Finite Difference Method

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Wendy Stevens
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1 Fte Dfferece Method MEL 87 Computatoa Heat rasfer --4) Dr. Praba audar Assstat Professor Departmet of Mechaca Egeerg II Deh

2 Dscretzato Methods Requred to covert the geera trasport equato to set of agebrac equatos Fte dfferece method Fte voume method Fte eemet method

3 Itroducto to Fte Dfferece aor seres epaso f ) f ) f f ) f )! gradet curvature

4 Dscretzato 6 ) ),

5 Represetato of a Dervatve ), O 6 ) ), Fte dfferece represetato 6 ) ), Forward dfferece

6 Bacward Dfferece 6 ) ) ), 6 ) ), ), O Bacward dfferece

7 Cetra dfferece 6 ) ), 6 ) ),,, ) O Subtractg, Cetra dfferece

8 Forward, bacward ad cetra dfferece stec ) O, ), O,, ) O

9 Stec drecto ) O ) O,, ) O

10 d Order ad Med Dervatve ) ) ),, ) ) ) ] ), ) [ 4,,,, O

11 Boudar Cosderato What d of dfferecg scheme s possbe at the boudares? ) O Cetra dfferece appromato s ot possbe as pot s beeath the boudar How we ca get a secod order accurate scheme? Possbt: Pooma approach

12 Pooma Approach Assume a b c At grd pot, a At grd pot where, a b c ) At grd pot where, a b c)

13 Pooma Approach cot d) Sovg these equatos, b 4 Dfferetato of gves At pot boudar), What s the order of appromato?? a b b c b c 4

14 Order of Appromato aor seres gves, Comparg wth the pooma epresso, we ca sa that our pooma s of O),, ca a be epressed terms of the pooma Represets oe-sded dfferece of d order accurac 6 ) ) ) 4 O O c b a

15 FDM for D dffuso Uses trucated aor seres epaso to appromate the dervatve of the DE Cosder -D dffuso equato d Γ S Epad aor seres about d pot Subtractg these equatos eds

16 FDM cot d) Addg the equatos d ) d ) ) Droppg the trucated terms d Γ d Γ ) Γ ) he fa dscretzed equato Γ Secod order trucato error S ) Γ ) S S )

17 FDM cot d) Γ ) Γ S ) Γ ) We ca wrte oe such equato for each grd pot Boudar codtos gves us boudar vaue Secod order accurate Need to fd a wa to sove the coupe agebrac equato set

18 D Stead State Coducto Cosder the stead state heat coducto a sab of thcess L, whch eerg s geerated at a costat rate of S W/m. he boudar surface at s mataed at a costat temperature o, whe the boudar surface at dsspates heat b covecto wth a heat trasfer coeffcet h to a ambet at temperature. Compute the temperature sde the sab for h W/m. C), 8 W/m. C), L. m, C, o 5 C, ad S o 5 C C 4 5. m

he boudar surface at s mataed at a costat temperature o, whe the boudar surface at dsspates heat

19 Souto.m L at h h d d) at ) S d d o B.C. 6 S L / 5) S ) S ) ) ) S ) ) )

20 reatmet of Boudar Codto o 5 C / cod cov 4 N5. m 6 Appcabe to ode -4 Appg t to ode 5 N N 4 N hn ) S h ) SN N h

21 reatmet of Boudar Codto Aother Wa) o 5 C 4 N5. m App to ode 5 B.C. gves h h h5 ) h 6 5 ) 4

22 Agebrac Equatos Ca be soved b homas agorthm Matr verso as show et sde

23 Matr Form [ ][ ] [ ] [ ] [ ] [ ] B A B A

24 Prescrbed Heat Fu q o X / N- N XL q N Eerg baace gves, q q N N N S SN

25 FDM represetato N q S ) q S ) N N N N for for N S ) S ) N N N for for For suated or smmetr boudar, Fu boudar,

26 Ustead Heat Coducto wth FDM Ustead heat coducto - D wth costat therma coductvt Epad the dvdua terms wth aor seres, t α t t t t ) ) 4 4

27 Ustead Heat Coducto cot d) t α ) )

28 Epct Soutos t α αt ) ) ) ) Fd -D ustead temperature dstrbuto t stead state w C w C 5 cm 5 cm 5 α 7 Ita temp C, t sec cm / s αt ) w ta ater

29 Resuts 8 C) 6 4 sec 9sec 6sec Nodes

30 D Stead State Heat Coducto ) S,,, ) D stead state wth heat geerato ) S 4,, where

31 Fu Boudar Codto,, q o W/m /,,- Nodes ) o a prescrbed heat fu boudar q, S After rearragemet, 4 S q o

32 Fu Boudar Codto aother wa), Appg the fte dfferece equato at the boudar ode ), q o W/m /, S,, 4,- B.C. qo qo,,, S qo,, 4

33 Covectve Boudar Codto Eerg baace gves,, h ) 4 S After rearragemet, h h,,, 6 S, -,, /, Covecto h,,-

34 Isuated Boudar 6b cos ) π b b Isuated Mataed at zero temperature Isuated Node : -4 Node : 4-4 Node : Node 4: Node 5: Node 6: Matr Form

Lecture 7. Norms and Condition Numbers

Lecture 7. Norms and Condition Numbers Lecture 7 Norms ad Codto Numbers To dscuss the errors umerca probems vovg vectors, t s usefu to empo orms. Vector Norm O a vector space V, a orm s a fucto from V to the set of o-egatve reas that obes three