Schwarz-Christoffel Mapping in the 1980's*
Lloyd N. Trefethen
TR 93-1381
September 1993
Department of Computer Science
Cornell University
Ithaca, NY 14853-7501
*Reprinted from Numerical Analysis Report 89-1, Dept. of Mathematics,
Massachusetts Institute of Technology, January, 1989.
Note
One day I hope to write a book on numerical Schwarz-Christoffel mapping. In the meantime.
this document is the closest I have come to preparing a survey of the field. It consists of the
transparencies (slightly edited) from my talk `?Schwarz-Christoffel mapping in the 19SO's?'
delivered at the Conference on Computational Aspects of Complex Analysis organized by Al
NIarden and Burt Rodin in Phoenix, Arizona. 11-14 January 19S9. The material is based
on ten years of experience with solving conformal mapping problems. often brought to me
by users or would-be users of my Fortran package SCPACK.
NIy chief purpose here is to outline the wide range of variations on the theme of Schwarz-
Christoffel mapping that arise in practical problems --H for only rarely does one encounter
precisely the ?standard" problem of mapping a disk or half-plane onto a prescribed poly-
gon. Details are omitted, but can be found in the references. NIy emphasis is entirely on
algorithmic and numerical matters, and the references are heavily biased in that direction.
Undoubtedly they are also biased towards my own contributions. and I apologize to others
whom I may have accidentally slighted.
I would like to highlight two themes that arise repeatedly in Schwarz-Christoffel mapping:
Afodifled Schwarz.Christoffel integrals. The standard Schwarz-Christoffel integrand is
a product f' = Hf? where each fk is an elementary conformal map (z--Hzk)?3?. By
modifying the choice of fk in appropriate ways, this same prescription can be adapted
to many different mapping problems, including exterior polygons. polygonal Riemann
surfaces, doubly-connected polygons, maps of an infinite strip, and the maps that arise
in ideal 2D free-streamline flows.
o+ Generalized parameter problems. Almost any Schwarz-Christoffel map requires the
solution of a parameter problem; in the simplest case one has to determine unknown
"prevertices" (zk? such that the corresponding vertices ?wk) are separated by the
correct side lengths I%L'k+i --Hwkl. and this amounts to a nonlinear system of equations to
be solved numerically. Surprisingly often, however, this paradigm needs to be enlarged
to a `?generalized parameter problem" --H which, fortunately. is often no harder to
solve. For example, in an inverse problem one may wish to delete one of the side
length conditions and replace it by a global condition involving the conformal module
(see p. 25). To put it generally: not all the conditions that define a Schwarz-Christoffel
map need be geometric. Often the geometry is only incompletely specified. or specified
partly in the domain and partly in the range; and indeed. sometimes polygons do not
enter into the problem at all except implicitly.
N?Iy work on Schwarz-Christoffel mapping began in 197S at the suggestion of Peter Henrici.
who died, too soon. in 1987.
Nick Trefethen
2) January 1989
s CHWARZ- CHRISTOFFEL
MAPPING IN THE 1980's
Lloyd N. Trefethen
1. Fundamentals of S-C mapping, p. 3
2. Generalizations of the S-C formula, p. 11
3. Applications, p. 23
Summary, p. 31
Dept. of Conip. Sci.
Cornell University
LNT?cs cornell.edu
Collaborators: Frederic Dias ?Vorcester Polytechnic Institute
Alan Elcrat NVichita State University
Peter Henrici ETH Zurich
Louis Howell, NI.I.T.
Ruth NVilliams, University of California at San Diego
2
1. Fundamentals of S-C Mapping
3
The Schwarz-Christoffel Idea
H = upper half-plane
P = polygonal region with n vertices
f = conformal rnap from H to P
fwk? = vertices of P? fzk? = `prevertices' Zk = f?'(wk)
H
Zk Zk+1
f
Wk?1
N\Pk7t
Wk
Idea: arg f' is piecewise constant on the real &xis.
Therefore ff can be wntten as a product of functions f?:
= C ?fl fk(z), fk(z) =
k=1
zk
fk
L
0 = fk(zk)
Each fk introduces a jump in arg f' at Zk
4
The Schwarz-Christoffel Formula
Integration gives
Reminder
f(z) = C?? fi (s--Hzk)?4ds
Pk7r = turning angle at kth vertex Wk --H known
zk = kth prevertex - unknown
The same formula works for a disk as well as a half-plane, and vertices
at cc are permitted.
For maps of the half-plane, one preveftex may lie at cc and is then
omitted from the S-C product (e.g. Z? = cc).
Refs: Z. Nehari, Conformal Alapping, 1952.
??. Koppenfels & F. Stalimaun? Praxis der konf. Abbildung 1959.
P. Henrici, Applied & Computational Complex Anai5rsis L 1974.
5
Numerical s-c Mapping
To apply the S-C formula two obstacles rnust be oN?rcome:
"PARAMETER PROBLEM9,
Given vertices fwkl, the prevertices fzk? are unknown.
1) Formulate the problem as a system of constrained nonlinear
equations involving the side lengths Wk+1 --H Wk
2) Change variables to eliminate the constraints Zk ? Zk+1:
3) Solve the system iteratively by standard optimization software
(e.g. NSO1A or NIINflNCN);
NUNIERICAL INTEGRATION
The S-C integral cannot be evaluated analytically
1) Gauss-Jacobi quadrature (for endpoint singularities):
2) Adaptive subdivision (to combat "`crnwding").
,Refs: K. Reppe, Siemens Forsch. u. Entwicld. Ber.,1979.
R.T. Davis, 4th ALAA Comp. Fluid Dynamics Conf., 1979.
L.N.T., SLAi\i J. Sci. Stat. Comp., 1980.
6
History
GAUSS (1820's) --H idea of conformal mapping
RIEMANN (1851) - Riemann mapping theorem
CHRISTOFFEL (1868)
SCHN\ARZ (1869?1870) --H independent
POLOZNII (1955)
NANTORON'ICH & NRYLOV (1958)
SAVENKOV (1963?1964)
GAIER (1964) --H book on numerical conformal mapping
FILCHANOV (1961?1968?1969.1975)
HAEUSLER (1966)
LAWRENSON & GUPTA (1968)
BEIGEL (1969)
HOFFMAN (1971?1974)
HO?E (1973)
N?ECHESLA?V, TOLSTOBROVA KOKOULIN (1973?1974)
CHEREDNICHENKO & ZHELANKINA (1975)
SQUIRE (1975)
NIEYER (1976,1979) --H comparison of algorithms
NICOLAIDE (1978)
HOPKINS & ROBERTS (1979) - solution by Kufarev?s method
BINNS, REES, & KAHAN (1979)
N?OLKOV (1979)
REPPE (1979) --H first fully robust algoritlim
DAN?IS SRIDHAR (1979?1982?1983) --H curved boundaries channel maps
TREFETHEN (1980--H1988) - Fortran package SCPACK
BRONVN (1981)
TOZONI (1983)
PROCHAZKA (1978,1982?1983)
HOEKSTRA (1983?1986) - curved boundaries annuli
FLORYAN ZEMACH (1985-1988) - channel maps. periodic domains
BJORSTAD & GROSSE (1987) - circular polygons
DIAS (1987--H1989) --H hydrodynamics applications
DAPPEN (1988) - doubly-connected S-C
HO?'ELL (1989) --H elongated polygons, circular polygons
SCPACK
Fortran package for S-C mapping (L.N.T. 1982)
Polygons may be unbounded (i.e.? vertices permitted at oc)
Available by tape or by e-mail (via ?`Netlib'?):
r
send index from conf ormal
send scpdbl from conf ormal
send sclibdbl from conformal
mail neti ib?research.att.com
or
mail netl i b?orn1 gov
The User's Guide can be obtained by contacting L.N.T.
Refs: L.N.T., SCPACK User's Owde MIT. report, 1989.
(Netlib:) J.J. Dongarra & E. Grosse, Commun. ACM, 1987.
8
Examples
Map to disk:
Map to half-plane:
Map to infinite strip:
Map to rectangle:
9
10
2. Generalizations of the S-C formula
Refs: Scliwarz and Christoffel, 1868--H1870.
J.NI. Floryan & C. Zemacli J. Comp. Phys., 1987
ii
Riemann Surfaces
Idea: f' is still piecewise constant on the real &xis. but may have
zeros ? in the upper half-plane.
Each ? introduces a factor
bk(z) =
in the S-C formula:
f(z) = c;z fi (s--Hzk)?4 11B bk(s) ds
k=1			k=1
no branch points
No numerical implementations as yet
one branch point
Refs: D. Gilbarg, Proc. National Academy of Sciences, 1949.
A. ?V. Goodman, Trans. Amer. Niath. Soc., 1950.
12
Exterior polygons
Problem: map the upper half-plane to the exterior of a
polygon, with f(i) = 00.
Solution: analogous to R,iemann surface case. f'(z) is piecewise
constant on the real ?xis with a double pole at z =
f(z) = c ;Z fl (s--Hzk)?4 (s2+1)?2 ds
k=1
j
A
--H? 0
Ref: L.N.T., unpublished memo 1987.
13
Doubly-Connected Polygons
Idea; wnte f' = Cllfk, with
k
Zk
fk
fk can be expressed in terms of theta functions.
Regions with higher connectivity: no S-C methods exist.
Refs: P. Henrici, Applied & Computational Complex Analysis III. 19S6.
NI. Hoekstra in Numerical Grid Generation, 1986.
H.D. Da?ppen, PhD thesis, ETH Zurich9 1988.
14
Free boundaries with 1f11 = const.
Standard S-C:
arg f' = piecewise constant on real &M5
S-C for free-boundarv problems:
arg f' = piecewise constant on [--H1,1],
ff = const. elsewhere on real axis
Applications: wakes, jets, cavities: area minimization (see p. 30)
Idea: write f' = C II fk, with
k
fk(z) =
3k
z --H zk
1 --H zkz + (l--Hz2)(l--Hzk)2
--H1			zk			1
fk
fk(0o)
0 = fk(zk)
Refs: AR. Elcrat & L.N.T., J. Comp. Appi. Math., 1986.
D. Gajer, Results in Alathematics, 1986.
F. Dias, A.R. Elcrat & L.N.T., J. Fluid Alech., 1987
15
fk(1)
Gearlike Domains
`Gearlike" domain G: bounded by radial line segments and concen-
tric circular arcs
Idea: P = log G is a polygon (possibly periodic) with
horizontal and vertical sides
Apply S-C type formula to P
capacity = 1.08184
capacity = 1.72442
Refs: AW. Goodman, Urnv. Nat. Tucuman, 1960.
L.N.T., unpublished memo, 1983
K.F. Jackson & J.C. Mason, in Algorithms for Appwximation 1987
K.			Pearce,			SlAM J. Sci. Stat. Comput., 1991
16
Channels and Elongated Polygons
Niapping elongated regions via a disk or half-plane is too ill-conditioned
to be feasible (the `crowding phenomenon?).
Idea: use an infinite strip instead as fundamental domain.
= C fi fk(z), fk(z) = (??sinh(z?zk))?Jk.
k=1
Zk + 7ri/2
Examples:
zk
fk
Refs: K.P. Sridhar & R.T. Davis, J. Fluids Engr. 1985.
J.NI. Floryan? J. Comp. Phys., 1985.
L.H. Howell and L.N.T., SLAAI J. Sci. Stat. Comp., to&ppcar.
L. Greengard, "Potential flow in channels," toappcar.
SlAM J. Sci. Stat. Conip., 1990
17
1990
Periodic Domains
Ref: J. M. Floryan, J. Comp. Phys., 1986.
18
Fractals
Example: (self-similar; map onto infinite strip)
Ref: L. H. Howell & L.N.T., SLAAf J. Sci. Stat. Comp.
19
Polygonal fractal H S-C map with n = cc
Applications in diffusion-limited aggregation. etc.
Algorithms not yet developed
1990
Circular Polygons
?Circular polygon": bounded by circular (or straight) arcs
The S-C integral becomes a 3rd-order o.d.e.
Fully robust implementations not ??t available.
Refs: P. Bj?rstad & E. Grosse SLAAI J. Sci. Stat. Comp., 1987.
L.H. Howell			.			0. Cornp. Appi. Math., 1993
20
General Curved Boundaries
Standard S-C turning angle 7rPk at veftex z?:
= c fi (z--Hzk)?4 --H Cexp[--H ??log(z--Hz?)].
k=1			k=1
Continuous S-C turning density function irP(z):
= Cexp[--Hfr%P(t)log(z--Ht)dt?
Integration gives:
f(z) = C fZ exp [--Hfr% p(t) log(s--Ht) dt 1 ds
$
(*)
There are dozens of integral equations for numerical conformal map-
ping besides (*), most of them simpler. Neveftheless (*) has proved
??ry useful for some problems.
Refs: L.C. Woods, The Theory of Subsonic Plane Flow, 1961
R.T. Davis, 4th ALAA Comp. Fluid Dynamics Conf.. 1978.
J.NI. Floryan, J. Comp. Phys. 1985.
NI. Hoekstra, in Numerical Grid Generation, 1986.
P. Henrici, Applied & Computational Complex Analysis III. 1986.
L.N.T., ed., Numerical Conformal AMppTh? 1986.
21
22
3. Applications
23
Electrical Resistance
Problem: find resistance (conformal module) of a `?quadnlateral?'
1) Niap the resistor conformally onto a rectangle
2) Resistance = Length/NVidth
v =0
Examples:
R = 1.11575250227
Refs: D. Gajer, Numer. Math., 1972.
L.N.T., Z. Angew. Math. Phys., 1984.
24
II
R = 4.55872841596
Inverse Problems, Side Conditions
Standard S-C:
geomet? fully specified (e.g.? n side lengths)
S-C with side conditions:
geometry partly specified (e.g. n--H1 side lengths)
? additional constraints (e.g.? specified conformal module)
Generalized parameter problem"
Example slit resistors with R = 2:
slit length = 0.727775151589
Ref: L.N.T., Z. Angew. Niath. Phys., 1984.
slit length = 0.330164529748
25
Piecewise Constant B.C.'s
Resistance problem (p. 24)			rectangle of unknown aspect ratio
Laplace problems with a larger number of piecewise constant h.c `s
rectilinear domain with slits of unknown dimensions
generalized parameter problem (linear hence easy)
Example:
II			-
Refs: R.F. Wick.9 J. Appi. Phys., 1954.
L.N.T. & R.J. Wilhariis, J. Comp. Appi. Math., 1986
26
Oblique Derivative B.C.'s
Refs: R.F. Wick., J. Appi. Phys., 1954.
L.N.T. & R.J. Williams, J. Gomp. Appi. Math., 1986.
27
4
Problem: V2u = 0 on a polygon. oblique Neumann b.c. `s
1) Find map f to another polygon with vertical b.c.?s
2) u := Ref
Examples:
Z3
f
w4
Zj
Classical Hall effect
M
Reflected
Brownian inotioy\
(tandem queues,
Vortex Methods in Fluid Dynamics
N'iscous flow at high Reynolds number
Simulation via point (or ?blob'?) vortices
Boundary conditions imposed by method of images
Conformal map to half-plane ensures one image per vortex
Example:
?-----H-???-?- %???o+??`?``			-			.-,.
?-- -.----,?,.- -.--
-			-			-?------ --H-.?=---------
--			?---- -
--H			--H--H			--H--H			-------H--H?-			--H--H
r			-			-? -			-
________			- - -			-
-?----??--%-:?-- =
o+ .?,			-,--H			--H-,.-
,- , -, - - ----H-,
 --H --H--H			--H
Ref: A.F. Glioniem & Y. Gagnon, J. Comp. Phys., ?987.
28
Re = 191
Complex
Soution
Approximation;
of Ax=b
Given: domain Q with boundar?r V
Fe;je'r Points on V: images of roots of unity under conformal map
of exterior of unit disk to exterior of Q. ?Uniformly distributed??
hence good for polynomial interpolation.
Application to iterative solution of Ax = b (nons??mmetric):
1) Determine estimate ? of spectrum of A
2) Calculate Feje'r points for Q via conformal map
3) Construct iteration based on interpolation of z?'
Refs: J.L. ?Va1sh, Interpolation and Approximation... 1935.
D. Gajer, Lectures on Complex Appwximation, 1987
B. Fischer & L. Reicliel, Numer. Alath., to&pp?ar.
H. Tal-Ezer, SLAXI J. Sci. Stat. Comp., subruitted.
L.N.T., Algorithms for Approximation IL toappzar.
29
1988
1990
Jets, Wakes, and Cavities
(Ideal 2D flow, no gra;?rjty.
see p. 15)
eotent?a? ???eS
??a??? s?acea
ecw??
--i
Refs: A.R?. Elcrat & L.X.T. J. Comp. Appi. Viath., 1986.
F. Dias A.R?. Elcrat & L.N.T., J. Fluid Mech., 1987.
30
Summary
o+ Niost S-C problems can be solved to full machine precision
in seconds or minutes (work = 0(n3))
o+ S-C variants ?modified S-C integrands'9 f' = II fk
k
o+ Often not all conditions are geometric
generalized parameter problems"
o+ All exactly-solvable conformal mapping problems are S-C!
(well almost all. .. sometimes in disguise)
31