A finite difference scheme is developed for constructing quasiconformal mappings for arbitrary simply and doubly connected regions. Computational grids are generated to reduce elliptic equations to canonical form. Examples of conformal mappings on surfaces are also included

Mastin, C. W.

Thompson, J. F.

English

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QUASICONFORMAL MAPPINGS
AND GRID GENERATION*
u. W. Mastin and J. F. Thompson
Mississippi State University
Mississippi State, MS. 39762
ABSTRACT
A finite difference scheme is developed for constructing quasiconformal mappings
for arbitrary simply and doubly-connected regions. Computational grids are
generated to reduce elliptic equations to canonical form. Examples of con-
formal mappings on surfaces are also included.
(NASA- R-173,920) OUASICON.FORMAL 'MAPPINGS
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Key words. quasiconformal mapping, grid generation, elliptic equation
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1, Introduction
The motivation for constructing quasiconformal mappings lies in their
3
application to the generation of curvilinear coordinate systems. Quasicon-
formal mappings may be used to reduce any second order linear elliptic partial
differential Equation to canonical form (i.e., the principal part of the
differential operator reduces to the Laplacian). Consequently, if one is
solving an elliptic boundary value problem, an appropriate quasiconformal
mapping could be used to simultaneously fit the boundary contours with coordi-
nate lines and simplify the original partial differential equation. The equation,
in canonical form, could possibly be solved more efficiently or by methods
which would not be applicable to the original equation in cartesian coordinates.
A related application of quasiconformal mappings is in the construction of con-
formal mappings on surfaces. Although they have been studied extensively by
complex analysis, very little work has been done on the numerical construction
of quasiconformal mappings. Of the methods which have been proposed, that of
Belinskii et al. [2] uses a fixed boundary correspondence which determines
the mapping parameters, and the method of Mastin and Thompson [4] would be dif-
ficult to implement on arbitrary regions. A finite element version of the
latter method developed by Weisel [9] appears promising. However, the class of
mappings and the type of regions presented in the examples are very limited.
In recent years a finite-difference method for constructing conformal
mappings developed by Allen [1] has been used by Mobley and Stewart [5], Pope [6],
and Yen and Lee [8] in the construction of orthogonal coordinate systems.
Although all of these authors use essentially the same numerical method, there
are differences in the way the boundary values and the conformal module of the
6
J4
2r
region are computed. This method is not as accurate or efficient as other
q
conformal mapping methods using integral equations or series expansions, but
n
it does have the advantage of simplicity since the module of the region, the
boundary correspondence, and the interior grid points are determined in
a single iterative procedure. Modeling this conformal mapping procedure, it
will be shown how quasiconformal mappings can be constructed and applied to
the reduction of elliptic equations to canonical form and the construction
of conformal mappings on surfaces. Except for the method of Godunov and
Prokopoy [3], this appears to be the only conformal mapping method which can
t
be easily adapted to handle the problem of constructing quasiconformal map-
pings,
2. Boundary Value Problem for uasiconformal Mappings
Let D be a bounded simply-connected region in the xy-plane whose boundary
C is a simple closed contour. Let z l , z2 , z 3 and z4
 be distinct boundary
points ordered by the orientation on C. There exists a unique quasiconformal
mapping of D onto the interior of a rectangle such that the points z i map to
the vertices which are also ordered by the orientation of the rectangle. The
ratio of the length to the width of the rectangle is a quasiconformal invariant
called the module of D and will be denoted by m. The quasiconformal mapping of
the region D onto a rectangular region can be obtained by constructing the
mapping of D onto a square region S which satisfies the linear system
9x = m(cny + b nx)
^y - -m(a nx + bny)
t
3JL
,
where ac - b 2 = 1. On setting (u,v) = (C,mn), it is obvious that u and v,
as functions of x and y, satisfy the Beltrami system and hence we arrive at
the desired quasiconformal mapping (see [4] for further details).
It is easily shgwn that x and y, as functions of g and n, satisfy
ax cc
2sxgn + yxnn J2(a x + by)
ay 	 2Rygn + Y
ynn = J 2 (bx + c y )
where
a = ayn
	  n- 2bxy + cx2
s = ay 9yn -
 b ( x Cyn + xny ) + cxExn
-y = ay  - 2bx y + cx2
J = xEyn - xnyC.
At each boundary point of 0 one of the functions, g or n, is constant
while the other satisfies an oblique derivative condition. This implies that
the condition S = 0 must be satisfied by x and y at each boundary point of
S (except for vertices). Note that in the solution of (1) we would have
s = 0 throughout S. Thus the computed value of S can serve as a test of the
accuracy of our solution. It also follows from (1) that m = Y'a—/y. Therefore,
the equations (2) can be written as
m2x99 + xnn = mJ(ax + by)
M2  += mJ(b + c )
u
(2)
n
^T	 1
i
(3)
ynn	 x	 y 
i
4i	 4
The same procedure is used for doubly-connected regions with a periodicity
r condition applied on two opposite sides of S.
The system (2) or (3) may be solved using an iterative procedure with
(i) the right-hand sides of the equations, (ii) the boundary values of x and
y, and (iii) either «,	 y, or an approximation of m, re-evaluated at each
iteration. As in the previously discussed conformal mapping methods, both
4	 (2) and (3) have performed equally well in numerical examples.
3. Reduction of E1j i^pti c Equations to Canonical Form
The application of quasiconformal mappings in the solution of elliptic
equations is well-known. An elliptic equation of the form
auxx + 2buxy + cuyy = f(u,ux) uy ), ac - b2 = 1
transforms to an equation of the form
m2 u	 + 
u
nn = g(u,ug,un)
under the transformation defined by (l). This transforniation^ will make'the
solution of many problems much easier. For example, if a, b, and c are con-
stants and f = 0, then it can be shown that g = 0 and we need only solve
m 2 
U + un,n = 0 on a square region. This is efficiently done by separation
of variables or a direct numerical method. Computational grids for solving
this problem on a simply and doubly-connected region are given in'Figure 1.
The practicality of using quasiconformal mappings , in solving elliptic
equations will be further examined in the following example. The function
u = cos(x - y) is a solution of the partial differential equation
uxx + uxy + uyy + cos(x - A = 0.
w
6
b
(4)
 J-4
(5)
RRR ^^
ca^
5
We will solve this equation numerically for 0 < x, y < w with Dirichet boundary
conditions prescribed by the known solution. In terms of curvilinear coordi-
nates, this equation can also be written as
m2u99 + unn + Um cus(x - Y) = 0.
VT
The quasiconformal grid for solving (5) is illustrated in Figure 2. For the
purpose of assessing the influence of the error in the iterative solution of
(2) on the error in the solution of (5), the iteration was stopped occasionally
and the solution of (5) was computed. A comparison of a normalized value of
S with the error in the solution of (5) is plotted in Figure 3. Note that
S has been normalized so that for the construction of conformal mappings
(a = c = 1, b = 0), the value, along the abscissa would represent the degree of
nonorthogonality. Equation (4) was also solved on a uniform rectangular mesh
with the same number of grid points. The error in this solution serves as an
approximation of the discretization error which would result in solving (5) with
the exact quasiconformal mapping. It also serves as a test of our method against
the traditional method for solving (4). Figure 3 indicates a nearly linear re-
lation between the plotted variables. This is to be expected since the major
part of the truncation error for larger S, is due to the omission of the mixed
derivative term, which is a linear function of s.
A few remarks concerning the numerical solution of (2) and (5) will be made.
The system (2) was solved using point SOR in the same way one would construct
a conformal mapping (see j7]). For the condition S = 0 on the boundary, a
form of one-side "upwind" differencing was necessary to maintain convergence
for the value of b = 1/ VY in this example. The elliptic equation (5) was solved
using a direct elliptic solver (see j.7]). After 35 iterations of (2) the
r
r
S
^'' f,
A
k
6
maximum error in the solution of (5) was within 125 per cent of what we
estimated to be the maximum discretization error. At this point the value of
(sl was still decreasing, but at a very slow rate. The exact value for m
in -this example is 1 due to symmetry. The computed estimate, which was
the root-mean-square value of vl'a—/y, was 1.00006 after 35 iterations.
This example does not illustrate an efficient use of quasiconformal
mappings. The method would be very efficient when one had to solve an
elliptic equation with many different boundary conditions or inhomogeous
terms. In that case the quasiconformal mapping would only have to be con-
structed once.
4. Conformal Mapping on Surfaces
A second area of application of quasiconformal mappings is in the con-
struction of conformal (isothermal) coordinates on a surface. Let M be
a smooth bounded surface in xyz - space which is defined by the parametric
equations
x = x4,e), y = y 4o), z = z(m).
The parameter region ^n the ^e - plane may be an arbitrarily shaped simply or
doubly-connected region but it is assumed that the boundary is composed of
simple closed contours and the mapping from the parameter region to the sur-
face has a nonvanishing Jacobian.
A conformal mapping of M onto a rectangular region can be constructed by
constructing a mapping from a square region of the gn - plane onto M which
sati fi es
P^	 Pn = 0 and miP E I = 1P 71	
t
Caj
ORIGINAL PAGV t)	 7
OF POOR QUALI"T"Y
where P = (x,y,z) and m is the module of the surface M. If these equations
i
are written in terms of the parametric variables 	 and e, we conclude that
e
and e satisfy
a
s
me c = ben - c4,n
M^^ = aen - bin
where
a = IPel2/d
b = P 	^ P e /d
C
	
IP^I2/d
i
x
	 d= I 	 PeI
f	 u
However, this is equivalent to (1) with (x,y) replaced by (^,e). In this case
the quantity which corresponds to 5/vCy would be the cosine of the angle be-
tween a E = constant and a n = constant coordinate line on the surface M.
Conformal grids have been constructed for several simply connected sur-
faces. Three surfaces are listed below. In the first two cases, the parametric
region was the projection of the surface onto the xy - plane.
i. Paraboloid: z	 1 - x2 - y2 , x2 + y2 < 1
ii. Bicubic: z = x 2y3 , - 1 < x,y < 1
iii. Torus: x = (2 + sin¢) cos e
z - (2 + sin¢) sin e,
_w/2<^<w/2,0 <e <7
s	 I J-11
Iri
Y
The plots of these surfaces appear in Figure 4. It is difficult to visualize
the orthogoisality from the plots, but the departure from orthogonality was
less than one degree except near vertices on the boundary where the ortho-
gonality condition was not imposed.
The advantages of conformal coordinates are well-known. Problems in-
volving heat conduction, ideal fluid flow, and electric fields can be so,^ed
as easily on the surface as they can on a rectangular region in the cartesian
plane.
6. Conclusions and Discussion
f,
A finite difference method, which has been widely used for the construction
of conformal mappings, has been generalized to construct quasiconformal mappings.. 	 j
This cevelopment will increase the class of problems which can be solved using
f
the currently available fast elliptic solvers developed by Swarztrauber and
Sweet [7]. Even when iterative methods are required, the absence of a mixed
w
f	
4
derivative and a rectangular region both would tend to give faster convergence
especially when optimal iteration parameters are known.
We will conclude this report with an open problem. It is known that if
a = c = 1 and b = 0, then the solution of (2) which satisfies s = 0 on the
boundary of the square S will also satisfy the system (1), and hence determines
a conformal mapping. This follows directly once it is noted that the quotient
n  
+ inx	
F
y + i gx
is an analytic function. Here it has been assumed, and numerical results tend
to verify, that the same result holds for arbitrary quasiconformal mappings.
However, no proof has been found.
C
_	
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^	 W
t
	
1
REFERENCES
1. C. N. de G, Allen, Relaxation Methods Applied to Conformal Transformation,
Quart. J Mech.	 Math. 15 (1962), 35-42.
2. P. P. Belinskii, S. K. Godunov, and I, K. Yanenko, The Use of a Class of
Quasiconformal Mapping: to Construct Difference Nets in Domains with
Curvilinear Boundaries, USSR Comp. Math. Math. P; s. 15 (1975), 133-144,
3. S K. Godunov and G. P. Prokopov, On the Computation of Conformal Trans-
formations and the Construction of Difference Meshes, USSR Comb.  Math.
Math. Phys. 7 (1967), 89-124.
4. C. W. Mastin and J. F. Thompson, Discrete Quasiconformal Mappings, Z.
An ew. Math. Phys. 29 (1978), 1-11.
5; C. D. Mobley and R. J, Stewart, On the Numerical Generation of Boundary -
Fitted Orthogonal Curvilinear Coordinate Systems, J. Comp. Phys. 34 (1980),
124-135.
6. S. B. Pope, The Calculation of Turbulent Recirculating Flows in General
Orthogonal Coordinates, J. Comp. Phys.  26 (1978), 197-217.
7. P. N. Swarztrauber and R. A. Sweet; Efficient, Fortran Subprograms for the
Solution of Separable Elliptic Partial Differential Equations, ACM Trans.
Math. Software 5 (1979), 352-364.
8. S. M. Yen and K. D. Lee, Design Criteria and Generation of Optimum Finite
Element Meshes, in Lecture Notes in Physics 90, pp. 579-586, Springer-
Verlag, Berlin, 1979.
9. J. Weisel, Numerische Ermittlung Quasikonformer Abbildungen mit Finiten
Elementen, Numer. Math. 35(1980), 201-222,
a
i
F
s
x
ORIGINAL PACs ^1'
OF POOR QUAI-1*9
0
110)
0
Figure 'I	 Quasiconformal grids for simply and doubl y-connected regions,
a = I, b	 1/2, c = 5/4.
Figure 2. Quasiconformal grid for 'the solution of elliptic equation.
a	 c - 2 / v73-, b--1/Y'-3 .
11 ^'w,
fco ,	 x) soo - n
4•
0
O
V)
41
r-
or-
S-0
0
rn
CL
8	 fri
E
ocl
	 4-
0 4-)
U co
-r- =
Ln cr
ra a)
=3
cr
ro
CL)
= 4-3
4-) C
a)
4-
L. 4-
O
L
CU
4—
0 -P
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w
4— cu
4J
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t• P 1
ORIGINAL PAGE 1,9
OF POOR QUALITY
Figure 4. Conformal grids on subsets of a paraboloid, torus, and bicubic surface.


Quasiconformal mappings and grid generation

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