A REGIONS* - Wichitadelillo/SISC_91_ext_maps.pdf ·...

SIAM J. ScI. STAT. COMPUT.Vol. 12, No. 2, pp. 399-422, March 1991

(C) 1991 Society for Industrial and Applied Mathematics010

A COMPARISON OF SOME NUMERICAL CONFORMAL MAPPINGMETHODS FOR EXTERIOR REGIONS*

THOMAS K. DELILLO" AND ALAN R. ELCRAT’:I:

Abstract. Several methods for conformally mapping the exterior of the unit disk onto the exterior ofa smooth curve are compared. These methods, which are known as the Timman, Friberg, Wegmann, andTheodorsen methods, are based on Fourier series and conjugation and are implemented using Fast FourierTransforms. The computations reported here indicate that Wegmann’s method converges faster and is morerobust than the others.

Key words, numerical conformal mapping, conjugation, the Theodorsen method, the Timman method,the Friberg method, the Wegmann method

AMS(MOS) subject classifications. 30C30, 65E05

1. Introduction. This paper is concerned with a comparison among severalmethods for computing the conformal mapping of the exterior of the unit disk ontothe exterior of a simple closed curve in the complex plane. The methods studied areoften associated with the names Timman, Friberg, Wegmann, and Theodorsen [3],10]. Timman’s method is also known as James’ method and appears to be the methodof choice in the aerodynamics community [1], [8], [9], [12]. Wegmann’s method isbased on solving a sequence of Riemann-Hilbert problems for the approximate map-ping functions; it has been shown theoretically to converge quadratically [15] undercertain hypotheses, and the suggestion has been made that it is superior to all knownmethods for a large class of problems [10, p. 422]. Theodorsen’s method is the oldestand most thoroughly studied of all these methods; we include it for completeness andas a bridge to earlier work. Friberg’s method may be thought of as a hybrid of that ofTimman and Theodorsen. The methods of Theodorsen, Timman, and Friberg convergelinearly.

Theodorsen’s method requires a star-shaped curve, but the others can be formu-lated for a general curve with arbitrary parameterizations. In many of the previousstudies arclength has been used; we use both arclength and other parameters and insome of our examples this has made a significant difference in the observed convergencebehavior. All of the methods considered here may be thought of as approximating theFourier series of the desired mapping, and they make use of Fast Fourier Transforms(FFTs) to perform conjugation of a function defined in the unit circle. For Timmanand Friberg the equation being solved iteratively is an integrodifferential equation andeach step requires integration of a function defined on the circle. Henrici [10] hassuggested using Fourier series to do this integration, and we have used this idea. Toour knowledge this has not been done in earlier computational studies.

The part of our work which compares Timman and Theodorsen expands onHalsey’s work [9], and is made in light of the crowding phenomenon [16]. For all ofthe methods we have given a more careful study of the discretization error than hasbeen presented before. For Timman and Friberg, Kaiser 13], [7] has given a theoreticalanalysis which implies convergence for curves near to a circle. Our experiments confirmthese results. Our general conclusion is that Wegmann’s method compares favorably

* Received by the editors August 7, 1989; accepted for publication (in revised form) January 16, 1990.

t Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67208-1595.$ The research of this author was partially supported by U.S. Air Force grant AFOSR-86-0274.

399

400 T. K. DELILLO AND A. R. ELCRAT

with the others in many situations. We have not included relaxation techniques in ourstudy. It is possible that they could significantly improve the range of performance ofboth Theodorsen [6] and Timman/James [12, p. 118].

An outline of the paper is as follows. Section 2 gives a derivation of the methodsbased on function conjugation, as in [7], and specifies the iterations. Section 3 treatssix examples.

2. Methods for exterior regions. Suppose that F is a smooth, simple, closed curvein the z-plane with the origin in its interior, and that z z(cr), 0 or=< L is a smoothparameterization of F with z’(cr) 0. We seek a conformal mapping g from the exteriorof the unit disk in the w-plane to the exterior of F in the z-plane, g is normalized byg(o) , and either g’(o)> 0 or g(1)= z(cro) for some fixed cro [0, L]. (The secondcondition makes sense by virtue of the Osgood-Carath6odory theorem [10] and thesmoothness of F.) We note that g has the Laurent expansion

g(w) cw+ ao+ aw- +"

with

C ’ye i’,where the positive number 3’ is the "capacity" of the curve F 10, Chap. 16]. We maywrite

g(ei)= z(cr(O))

and, since g is determined by its boundary values, our problem reduces to finding r(0).The unit disk in the w-plane is denoted by D, the exterior by DC, and the circle

by S.Each of our methods is associated with an auxiliary function h of g defined on

D [3], [7], [10]. We need the following proposition.PROPOSITION. For h HI(D):(a) Im h(e’)-Im h()=-K [Re h(e’)],(b) Re h(e’)-Re h(o)= K Jim h(e’)].

Here K is the conjugation operator,

K F-tifFwhere F takes a function to its Fourier coefficients, a, and

/’a, -i sgn (n)a,,.

The discrete version of K is

where Fn is the discrete Fourier transform, which can be realized in O(N log N)multiplications using the FFT algorithm (see [5]). H(Dc) is the Hardy space associatedwith L(S). Note that K maps constant functions to zero.

The methods to be discussed are now derived from the above proposition usingthe appropriate auxiliary function.

(i) Theodorsen [3], [6], [7], [9]. We assume here that F is given by

z(cr) p(cr) ei, 0=< o’<27r

for a smooth positive function p. Then

CONFORMAL MAPPING FOR EXTERIOR REGIONS 401

and

h(ei) log Iz(cr(0)) + i(arg z(cr( 0))- 0)

-log p(cr(0)) + i(cr(O)- 0).

Using the normalization g’()> 0 (a 0), Proposition (a) implies

tr(O) 0 -K[log p(tr(O))].

For Theodorsen’s method

o-"+1(0)- 0 -K[log p(cr" (0))].

It is known that the "e-condition"

e =max 1 <1

implies that r"(0) converges linearly to r(0) in L2; there is a corresponding theoryfor the discrete problem and a bridge between the discrete and continuous.

(ii) Timman [3], [7], [9], [10], [13]. We define

h(w)=logg’(w).

Since g’(e io) -z’(cr( O))cr’(O) e-i, we have

h(ei)=log or’(0)+ log Iz’(o’(O))l+ (arg z’(cr(O))-O-).Proposition (b) implies

log o-’(0)+ log Iz’(cr(0))l-log y= K[arg z’(cr(O))-0-].Suppose we use g(1)= z(cro) as our normalization. Then

a Im h() -- (arg z’(tr(O))-O) dO-2since Im h(w) is harmonic. The iteration is

n+l(cr 0)= y, exp (K[arg z’(o" (O))-O])/[z’(o’"(O))] dO

where LT- is given by the integral over [0,2r]. We perform the integration byintegrating the Fourier series of the integrand term by term as suggested by Henrici10]. Since

h(w) log (c-O(w-2)) O(w-2) +log c

at infinity, we set the first two Fourier coefficients of K[arg z’(o’"(O))-O] equal tozero [10], [1].

Kaiser [13] has shown that r" converges linearly to o- if F satisfies certainhypotheses. In particular, if o" is arclength and F is close to a circle the convergencefactor is .5.

(iii) Friberg [3], [7], [13]. Here

g(w)h(w)=logg’(w)-log.

W


Note that h(w)= O(w-1) at infinity and that h(oo) =0. Proposition (b) implies

log tr’(0) K[arg z’(er(0))- arg z(tr(0))] +log Iz(tr(0))l-log Iz’(o-(0))l;

the iteration is

| ’("tr"+’(O) Iz (0)) I- z(tr"(O))l exp (K[arg z’(o’"(0))-arg z(er"(0))]) dO./o

and o’"+l(O)=d-"+l(O)L/d-"+l(2cr). We note that a, y can be found as a part of thisiteration using the Timman auxiliary function.

Kaiser [13] also proved a convergence theorem for this method; in this case theconvergence factor for a curve close to a circle is nearly zero.

(iv) Wegmann [10], [15]. This method is described by giving a correction to acurrent approximation to o-(0). We want to compute r(0) so that

g(ei) z(cr(O)+ r(0));

linearizing

z(o-(0)) + z’(o’(O))r(O)= g-(ei)

where g-(ei) are the boundary values of a map to a nearby curve. We normalize byassuming g’(oo) is real. Since r(0) is real g-(e) must be a solution of the Riemann-Hilbert problem

Re (iU(s)g-(s)): Re (ff’(s)z(s))iowhere s e z(s)= z(cr(O)), z’(s)= z’(er(O)). It can be shown that

g-(w)=[e-ig’k,(w)-./sin g]w e l(w

where (note the correction of a misprint in [10] in the first of the following and theuse of a general parameterization)

Then

___1 arg z’(s) 0ds,l(w)

"/r ,J]s]=l S-- 14/

A(s) exp (-K[arg z’(s)-O]) Im (z’(s)z(s))lz’(s)l-’,1 A(s)

k(w) | ds,7r .[sl=l

X A (e’) dO

2rr[arg (z’(cr(0)) 0)] dO.

g-(s) -exp (i arg z’(s) + K[arg z’(s) 0]){K[A (s)] + cot g+ iA (s)}

z((0)) + ’((0))(0).The iteration is given by

and

g-( ei) z( cr" O

"+(0) "(0)+ "(0).


Wegmann [15] has shown that for smooth curves and a good enough initial guess,converges to cr quadratically in W1’2(S).

3. Numerical experiments. We have done numerical experiments with maps to theexterior of the following regions: (a) ellipse, (b) sports ground, (c) inverted ellipse,(d) cosine airfoil, (e) a perturbed circle, and (f) a general spline curve. Our implementa-tion uses the radix-2 N point FFT routine listed in [2, p. 416] where N--2M.

There are two errors associated with the iterations"

I=j=N

and

max I r(Oj) 0+)1t_<j- N

where 0j 2r(j- 1)/N are the Fourier points, o- is the exact boundary correspondence,and cr(")(0j) are the values at the nth iterate. The first error can generally be iteratedto zero in machine precision. Exceptions to this occur for Wegmann’s method, whereit may cease to decrease much below the level of the discretization error and oftenexhibits a convergence/divergence behavior, to be discussed below. The second erroris our computational measure of the discretization error. We have o-(0) only in the

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32 64 96 128 192 256 320 384 448 512

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FIG. 1. maxl__<j=<v ICK-KN)logpCoCOj))lfor ellipse.

alpha

404 T.K. DELILLO AND A. R. ELCRAT

TIMHAN FOR ELLIPSEN- 2**H

7.$0

FIG. 2(a). Discretization error for Timman for ellipse.

o 1E-07-

1E-08-

448 512

I .2 SPLINE

-e...4 ANALYTIC

.8 ANALYTIC

case (a), so in the other examples we estimate the discretization error by computingan approximation to

max dist (g(")(ei), F),0-< 02-n-

the Hausdorff distance between g(’)(S) and F. Here g(’) is the approximation to theTaylor series of g found using trY")(0) and N’> N Fourier points; the values ofg"(ei) at these N’ points are found using the N’-point FFT and the Taylor seriesof g(") zero padded from N to N’ points. (For example, (b) dist (g(’)(ei), F) can befound exactly and it is approximated by

Ig(’)(ei) z(arg g(")(ei))

for (c) and (d).) We do not compute the discretization error for (e) and (f).The curves (a), (b), (c), (d) are all given by piecewise analytic expressions. We

also consider periodic cubic splines parameterized by the chordal distance [11] fittedto points on these curves. The chordal distance approximates the arclength so [z’(r)[ 1.Usually we use NN 1,000 knots evenly distributed in the original parameterization.In this case the splineapproximates the analytic curves to within about 10-7 to 10-1

in most of our examples, so the discretization error can never go below that. However,the spline is needed in example (d) where a parameterization is needed for the nearcircular region generated by the Karman-Trefftz map. (To treat this case withTheodorsen’s method we would have to use a spline parameterized by polar angle;


FNBERG FOR ELLIPSEN- 2**M

32 64 96 1281E+O0],, ,,,,

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192 256 ,:320 384 448 512

alpha

.2 SPLINE

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FIG. 2(b). Discretization error for Friberg for ellipse.

we have chosen to omit this case.) In practice, we usually have a finite number ofpoints to define a boundary. This can be done conveniently with a spline as illustratedin example (f).

The initial guess for the examples below is always o-()(0) LO/27r, where 06[0, 2].

The numbers near the data points on the figures are rough timings of CPU second(IBM 3081, FORTVS compiler, double precision arithmetic).

(a) Ellipse [10, p. 412]. F is given by

z(O) =cos 0+ ia sin 0, 0 <- 0-< 27r.

r is a "thinness" parameter, 0 < ce _-< 1. The exact solution, if g’(oo)> 0 or g(1)= 1 isimposed, is

g(w) 1/2((1 + c)w+ (1 ce) w-l).

We can also represent F in polar coordinates, z(o’)=p(o’)e i’, 0=< o’=<2r, with

p(r) a//1 -(1 c 2) cos2o’.

In this representation the exact boundary correspondence is

r(0) arc tan (a tan 0).


WEGllANN FOR ELLIPSE

1E+O0,

1E-01

1E-O2-

1E-O3-

1E-O4-

1E-O5-

1E-06-

o 1E-07-

1E-O8-

1E-Og-

1E-10-

1E-11-

1E-12-

1E-13-

1E-14-

1E-15-

N- 2.*132 64 96 128 192 256 320 38’1. 448 512

’1

FIG. 2(C). Discretization error for Wegmann for ellipse.

We note that the Laurent series of g converges (in fact has only two terms) for allw 0 but that g’(w)= 0 at

From the expression for g(w) we derive the Fourier coefficients ofthe auxiliary functionfor Theodorsen’s method:

h(ei) log (g(e’)/e’)log p(cr(0)) + i(o’(0) 0)

(l+a) f12log2

+/3 cos (20) -- cos (40) +.

+ -fl sin (20) +-- sin (40)

where fl (l-a)/(1 +). The N-point FFT gives an approximation to the first N/2terms of the sin/cos series (see [5]). Thus for the above series we have

logp(tr(0)) log(l+c) 22

+/3 cos (20) cos (40) + +/- flN/4N/4 cosN 0.


FIG. 3(a). Ellipse .8 with Theodorsen N =64.

FIG. 3(b). Ellipse a .4 with Friberg N 256.


FIG. 3(c). Ellipse a .2 with Wegmann N 512, spline parameterization.

FIG. 4. Sports ground a .2 with Timman N 512.

We see that the error committed in truncating the series is

(K-Kv)[log p(o-(0))] O(/4)in agreement with Gaier [4]. This is also in agreement with Wegmann 15, Ex. b, 9];there, bounds of order R-N/2 are exhibited for the interior problem where R > 1 isthe modulus of the nearest singularity or zero of the derivative. Here R2=/3 < 1.

Similarly for Timman’s method,

h(e’) log (g’(ei))

2/3 cos (20) +-- cos (40)+...

+i fl sin (20)+--sin (40)+....


CONVERGENCE/DIVERGENCE FOR TIMMAN FOR .3 ELLIPSENUMBER OF ITERATIONS

30 50 70 90 110 130 150 170 190

N= 2,56

FIG. 5(a). Convergence/divergence for Timman for .3 ellipse.

As observed previously,

log (or’(0)) + log Iz’(o’(0)) Re h(ei)

and

arg (z’(tr(O)))-O -= Im h(e’),2

so that the errors here exhibit the same order as above.We also recall that the e-condition in this example can be computed exactly, and

e < 1 if and only if a > x/- 1 .4142 .Our computations of the discretization error are given in Fig. 1. Gaier [4] shows

that the aliasing error of the conjugated trigonometric interpolant is of the same orderas the error in truncating the Fourier series of the conjugate function, as exhibitedabove. The errors shown here follow the trend predicted by the theory. The discretizationerror for Theodorsen follows Fig. 1 for a >.41.... The discretization errors forTimman, Friberg, and Wegmann for the ellipse are given in Figs. 2(a), 2(b), and 2(c).The data there are for analytic parameterization for a .8, .4, and for a .2 a splinewas used. Often the spline gave a smaller discretization error. Here for a .2, largerN was required for convergence in the analytic case. These errors follow Fig. 1 for


CONVERGENCE/DIVERGENCE FOR FRIBERG FOR .4 ELLIPSE

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NUMBER OF ITERATIONS10 20 30 40 50 60 70 80 90 100

....I I--/" .oO

\

o

ii

FIG. 5(b). Convergence/divergence for Friberg for .4 ellipse.

N =512

N = 256

N 128

larger a, but degrade for smaller a (say a <.4). Some computed regions are shownin Fig. 3.

(b) Sports ground. This region of length two is bounded by two semicircles ofradius a connected by horizontal line segments. See Fig. 4. We use arclength as aparameter; z’(r) is not differentiable at the points connecting the semicircles and lines.Here Iz’(tr)l 1 for all r, in contrast to example (a) where Iz’(tr)l varies between a

and 1.We have observed a convergence/divergence phenomenon (further addressed in

[16]) in which errors decrease up to a certain point and then increase, for Timman,Friberg, and Wegmann. This occurs only in certain situations. For Timman it neveroccurs for a spline, never for the sports ground, and only for small a for an ellipsewith analytic parameterization. For Friberg it occurs for an ellipse with small a forboth the spline and the analytic parameterizations. For Wegmann it occurs to someextent in all cases.

These results indicate that the division by Iz’(tr(0))l in Timman or Wegmann, andalso the multiplication by ]z’(tr(O))l in Friberg has some effect on the convergencebehavior, and may trigger this phenomenon. It may also be associated with discreti-zation error. It does not occur for Theodorsen where the discrete equations can besolved to machine accuracy for e < 1. Wegmann [15, p. 457] noticed the start of thisbehavior for the interior problem and attributed it to roundoff error. Examples of


CONVERGENCE/DIVERGENCE FOR WEGMANN FOR .6 ELLIPSE, N : 32

NUMBER OF ITERATIONS30 50 70 90 110 130 150 170 190

I’ 1

SUCC. ITERo ERROR

DISCR. ERROR

FIG. 5(C). Convergence/divergence for Wegmann for .6 ellipse N 32.

convergence/divergence are shown in Fig. 5 for the ellipse; Figs. 5(a) and 5(b) givediscretization error versus number of iterations for Timman and Friberg, respectively(error between successive iterates is roughly the same); Fig. 5(c) shows discretizationerror and that between successive iterates for Wegmann. It is interesting to observethat convergence rates are independent of N.

(c) Inverted ellipse. If we write z(tr)= p(tr)e i, then

p(r) =x/l- (1- a :) cos (r).

The e-condition is the same as in (a). (See Figs. 7(a)-(c).)The discretization error for Timman’s method with this parameterization is given

in Fig. 6(a). Friberg yields almost indistinguishable results. Using a spline gives resultswhich are often an order of magnitude more accurate. The timings given in Fig. 6(a)clearly indicate that Friberg is faster than Timman for larger c. This confirms Kaiser’sanalysis 13], [7, p. 62], which says that the linear convergence rate for Friberg’s methodis nearly zero for nearly circular regions. We did not succeed in getting convergedsolutions for a smaller than .5 with Friberg or Timman.

Figure 6(b) gives the discretization error for Wegmann’s method. The results forTheodorsen’s method were nearly the same, except that Theodorsen’s method failedto converge for a < x/- 1 .41 (e > 1). The use of a spline fit produced nearly identicalresults down to the level of accuracy of the spline. Some timings are again indicatedas above.


TIMMAN(FRIBERG) FOR INVERTED ELLIPSEN =

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FIG. 6(a). Timman (Friberg) for inverted ellipse.

Wegmann’s method converged for N =2,048, a =.3, also (see Fig. 7(c)). It isclearly seen to be the best of the four methods here both in timing and in range of a.

Since it converges quadratically Wegmann’s method usually only required three orfour iterations to reach minimum dicretization error. For a .6, N 512, for instance,Theodorsen required 49 iterations. See [14] for similar observations.

Note that the exterior problem for the inverted ellipse is, like the interior problemfor the ellipse, a highly ill-conditioned problem as aS0. (The inverse map exhibitssevere crowding; see [6] and [17].) We may expect such difficulty for any methodwhich has an elongation or fingering of a portion ofthe boundary between the canonicaland the target regions. Examples (a) and (b) are easier in this sense, i.e., maps to theinterior of slender regions are harder than maps to the exterior of slender regions.

(d) cosine airfoil. We use the following curve to illustrate the map to the exteriorof a curve with a corner:

z(r) -cos (kr) ei-’n" 7r

2k =2k"The exterior angle 2r- h is removed by the Karman-Trefftz transformation

" ’1 Z,-Z

"- ’ z- z


WEGIIANN(THEODORSEN) FOR INVERTED ELLIPSE

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FIG. 6(b). Wegmann (Theodorsen) for inverted ellipse.

where zl, z2 map to 1, ’2, and

We map NN 1,000 points to the ’-plane and interpolate with a periodic cubic spline;this yields a smooth curve which is more nearly circular for a good choice of z2. (Thiscurve is not parameterized by argument, so we omitted Theodorsen’s method.) Timman,Friberg, and Wegmann give the map to the region in the ’-plane. This is composedwith the inverted Karman-Trefftz map. Below zl ’1 0, z2 ’2 < 0 for Timman andWegmann. (Since Friberg requires the curve to surround the origin we translated thespline to the right by -’2/2 and then back by -g(1) after the iteration.) Plots of theimage in the z and sr planes are shown in Fig. 8.

Figures 8(a)-8(d) show the maps for two cosine curves and their images underthe Karman-Trefftz transformation. Figure 9 shows the discretization error in the planeof the cosine curve plotted against M for Wegmann with z2 -.9 and k 2 (Figs. 8(a)and (b)). An error of 10-9 was reached in successive iterates of the map to the splinecurve. The discretization errors for Timman and Friberg were similar but somewhaterratic; however, the errors in the successive iterates could be decreased to 10-15

Timings for N 128 were" Wegmann, 1.3 sec; Friberg, 1.8 sec; and Timman, 4.0 sec.


FIG. 7(a). Inverted ellipse ct .8 with Timman N 64.

(e) Perturbed circle. Here z(cr)=(l+.03sin(16o-))e i. The methods ofTheodorsen and Friberg worked best here converging to machine precision in onlytwo iterations (see Fig. 10(a)). For N 128 Wegmann reached an error of 10-11 in thesuccessive iterates. For N 64 Timman’s method converged to machine precision toa map with loops exhibited in Fig. 10(b). In this case the approximate map must havezeros of its derivative just outside the unit circle. For N 256 Timman converged tothe correct map. Similar loops are shown in [6] for Theodorsen’s method for maps tothe interior of a thin ellipse and to regions with corners.

(f) Spline curves. Here our boundary curves were generated by selecting a fewpoints in the plane and interpolating them with our periodic cubic spline routine. Forthe region in Fig. l l(a), Timman, Friberg, and Wegmann all converged. Machineprecision was achieved in the successive iteration error for Timman and Friberg andmost N tried. Wegmann achieved the successive iteration errors given in Fig. l l(c) ina few iterations and then diverged. This convergence/divergence behavior may besomewhat disadvantageous in practice without a good procedure to stop the iterations.However, Wegmann again is more robust. It converged for Fig. l l(b) where Timmanand Friberg failed. The effect of the curve geometry is evident again: as in thecomparison of inverted ellipse and the ellipse, the fingering in Fig. 1 l(b) causes moreinaccuracy than the slenderness in Fig. l l(a). Note also that the errors in Fig. ll(c)are roughly O(N-3) as expected with cubic spline boundary, z(tr) W3’. Sampletimings for Fig. l l(b) are 3.7 sec for N= 1,024 and 7.5 sec for N=2,048. Thus


FIG. 7(b). Inverted ellipse a =.6 with Friberg N 512.

Wegmann’s method seems quite promising in this practical case. In future work wehope to report on computations where the spline curve is generated by explicit maps,such as the osculation maps in [10].

4. Conclusion. In comparing the methods of Wegmann, Timman, Friberg, andTheodorsen our general conclusion is that Wegmann is the most efficient and mostrobust. Friberg is superior to Timman for nearly circular domains as the theory suggests;for extreme regions Timman is preferable if the arclength is used as the parameter.

We have observed a curious convergence/divergence phenomenon. For Timmanit can be correlated with the use of a general parameterization; it did not occur forTimman when the arclength was used. It may also be associated with the relationbetween the discrete and continuous problems; it does not occur for Theodorsen withe < 1 where the discrete equation can be solved to machine precision.

A careful analysis has been made of discretization error; for extreme regions theerror is worse than the theory indicates for Timman, Friberg, or Wegmann. If a remedyfor this could be found, the range of usefulness of Fourier methods might be extendedsignificantly.

Acknowledgments. We thank Professor Mike Papadakis of our AeronauticalEngineering Department for assistance with the graphics and one of the referees forsuggesting some corrections and improvements.


FIG. 7(c). Inverted ellipse ce .3 with Wegmann N 2,048.

FIG. 8(a). Cosine curve k 2 with Wegmann N 128.


FIG. 8(b). Image of Fig. 8(a) under Karman-Trefftz with z2 =-.9.

FIG. 8(c). Cosine curve k =4 with Timman N 512.


FIG. 8(d). Image of Fig. 8(c) under Karman-Trefftz with z -"-.99.

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WEGMANN COSINE CURVE, Z2 : ".I, K,: 2

2 3 4 5 6 7 8 9 10 11 12 13 14 15’1

FIG. 9. Discretization error for Wegmann for cosine curve, z --"-.9, k 2.


FIG. 10(a). Perturbed circle with Theodorsen N =64.

FIG. 10(b). Loops produced by perturbed circle with Timman N =64.


FIG. l(a). Spline curve with Friberg N 128.

FIG. 11 (b). Spline curve with Wegmann N 1,024.


WEGMANN FOR SPLINE CURVES

E+O05

IE-01

o 1E-02-

o 1E-03-

1E--

r 1E-05-

IE-06-

1E-07-

IE-08-

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20’1 I’

I FIGURE 11 B-- FIGURE 11A

FIG. ll(c). Successive iteration error for Wegmann for spline curves in Figs. ll(a) (’) and ll(b)

REFERENCES

[1] F. BAUER, P. GARABEDIAN, D. KORN, AND A. JAMESON, Supercritical Wing Sections II, LectureNotes in Econ. Math. Syst., Vol. 108, Springer-Verlag, New York, 1975.

[2] G. DAHLQUIST AND A. BJ6RCK, Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974.

[3] D. GAIER, Konstruktive Methoden der konJbrmen Abbildung, Springer-Verlag, Berlin, 1964.

[4], Ableitungsfreie Abschiitzuugen bei trigonometrischer Interpolation und Konjugiertenbestimmung,Computing, 12 (1974), pp. 145-148.

[5] M.H. GUTKNECHT, Fast algorithmsfor the conjugateperiodicfunction, Computing, 22 (1979), pp. 79-91.

[6] , Numerical experiments on solving Theodorsen’s integral equation for conformal maps with theFFT and various nonlinear iterative methods, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 1-30.

[7], Numerical conformal mapping methods based on function conjugation, in Numerical ConformalMapping, L. N. Trefethen, ed., North-Holland, Amsterdam, 1986, pp. 31-78.

[8] N. O. HALSEY, Potential flow analysis of multielement airfoils using conformal mapping, AIAA J., 17(1979), pp. 1281-1288.

[9 ,Comparison ofconvergence characteristics oftwo conformal mapping methods, AIAA J., 20 (1982),pp. 724-726.

[10] P. HENRICI, Applied and Computational Complex Analysis, Vol. III, John Wiley, New York, 1986.

11 W. D. HOSKINS AND P. R. KING, Periodic cubic spline interpolation using parametric splines, Algorithm73, Computer J., 15 (1972), pp. 282-283.

12] D. C. IVES, Conformal grid generation, in Numerical Grid Generation, J. F. Thompson, ed., North-Holland, Amsterdam, 1982, pp. 107-135.


[13] A. KAISER, L-Konvergenz verschiedener Verfahren der sukzessiven Konjugation zur Berechnung kon-former Abbildungen, Ph.D. thesis, Mathematics Department, ETH, Zurich, Switzerland, 1986.

14] P. LUCHINI AND F. MANZO, Flow around simply and multiply connected bodies: a new iterative scheme

for conformal mapping, AIAA J., 27 (1989), pp. 345-351.15] R. WEGMANN, Convergence proofs and error estimates for an iterative method for conformal mapping,

Numer. Math., 44 (1984), pp. 435-461.16] , Discretized versions of Newton type iterative methods for conformal mapping, J. Comput. Appl.

Math., 29 (1990), pp. 207-244.

[17] C. ZEMACH, A conformal map formula for difficult cases, in Numerical Conformal Mapping, L. N.Trefethen, ed., North-Holland, Amsterdam, 1986, pp. 207-216.

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