b Laren DeDe Advsor: George Chen
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
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.
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.
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
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.
In wo dmensonal cases a cell has for verces: A B C D D C A B The wo dagonal lenghs are:
The mesh can be generaed b mnmzng he followng problem: where M s he monor fncon 4 c c ] [ n I
Mnmzng hs problem s eqvalen o solvng he followng lnear algebra ssem: I 0
and I 0 where. n 3
Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa.
Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa. For eample: I large small 0
Noe: When mnmzng eqaons f he coeffcen s large hen he varable wll be small and vce versa. For eample: I small large 0
Le s see a smple eample: Mnmze f z a b cz where z 0 a z c b 3 and.
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
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.
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.
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. 7 4 7 3 0 0
z Mnmze 7 a a a c f z a b cz where z 0 7 6 7 6 4 3 7 a z c b 3 and. z 7 3 7 3 7
z a z z c b 3 Mnmze where 0 and. 7 a a a c f z a b cz 7 6 7 6 4 3 7 z 7 3 7 3 7
] [ n M c In order o make he mesh more smooh and orhogonal we add hgher order dfference erms: ] [ n I
We can se he followng sem-dscrezed dfferenal eqaons o replace & 3: 4 * And smlarl for.*
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.*
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.
Mnmzng hs problem s eqvalen o solvng he followng lnear algebra ssem: I 0
and I 0 where. n 3
We can se he followng sem-dscrezed dfferenal eqaons o replace & 3: 4 * And smlarl for.*
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.
Consder he hea eqaon: f 0 0 where Ω=[ab] [ab].
In he compaonal doman we have: [ ]
In he compaonal doman we have: ] [ fed fed fed d d
fed fed fed d d In he compaonal doman we have: ] [ Chan Rle
In he compaonal doman we have: [ ] Chan Rle
Le J J J J
In he compaonal doman we have: fed fed
f J In he compaonal doman we have: fed fed J where
In he compaonal doman we have: J f Second Dervave
In he compaonal doman we have: J f J
f J In he compaonal doman we have: J Le w =
f J In he compaonal doman we have: J w w w w w w Le w =
In he compaonal doman we have: w J w w Fond sng mar mehod lke las me. Sb back n J J J
So J J J and smlarl J J J
f 0 0 where Ω=[ab] [ab]. Le f 3 0sn sn where Ω=[0] [0].
4 c c c c c 3 ma 3/
a Global Vew n Phscal Doman
b Solon n Phscal Space = 0.0066
c Solon ξ η n Compaonal Space
d Solon near one of he peaks n he phscal doman n he range 0.5 7 0 5 0.5 7 0 5.
f 0 0 where Ω=[ab] [ab]. Le f 4 5 0sn sn where Ω=[0] [0].
a Global Vew n Phscal Doman
Ren & Wang000: 3.0 5 0.4 0 Cenceros & Ho 00: 7 =0.0058 Chen Rsell Sn 004: 8 4.5 0 = 0.0049 = 0.005 b Solon n Phscal Space
c Solon ξ η n Compaonal Space
d Mesh near one of he peaks n he phscal doman n he range [0.504667 0.5046677] [0.49999998 0.5000000].
e Solon near one of he peaks n he phscal doman
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.
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.