Introduction of Curvilinear Coordinates into Numerical Analysis

Transcription

1 Inroon of Crvlner Coornes no Nmerl Anlss Dee o Professor Emers Yosk Ymmoo Te Unvers of Toko Hros Issk Inse of Meml Anlss Osk Jpn ssk@.-o.ne.jp Dske Kzw Inse of Insrl Sene Te Unvers of Toko k@s.-oko..jp Asr Inroon of rvlner oornes m e ver onvenen n mn ses. Teorell ensor nlss wol e mos se. However ensor noon n e se n nmerl proere. For emple e sr srmnon of pper n lower sffes s mpossle n e presen omper lnes. In e presen pper rer elemenr ppro more se o wre oes of omper prormmn s pe. If we nroe rvlner oornes we wol e le o re n more nrl w o nle fe sonn movn sonn n rve onr. We ol enere fne mes n e neoroo of fe n movn sonnes. Frermore f we nroe rvlner oornes we n rnsform non-sqre reon no sqre one n n se relr mes. Usll n nmerl llon rve onr s pprome je or non-smoo onr. An nroon of rvlner oornes ol solve mn prolems n e nmerl nlses.. Inroon Inroon of rvlner oornes m e ver onvenen n mn ses. Teorell ensor nlss wol e mos se for e prpose. However ensor noon n e se onvenenl n nmerl proere. For emple e sr srmnon of pper n lower sffes s mpossle n e presen omper lnes ~. In e presen pper rer lssl ppro s pe. Te meo m e more se o wre oes of omper prormmn sn omper lne vlle now. In orer o re rp ne of e solon fnon we sll se fner mes o follow e rp ne. If we nroe rvlner oornes we wol e le o re n more nrl w o nle fe sonn movn sonn. Nmel If we nroe rvlner oornes we ol enere fne mes n e neoroo of fe n movn sonnes n nrese e r of e solon. Frermore f we nroe rvlner oornes we n rnsform non-sqre reon no sqre one n n se relr mes n e rnsforme oornes. Frermore n nmerl llon rve onr s sll pprome je or non-smoo onr. In seons n fe n movn sonnes re ssse respevel n n seon 5 rnsformon of non-sqre reon no sqre reon s ssse. An nroon of rvlner oornes ol solve mn prolems n e nmerl nlses.. D oornes rnsformon Frsl we sss D se for smpl. A Crvlner Coornes CC s nroe ro oorne rnsformon:. Te fferenon eween oornes s smmrze elow:

2 Coornes rnsformon of ervves of fnon s smmrze s For emple we ppl e ove resls o eqon of Eler em: f EI were E I n f re e efleon Yon s mols seon momen of re n eernl fore respevel. Eqon s rnsforme no f EI. Te ervves n re ven Eqs. n respevel.. Fne fferene solon of D rers eqon w fe sonn Te nl n onr vle prolem of Brers eqon s ven

3 n for for n for. In s prolem sonn s forme wen me nreses. If we nroe Crvlner Coornes CC ven Eq. Eq. s rewren from Eqs. 6 n 7 s. 5 Wen s ven s n epl fnon of Eq. 5 n e se for nmerl llon. However wen s ven s n epl fnon of we sse Eqs. n no Eq. 5 n on. 6 In e nl n onr vle prolem ven Eqs. n e sonn s forme s me nreses. Ten we se for emple e follown CC: n p 7 were p n s n o neer n s ven. 8 Ssn Eq. 7 no Eqs. n we ve 6. 9 Ssn Eq. 9 no Eq. 6 we on 6. From Eq..9 we on e follown fferene eqon: 6. Te onr n nl onons re ven for n for respevel. In e nmerl llons prmeers re ven s 8 8 M.. n.5. Fre sows oornes rnsformon eween n ven Eq. 7. Nmerl resls re sown n F. were CC mens Crvlner Coornes. If we se e enrl fferene for eql spn e solon vere wen =.. A eer resl s one n CC sn enrl fferene s sown n F..

4 F.. Coornes rnsformon. E solon Nm. sol. n CC Upwn =. n p = Nm. sol. n CC Cenrl fferene =. n p = Nm. sol. n Eql spn Upwn = n p = F.. Fne Dfferene solon of D rers eqon w fe sonn M=8. Fne Dfferene solon of D onveve ffson eqon w movn sonn We onser e follown onveve ffson prolem: U n for 5 n for 6 n n n for. 7 In s nl n onr vle prolem e sonn moves w velo U. We nroe Crvlner Coornes CC s 8

5 . 9 From Eqs. 8 n 9 we ve. From Eq. we on or. From Eq. we on or. From Eq. e ervves of n e wren s. Ssn Eqs. n no Eqs. n we ve 5. 6 Ssn Eqs. 5 n 6 no Eq. 5 we on U. 7 Rewrn Eq. 7 e onveve ffson eqon 5 n ssem n e wren s U. 8 Te me of e oornes rnsformon s llsre n F.. =ξτ ξ = ξ ξτ ξ ξ ξ

6 F.. Crvlner Coornes CC e e e poson of e sonn me. Wen n e follown epresson s e oornes rnsformon ol e ven e 9 sne n p e e e p. Inern Eq. 9 we on erf erf C C p e p e C p e C e p p p were erf s e error fnon n C s n rrr fnon of. We ol oose C s erf C. Hene we ve erf. Wen s ven s one solvn Eq. sn Newon-Rpson meo. Te orreon of s ven ep erf. From Eq. we ve ep. 5 From Eq. we on e e e. 6 Te poson of e sonn s one s follows. e e e nl poson of e sonn. Ten ssfes erf U. 7 Sne we rewre Eq. 7: U. 8 From Eq. 8 we ve

7 U. 9 In e nmerl llons prmeers re ven s 8 M 8 or 6.. n.5. Nmerl resls re sown n Fs. n 5 were CC mens Crvlner Coornes. A m eer resl s one n CC. Eql spn n Upwn M=8 = Eql spn n Cenrl fferene M=8 = Eql spn n ξ or CC Cenrl fferene M=8 =.9 = F.. Trnson of sonn M=8

8 = = = =6 e =8 f = F. 5. Trnson of sonn Eql spn n ξ or CC Cenrl fferene M=6 =.99 =5 5. Poenl flow ron rle We now ppl Crvlner Coornes CC o solve poenl flow ron rlr lner n nform flow. In e onvenonl nmerl proere e rlr onr s pprome je or non-smoo onr. However f we se CC we ol nle e rve onr more resonl. Te se onr vle prolem of poenl flow ron rlr lner n nform flow w velo U s ven U 5 n n 5 Un n n on n 5 s. 5 were s Cresn oornes w e orn e ener of e rle n re velo poenls n n n s n norml veor on e rle nwr o fl. We efne CC s 5. 5 If we ppl 55 n 56 we ve

9 . 57 Eqon 57 s one kn e rnspose of mr epresson erve ssn n no Eq. 55. From Eq. 57 we on 58 or were. 6 Smlrl from Eq. 57 we on we on 6 or 6 6 were. 6 W respe o mres n we n erve e follown properes:

10 6 6 n From Eq. 57 we ve I. 66 Wen s epresse epll s fnon of mr n Eq. 59 eome mporn. On e onrr f s epresse epll s fnon of mr n Eq. 6 eome mporn. Wen s epresse epll s fnon of e seon ervves of w respe o re ven Dfferenons of re ven s follows: 68

11 Ssn Eqs. 68 n 6 9 no Eq. 5 we on. 7 Wen s epresse epll s fnon of e seon ervves of w respe o n Eq. 7 ms e lle sn Eqs. 59 n Emple : Inroon of polr oornes Te onr vle prolem s efne Eqs. 5 5 n 5 n s sown n F. 6. In e follown we onser e prolem n e pper lf plne sn e smmer of e prolem. For e prpose we onon on : on. 7

12 η η=π R ξ= ξ=r U η= ξ η =-π plne ξ η plne F. 6. Polr oornes Frsl we nroe e polr oornes s sown n F. 6. Te Cresn oornes s epresse epll sn e polr oornes s os sn 7 os sn sn os 7 sn os os sn. 7 In s se e nverse rnsformon n e one esl n e polr oornes n e ven e Cresn oornes s os. 7 Te ervves of e polr oornes w respe o e Cresn oornes ol e one lso e one esl. In s se e oeffen of e fferenl eqon 7 ol e erve rel from Eq. 7. However f we se Eqs. 7 n 7 e oeffens re one Eqs n 67. In e follown nmerl llons we op meo sn Eqs. 7 n 7. Te e solon s ven elow: U os U os U U 75 U U U v U. 76 In e nmerl llons prmeers re ven s R M N n U. Nmerl resls re sown n Fs. 7 8 n 9 were CC mens Crvlner Coornes. Te nmerl resls sow resonle onene w e nll solons.

13 Mppe plne Psl plne F. 7. Coornes rnsformon Velo poenl on r = Tnenl velo on r =

14 F. 8. Comprson of velo poenl n enenl velo w e ones on r = Velo poenl p velo velo v F. 9. Ar of e solon 5.. Emple : Inroon of Jokowsk rnsformon We solve e sme prolem s ssse n seon 5. sn rvlner oornes one from Jokowsk rnsformon: z. 77 z were s e rs of e rlr lner n z n re e psl n mppe plnes respevel. Te Jokowsk rnsformon mps e rle n e z -plne s sown F. no e lne semen. Te relr mes n e -plne s sown n F.. R U η=ons. plne ξ=ons. ξ = ons. n η = ons. lnes F. 7. Jowkowsk rnsformom From Eq. 77 e rnsformon eween n s ven. 78 Also from Eq. 77 e nverse rnsformon of Eq. 77 s ven. 79 However n e presen pper Newon-Rpson meo s pple o Eq. 78 o on from. Te erve proere w e nl vle s sown elow: 8. 8

15 From Eq. 77 e ervves of w respe o re ven 8. 8 In e solon of e prolem e reon ws pprome B. Te follown onr onons were pple: on & ; 8 Un n n on ; 8 U ; ; 8 on B ; 8 on B. 8 In e nmerl llons prmeers re ven s B M N n U. Nmerl resls re sown n Fs. n. Te nmerl resls sow resonle reemen w e nll solons. However e r s lower n e resls n seon 5.. Te fferene eween F. 7 n F. wol epln e reson. Te mes n F. s no fne eno. Mppe plne

16 Psl plne F.. Coornes rnsformon F.. Norml velo n = φ/n n nenl velo s = φ/s on lner srfe Velo poenl p velo velo v F.. Ar of e solon 6. Conlsons Inroon of rvlner oornes no nmerl ompon ws ssse n e mporne ws sown. Alo ensor nlss wol e mos se for s kn of ssson ensor noon n e se n e presen omper lne. For emple e sr srmnon of pper n lower sffes s mpossle n e presen omper lnes. In e presen pper rer elemenr ppro more se o wre oes of omper prormmn s pe. Te eor s evelope no spefll enerll. Hene e eor ol e pple wel.

17 In e presen pper we se ree prolems nmel fe sonn movn sonn n rve onr. We n enere fne mes n e neoroo of fe n movn sonnes sn Crvlner Coornes CC. Frermore f we nroe rvlner oornes we n rnsform non-sqre reon no sqre one n n se relr mes. Usll n nmerl llon rve onr s pprome je or non-smoo onr. Te ppropreness n mporne of nron rvlner oornes no nmerl nlses were sown resonl ro eorel onserons n nmerl verfons. An nroon of rvlner oornes ol solve mn mporn prolems n e nmerl nlses. In e presen pper e rnsformon fnons were ven nlll. However we wol e le o efne oornes rnsformons sn non-nll meos. For emple we ol rw pre of mes vson n enere sree for e rnsformon. If we nerpole e sree sn e proper se fnons n nerpolon sn for emple es Sqre Meo SM we wol e le o on onnos n fferenle rnsformon fnon. Referenes. Tensor - Wkpe ps://en.wkpe.or/wk/tensor.. Crvlner Coornes - Wkpe ps://en.wkpe.or/wk/crvlner_oornes.. Erwn Kresz Dfferenl eomer Dover Books on Mems.