Papamichael, N

Sideridis, A

English

Brunel University Research Archive

                       TR/72                                                      SEPTEMBER1977   FORMULAE FOR THE APPROXIMATE  CONFORMAL MAPPING OF SOME  SIMPLY CONNECTED DOMAINS                                                       BY                             N.PAPAMICHAEL and A.SIDERIDIS 1. Introduction The problem of conformally mapping a given simply-connected domain onto the unit disc is of great physical interest, having important applications for example in the fields of fluid mechanics, electrostatics and steady-state heat flow. Although Kober [9] provides an excellent dictionary of special conformal transformations, there are many domains that occur in practice for which the conformal map can be obtained only by numerical means. In the present paper we give explicit formulae for the approximate conformal mapping of a selection of simply-connected domains. These approximations were derived by means of the Bergman kernel method which has been recently proposed by D. Levin and the present authors in [10]. Some of the formulae given have been used by the authors in [11] to obtain, by means of a conformal transformation method, accurate numerical solutions to certain elliptic boundary value problems involving boundary singularities. They are presented here as they might be of value in other applications.   2. Approximate Formulae Let Ω be a simply-connected domain with boundary ∂Ω in the complex   z-plane and assume, without loss of generality, that the origin of coordinates 0 lies in Ω. Let  w = f(z) be the mapping function which maps Ω U& Ω conformally onto the unit disc      | w | ≤ 1 in such a way that  f (0) = 0 and f'(0) >1. We consider a selection of simply-connected domains and, for each domain, we give an explicit formula approximating the mapping function f(z). All formulae are derived by means of the Bergman kernel method of [10] and are of the form   fN(z) = c  .(z)vaN1nnn∑= The criteria for selecting the functions vn (z) and the technique for computing the coefficients an are described fully in [10]. Here, for each domain considered, we only list the functions vn (z), n = l,2,...,N and tabulate the complex coefficients an n= l,2,...,N and the real 2  constant   c = 21N1nnn (0)'vaRe ⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛∏∑=. In each case the positive integer N, i.e., the number of functions vn (z) used in the approximation, is the "optimum number" which gives maximum accuracy in the sense explained in [10]. An estimate of the maximum error in the modulus of fN(z) is given by the quantity EN. This is obtained, as described in [10], by computing  eN(z) =1-|fN(z)| . at a number of "boundary test points" ,zj∈∂Ω , and then determining,  EN = |ejmax N(zj)| . Each example heading is followed by a list of references. These indicate the publications in which the domain under consideration was the domain of definition of a problem which has been or may be solved by conformal transformations. In obtaining the approximations presented here, all computations were carried out, in single length arithmetic, on a CDC 7600 computer. 3 2.1    Quadrilateral domain of Fig.1 ([11],[16]).          Figure 1      ⎩⎨⎧ ++−−++−= ∑∑=++=31n1n6n41n2/3c2/3c65nn17 1)/(nza})z()z{(zaza)pz/(zac(z)f  .4)/(nza})z()z{(za71n4n10n2/9c2/9c10 ⎭⎬⎫++−−−+ ∑=++4        c=0.398146533868E+01 COEFFICIENTS an   0.791694815026E-01 -.404577265524E-03  0.427384527935E-05 -.159158948771E+00 -.356087929037E-05 0.159160208427E+00 -.570240876737E-01 -.216619859008E-01 -.184900259407E+00  -.320036339889E+00 0.482506039583E-01 0.582610812361E-02 0.512212016237E+00  0.466945150131E+00 -.383000552846E+00 -.116177383540E+00 0.871665460256E-01 -.143531150834E-01 -.232490033139E-02 -.553110980205E-02 -.388375067171E-02 0.299344559392E-02 -.972207514203E-04 0.208699226279E-03 0.657585933207E-05 0.243954703546E-04 0.154351240627E-05 0.223160629975E-05 0.455843972998E-06 -.692510627560E-07 0.572694620456E-07 -.802076687538E-07 -.313898187880E-08 -.550573131362E-08        The estimate of the maximum error in |f17(z)| determined by computing e17(z) at a selection of boundary points (see [10], example 2) is,  E17 =| e17(zB)| = 4.7×10-7. 5         2.2.    Domain of Fig.2 ([7],[11]).     Figure 2 The coordinates of the point p4 are    x = -.312000000000E+00  y = 0.125549151733E+01   f22(z) = c  ./nza)pz/(za41n181nn4nnn ⎭⎬⎫⎩⎨⎧ +−∑ ∑= =+6 c =0.172185936062E+01   COEFFICIENTS an    -.939969248427E+02 0.122329979451E+04  -.551978734456E+03 -.174377012616E+03  -.261501992446E-03 0.477495901722E+00  0.531833825380E-01 -.251139228085E+00  -.114350378084E+03 -.349412672348E+03  -.807660885199E+02 0.131147734849E+03  -.214863408023E+02 -.6553 65881560E+02  -.101033444392E+02 0.1 63810803570E+02  -.223119413537E+01 -.681137279041E+01  -.931403212463E+00 0.153119094936E+01  -.200042800284E+00 -.597 686794743E+00  -.882079024052E-01 0.133843303025E+00  -.135210164450E-01 -.516934917590E- 01  -.220551777920E-02 0.725856779886E-02  -.176278084664E-02 -.130894577392E-02  -.157003559571E-02 0.151609606350E-02  -.292606857151E-04 -.987322440601E-03  0.156204714583E-03 -.678079650308E-04  -.347686237615E-05 0.963751336872E-04  -.193257751761E-04 0.127605650111E-04  -.151735706256E-05 -.999870734112E-05  0.271991501277E-06 0.329824918668E-06  The estimate of the maximum error in |f22(z)| determined by computing e22 (z) at 50 suitably chosen boundary points is,   E22 = | e22 (zD)| =4.9×10-5. 7 2.3 L-shaped domain of Fig.3 ([1]- [3], [8], [11]- [13], [15], [17]).  ⎩⎨⎧ −−−++−−−+−= ∑=41n4/3c4/3c762/3c2/3c5nn26 })z()zz{(aza})z()zz{(a)pz/(zac(z)f  })z()zz{(a3/za})z()zz{(a2/za 3/10c3/10c113103/8c3/8c928 −−−++−−−++.)4n/(za})z()zz{(a4/za131n4n13n3/14c3/14c13412 ⎭⎬⎫++−−−++ ∑=++             8     c =0.410656087112E+01    COEFFICIENTS an    0.159612267043E+00 0.335646846333E-04  -. 256039962417E-04 -.159613596509E+00  -.312878304089E+01 0.343839952158E+01  -.355172625260E+01 0.317083605565E+01  0.122695459141E+00 0.122695858537E+00  -.505064867897E+00 -.192434451322E-01  -.463392143878E-01 0. 463423955230E-01  -.390325634275E+00 -.398067693316E+00  0.677359626098E-01 -.677611692060E-01  0.973540694394E-03 0.112158829299E+00  -.152126322197E-01 - .151810657862E-01  -.821491781895E-01 0.817879859362E-01  -.394581627595E-02 -.395401615778E-02  0.598783682967E-02 -.639779898442E-04  0.4063180 68481E-03 0.387700851936E-03  0.110158129158E-05  0.215408946060E-03  -.239529581917E-04 0.237520089010E-04  -.194808589058E-04 -.204762611350E-06  0.389998066780E-05 0.394686114515E-05  0.186535402646E-07 -.169222348084E-05  -.343672119284E-06 0.337774475874E-06  0.723026529702E-07 0.414316531930E-09  -.563957261560E-08 -.568917941216E-08  0.689479871346E-11 0.963180754574E-09  -.274511826388E-10  0.266789197888E-10  -.209894966791E-10 -.344489230867E-12  The estimate of the maximum error in |f26(z)| determined by computing e26(z) at a selection of boundary points  (see [10], example 3) is,  E2 6 = |e2 6(zC) |  =2.2×l0-5.  9      2.4  L-shaped domain of Fig.4 ([11],[12],[15]).     Figure 4     f17(z) = c  ⎩⎨⎧ +−−−+−∑=41n62/3c2/3c5nn za})z()zz{(a)pz/(za  })z()zz{(a2/za})z()zz{(a 8/3c8/3c9284/3c4/3c7 −−−++−−−+  4/za})z()zz{(a3/za 41210/3c10/3c11310 +−−−++   .)4n/(za})z()zz{(an1n4n31n14/3c14/3c13 ⎭⎬⎫++−−−+ ∑=++10       c =0.149098601250E+02 COEFFICIENTS an    -.171147593328E-01   -.945691444885E-03 0.953821713728E-05  -.5301040 68099E-01 0.955178615453E-03  0.233189952548E-01 0.362715490460E-01  -.258868194269E-02 0.216876372822E-01  0.157865852619E-01 0.857275037075E-01  0.855965062994E-02 -.408292565825E-02  0.407932002784E-02 -.231704817365E-01  -.288822137143E-01 0.142602603603E-02  -.133972804815E-02 -.809913758946E-03  0.315437414610E-02 -.187405614369E-03  -.234230541311E-03 -.272980529148E-04  0.121172385031E-03 -.415596940524E-05  -.156407058589E-05 0.549528008224E-05  -.212981873915E-05 0.767447829679E-07  0.379621311941E-07 0.182068289580E-08  0.186980825904E-08 0.330153679120E-10  0.318968182170E-10 The estimate of the maximum error in  |f17(z)| determined       by computing e1 7(z) at  60 uniformly distributed boundary points is ,   E1 7 = |  e1 7(z) |  =3.6×l0-5.  11       2.5 Domain of Fig.5 ([14]).    .)4n/(za})z()zz{(a4/za})z()zz{(a3/za})z()zz{(a2/za})z()zz{(a})z()zz{(aza})z()zz{(a)pz/(zac)z(f31n4n17n21n2/91n2/91n15n4153/1013/1011431321n3/813/81122112/31n2/31n8n3/413/41873/213/216n51nn20⎭⎬⎫++−−−++−−−++−−−++−−−+⎩⎨⎧ −−−++−−−+−=∑∑∑∑=++=+++=+++=  12 c=0.384813039878E+01  COEFFICIENTS an   0.159284421298E+00 -.159220301003E+00    -.101634739203E+00 -.310258109558E+00   0.265290450317E+00 0.126714109892E+00    -.179487385242E+00 0.618115987237E+00    -.647477350924E+00 0.165720704406E+00  0.140781101274E+00 0.140753194463E+00  -.884229280592E+00  -.115245672970E-01  -.504483222227E-01   0.503380232638E-01  0.183062993522E-01  -.557065867636E-01  -.260616373469E-01  0.520909654621E-01 0.265009406749E+01 0.328745759485E+01 0.702148907135E-01   -.700259989674E-01 -.181970916547E+00  0.385046440527E+01  -.245095003698E-01  -. 246046221971E-01 0.150427012012E+00 -.105778520099E+00  -.139780891609E-01   0.262235102543E-01  -.283704894445E-01   -.128636021221E-01 0.511893386102E-01  0.361885087423E-02 -.121278584821E-02  -.136056722742E-02 -.335846636037E-05  0.832372793781E-04 The estimate of the maximum error in |f20(z)| determined by computing e20(z) at 84 uniformly distributed boundary points is,  E20 =| e20(zF)| =2.7×10-4. 13       2.6    Domain of Fig.6    ([6],[17])       Figure 6      f24(z) = c  ./nza)pz/(za191nn5n51nnn ⎭⎬⎫⎩⎨⎧ +− ∑∑=+=14 c =0.228415919526E+01 COEFICIENTS an0.159589174051E+01 0.616641145244E+01 -.404332665648E-01            0.117754814924E+00 -.117976206660E+00  0.401865912192E-01 -.616872184369E+01  -.159753804485E+01 -.159064714691E+00  0.159064726161E+00 0.192103490460E+01 -.104448828542E-02 0.393062784114E+01 0.393030848454E+01  -.524925366026E-03  0.455875556034E+01 -.122442838252E+01  0.122332203416E+01                       0.561997919277E+00  -.326578297926E-03 0.706679205511E+00 0.706618976768E+00  -.762338109170E-04  0.632026838106E+00 -.136131717704E+00  0.135993954577E+00 0.738094509382E-01  -.363370940083E-04 0.742978269656E-01  0.742921098542E-01 -.782995047545E-05  0.565753236997E-01 -.629780866930E-02  0.628453632281E-02 0.168976932960E-01  -.342843855288E-05 0.132218375855E-01  0.132212897643E-01 -.731629360750E-06  0.125839534572E-01 -.453494793588E-02  0.453366655983E-02 -.234770689984E-02  -.436508702388E-06 -.480114222841E-03  -.480361182082E-03 0.503308162460E-07  -.128238705740E-03 The estimate of the maximum error in |f24(z)| determined by computing e24(z) at 35 suitably chosen boundary points is, E24 = | e24(zC)| = 1.6×10-5. 15 2.7  Domain of Fig.7 ([17]-[19]).    Figure 7      f20(z) = c  .n/za)pz/(za151nn5n51nnn ⎭⎬⎫⎩⎨⎧ +− ∑∑=+=    16 c = 0.238602453472E+01 COEFFICIENTS an 0.338244334162E+00 -.938002569877E+00 0.142856383390E+02 -.922931228109E+01  -.100961839966E-01  -.147702977428E+00 -.274463761715E+00  0.601056362376E-01 -.158938315388E+00  0.159375794676E+00 0.327268249832E+01 0.170658047576E+01 0.217586460829E+01 0.159884062311E+01 -.507028864734E+00  0.197312115584E+00 0.266260234935E+00  -.351400849349E+00  -.164655269553E-01  -.107753511927E+00 0.428578583123E-01  -.966271278185E-01 -.165800199874E-01  -.657251368831E-01 0.613672240328E-02  -.702520658596E-01 0.269169252747E-01   -.430475686862E-01 0.265486218268E-01   -.126796315567E-01 0.128937127375E-01   0.181705537018E-02 0.353341167368E-02  0.307333883407E-02 0.436805808621E-03   0.123155300945E-02 -.115955371853E-04  0.242540950876E-03 -.778553068063E-05  0.209687291836E-04  The estimate of the maximum error in |f20(z)| determined by computing e20(z) at 36 suitably chosen boundary points is,  E20 = | e20(zC)| = 9.0×10-5. 17     2.8 T-shaped domain of Fig.8 ([17],[19]).     Figure 8   .3)/(nza})z()zz{(a/3za})z()zz{(a/2za})z()zz{(aza})z()zz{(a)pz/(za  c  (z)f21n3n71n15n10/3n10/3n13n31321n8/3n8/3n10n21n2104/3n4/3n7n21n72/3n2/3n4nn41nn22⎭⎬⎫++−−−++−−++−−−++−−−+⎩⎨⎧ −=∑ ∑∑∑∑∑=+=++=+=+=+=  18 c = 0.644171861702E+01 COEFFICIENTS an -.139330408186E+00  0.292849207151E-01 0.139325986066E+00 0.292808688565E-01 0.483462401586E-07  -.783483097128E-01  -.459122784053E-05  0.264644492158E+00  0.993496867376E-01 0.115358319092E+00 0.149578177198E+00 0.283602158679E-01 0.190168028116E+01 0.349646913855E-06  -.570392561084E-01  0.587949732079E-01  -.794371512625E-01  0.199998284021E-01  -.789264457332E-05  -.183952550387E+01 0.849926982794E-01  -.872347496606E-01  -.330509646063E-01  0.117222306826E+00 0.856794253736E-02  0.435622877023E-05 -.339908165943E-01  -.334777708563E-01 0.119970293070E-01  0.461752980875E-01 0.878434670236E-07  -.157918935838E-01 0.946928665881E-03  0.183287478603E-10 0.826932157977E-09  0.662089384249E-04 0.108990075613E-05  0.455191540715E-09 -.693802832716E-10  0.310843458394E-05                  -.7565488 60137E-06       -.233774906093E-11 -.126155586361E-11  -.842157925650E-07 The estimate of the maximum error in |f22(z)|  determined by computing e22(z) at 76 uniformly distributed points on the boundary is, E22 = | e22(zB)| =7.6×10-5. 19 2.9 Octagon of Fig.9 ([2],[8],[11],[12],[15]).   Figure 9   .1)/(2nza})z()zz{(a})z()zz{(aza})z()zz{(a100}1)(z/{za c  (z)f12n141n9n8/3n8/3n21n7n4/3n4/3n21n5n52/3n2/3n21n2nn221nn23⎭⎬⎫++−−−+−−−++⎩⎨⎧ −−−+−+=+=+=+=+=+=∑∑∑∑∑  20 c = 0.161338731615E+02 COEFFICIENTS an -.502221568821E+01            -.271655780489E+02 0.575144495893E+01 -.273275962284E+02 0.157961627689E-01  0.157961092356E-01 0.215779132689E-01  -.578182362464E-02  -.117027339008E-01  0.162858580467E-02  -.285568077824E-02  0.285548250042E-02  -.390076077118E-02  -.104535084836E-02 0.101280737002E-02  -.101262212726E-02  -.370552801642E-03  -.138342797524E-02 0.218532392472E-03  -.139358736891E-01  -.499309982568E-04  0.809660612957E-06 0.510513062858E-07  -.383955355692E-05  -.811961965522E-08  0.145915691135E-09 0.801915064071E-11  -.591943535002E-09  -.134678095638E-11  0.210100502776E-13 0.109470114287E-14  -.831910355335E-13  -.140124034777E-15  0.281583928368E-17 0.135982628640E-18  -.997112188560E-17  -.240106737718E-19  0.353288221087E-21 0.156350345451E-22  -.140067659335E-20 0.589916787888E-23  0.918065754649E-25 0.163723265798E-27  -.106784710352E-25 -.592414723668E-28  -.795542066066E-30 The estimate of the maximum error in |f23(z)| determined by computing e2 3(z)  at  a  selection of  boundary points (see [10], example 4) is,  E23 = | e23 (zA) | = 5.7×10-6. 21    2.10. Cross-shaped domain for Fig.10 ([4],[5]).    Figure 10  ⎪⎩⎪⎨⎧ −−−+−+= ∑∑=+=41n2/3n2/3n2n21nn2n31 })z()zz({a144}1)(z{/zac(z)f    /3za})z()zz{(aza 31641n4k/3n4k/3n3n4k21k7 +⎥⎥⎦⎤⎢⎢⎣⎡ −−−+++ ∑∑=++=  ⎥⎥⎦⎤⎢⎢⎣⎡ −−−+ ∑∑=++++=41n6)/3(4kn)/36(4knn3)4(k21k})z()zz({a  .)3n2/(za71n3n224n ⎪⎭⎪⎬⎫++∑=++ 22     c=0.110654953585E+02  COEFFICIENTS an                          -.113547022434E+03 0.402422485026E+02                        0.400289588727E+02 -.207267503800E+03  0.240508211282E-01 0.3109 28150582E-01  0.516711472991E-02  0.602533014794E-01 0.389526843128E-01  -.528223562672E-02 0.547645283394E-01  0.256518129974E-01 -.972995934216E+00 0.222881525546E+01 -.814505415142E-02  0.299070129942E-02 -.137121226750E-01  0.173543085862E-01 -.666237060585E-02  -.555839096574E-02 -.218852006291E-01  -.319797355990E-02 0.322603704159E-02 0.179420381890E-03 -.129475994397E-01  -.453105050884E-02 0.176838702364E-02  -.270405385863E-02 -.103978386334E-01  0.894736536272E-02 0.296733986385E-01 -.117028969715E+00 0.122459991609E-02 0.326804521153E-03 0.113376127423E-02 0.275668175064E-02 0.329266902964E-03 0.122394289036E-02  -.182049401568E-02 0.236020322521E-02  -.141928705109E-03 -.144420582969E-03  -.361762736184E-04 -.148590025862E-03  -.196036049499E-03 0.507034917087E-04  -.146770771147E-03 -.429653402343E-04 0.759304019281E-03 0.429105019088E-03 -. 110009448857E-05 -.292378113692E-05  -.301644020385E-07 0.367056995633E-07  -.945621364427E-10 -.164571343843E-09  -.397839212546E-12 0.215236399098E-11  -.728964203330E-14 -.217919914769E-13  -.255784943886E-15 0.258414598822E-15         The estimate of the maximum error in  |f31(z)|determined          by computing e31(z) at 96 uniformly distributed points on the          boundary is,     .105.7)(ze 5D31−×=   REFERENCES   [1]    G.E. Bell and J. Crank, A Method of treating boundary singularities             in time-dependent problems, J.Inst.Maths Applies.12 (1973),37-48. [2]    G.E. Bell and J. Crank, Influence of imbedded particles on steady-state diffusion, Journal of the Chemical Society, Faraday Transations II 70 (1974), 1259-1273. [3]    G.E. Bell and J. Crank, A simple finite-difference modification for improving accuracy near a corner in heat flow problems, Inter. J.Num.Meth.Eng. 10 (1976), 827-832. [4]    K.J. Binns, The magnetic field and centring force of displaced ventilating ducts in machine cores, The Inst, of Electrical Engineers , Monograph No.394U (1960), 64-70. [5]    K.J. Binns and F.J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems, Fergamon Press, London, 1963. [6]    B.L. Buzbee, F.W. Door, J.A. George and G.H. Golub, The direct solution of the discrete Poisson equation on irregular regions, SIAM J.Num.Anal. 8 (1971), 722-736. [7]    L. Collatz, Discretization and chained approximations, in: Conference on the Numerical Solution of Differential Equations, Lecture notes in Mathematics 363, Springer-Verlag, Berlin, 1974. [8]    J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1975. [9]    H. Kober, Dictionary of Conformal Representations, Dover, New York, 1957. [10]  D. Levin, N. Papamichael and A. Sideridis, The Bergman kernal method for the numerical conformal mapping of simply connected domains, Report TR/71, Department of Mathematics,  Brunel University (1977). [11]  N. Papamich el and A. ideridis The use of conformal transformations for a S , the numerical solution of elliptic boundary value problems with boundary singularities, Report TR/73 epartment of Mathematics, Brunel University (1977), to appear. , Di n, A nummethod for harmonic mixed boundary value problems in polygonal domains, 2.angew.Math.Phys. [12]  N. Papamichael and J.R. Wh tema erical conformal transformation 24, (1973), 304-316. [13]   G.F.C. Rogers and Y.R. Mayhew, Engineering Thermodynamics Work and Heat Transfer, Longmans, London, 1963. [14]  G.D. Smith, Numerical Solution of  Partial Differential Equations, Oxford University Press, London, 1965. [15]  G.T. Symm, Treatment of singularities in the solution of  Laplace's equation by an integral equation method, Report NAC 31, National  Physical Laboratory (1973). [16]  R.W. Thatcher, The use of infinite grid refinement at singularities in               the solution of Laplace's equation, Numer. Math. 25 (1976),163-178. [17]  A. Thom and C.J. Apelt, Field computations in Engineering and Physics, D.Van Nostrand, London, 1961. [18]   H.R. Vallentine, Applied Hydrodynamics, Butterworth & Co, London, 1969. [19]  D. Vitkovitch, Field analysis, D.Van Nostrand, London, 1966. 

A Method of treating boundary singularities in time-dependent problems, J.Inst.Maths Applies.12

A simple finite-difference modification for improving accuracy near a corner in heat flow problems,

Analysis and Computation of Electric and Magnetic Field Problems,

Applied Hydrodynamics,

Dictionary of Conformal Representations,

Discretization and chained approximations, in:

Engineering Thermodynamics Work and Heat Transfer,

Field analysis, D.Van Nostrand,

ideridis The use of conformal transformations for a S , the numerical solution of elliptic boundary value problems with boundary singularities,

Influence of imbedded particles on steady-state diffusion,

Numerical Solution of Partial Differential Equations,

The Bergman kernal method for the numerical conformal mapping of simply connected domains,

The direct solution of the discrete Poisson equation on irregular regions,

The magnetic field and centring force of displaced ventilating ducts in machine cores,

The Mathematics of Diffusion,

The use of infinite grid refinement at singularities in the solution of Laplace's equation,

Treatment of singularities in the solution of Laplace's equation by an integral equation method,

Wh tema erical conformal transformation 24,

Formulae for the approximate conformal mapping of some simply connected domains

Formulae for the approximate conformal mapping of some simply connected domains

Abstract

Similar works

Full text

Available Versions

Brunel University Research Archive