 cm C h s 1.5 S 3

31 Composit wall For lmnt, L L W C 1D Hat ransr Problms W.1 cm h C C W. cm C C W.6 cm C Upon assmbly, 5 cm ( ) Q (1) () 1 3 (Unnown but rquird to balanc quations) 31

32 Composit wall 1D Hat ransr Problms W.1 cm h C W. cm C C 5 C W.6 cm C (.1) Solving: cm (1) () 1 3 3

33 D Hat ransr Problms Elmnt quations D D g Q y y 1 3 For on lmnt, R N ( D D )d y g Q y Not: w = N : Galrin approach 33

34 Elmnt quations R N ( D D )d y g Q y D Hat ransr Problms (Nd to us divrgnc thorm to valuat intgral in rsidual.) N N N (Product rul o dirntiation) hror, N N N d ( )d D D D d Divrgnc thorm: ( N )d cosd N d cos d N N D d D D N 34

35 Elmnt quations D Hat ransr Problms nd intgral: d sin d N N D d y D y Dy y N y y y hror, R N D cos Dy sin d y N N D Dy d y y gn d QN d 35

36 Elmnt quations D Hat ransr Problms ( ) N( ) R N D cos Dy sin d y b N N N N D Dy d y y D ( N Nd ) N d g Q g Q 36

37 Elmnt quations D Hat ransr Problms R b [ ] ( ) ( ) ( ) ( ) ( ) ( ) D g Q whr b N D cos Dy sin d y N N N N D D d y y D y g g N Nd Q Q N d 37

38 Elmnt quations D Hat ransr Problms N N N N D D d y y D y D Din D D, y B N N N N 1 N n d B N N N 1 N n d y y y y N y y (Strain matri) B DB D D y N N N N y y D B DB d 38

39 riangular lmnts 3 1 N [ N1 N N3] Not: constant strain matri D 39 D Hat ransr Problms y, B DB d B DB d B DB 1 ( 1, y 1 ) 1 b b b b b c c c c c i i j i i i j i D D y D b i b j b j b j b c i c j c j c j c 4 4 bib bjb b cic c jc c 3 ( 3, y 3 ) 3 (, y ), Ni ai bi ci y 1 ai ( j y y j ) 1 bi ( y j y ) 1 ci ( j ) (or N i = L i )

40 riangular lmnts g N1 gn Nd g N [ N1 N N3]d N 3 N1 N1N N1N 3 g 1 3 d N N N N N N1N3 NN3 N 3.g. 1 1 D Hat ransr Problms Not: L m 1 L n L p 3 d m! n! p! ( m n p (ra coordinats) 1!1!! 1 N1 Nd L1 LL3d (1 1 )! )! hror, g 1 1 g

41 riangular lmnts D Hat ransr Problms Similarly, N L 1 N d d Q i 1 Q Q d Q N j Q L 1 3 N L 3 1 Not: b () will b discussd latr 41

42 Rctangular lmnts N N N N N [ 1 3 4] 3 N N N N (1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 ) D Hat ransr Problms 1 4 B a a a a b b b b y 4 ( 4, y 4 ) 3 ( 3, y 3 ) b a 1 ( 1, y 1 ) (, y ) 4 (1, +1) 3 (1, +1) b a 1 (1, 1) (1, 1) a, y b 4

43 Rctangular lmnts D Hat ransr Problms 1 1 D d ab 11 B DB B DBdd Db 1 1 Da y 1 1 6a 1 1 6b g g d abg 11 N N N Ndd abg N N N N N N N N N N N N N N N N N N N N N N N N N N N N dd g g

44 Rctangular lmnts N1 1 N Q 1 Q Q N d Q N d N 4 1 D Hat ransr Problms Not: In practic, th intgrals ar usually valuatd using th Gauss intgration schm 44

45 Boundary conditions and vctor b () D Hat ransr Problms b b b I B Intrnal Boundary (3) b B b B () nds to b valuatd at boundary p 3 i r q (3) b I () b I 1 j () b I n m Vanishing o b I () 45

46 Boundary conditions and vctor b () D Hat ransr Problms Nd not valuat b B Essntial boundary is nown 1 n n Nd to b concrn with b B () Natural boundary: th drivativs o is nown 46

47 Boundary conditions and vctor b () b D Hat ransr Problms N D cos Dy sin d y Essntial boundary is nown 1 D cos Dy sin Mb S y n n on natural boundary D cos D y sin Mb S y n Hat lu across boundary Natural boundary: th drivativs o is nown 47

48 Boundary conditions and vctor b () Insulatd boundary: D Hat ransr Problms M = S = b B Convctiv boundary condition: q n q h h b n n h b h n M S q c q n q h h b n h b h M S M h S h, 48

49 Boundary conditions and vctor b () Spciid hat lu on boundary: D Hat ransr Problms q n b s M M, S qs S n q n q s n q q q s M= S=q s s n S Positiv i hat lows into th boundary Ngativ i hat lows out o th boundary insulatd 49

50 Boundary conditions and vctor b () For othr cass whrby M, S b N D cos D sind y B y N ( M + S)d b D Hat ransr Problms b N b N ( MN + S)d B ( N MNd ) N Sd M B 5

51 Boundary conditions and vctor b () b B M S D Hat ransr Problms whr N MN d, M S N Sd For a rctangular lmnt, S S N1 N 1 N SN d S a 1 N3 d Sa 1 1 d Sa y b 4 ( 4, y 4 ) (Equal sharing btwn nods 1 and ) 3 ( 3, y 3 ) a 1 ( 1, y 1 ) (, y )

52 D Hat ransr Problms Boundary conditions and vctor b () Equal sharing valid or all lmnts with linar shap unctions S,3 1 Sb 1 S,34 Sa 1 1 S,1 4 1 Sb 1 y, 3 ( 3, y 3 ) 3 pplis to triangular lmnts too 1 ( 1, y 1 ) 1 (, y ), S,1 SL 1 S,3 SL S,13 SL 1 1 5

53 Boundary conditions and vctor b () M or rctangular lmnt N1N N NN3 NN4 M M N 1 N 3 N N 3 N 3 N 3 N 4 N1N4 NN4 N3N4 N4 D Hat ransr Problms N N N N N N N d N N N d NN1 N M,1 am 1 53

54 Boundary conditions and vctor b () N d NN d Nd 1 1 M,3 d d d M b 6 1 M a M,34 M, D Hat ransr Problms M,1 1 M b 6 1 am ( am ) Shard in ratio /6, 1/6, 1/6, /6

55 Boundary conditions and vctor b () Similar or triangular lmnts D Hat ransr Problms ML 1 ij M, i j 1 M, j 6 ML 6 j 1 1 M, i ML 6 i

56 Point hat sourc or sin Prrably plac nod at sourc or sin D Hat ransr Problms F i j Q * 1 * j Q 56

57 D Hat ransr Problms Point hat sourc or sin within th lmnt y Point sourc/sin Q Q N d X y Q Q Q Y (Dlta unction) Ni Q N j X y Y dd y N i Q * (X, Y ) j Q N X, Y i Q N j X, Y N X, Y 57

58 Fild problms SUMMRY D D y g Q y t t y h h Sourc and sin R b [ ] ( ) b N D g M Q * S Only or lmnts on th drivativ boundary D cos Dy sind y h M b h S Q y n 58