by Lauren DeDieu Advisor: George Chen

Transcription

1 b Laren DeDe Advsor: George Chen

2 Are one of he mos powerfl mehods o nmercall solve me dependen paral dfferenal eqaons PDE wh some knd of snglar shock waves & blow-p problems. Fed nmber of mesh pons Moves mesh adapvel so errors are dsrbed nforml. Unform Mesh Movng Mesh

3 Wnslow 967 Thompson 985 and ohers proposed varos ellpc eqaons and varaonal mehods for defnng adapve mappng. Bdd Hang and Rssel 009 n her paper Adapv wh Movng Grds smmarze radonal hp-refnemen mehods n whch mesh s added/deleed based on esmaes abo he solon error. The conclde r-adapve mehods whch nall se nform mesh whch laer becomes concenraed where he solon has neresng behavor have enormos poenal alhogh he are crrenl n her earl sages of developmen.

4 Cenceros and Ho 00: -developed a smple e effcen dnamcall adapve mesh generaor for me-dependen problems. -Mesh s generaed drecl from he phscal doman so he mesh eqaons become qe smple.

5 Cenceros and Ho 00: -developed a smple e effcen dnamcall adapve mesh generaor for me-dependen problems. -Mesh s generaed drecl from he phscal doman so he mesh eqaons become qe smple. Compaonal Doman ξη Phscal Doman

6 Cenceros and Ho: -developed a smple e effcen dnamcall adapve mesh generaor for me-dependen problems. -Mesh s generaed drecl from he phscal doman so he mesh eqaons become qe smple. We focs on blow-p problems and smplf and mprove Cenceros and Ho`s resls b addng an addonal erm o make he mesh more orhogonal.

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8 In wo dmensonal cases a cell has for verces: A B C D D C A B The wo dagonal lenghs are:

9 The mesh can be generaed b mnmzng he followng problem: where M s he monor fncon 4 c c ] [ n I

10 Mnmzng hs problem s eqvalen o solvng he followng lnear algebra ssem: I 0

11 and I 0 where. n 3

12 Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa.

13 Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa. For eample: I large small 0

14 Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa. For eample: I small large 0

15 Le s see a smple eample: Mnmze f z a b cz where z 0 a z c b 3 and.

16 Mnmze f z a b cz where z 0 a z c b 3 and. f z a z b cz f a z b a a az b 0 a a az b 0 f z a z cz a a az cz a a z a c 0

17 Mnmze z a a z a c f z a b cz where z 0 f a a az b 0 a z c b 3 and.

18 Mnmze z a a z a c f z a b cz where z 0 f a a az b 0 a a a a a b 0 a c a z c b 3 and.

19 z Mnmze f a a a c f z a b cz where z 0 z a a az b a a a a a b a c 0 0 a z c b 3 and

20 z Mnmze 7 a a a c f z a b cz where z a z c b 3 and. z

21 z a z z c b 3 Mnmze where 0 and. 7 a a a c f z a b cz z

22 ] [ n M c In order o make he mesh more smooh and orhogonal we add hgher order dfference erms: ] [ n I

23 We can se he followng sem-dscrezed dfferenal eqaons o replace & 3: 4 * And smlarl for.*

24 We can se he followng sem-dscrezed dfferenal eqaons o replace & 3: c c c c c 4 4 nd Order Dfference * And smlarl for.*

25 Dsadvanages of -3:. Dependen on nal vales.. The convergence rae s ver slow so akes a long me o compe. 3. When s large ma be nsable small oscllaon. Advanages of 4:. Alwas has a solon and s smooh.. Easer o conrol he movemen of mesh.

26 Mnmzng hs problem s eqvalen o solvng he followng lnear algebra ssem: I 0

27 and I 0 where. n 3

28 We can se he followng sem-dscrezed dfferenal eqaons o replace & 3: 4 * And smlarl for.*

29 Dsadvanages of -3:. Dependen on nal vales.. The convergence rae s ver slow so akes a long me o compe. 3. When s large ma be nsable small oscllaon. Advanages of 4:. Alwas has a solon and s smooh.. Easer o conrol he movemen of mesh.

30 Consder he hea eqaon: f 0 0 where Ω=[ab] [ab].

31 In he compaonal doman we have: [ ]

32 In he compaonal doman we have: ] [ fed fed fed d d

33 fed fed fed d d In he compaonal doman we have: ] [ Chan Rle

34 In he compaonal doman we have: [ ] Chan Rle

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40 Le J J J J

41 In he compaonal doman we have: fed fed

42 f J In he compaonal doman we have: fed fed J where

43 In he compaonal doman we have: J f Second Dervave

44 In he compaonal doman we have: J f J

45 f J In he compaonal doman we have: J Le w =

46 f J In he compaonal doman we have: J w w w w w w Le w =

47 In he compaonal doman we have: w J w w Fond sng mar mehod lke las me. Sb back n J J J

48 So J J J and smlarl J J J

49 f 0 0 where Ω=[ab] [ab]. Le f 3 0sn sn where Ω=[0] [0].

50 4 c c c c c 3 ma 3/

51 a Global Vew n Phscal Doman

52 b Solon n Phscal Space =

53 c Solon ξ η n Compaonal Space

54 d Solon near one of he peaks n he phscal doman n he range

55 f 0 0 where Ω=[ab] [ab]. Le f 4 5 0sn sn where Ω=[0] [0].

56 a Global Vew n Phscal Doman

57 Ren & Wang000: Cenceros & Ho 00: 7 = Chen Rsell Sn 004: = = b Solon n Phscal Space

58 c Solon ξ η n Compaonal Space

59 d Mesh near one of he peaks n he phscal doman n he range [ ] [ ].

60 e Solon near one of he peaks n he phscal doman

61 The smple movng mesh mehod s ver good a solvng parabolc eqaons wh blowp properes. Nmercal compaons show ha hs mehod s mch faser han radonal mehods. I akes onl half a da o compe on a normal PC.

62 We wll selec more effcen monor fncons o mprove crren resls. We wll r o appl hs mehod o oher real problems. In parclar we wll appl or mehod o a nonlnear damped p-ssem wh an nbonded doman.