PART 8. Partial Differential Equations PDEs

Transcription

1 he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef Cvl Engneerng Department he Islamc Unverst of Gaza

2 PAR 8 Partal Dfferental Equatons PDEs CH 9; Fnte Dfference Method Ch3; Fnte Element Method Chapter 9

3 Partal Dfferental Equatons Chapter 9 3

4 Partal Dfferental Equatons. Laplace equaton. Dffuson equaton 3. Wave equaton A partal dfferental equaton (PDE) nvolves two or more ndependent varables. For eample: t c t f f k 0 f f Chapter 9 4

5 Fnte Dfference: Ellptc Equatons Smlar to the ODE central dvded dfferences are substtuted for the partal dervatves n the orgnal equaton. hus a partal dfferental equaton s transformed nto a set of smultaneous algebrac equatons that can be solved b the methods descrbed earler. Because of ts smplct and general relevance to most areas of engneerng we wll use a heated plate as an eample for solvng ellptc PDEs. 5

6 Fnte Dfference: Ellptc Equatons he Central dvded dfferences

7 Fnte Dfference: Ellptc Equatons 7

8 Fnte Dfference: Ellptc Equatons 8

9 he General steps. Choose ntegers n and m to defne step szes h and k n and drecton respectvel.. Partton the nterval [a b] nto m equal parts of wdth h and [c d] nto n equal parts of wdth k n =d b a d c h k m n o =c o =a m =b 9

10 he General steps 3. Place a grd b drawng vertcal and horzontal lnes through the ponts of coordnates ( ). 4. For each mesh ponts; h k n =d o =c o =a m =b Chapter 9 0

11 he Laplacan Dfference Equatons/ k h Chapter 9 Laplacan dfference equaton. Holds for all nteror ponts Laplace Equaton O[() ] O[() ] -4

12 Fnte Dfference: Ellptc Equatons he boundar condtons along the edges must be specfed to obtan a unque soluton. he smplest case s where the temperature at the boundar s set at a fed value ths s called; Drchlet boundar condton. Suppose that at the four edges are; 0 o C 50 o C 00 o C and 75 o C he Drchlet (or frst-tpe) boundar condton s a tpe of boundar condton named after Peter Drchlet ( ). When mposed on an ordnar or a partal dfferental equaton t specfes the values that a soluton needs to take on along the boundar of the doman. In engneerng applcatons a Drchlet boundar condton also referred to as a fed boundar condton.

13 Fnte Dfference: Ellptc Equatons A balance for node () s: Smlar equatons can be developed for other nteror ponts to result a set of smultaneous equatons. 0 Chapter 9 3

14 Fnte Dfference: Ellptc Equatons he result s a set of nne smultaneous equatons wth nne unknowns: Chapter 9 4

15 he Lebmann Method/ Most numercal solutons of Laplace equaton nvolve sstems that are ver large. For larger sze grds a sgnfcant number of terms wll be zero. For such sparse sstems most commonl emploed approach s Gauss-Sedel whch when appled to PDEs s also referred as Lebmann s method. Chapter 9 5

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17 Eample 9. Cont. Chapter 9 7

18 Fg 9.5 Chapter 9 8

19 Man engneerng problems ehbt rregular boundares. Irregular Boundares Chapter 9 9

20 0 Frst dervatves n the drecton can be appromated as: ) ( ) ( Irregular Boundares

21 A smlar equaton can be developed n the drecton. Substtute n Laplace Equaton ) ( ) ( Irregular Boundares 0 0 ) ( ) ( ) ( ) ( Chapter 9