45t AIAA Aeropae See Meetg ad Exbt 8 - Jauary 007 eo evada AIAA 007-33 Effet Implt LU-SGS Algortm for Hg-Order Spetral Dfferee Metod o Utrutured Hexaedral Grd uz Su ad.j. Wag Departmet of Aeropae Egeerg Iowa State Uverty Ame IA 500 e Lu ASA Ame eear Ceter Moffett Feld CA 94035 C.L. Ce owell Setf Touad Oa CA 9360 A effet mplt lower-upper ymmetr Gau-Sedel LU-SGS oluto algortm a bee developed for a g order mult-doma petral dfferee metod o utrutured exaedral grd. Te LU-SGS olver preodtoed by te blo elemet matrx ad te ytem of equato te olved wt a exat LU deompoto approa. I addto te effet of everal parameter o te overgee rate te mplt eme ave bee vetgated for bot exteral ad teral flow. Te mplt eme a ow a peed up fator of more ta a order of magtude relatve to te mult-tage explt uge-kutta eme for everal demotrato problem. I. Itroduto e advatage of g-order metod order of auray > over frt ad eod-order oe well ow te T CFD ommuty. Geerally peag wt te ame umber of degree-of-freedom DOF or oluto uow g-order metod are apable of produg mu more aurate reult. For problem requrg very g auray e.g. wave propagato problem omputatoal aeroaout g-order metod ave bee te ma oe. May g-order metod were developed for trutured grd e.g. EO/WEO metod ompat metod -3 optmzed metod 4 to ame ut a few. I te lat two deade tere ave bee teve reear effort o g-order metod for utrutured grd e may real world applato ave omplex geometre. A omplete lt of otable example lude te petral elemet metod 5 mult-doma petral metod 6-7 - exat fte volume metod 8 WEO metod 9 dotuou Galer DG metod 0- g-order redual dtrbuto metod 3 petral volume SV 4-7 ad petral dfferee SD metod 8-. Amog toe metod ome are baed o te wegted redual form of te goverg equato for tae te DG metod 0-. Some are baed o te tegral form of te goverg equato e. g. te -exat fte volume metod 8 ad SV metod 4-7. Oter u a te taggered grd mult-doma petral metod 6-7 ad te SD metod 8- are baed o te dfferetal form. We oe ooe a partular metod for tree-dmeoal applato te ot ad te omplexty mplemetg te metod ofte a mportat fator. It obvou tat metod baed o te dfferetal form are te eaet to mplemet e tey do ot volve urfae or volume tegral. T partularly true we gorder urved boudare eed to be dealt wt. We reetly developed a g order SD metod for te tree dmeoal aver-stoe equato o utrutured exaedral grd. Hg order auray ad petral overgee are aeved for everal bemar problem. It wa alo ow tat te wall boudare mut be approxmated wt g-order urfae. A explt uge-kutta tme tegrato eme wa ued te mplemetato. Altoug te explt eme eay to mplemet ad a g-order auray tme t uffered from low overgee epeally for vou grd w are lutered te vou boudary layer. It wellow tat g-order metod are retrted to a maller CFL umber ta low order oe. I addto tey alo Potdo eear Aoate of Aeropae Egeerg 7 Howe Hall uyuz@atate.edu AIAA Member. Aoate Profeor of Aeropae Egeerg 7 Howe Hall zw@atate.edu Aoate Fellow of AIAA eear Sett e.lu@aa.gov Mal Stop T7B-. Maager of Appled Computatoal Py Departmet owell Setf e@rw.om Amera Ittute of Aeroaut ad Atroaut Copyrgt 007 by. Su.J. Wag. Lu ad C.L. Ce. Publed by te Amera Ittute of Aeroaut ad Atroaut I. wt permo.
poe mu le umeral dpato. Terefore t tae exeve CPU to rea a tate-teady oluto wt explt g-order eme. Te omputato ot of g-order explt metod for may teady-tate problem o g tat tey beome le effet ta low-order mplt metod term of te total CPU tme gve te ame level of oluto error. It terefore mperatve to develop effet mplt oluto approae for gorder metod to fully realze te potetal w te obetve of te preet tudy. Implt tme-tegrato eme are gly dered for mproved effey e tey a advae te oluto wt gfatly larger tme tep omparg wt te explt metod. May mplt eme ave bee developed ad appled uefully to utrutured grd to aelerate overgee to teady tate 3-8 te lat oe ad a alf deade. I t paper a effet mplt lower-upper ymmetr Gau-Sedel LU-SGS oluto algortm a bee developed for te g order petral dfferee metod o utrutured exaedral grd. Te paper orgazed a follow. I te ext eto te formulato of te 3D petral dfferee metod ludg bot explt ad mplt eme derbed for a exaedral elemet. I Seto 3 everal repreetatve tet ae are eleted to demotrate te effey of te mplt eme ad tudy te effet of everal parameter o te overgee rate. Coluo ad poble future wor are ummarzed Seto 4. II. 3D Formulato of Hg-Order Mult-doma Spetral Dfferee Metod We aume te omputatoal doma dretzed to utrutured exaedral ell w are traformed to te tadard ube te omputatoal doma for effet mplemetato a ow Fgure. I ea elemet two et of pot are defed amely te oluto pot ad flux pot a ow Fgure. Soluto uow are defed at te oluto pot wle fluxe are omputed at te flux pot. Soluto pot D are te Gau quadrature pot ad te flux pot are te Gau-Labatto pot. Te oluto are dotuou aro elemet boudare ad fte-volume type ema olver are ued to ompute a ommo terfae flux tu provdg elemet-we ouplg. Fgure. Traformato from a pyal elemet to a tadard elemet Fgure. Dtrbuto of oluto pot rle ad flux pot quare a tadard elemet for a 3 rd order SD eme. After te traformato te aver-stoe equato te tadard elemet a be wrtte a F G H t 0. Te redual defed a F G H. Tee equato a be vewed to be pot-we or ell-we elemet-we equato. Te otruto of te oluto ad flux te ut ube a be expreed a follow: Amera Ittute of Aeroaut ad Atroaut
Amera Ittute of Aeroaut ad Atroaut 3 l F F l G G l H H were te gve umber of Gau quadrature pot a oordate dreto - te degree of otruto polyomal ad K o π l o K π For detal of te umeral metod ludg te omputato of te vd ad vou fluxe refer to eferee. A. Explt eme At ell ug te forward Euler dfferee a be wrtte a dt. Smlarly a tree-tage TVD uge-kutta eme a be wrtte a dt. [ ] 4 4 3 dt. [ ] 3 3 dt. B. Implt eme At ea ell ug te baward Euler dfferee a be wrtte a [ ] t 3 Let ad learzg te redual we obta b b b 4 were b date all te egborg ell otrbutg to te redual of ell. Terefore te fully learzed equato for 3 a be wrtte a b b b t I. 5 However t ot too mu memory to tore te LHS mplt Jaoba matre. Terefore we employ a LU-SGS eme to olve 5.e. we ue te mot reet oluto for te b ell b b b t I. 6
Te matrx I D 7 t te elemet or ell matrx w ot too large for d to 4 t order SD eme. 6 te olved wt a dret LU deompoto olver. Se we do ot wat to tore te matre 6 furter mapulated a follow. b ote tat { } b b b b b { } { } b b b b. 8 Let. Te ombg 6 ad 8 togeter we obta I t. 9 t Eq. 9 te olved wt multple ymmetr forward ad baward weep wt a prerbed overgee tolerae ε. ote tat f 9 olved to mae zero te uteady redual zero at ea tme t tep. Te tal gue for a be et to. Terefore te tal uteady redual te ame a te teady redual at te lat tme tep.e.. Te uteady redual te motored for overgee. For teady tate problem t ot eeary to drve te uteady redual to mae zero. I fat t may be more effet to et a maxmum umber of weep for 9 er_weep to ut a few 3 0. III. umeral eult ad Duo I order to demotrate te effey of te mplt LU-SGS eme ad aalyze te effet of varou parameter o te overgee rate tree flow problem are oe a te demotrato problem. All of te mulato tart from te free tream ad overge to mae zero. A. Ivd flow over a pere A vd flow over a pere wt a free tream Ma umber of 0.535 eleted a te frt tet to demotrate te effey of te mplt eme ad alo to tudy te effet of CFL umber ad er terato otrol parameter o overgee aratert. Fgure 3 ow te omputatoal grd ued te mulato w lude 768 exaedral ell. Fgure 4 dept te Ma otour dtrbuto omputed wt te 4 t order SD eme. Bot te tree-tage uge-kutta explt ad te LU-SGS mplt eme wt d 3 rd ad 4 t order patal auray were employed te mulato. Te mplt eme dramatally aelerate te overgee rate to te teady tate for t exteral flow. T llutrated te Fgure 5 w dplay te 4 Amera Ittute of Aeroaut ad Atroaut Fgure 3. Grd for vd flow over a pere
overgee tore term of CPU tme. Te overgee rate wt te mplt eme more ta a order of magtude fater ta te explt eme. ext we tudy te effet of everal parameter o te overgee rate of te mulato. Obvouly te CFL umber a mportat overgee parameter. I te preet tudy te CFL umber omputed baed o te followg power law form ad bouded by te mmum ad maxmum CFL umber: CFL MI CFL m α CFLmax were α te amplfato fator ad te terato umber. I te frt tet CFL ad α are m Fgure 4. Ma otour for flow over a pere Fgure 5a. edual tory of vd flow over a pere wt te d order patal auray Fgure 5b. edual tory of vd flow over a pere wt 3 rd patal auray fxed at erta value wle CFL max a varable. Te effet of CFL max o te overgee rate are owed Fgure 6a wt CFL m. 0 for te 3 rd order SD eme ad Fgure 6b wt CFL m 0. 5 for te 4 t order eme. Te amplfato fator et to be.5 for bot eme. It ealy oberved tat te overgee rate trogly deped o te CFL umber. Te larger CFL umber reult ger overgee rate. However we alo wat to empaze tat a too large CFL umber a aue te mulato to dverge. Te eod parameter o te overgee rate te umber of te er terato.e. te umber of forward ad baward Gau-Sedel weep te LU-SGS approa. Oe weep defed to lude bot te forward ad baward weep ere ad deoted by er _ weep. Te uteady redual ea tme terato tep a be drve to mae zero f er _ weep bg eoug. I te preet mulato er _ weep 3 te mallet umber to guaratee tablty ad overgee to te teady tate. I t tet we 5 Amera Ittute of Aeroaut ad Atroaut Fgure 5. edual tory of vd flow over a pere wt 4 t order patal auray
let er _ weep vary ad fx te oter parameter. From Fgure 7 t eem tat te umber of er terato doe t trogly fluee te overgee rate for te vd flow over a pere. Fgure 6a. Effet of CFL umber 3 rd order Fgure 6b. Effet of CFL umber 4 t order I te LU-SGS approa te omputato of te elemet Jaoba matrx qute tme oumg e t ze qute large. Oe dea to mprove te effey to freeze t matrx for everal tme tep. Te frequey w t matrx updated deoted by F _ ITIME. For example F _ ITIME 5 mea te elemet Jaoba matrx omputed every 5 tme tep. Fgure 8 ow te overgee tore wt dfferet F _ ITIME wt a 3 rd order SD eme. I t tet te er_weep et to be 5 ad CFL rage from to 000000. It a be oberved tat te bgger F _ ITIME reult ger effey. Fally te effet of te amplfato fator o te overgee rate tuded. I t tet α vare from.5 to 3 wt a 0.5 terval. Oter parameter are et a follow: CFL m CFL max 000 000 er_weep 5 ad F_ITIME 40. I Fgure 9 te overgee rate are plotted togeter for.5 α. 5. It appear te overgee rate doe ot trogly deped o te amplfato fator we.5 α. 5. However te mulato dverged we α. 5 or α 3. 0. Fgure 7. Effet of er terato 3 rd Fgure 8. Effet of matrx freezg frequey 6 Amera Ittute of Aeroaut ad Atroaut
Fgure 9. Effet of te amplfato fator B. Ivd flow over a 3d bump Ivd flow over a 3D bump wa eleted to repreet teral flow problem. Fgure 0 ow te omputato grd wt 307 exaedral ell. Fgure ow te teady tate preure otour at te mddle uttg plae omputed wt te 4 t order SD eme. Bot te tree-tage uge-kutta explt ad LU-SGS mplt eme were employed for t problem wt d 3 rd ad 4 t order patal auray. Fgure ow te overgee tore. From Fgure a we a oberve tat te overgee to te teady tate aelerated by more ta 0 tme ug te d order SD eme. For 3 rd ad 4 t order SD eme te tree-tage uge-kutta eme faled to overge te mulato a ow Fgure b ad. Te effet of te maxmum CFL umber are llutrated Fgure 3. For tee mulato α te power law et to be.5 ad CFL m. ote aga tat te larger CFL max reult ger overgee rate w otet wt te pere ae. Fgure 0. Grd for vd flow over a 3D bump Te effet of te umber of er terato o te overgee rate te ame a te pere ae.e. te dfferet value of er _ weep do t mae mu dfferee o te overgee rate w ow te Fgure 4. ext te effet of matrx freezg frequey ow Fgure 5. ote alo tat te larger F _ ITIME te ger te effey. Fgure. Preure otour at te mddle uttg plae Fgure a. edual tory of vd flow over a 3d-bump wt d order patal auray 7 Amera Ittute of Aeroaut ad Atroaut Fgure b. edual tory of vd flow over a 3d-bump wt 3 rd patal auray
Fally te effet of te amplfato fator o te overgee rate ow Fgure 6. Aga te overgee rate ot trogly depedet o t parameter a te ae of flow over te pere. C. Ivd flow over a ACA wg Te ae of vd flow over a ACA wg oe to furter aalyze te effet of dfferet parameter o te overgee rate. Fgure 7 ow te omputatoal grd wt 48 exaedral ell ad Fgure 8 dplay te teady tate Ma otour omputed wt te 4 t order SD eme. To tet te effet of te maxmum CFL umber all oter parameter are fxed a follow: CFL m 0. 5 α.5 ad te elemet Jaoba matrx updated every 40 tme tep. Te overgee tore are plotted Fgure 9 for te dfferet value of a bgger value of maxmum CFL reult a ger effey. Fgure. edual tory of vd flow over a 3d-bump wt 4 t order SD eme CFL max. Aga we ee tat Fgure 3. Effet of maxmum CFL umber Fgure 4. Effet of umber of er terato Fgure 5. Effet of matrx freezg frequey Fgure 6. Effet of amplfato fator 8 Amera Ittute of Aeroaut ad Atroaut
Fgure 7. Grd for vd flow over a ACA wg Fgure 8. Ma otour tart at Ma 0. wt a 0.05 terval for ACA wg Fgure 9. Effet of maxmum CFL umber Te effet of te umber of er weep ow Fgure 0. Obvouly for t ae te le te umber of er weep te more effet te mulato. Fally te effet of te matrx freezg frequey o te overgee rate ow Fgure. I t mulato te oter parameter are et a follow: CFL m 0.5 CFL max 00 ad α.5. Te fgure ofrm tat te le frequetly te matrx updated te more effet te mulato. IV. Coluo ad Future Wor I t paper a effet mplt lower-upper ymmetr Gau-Sedel LU-SGS oluto algortm a bee developed for a g order mult-doma petral dfferee metod o utrutured exaedral grd. Te mplt eme a ow more ta a order of magtude of peed-up relatve to te mult-tage uge- Kutta explt tme tegrato eme for everal Fgure 0. Effet of umber of er terato Fgure. Effet of matrx freezg frequey 9 Amera Ittute of Aeroaut ad Atroaut
demotrato problem. I addto te effet of everal parameter o te overgee rate ave bee vetgated umerally for bot exteral ad teral flow. Geerally peag larger CFL umber le frequet matrx update ad maller umber of er terato reult te fater overgee. We are urretly extedg te approa to vou flow problem ad te reult wll be preeted a future publato. Aowledgemet Te tudy a bee fuded by owell Setf/DAPA uder otrat W9F-04-C-00 ad partally upported by te Ar Fore Offe of Setf eear. Te vew ad oluo otaed ere are toe of te autor ad ould ot be terpreted a eearly repreetg te offal pole or edoremet eter expreed or mpled of DAPA AFOS or te U.S. Govermet. eferee Su C.-W. Eetally o-ollatory ad wegted eetally o-ollatory eme for yperbol oervato law Advaed umeral Approxmato of olear Hyperbol Equato edted by A. uartero Leture ote Matemat Sprger-Verlag Berl/ew or 998 Vol. 697 P. 35. Lele S.K. Compat Fte Dfferee Seme wt Spetral-Le eoluto Joural of Computatoal Py Vol. 03 99 pp. 6-4. 3 Vbal M. ad Gatode D. So Capturg Ug Compat- Dffereg- Baed Metod AIAA-005-65. 4 Tam C.K.W. ad Webb J.C. Dpero-elato-Preervg Fte dfferee Seme for Computatoal Aout Joural of Computatoal Py Vol. 07 o. 993 pp. 6-8. 5 Patera A.T. A Spetral elemet metod for flud dyam: Lamar flow a ael expao J. Comput. Py. 54 468984. 6 Koprva D.A. ad Kola J.H. A oervatve taggered grd Cebyev multdoma metod for ompreble flow J. Comput. Py. 5 44996. 7 Koprva D.A. A Staggered-Grd Multdoma Spetral Metod for te Compreble aver Stoe Equato Joural of Computatoal Py Volume 43 pp. 5-58 998. 8 Bart T.J. ad Fredero P.O. Hg-Order Soluto of te Euler Equato o Utrutured Grd ug uadrat eotruto AIAA Paper o. 90-003990. 9 Hu C. ad Su C.-W. Wegted eetally o-ollatory eme o tragular mee J. Comput. Py. 50 97 999. 0 Cobur B. ad Su C.-W. TVB uge-kutta loal proeto dotuou Galer fte elemet metod for oervato law II: Geeral framewor Mat. Comput. 54989. Cobur B. ad Su C.-W. Te uge-kutta dotuou Galer metod for oervato law V: multdmeoal ytem J. Comput. Py. 4 99-4 998. Ba F. ad ebay S. Hg-order aurate dotuou fte elemet oluto of te D Euler equato J. Comput. Py. 38 5-85 997. 3 Abgrall. ad oe P.L. Hg Order Flutuato Seme o Tragular Mee Joural of Setf Computg Volume 9 pp. 3 36 003. 4 Wag.J. ad Lu e Spetral Fte Volume Metod for Coervato Law o Utrutured Grd III: Exteo to Oe-Dmeoal Sytem J. Setf Computg Vol. 0 o. pp.37-57 004. 5 Wag.J. ag L. ad Lu e Spetral fte volume metod for oervato law o utrutured grd IV: exteo to two-dmeoal ytem J. Comput. Py. 94 76-74004. 6 Lu. Vour M. ad Wag.J. Spetral Fte Volume Metod for Coervato Law o Utrutured Grd V: Exteo to Tree-Dmeoal Sytem Joural of Computatoal Py Vol. pp. 454-47 006. 7 Su uz Wag.J. ad Lu e Spetral Fte Volume Metod for Coervato Law o Utrutured Grd VI: Exteo to Vou Flow Joural of Computatoal Py Vol. 5pp.4-58 006. 8 Lu. Vour M. ad Wag.J. Dotuou Spetral Dfferee Metod for Coervato Law o Utrutured Grd Proeedg of te 3 rd Iteratoal Coferee CFD Toroto Caada July 004. 9 Lu e Vour M. ad Wag.J. Mult-Dmeoal Spetral Dfferee Metod for Utrutured Grd AIAA- 005-030. 0 Wag. J. ad Lu e Te Spetral Dfferee Metod for te D Euler Equato o Utrutured Grd AIAA- 005-5. Huag P.G. Wag.J. ad Lu e A Implt Spae-Tme Spetral Dfferee Metod for Dotuty Capturg Ug Adaptve Polyomal AIAA-005-555. Su uz Wag.J. ad Lu e Hg-Order Mult-doma Spetral Dfferee Metod for te aver-stoe Equato o Utrutured Hexaedral Grd Commuato Computatoal Py Vol. o. pp. 30-333 007 ad alo AIAA-006-30. 3 Veatara V. ad Mavrpl D.J. Implt Solver for Utrutured Mee Joural of Computatoal Py Vol. 05 o. 993 pp.83-9. 4 Veatara V. ad Mavrpl D.J. Implt Metod for te Computato of Uteady Flow o Utrutured Grd Joural of Computatoal Py Vol. 7 o. 996 pp.380-397. 0 Amera Ittute of Aeroaut ad Atroaut
5 Soetro M. Imlay S.T. ad obert D.W. A zoal Implt Proedure for Hybrd Strutured-Utrutured Grd AIAA Paper 94-0645 Ja. 994. 6 Fr.T. Aemet of a Utrutured-Grd Metod for Predtg 3-D Turbulet Vou Flow AIAA Paper 96-09 Ja. 996. 7 Blao M. ad gg D.W. A Fat Solver for te Euler Equato o Utrutured Grd Ug a ewto-gmes Metod AIAA Paper 97-033 Ja. 997. 8 Ce.F. ad Wag.J. Fat Blo Lower-Upper Symmetr Gau-Sedel Seme for Arbtrary Grd. Amera Ittute of Aeroaut ad Atroaut