OTHER TPOICS OF INTEREST (Convection BC, Axisymmetric problems, 3D FEM)

Transcription

1 OTHER TPOICS OF INTEREST (Convction BC, Axisymmtric problms, 3D FEM) CONTENTS 2-D Problms with convction BC Typs of Axisymmtric Problms Axisymmtric Problms (2-D) 3-D Hat Transfr 3-D Elasticity Typical 3-D Finit Elmnts 2-D Problms: 1

2 CONVECTION HEAT TRANSFER u u a a f x x y y x y n u u a n a n u u q x y ( ) i h i h h h i x y i h i h i n w u w u a a wf dxdy x x y y u u w a n a n ds x y w u w u a a wf dxdy w q x x y y ( ) ( ) h i h i h i h i n u u ds w u w u a a wf dxdy w u ds w q u ds x x y y 2-D Problms: 2

3 CONVECTION HEAT TRANSFER n 0 Ku ij j fi Qi Ku ij j Fi or Ku F j1 j1 K a a dxdy ds i j i j ij i j x x y y n i i i n i i i F f dxdy ( q u ) ds f Q P H ij Convctiv hat transfr contributions ( H ) ij and Pi ar only from lmnt boundary Th contributions nd to b calculatd only for lmnts with convctiv boundary 2-D Problms: 3

4 CONVECTION HEAT TRANSFER (continud) y y _ y y _ b 1 a 2 x _ x b 1 5 a 2 x _ x H i j ds i j ds ds i j ds i j ds ij i j D Problms: 4

5 Conditions for Solution Symmtry Th solution of a problm may b symmtric about a lin or plan, allowing on to modl only a part of th domain and thrby rducing th computational ffort. Th solution is symmtric about a lin or plan, if and only if (a) gomtry is symmtric, (b) matrial proprtis ar symmtric, (c) loads ar symmtrically applid, and (d) boundary conditions ar symmtric about th lin or plan. Us of th solution symmtry allows us to idntify a subdomain whos analysis yilds th solution in th ntir domain. Th subdomain ncssarily will hav boundaris that coincid with th lins or plans of symmtry, and on must idntify th boundary conditions along ths lins or on th plans of symmtry Axisymm. & 3-D Problms 5

6 Rduction of Problm Siz from 3-D using solution symmtry z 3-D Modls Gnral loading, boundary conditions, and matrial proprtis, all of which may chang along th lngth and around th circumfrnc (i.., with z and θ) 2-D Modls z ( r, θ, z) θ r θ ( r, θ ) r rz, ) Th loading, boundary r conditions, and matrial proprtis do not chang along th lngth Th loading, boundary conditions, and matrial proprtis do not chang around th circumfrnc Axisymm. & 3-D Problms 6

7 Rduction of Problm Siz from 3-D 1-D Modls Th loading, boundary conditions, and matrial proprtis do not chang around th circumfrnc as wll as th lngth r r Axisymm. & 3-D Problms 7

8 AXISYMMETRIC PROBLEMS (2-D) Govrning Equation 1 u u ra a f ( r, z) r r 11 r z 22 z L R 0 z Typical axisymmtric plan r Wak Form 1 u u 0 w i ra a f ( r, z) rdrdz r r 11 r z 22 z wi u wi u a a wif rdrdz qnwirds r r z z dv rd dz dr rdr d dz u u q qnˆ a11 n a22 n r z n r z Axisymm. & 3-D Problms 8

9 AXISYMMETRIC PROBLEMS (cont.) Finit Elmnt Modl u u ( rz, ) n h j j j1 0 n i j i j u j a a rdrdz j r r z z 1 n KufQ j1 f rdrdz q w rds ij j i i i n i i j i j Kij a11 a22 rdrdz f f rdrdz, Q q w rds i i i n i r r z z Axisymm. & 3-D Problms

10 SINGLE-VARIABLE PROBLEMS IN 3-D Govrning Equation u u u a a a f( xyz,, ) x 11 x y 22 y z 33 z Boundary Conditions Spcify: u or q q q u u u q qnˆ a11 n a22 n a33 n x y z q ( uu ) cnd cnv n cnd x y z cnv Axisymm. & 3-D Problms 10

11 3-D HEAT TRANSFER (continud) ds n z nˆ n y Parts of th boundary z y n x Domain Ω Surfac Γ Ω Γ x A six-fac 3-D finit lmnt Wak Form wi uh wi uh wi u h 0 a11 a22 a33 dxdydz wiuhds x x y y z z f w dxdydz q u w ds i n i Axisymm. & 3-D Problms 11

12 3-D HEAT TRANSFER (continud) Finit lmnt approximation Finit lmnt modl n uh ujj( xyz,, ) n j1 0 KufQ j1 ij j i i K a a a dxdydz ds i j i j i j ij i j x x y y z z f f dxdydz, Q q u ds i i i n i Axisymm. & 3-D Problms 12

13 Equations of Motion 3-D + f x = f y = f z = 2 Strain-Displacmnt Rlations " xx ; 2" xy ; 2" yz " yy ; " zz 2" xz 3-D Elasticity 13

14 3-D ELASTICITY (continud) Constitutiv Rlations 8 >< >: ¾ xx ¾ yy ¾ zz ¾ xz ¾ yz ¾ xy >= >; = 2 c 11 c 12 c c 12 c 22 c c 13 c 23 c c c c >< 7 5 >: " xx " yy " zz 2" xz 2" yz 2" xy Th matrial axs ar assumd coincid with th global axs and th matrial is orthotropic with rspct to th global axs. Boundary Conditions t x ¾ xx n x + ¾ xy n y + ¾ xz n z = ^t x t y ¾ xy n x + ¾ yy n y + ¾ yz n z = ^t y t z ¾ xz n x + ¾ yz n y + ¾ zz n z = ^t z = >= >; ; on ¾ or u = ^u on u 3-D Elasticity 14

15 3-D ELASTICITY (continud) MATRIX FORM OF THE GOVERNING EQUATIONS Notation ¾ = D T = 8 >< >: @=@y >< >= < f x = < u x = " = ; f = f : y ; ; u = u : y ; f z u z >: >; ¾ xx ¾ yy ¾ zz ¾ xy ¾ xz ¾ yz Govrning quations D T ¾ + f = ½Äu ¾ = C" " = Du; 3 " xx " yy " zz 2" xz 2" yz 2" xy >= ; >; 3-D Elasticity 15

16 3-D ELASTICITY (continud) Principl of virtual displacmnts (in matrix form) 0 = Z (D±u) T C (Du) + ½u T Äu Z I dx (±u) T f dx (±u) T t ds Finit lmnt approximation (in matrix form) u = ª = 8 < : u x u y u z = ; = ª ; w = ±u = 8 < : ±u x ±u y ±u z = ; = ª± à à : : : à n à à 2 0 : : : à n à à 2 0 : : : 0 à n = f u 1 x u 1 y u 1 z u 2 x u 2 y u 2 z : : : u n x u n y u n z g T D Elasticity 16

17 3-D ELASTICITY (continud) Finit Elmnt Modl whr K = F = M Ä + K = F + Q Z Z h B T CB dx; M = ½h ª T ª dx Z I ª T f dx; Q = ª T t ds At ach nod ( uvw,, ) D Elasticity 17

18 TYPICAL 3-D FINITE ELEENTS nods Axisymm. & 3-D Problms 18

19 TYPICAL 3-D FINITE ELEMENTS Linar ttrahdral lmnt L 3 = fª g = 2 8 >< >: L 1 L 2 L 3 L 4 L 1 = 0 3 L 4 = 0 >= >; Quadratic ttrahdral lmnt fª g = Axisymm. & 3-D Problms 1 8 >< >: 3 L 1 (2L 1 1) L 2 (2L 2 1) L 3 (2L 3 1) L 4 (2L 4 1) 4L 1 L 2 4L 2 L 3 4L 3 L 1 4L 1 L 4 4L 2 L 4 4L 3 L 4 >= >;

20 TYPICAL 3-D FINITE ELEMENTS Linar prism lmnt ζ L 3 = 0 η ξ L 2 = 0 4 ζ = L 1 = ζ = 1 2 u (,, ) h c c c c c c fª g = >< >: L 1 (1 ³) L 2 (1 ³) L 3 (1 ³) L 1 (1 + ³) L 2 (1 + ³) L 3 (1 + ³) Axisymm. & 3-D Problms 20 >= >;

21 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic prism lmnt fª g = >< >: L 1 [(2L 1 1)(1 ³) (1 ³ 2 )] L 2 [(2L 2 1)(1 ³) (1 ³ 2 )] L 3 [(2L 3 1)(1 ³) (1 ³ 2 )] L 1 [(2L 1 1)(1 + ³) (1 ³ 2 )] L 2 [(2L 2 1)(1 + ³) (1 ³ 2 )] L 3 [(2L 3 1)(1 + ³) (1 ³ 2 )] 4L 1 L 2 (1 ³) 4L 2 L 3 (1 ³) 4L 3 L 1 (1 ³) 2L 1 (1 ³ 2 ) 2L 2 (1 ³ 2 ) 2L 3 (1 ³ 2 ) 4L 1 L 2 (1 + ³) 4L 2 L 3 (1 + ³) 4L 3 L 1 (1 + ³) >= >; Axisymm. & 3-D Problms 21

22 TYPICAL 3-D FINITE ELEMENTS (cont ) Linar brick lmnt ξ 6 u ( h,, ) c0 c1 c2 c3 c4 c5 c6 c7 8 (1»)(1 )(1 ³) (1 +»)(1 )(1 ³) fª g = 1 > (1 +»)(1 + )(1 ³) < >= (1»)(1 + )(1 ³) 8 (1»)(1 )(1 + ³) (1 +»)(1 )(1 + ³) > : (1 +»)(1 + )(1 + ³) > ; (1»)(1 + )(1 + ³) ζ η Axisymm. & 3-D Problms 22

23 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic Brick Elmnt ζ ξ η fª g = >< >: (1»)(1 )(1 ³)(» ³ 2) (1 +»)(1 )(1 ³)(» ³ 2) (1 +»)(1 + )(1 ³)(» + ³ 2) (1»)(1 + )(1 ³)(» + ³ 2) (1»)(1 )(1 + ³)(» + ³ 2) (1 +»)(1 )(1 + ³)(» + ³ 2) (1 +»)(1 + )(1 + ³)(» + + ³ 2) (1»)(1 + )(1 + ³)(» + + ³ 2) 2(1» 2 )(1 )(1 ³) 2(1 +»)(1 2)(1 ³) 2(1» 2 )(1 + )(1 ³) 2(1»)(1 2)(1 ³) 2(1»)(1 )(1 ³ 2 ) 2(1 +»)(1 )(1 ³ 2 ) 2(1 +»)(1 + )(1 ³ 2 ) 2(1»)(1 + )(1 ³ 2 ) 2(1» 2 )(1 )(1 + ³) 2(1 +»)(1 2)(1 + ³) 2(1» 2 )(1 + )(1 + ³) 2(1»)(1 2)(1 + ³) >= >;

24 SUMMARY In this lctur, th following topics wr covrd: 2-D problm with convction Conditions for solution symmtry Typs of axisymmtric problms FEM of axisymmtric problms (2-D) 3-D hat transfr 3-D lasticity Typical 3-D finit lmnts Axisymm. & 3-D Problms 24