On composite conformal mapping of an annulus to a plane with two holes

Transcription

1 O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy cicula hole. It is show that such tasfomatio is good if the distace betwee cetes of holes ae lage o adius of cicula hole is small. Examples fo biliea-hypotochoids mappig ad biliea- Schwaz-Chistoffel mappig ae peset. Keywods: Cofomal mappig, composite mappig, biliea mappig, Schwaz- Chistoffel mappig Itoductio I thei ecet aticle Lu ad cowokes [] coside a composite mappig fuctio which maps a aulus to a ifiite egio with elliptic ad ealy cicula hole. I this shot aticle, we will use thei idea to map aulus to a ifiite egio with polygoal hole ad ealy cicula hole. To best of ou kowledge such tasfomatio is ot discussed i the liteatue [-9] tough it may be useful i some applicatios i potetial theoy, liea elasticity ad mesh geeatio. Figue. Composite mappig

2 The mappig We fist coside a mappig fom complex plae to the complex plae w by the biliea fuctio [4, 6, 0] i the fom whee. If ad w f () the this fuctio cofomally maps cocetic aulus o ζ plae oto ifiite egios with two o-ovelappig cicula holes o w plae. Cicle is mapped oto the cicle w while cicle w is map oto the cicle w e whee e fo o-ovelappig cicles obey the followig iequality We ote that w f has pole o aulus i.e. f e (). If ad e ae give the ad ae calculated by [0]: e e e e e e (3) (4) Let us ow coside a fuctio c z F w C w w (5) zw,, Cc, that is aalytic i the domai w. C is scale facto ad ca be, uiquely detemiate fom omalizatio e.g. legth. Combiig () ad (5) we obtai F a, whee a is some chaacteistic z F f C c (6)

3 Because f ad F ae aalytic with o-zeo deivatives o thei domai of defiitio the mappig of aulus fom ζ plae oto z plae is cofomal. If o z-plae we have some simple oitesectig cuve L eclosig coodiate oigi the oe ca fid coefficiets of the fuctio [0]. Obviously the cicle o ζ plae will be by the composite mappig F f mapped oto cuve L. The cicle will be mapped oto the cicle w e ad Fw will map this cicle oto a cuve give by z C e Expadig this ito powe seies of e i whee we deote i e we obtai i e Fw that will map cicle w to L c (7) i e e z h R O (8) h c C e e (9) c R C e (0) e s 0, () s is miimal spacig betwee the cicles. Thus if is small i.e. is small o s is lage, the the cicle w e is appoximately map oto the cicle z h R. I geeal, we caot select locatio of cicle h ad compute e, becouse e is eal umbe ad theefoe if h is give the (9) leads to two equatios fo oe ukow. Howeve if cuve L is symmetic with espect to x axis the the coefficiets c of expasio (5) ae eal umbes ad theefoe ae h ad R. If i this case h is give the (9) become the equatio fo calculatig e

4 c h e 0 () e C Pactically we deal with fiite umbe of tems N i seies. Theefoe above educe to polyomial equatio of ode N. Howeve, because by assumptio coefficiets c foms coveget seies ad because e the equatio ca be moe effectively solved by some iteatio method. Fo example, oe may use diect iteatio scheme i the fom e e 0 k h C h C c N ek (3) Oce we kow e we ca fo give R calculate usig (0) R N c C e (4) Kowig e ad we ca the calculate ad of biliea mappig () by meas of (3) ad (4). I this way we obtai a mappig which cofomal maps aulus to ifiite egio bouded by the cuve L ad ealy cicula hole z h R. 3 Examples 3. Composite biliea-hypotochoids mappig Coside the mappig [0] m z Fw C w (5) w whee 0 m, ad. By this fuctio the cicle w is mapped oto a cuve which is bouded i a egio i e i C m F C m (6)

5 The cuve miimal adius of cuvatue at mi 0 ad maximal adius of cuvatue at max ae m C m mi (7) If max the m C m max (8) m (9) I wods, the figue has locally staight edges. I Figues, 3, 4, 5 ae show some of these maps. Figue. Composite biliea-hypotochoids mappig fo the case, d 0.5, ad. The maximal discepacies fom cicle ae 0.0 ad Let us ow coside a discepacy of the mappig to the give cicle. Usig (9) ad (0) fo e ad ca wite the diffeece betwee the cicle z h R ad the mappig cuve as i i C i f e e h Re e e Fo we have e i (0)

6 C i 3 e O e () As was aleady show the diffeece becomes smalle with deceasig, howeve it also decease by iceasig. Coside ow a case whe s 0 i.e. the cicles o w plae almost touch. I this case by usig () fo we ca wite (0) as i i e e C Os we obtai maximum diffeece At e i () So C max Os (3) C lim (4) max i.e. the diffeece is bouded ad decease with iceasig. This popety ca be also obseved fom umbes i Table. Table. Diffeece max fo the case m, hole R ad spacig d. ( ) e,, ad diffeet size of d R max max max

7 Figue 3. Composite biliea-hypotochoids mappig. Effect of hole size fo m,,, , , d 4,,. The maximal diffeeces fom the cicle ae Figue 4. Composite biliea-hypotochoids mappig. Effect of hole positio fo m,,, the cicle ae , ad ad d 0, 0.,. The maximal diffeeces fom

8 m Figue 5. Composite biliea-hypotochoids mappig. Effect of hole shape fo, d,, ad, 3, 4 the cicle ae 0.084, 0.00, The maximal diffeeces fom 3. Biliea - Schwaz Chistoffel mappig Fo egula polygo of sides the Schwaz-Chistoffel mapig has the fom [] z Fw C w d (5) The itegal ca be evaluated by seies expasio k k w z Fw C k0 k k (6) Fo a figue otated by agle the mappig is slightly diffeet k w z Fw C k0 k k (7) I Figues 6 ad 7 ae show some of these maps.

9 Figue 6. Composite biliea-schwaz-chistoffel mappig with five tems. Effect of shape fo C, d ad ad fom the cicle ae 0.054, , fo 3, 4, 5. The maximal diffeeces Figue 7. Composite biliea-schwaz-chistoffel mappig with thee tems. Fo tiagula hole C, d, R, fo squae hole a, d 0.5, R 0.5, ad fo octagoal hole a, d 0.5, R. The maximal diffeeces fom the cicle ae 0.088, 0.08, 0.07 Refeeces [] A.-z. Lu, Z. Xu, N. Zhag, Stess aalytical solutio fo a ifiite plae cotaiig two holes, Iteatioal Joual of Mechaical Scieces, 8 (07) [] V.I. Ivaov, V.J. Popov, Cofomal mappig with applicatios, Moskow, 00. [3] H. Kobe, Dictioay of cofomal epesetatios, d ed., Dove, s.l., 957. [4] S.G. Katz, Hadbook of complex vaiables, Bikhäuse, Bosto, 999.

10 [5] V.I. Lavik, V.N. Savekov, Hadbook o cofomal mappig, Kiev, 970. [6] Z. Nehai, Cofomal mappig, Dove Publicatios, New Yok, 975. [7] R. Schizige, P.A.A. Laua, Cofomal mappig methods ad applicatios, Elsevie, Amstedam etc., 99. [8] M.R. Spiegel, Schaum's lie of theoy ad poblems of complex vaiables with a itoductio to cofomal mappig ad its applicatios [icludig 640 solved poblems], Schaum Publishig, New Yok, 964. [9] V.I.A. Ivaov, M.K. Tubetskov, Hadbook of cofomal mappig with computeaided visualizatio, CRC Pess, Boca Rato, 994. [0] N.I. Muskhelishvili, Some basic poblems of the mathematical theoy of elasticity; fudametal equatios, plae theoy of elasticity, tosio, ad bedig, P. Noodhoff, Goige,, 963. [] T.A. Discoll, L.N. Tefethe, Schwaz-Chistoffel mappig, Cambidge Uivesity Pess, Cambidge, 00.