Harmonic grid generation methods for multiply connected plane regions and regions on curved surfaces are discussed. In particular, using a general formulation on an analytic Riemannian manifold, it is proved that these mappings are globally one-to-one and onto

Smith, P. W.

Sritharan, S. S.

English

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NASA ContractorReport 172331
IC_E REPORT NO. 84-12 NASA-CR-172331
19840014259
ICASE
ON THE GRID GENERATION METHODS BY HARMONIC M_.PING
: J("_T_ _ ,'T'-_Tn';'-_T_ ".:_-_, _ ""_ < "
ON PLANE AND CURVED SURFACES .E M-)-£_- ........ " "-:" _':k_._-_
S. S. Sritharan
and
Philip W. Smith
Contract No. NASI-17070
March 1984
INSTITUTEFOR COMPUTERAPPLICATIONSIN SCIENCEAND ENGINEERING
NASA LangleyResearch Center,Hampton,Virginia 23665
Operatedby the UniversitiesSpace ResearchAssociation
NationalAeronauticsand
.. , . y- . ,SpaceAdministration ' ;" 9' c-=IS_
Langley Research Center
Hampton,Virginia 23665 IZ,NGLEYRESEARCHCEr_TEP,
UB,_ARY.NASA
H,_.... iO,l, VIRG!i'IIA
https://ntrs.nasa.gov/search.jsp?R=19840014259 2020-03-20T23:55:09+00:00Z

ENTER"

ON THE GRID GENERATION METHODS BY HARMONIC
MAPPING ON PLANE AND CURVED SURFACES
S. S. Sritharan
Institute for Computer Applications in Science and Engineering
Philip W. Smith
Old Dominion University
Abstract
Harmonic grid generation methods for multiply connected plane regions and
regions on curved surfaces are discussed. In particular, using a general
formulation on an analytic Riemannian manifold, it is proved that these
mappings are globally one-to-one and onto.
Research was supported by the National Aeronautics and Space Administration
under NASA Contract No. NASI-17070 while the first author was in residence at
ICASE, NASA Langley Research Center, Hampton, VA 23665.
i

INTRODUCTION
One of the most important tasks of computational physics is the generation
of boundary conforming grids in the computational domain. This problem can be
framed mathematically as the construction of certain coordinate charts on a
given Riemannian manifold. In this paper the method of harmonic mappings for
two-dimensional manifolds is considered. The method of two-dimensional
harmonic mappings has been used, for example, to generate grids in multiply
connected plane regions [I] and to generate coordinate surfaces for three-
dimensional domains using the Gauss equations [2]. We will see that the
harmonic mapping method can also be used to generate grids on a portion of a
given analytic curved surface. We will describe and analyze these methods by
unifying them to a general mapping problem on a simply connected Riemannian
manifold.
2. HARMONIC NAPPING METHOD FOR AN ANALYTIC RIEHANNIANNANIFOLD
The slmplest grid generation problem of this kind can be formulated as
follows. Let Q be a simply connected region on a two-dimenslonal manifold
M 2 with compact closure (see Figure I). Suppose (8,$) is a glven (natural)
coordinate system on this manifold. The problem is to find a boundary
conforming coordinate system (_,_). This is accomplished by first mapping
_Q0 in a continuous i-I manner to the boundary of the rectangle 01 in the
(_,_) plane. To map the interior of Q0 to the interior of QI' we use the
following conditions:
(2.1) A_ = 0, A_ = 0
where A is the Laplace-Beltrami operator.
We will see later that more complicated grid generation problems can be
treated in this manner. Let us investigate more closely this technique. We
set (_i,_2) = (e,_) and (xl,x 2) = (_,_) for ease of notation. Then g--_8
and ge8 are respectively the metric tensors of these coordinate systems.
Then from the elementary theory of Riemannian geometry we have the following
explicit expression for the Laplacian (Beltrami) of any scalar function _ on
M2
2
(2.2) A! = geS[_ i F% _]8 _8 _x
where e,8,% = 1,2. In this context we have used the Christoffel symbol of
the second kind in the x_ coordinates. It is well known that this symbol
obeys the transformation laws,
where, of course, _6 is the Christoffel symbol of the second kind for the
coordinate system x .
2.2. Formulation on a Given Manifold
We note here that if the manifold M2 E E2 (Euclidean space) then x
can be chosen as the Cartesian coordinate system yielding _y m 0 [5].
-2-
This is not possible for a general manifold• However, since any two-
dimensional Riemannian manifold is conformally flat [5], it is always possible
--2
to choose (locally) a coordinate system x known as the isothermal
coordinate system which satisfies
(2.4) g--aB= X2(xl'x2) 6aB"
In this case, one can verify that
_-XaB_B = 0 yielding
(2.5)
2 _xX _ aB _2-_y
A¢ = gab a ¢ X g 17 "ax_ axe a_x ax ax_ax
If we choose _ = x (one of the coordinate functions) then
considerable simplification takes place and we have
Ax_ 3x_ aB 22 xY
-- j g •
_xx _x_ axe
By imposing the condition Ax _ = 0, we in effect impose conditions on the
resulting metric tensor. Yielding the equation
2--X
--gab_ x - O, X = 1,2
_x_ _
as the generating equations to be solved in the (xl,x 2) plane. For example,
we consider a portion of a spherical surface• In this case we can obtain an
--X
isothermal natural coordinate system x using the stereographic projector
-3-
tanI_/2) exp(i8)X + iX =
where (O,O) are the spherical coordinates. Imposing hx _ = 0, 6=1,2 yields
as before
2 --_
g_6 8 x - 0 I = 1,26
with Dirichlet boundary conditions. A practical grid system for a conical
wing shape [3] is given in Figure 2. Similar computations could be performed
for other simple surfaces where some isothermal coordinates are explicitly
known. For certain simple surfaces, isothermal coordinates are given in
[4,5].
2.3. Surface ConstructlonUsingHarmonlcMappings
We now consider the problem of generating a coordinate surface in E3
connecting two one-dimensional boundaries (lying in two boundary surfaces).
Again, the idea is to place implicitly restrictions on the metric tensor (see
Figure 3). Let the unknown manifold M2 be immersed in E3 with Cartesian
coordinate system xi, i = 1,2,3. The tangent vectors are
Bi "- 8xl i = 1,2,3, a = 1,2
where (_I,_2) are some coordinate system for M2. The Gauss [7] equations
are
Bi Ni
-4-
where "If" denotes covariant differentiation, Q=_ is the second fundamental
form and Ni is the normal to the manifold. Multiplying by g=_ yields
=_ 82 xi _ Fk i i Bh k
g g _ Bk + Fhk _ B_ g_ = (kI + k2)N i5__ 5__
where Fk
_ and F k are respectivelythe Chrlstoffelsymbolsof the second
kind of the _ coordinate system on the manifold and the xi coordinate
system on E3. Also (kI + k2) is twice the mean curvature. If xi are
i = 0 and recallingour earlierdevelopment,weCartesiancoordinatesthen Fhk
obtain
gU_ 82 xi + BkiA_k = (kI + k2)Ni"
5__ 5__
If we impose A_k = 0 for k = 1,2,we get
=_ 82 i
g x = (kI + k2)Ni
which could be used as generatingequationsfor M2. This techniquehas been
exploitedby Warsl [2], [6].
3. MULTIPLY CONNECTEDDOMAINSAND THE CONSTRUCTIONOF AN ABSTRACTRIEMANN
SURFACE
We will first describe briefly the constructionof an abstract Riemann
surface for an algebraic function as a compact and orlentable surface that
could be realizedin the Euclideanspace E3. Considerthe function
-5-
2N
wz--[I (z- ri).i=l
Then w will be single-valued in the two-sheeted cut plane as shown in Figure
4. Projecting each of these planes stereographically to their Riemann spheres
and joining the spheres along the cuts after appropriate rotations, we get a
closed surface that is topologically equivalent to a surface of genus N-I.
We have thus obtained a surface on which w is single-valued, in a form which
could be realized in E3.
If we are now interested in generating grids by harmonic mapping on a
mulitply connected domain _ with appropriate cuts as shown in Figure 5, we
see that this domain will have a corresponding image on a suitable abstract
Riemann surface S. We will now introduce the Riemannian metric in this
surface to convert it to a metric Riemann surface (conversely, an analytic
Riemannian manifold with isothermal coordinates will become an abstract
Riemann surface). Thus, we see that the problem of generating grids on a
multiply connected plane domain and that on a portion of an analytic curved
^
surface are equivalent. Let S be the universal covering surface (simply
connected) of our Riemann surface S. On this covering manifold the domain of
interest fl will have many (disconnected) images and these images can be
transformed to each other by what is called the covering transformation. We
will thus work with the fundamental domain on this manifold that is left
invariant by the group of covering transformations. Since any simply
connected Riemann surface can be conformally mapped to a plane [4], the
A
covering surface S will have a one-to-one conformal transformation with a
domain G on the plane (see Figure 6).
-6-
ALet P be a point on the Riemann surface S and P be one of its images
A
on the universal cover S. The projection mapping _ is defined by
=P,
^
(note _-I is not single-valued). Let f map S to G, then f is
univalent and
--I _2 ^= x + i = f(P)
or
p = (= o f-l)(_) =_(_).
Thus, _ = _-l(p) is the uniformizer for the Riemann surface S. This
procedurehas providedus with a global familyof isothermalcoordinates
(_I,_2) on the manifold S. We have to note that_ -I is not single-valued
if S is not simply connected and the potential _ may have stationary
points (for a surface of genus g, a potential with m poles will have
2m + 2g - 2 stationary points). However, it is to be expected that a
suitable _ can be found with no singularitiesin the domain of interest.
We will now consider the problemof producinga grid on Q as above using
harmonic mapping techniques. This method will provide non-isothermal
coordinates in general. However, we will use the fact that the domain of
interest Q on the manifold could be covered by a global isothermal
coordinate system to develop the rest of the theory. Thus, the harmonic
mapping problem on a curved surface or a multiply connected plane domain with
multivalued coordinates (xl,x 2) has been reduced to a problem of finding
slngle-valued coordinates (xl(_l,_2),x2(_I,_2)) on a simply connected
domain in the unlformlzlngplane G.
--7--
We will now express the Laplace-Beltrami operator in these uniformizing
variables,
1 _2 _2
A = 7 (_(_I)2 + _(_2)2 )' % = %(_i,_2).
Thus, we have Axe = O, e = 1,2, in the image _0 on the plane region G.
Let us now show that this grid generation procedure does map the interior
of _0 to the interior of _I in a one-to-one manner. The problem has been
reduced to the uniformizing plane and therefore the following theorem applies
[8], we include the proof for completeness.
TREOP,_ 1: The coordinate functions (xl,x 2) obtained by harmonic
mapping techniques map _0 into the rectangle _I have non-vanishing
gradients in _0"
Proof: Suppose Vx I = 0 at _ s _0' then
1"
W(o) = xI + i x ,
where x I* is an harmonic conjugate of x 1, must satisfy
_W
_--_= 0 at °O'
which implies that Z(o) = W(o) - W(o O) should have a zero of order greater
than or equal two. If this is true, then the argument of Z(o) around a_0
is at least 4_ which means that x1(o) - xl(o0) should vanish at least four
times on a_0" But we chose RI to be a rectangle and therefore this
-8-
function has exactly two zeroes on _0" Therefore,we conclude that Vx1
o
Vx2 are never zero in _0" We remark that if _I had been a more
and
complicated region, then this proof would be no longer valid.
THEOREM 2: Le____tT be the harmonicmapping of _0 into _I" That is,
T : 8f10+ Bill homeomorphically
and
o
AT = 0 on _ .
-- 0
Then if T __is C1 __°n _0\{Pi}_=l where {Pi }Ni=lC 8_0' then the Jacobian
O
of T never vanishes in _ .
-- 0
Proof: Let
P _ _n0\{Pi}_=I.
Then 8_0 is smooth at P and we may unambiguously define an orthogonal
coordinate system _(P), _(P) where _(P) is the tangent vector on 8_ 0
obtained by traversing _0 at constant velocity and _(P) points into _0"
o
This yields an orientation of _a 0. Now let {Rn}_= I C a with Rn + p.
Then computing the Jacobian of T at Rn in the coordinates _(P), _(P), we
have for C # 0,
1 2 2 l)(Rn ) = J(TI(Rn)"C.(Xs Xn - Xs Xn
Since T is CI, these values converge to the value at P, namely
-9-
1 2 2 I)( x -X x (P).C. xs n s n
But since P is a boundary point and _i is a rectangle, then one of the two
terms vanishes and the remaining term is nonnegative if both curves have been
oriented counter-clockwise. Thus, we conclude that the Jacobian of this
mapping is well-deflned at the boundary and nonnegative.
Now consider
X(C) = (3x2)I(3_l) + i[(3x2")I(_I)]
( xl)IC x2) + '
2_
where x is the conjugate harmonic function to x2. Thus, making use of
the Cauchy-Riemann equations we have
Re(x(c)) --_(xl,x 2) I lVxll2)
o
which is harmonic in _ and hence, is positive in _ by the maximum
0 0
principle. Here we have used the fact that Vx I # 0 and the nonnegativity of
the Jacobian in the boundary.
Thus, the harmonic mapping method produces for multiply connected plane
domains as well as portions of analytic Riemann manifolds, a locally i-i
mapping function. We will now prove that these mappings are globally I-1.
Although our present focus is on mappings in R2, the next two lemmas are
presented in RN for future extensions to mapping problems in higher
dimensions.
-i0-
m
o
LEMMA I: Let _0 and _I be compact subsets of _ with _ =0 0
and _i convex. Let T be a continuous mapping from _0 into _ which
o
is differentiable in _ and satisfies
i. T(_0) = _1
o
2. IT'(x)I_ 0 for all x _
-- 1
then T maps _0 onto _i"
o
Proof: The first thing to notice is that T is an open map on _ and
0
o
hence, T(a ) is open. We claim that T(_ O) C _1" If not by compactness,0
_ o
there is an x _ _ satisfying
-- 0
0 < distIT(x ), _i) = max dist(T(x), _1) ,
O
But T is open on _ and _ is convex, which implies that
0 1
y + dist(y, _i ) has no local maxima. This contradicts the assumption that
o o
T(_0) _ _i" Thus, we have T(_ 0) C _i and even T(_ 0) C _i. If we fail
O
to have equality, there is a y _ _I\T(_0). But for any such y, we must have
inf liT(x)- yll = inf liT(x) - yn,
x _ a0 x € _0
o o O
since T is open on _ . However, one can always produce a y _ _ \T(_o)I I
o
arbitrarily close to T(_ ) and relatively far from _I = T(_0) to violate0
this condition. This completes the proof of this lemma.
-II-
LEMMA 2: Let _0 and _1 be compact homotopic subsets of _ with
_I convex. Let R , 0 < e < I be the homotopy with
R (R1) = Ru
m
o
=_
Rl(x) = x for x 8 _I
o
]R'(x)l _ 0 for x € _ .
a -- -- 1
Let T, 0 _ _ < 1 be the mapping from fl . satisfying
TI(_) = _ for x £ _I
o
[T'(x) I # 0 for x _
1
T(x(3_a) = 3_I
Tu o Ru is a homotopy on _I" Then Ta is a one-to-one and onto
transformation from _ into _0" In particular, TO maps _0 one-to-one
and onto _I"
Proof: Recall that for a differentiable map T : _ . _ (for _ compact
subset of _) we can define
-12-
(a) d = degree (T,y,_) = _ signlT'(_)l
T(x) = y
0
o
provided y _ T(8_) and for each x € _ above, IZ'(_){ # O. Furthermore,
this integer d is invariant under homotopy (homotopy invariance theorem)
provided no solutions are introduced on the boundary [9]. Now to prove the
lemma, set
S =T oR.
(I (% 5
o
Thus S is a homotopy on _i and for each x _ _ ,5 -- 1
I I 0
and hence, IS_(x__)l is always positive or always negative. But IS_(_)I = 1
o
and hence, IS'(x) l > 0 for all x € _ . We know from Lemma I that T
-- -- I 5
o o
maps _ onto _ and _ onto _I" Thus, S maps _I onto _I ando 5 o 1 o 5 5
_1 onto _1" Let y € _1 and note
degree(SI, y, _i ) = 1
since SI is the identity. Both S5 is a homotopy which introduces no
solutions on the boundary and hence, by degree theory
degree(S5, y, _i ) = I,
o
since {S'(x){ > 0 for all x € _ . Thus, (a) actually counts the number of
5-- -- 1
-13-
x s, so that
Sa(x) = y.
o
Since this number is one we conclude that, S maps _ , one-to-one and onto
a I
o
, and hence T maps _ one-to-one and onto _I and finally _0 one-to-1 a a
one onto _I as was to be shown.
CONCLUSION
Grid generation methods using harmonic mappings have been analyzed in a
unified framework and a rigorous justification for these methods is given. We
have proved that for these mapping methods, mesh intersection, or overspill
cannot occur. Our conclusions are limited to two-dimensional manifolds,
although substantial portion of the theory developed is applicable for higher
dimensions. An extension of this theory should include the generalization to
higher dimensions and to the method of nonharmonic mappings.
Acknowledgment
The authors would like to thank B. Moss, Old Dominion University for
stimulating discussions and A. Sherif for part of the computations.
-14-
References
[i] Thompson, J. F., Thames, F. C., and Mastln, C. W., J. Comput. Phys. 15
(1974),pp. 299-319.
[2] Warsi, Z. U. A., Quart. Appl. Math., July (1983), pp. 221-236.
[3] Mason,W. H. and Miller, D. S., AIAA Paper No. 80-1421,July 1980.
[4] Springer, G., Introduction to Riemann Surfaces, Chelsea Publishing
Company, New York, 1981.
[5] Kreyszlg, E., Introduction to Differential Geometry and Riemannian
Geometry,Universityof ToronotPress, 1975.
[6] Warsi, Z. U. A. and Ziebarth, J. P. in Numerical Grid Generation_ J. F.
Thompson, ed., North-Holland, 1982.
[7] Lovelock, D. and Rund, H., Tensors_ DifferentialForms and Variational
Principles,John Wiley & Sons, 1975.
[8] Mastin, C. W. and Thompson, J. F., J. Math. Anal. Appl., 62 (1978),
pp. 52-62.
[9] Ortega, J. M. and Rheinboldt, W. C., Iteratlve Solution of Nonlinear
Equationsin SeveralVariables_AcademicPress, 1970.
-15-
_Q
M2Q
Figure 1
-16-
Figure 2
-17-
Outer
surface
M2
Inner
surface
E3 _ ,
Figure 3
-18-
/r4 rnJ
r3 2_
,I
J Two-sheetedplanewith N branch cuts
ClosedRiemannsurface
of genus g-- N - 1
Figure 4
-19-
42
3 7
9
I0
Figure 5
-20-
AP2 I_3 ^" pPI n
S
Figure 6
=21-
1. ReportNo. 2. GovernmentAccessionNo. 3. Recipient'sCatalogNo.
NASA CR-172331
4. Title and Subtitle 5. Report Date
March 1984
On the Grid Generation Methods by Harmonic Mapping on
Plane and Curved Surfaces 6. PerformingOrganization Code
7. Author(s) 8. Performing Organization Report No.
S. S. Sritharan and Philip W. Smith 84-12
10. Work Unit No.
9. Performing Organization Name and Address
Institute for Computer Applications in Science
and Engineering 11. Contract or Grant No.NASI-17070
Mail Stop 132C, NASA Langley Research Center
Hampton, VA 23665 13, Type of Report and Period Covered
12. Sponsoring Agency Name and Address
contractor report
National Aeronautics and Space Administration 14. SponsoringAgencyCode
Washington, D.C. 20546
15. Supplementary Notes
Langley Technical Monitor: Robert H. Tolson
Final Report
16. Abstract
Harmonic grid generation methods for multiply connected plane regions and regions
on curved surfaces are discussed. In particular, using a general formulation on an
analytic Riemannian manifold, it is proved that these mappings are globally one-to-one
and onto.
T
17. Key Words(Sugg_ted by Author(s)) 18. Distribution Statement
grid generation 64 Numerical Analysis
harmonic mapping 67 Theoretical Mathematics
Unclassified-Unlimited
19. S_urity Oa_if.(ofthisreport) 20. S_urityClassif.(ofthis_ge) 21. No. of Pa_s 22. Dice
Unclassified Unclassified 23 A02
ForsalebytheNationalTechnicalInformationService,Springfield,Virginia22161 NASA-Lang'ley,1984




On the grid generation methods in harmonic mapping on plane and curved surfaces

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