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Grid generation via deformation
089%9859/92 $5.00 + 0.00 Copyright@ 1992 Pergamon Pnss Ltd
Appl. Math. Lett. Vol. 5, No. 3, pp. 27-29, I992 Printed in Great Britain. All rights reserved
GRID
GENERATION
VIA DEFORMATION*
G. LIAO AND J. SU** Department of Mathematics, University of Texas at Arlington Arlington, TX 7601~0408,
(Received
U.S.A.
October 1991)
Abstract-The main results of the deformation technique are presented in the physical domain. A derivation of the formula for the deformation vector field is obtained. A technical constraint on the prescribed Jacobian determinaut is removed.
1. INTRODUCTION A deformation method was first used by J. Moser [l] in his study of volume element of a Riemannian manifold. The technique is improved and applied to grid generation problem in [2] and [3]. The method gives rise to a numerical construction of a diffeomorphism u : Q 4 R C R” such that u maps a grid of R into a new grid whose mesh sizes are equal to a prescribed characteristic function f. The method works in theory for all dimensions. The theorem stated below is new compared with those in [2] and [3]. The main difference is that the characteristic function f is evaluated at the new grid points. Note also that the constraint f = 1 on I~Q [2] for a cubic domain Q is removed. 2. THE
DEFORMATION
TECHNIQUE
MAIN THEOREM. Let R be a bounded open set in R”, B be a ball (or an a.unuIus, a cube) in R”. Suppose there exists p : s2 -+ B, an orientation preserving CK+3 diffeomorphism (i.e., det VP > 0 in R ), IC 2 1. Let f > 0 be in CK(fi) and sn l/f(x) dx = ICI]. Then there exists 21E oiffK(b) satisfying
det Vu(x)
= f(u(x)),
PROOF. It suffices to show the theorem w : Q -+ R is a solution to
u(x) = x,x E an.
x E a;
for the case when
det VW(X) = g(x), x E R;
w(x)
n = B (cf. [3,4]).
by u(x) = w-l(x),
STEP
v solves the linearized
v E Ck+‘(R,
Fan) such that
div v(x) = g(x) - 1,x E R; STEP 2.
Solve, at fixed x E a, the evolution $(x,t)
Observe
=x,xEai2,
where g(x) = l/f(x), then u : Cl + 52, defined we follow the two steps below. 1. Construct
(2.1)
satisfies
that
if
(2.2) (2.1).
problem
To solve (2.2),
of (2.2)
v(x) = 0, x E 8R.
(2.3)
equation
= rl(P(x,t)),t
> 0;
cp(x,O) = x,
(2.4)
*This work is partially supported by a grant from the National Science Foundation. **The authors would like to thank Mrs. Leslie Wagner for the excellent typing of the manuscript. Typeset by A,&-T&X 27
G.
28
where the deformation
LIAO,
J. Su
vector field q : d ---i R” is determined
??(Y) =
by
V(Y) t + (1- MY) *
Then the solution to (2.4) at t = 1, W(Z) = cp(z, l), satisfies shows that, H, defined by
(2.2). In fact, the direct computation
(2.6)
H(t, ~1= 0 + Cl- Mcp(x, t))) detVv(+, t), is time indepenent We conclude
(cf. [4]). Thus,
this section
H( 1, z) = H(0, z), which is equivalent
by a derivation
to (2.2).
PROOF OF (2.5). In this case, detV+$z,t) = &p(z,t), div v(z) = v. with respect to t, we get from (2.3) and (2.4), setting Y = cp(z,t), that
dH dt = dA(y)(t dy Thus,
%
I
of (2.5) in the case n = 1.
+ (1 - My))
Differentiating
(2.6)
- V(Y)]*
(2.7)
- v(y) = 0, which gives rise to (2.5).
= 0 if ~(y)(t + (1 - t)g(y))
3. DETERMINATION
OF
Much of our work is devoted to the numerical algorithm typical domains for which v can be written explicitly.
v of solving
(2.3).
Here we list two
(a) Q = [0, 113. Let h(z) be an arbitrary smooth function with h(0) = h'(0)= h’(1) = h(1) = 1. Then v = (VI, VZ,2)~) can be determined by [2] Vi(Z) =
0=i7
- 1)V 1,22,23)&
- h(zl)
/
'Cs-
(9 - l)(h,t2,~3)dtl~tt
1 JJ
l)(h,~2,~3)dh,
J0
v2(z) = h'h)
- h(Z2)
0
v3(4
= h'(q)h'(z2)~='1lI'(l-
0,
1
(3.1)
(9 - l)(w2,~3)&%
0
I
l)(tl,t2,23)dtldt2dt3.
For an alternative (b) R = B(R2)
formula which works for all n, see [3]. an annulus in R2. Let (r, 0) be polar coordinates. \B(&),
Set
sl(r, 4) = (g(r,e) - 1)- p(r) JR: s(s(s,e)- l)ds, Ra
g2w =
J RI
where p(r) 2 0 is any smooth be determined by
4ds, 0) - 1)ds, function
such that
Jil’ rp(r) dr = 1. Thea
u = (Vi, va) CZJII
(in:)+v(~$z4a)d~ (12:). (3.2) J sgl(s,B)ds r
v(r,e) = 1 r RI For n 13,
see [3].
Grid generation
4. RELAXING
2S
OF g = 1 ON BR
For the balls in R”, the method in [3] does not require g = 1 on BR. Thus, we can treat other domains without the requirement g = 1 on 852 as well. For example, we deform the cube Q to a ball B with ]B] = 1 by a radial mapping \E : Q + B such that p E BQ is mapped to q(p) E 6B and the interior points are mapped proportionally in the radial direction. Next, construct a :B --+ B and p :B --) B by polar coordinates [3] such that det V&(z) = det V!P-l(z), detV/?(z)
= g(W’(z))
z E B;a(z)= z,z E 8B;
-detVW’(z),z
E B;P(x)
= 2,~ E aB.
(4.1) (4.2)
Then u = \E” o o-l o ,f3o 9 clearly satisfies det VU(Z) = g(z), t E 60;
u(x) =
x,xEac2.
(4.3)
For other methods in grid generation, we refer to [5] and [S]. For related work in fluid mechanics, we refer to [7] and [8]. REFERENCES 1. J. Moser, On the volume elements on a manifold, IPrcrnr. A.M.S. 120 (November), 288-294 (1966). 2. G. Liao and D. Andemon, A new approach to grid generation, Applicable Analyris (to appear). 3. G. Liao and J. Su, A direct method in Dacorogna-Moser’s approach of gxid generation, Applicable Analyrir 4. 5. 6. 7. 8.
(to appear). B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. JuJ~. Htnri Poincar, Analyre Non Lindire 7 (l), 1-26 (1990). J.F. Thornwon, Z.U.A. Warsi and C.W. Mastin, Numerical Grid Generation, North-Holland, New York, (1985). J. Cantillo, S. Steinberg and P.J. Boa&e, Parameter estimation in variational grid generation, Applied Mafh. and Computers 28,156177 (1988). D. Anderson, Grid cell vohune control with an adaptive grid generator, Applied Math. and Computers 98, l-9 (1990). P. E&man, Grid generation for fluid mechanics computation, Annual Review of Fluid Mechanicr 17, 487-522 (1985).
Appl. Math. Lett. Vol. 5, No. 3, pp. 27-29, I992 Printed in Great Britain. All rights reserved
GRID
GENERATION
VIA DEFORMATION*
G. LIAO AND J. SU** Department of Mathematics, University of Texas at Arlington Arlington, TX 7601~0408,
(Received
U.S.A.
October 1991)
Abstract-The main results of the deformation technique are presented in the physical domain. A derivation of the formula for the deformation vector field is obtained. A technical constraint on the prescribed Jacobian determinaut is removed.
1. INTRODUCTION A deformation method was first used by J. Moser [l] in his study of volume element of a Riemannian manifold. The technique is improved and applied to grid generation problem in [2] and [3]. The method gives rise to a numerical construction of a diffeomorphism u : Q 4 R C R” such that u maps a grid of R into a new grid whose mesh sizes are equal to a prescribed characteristic function f. The method works in theory for all dimensions. The theorem stated below is new compared with those in [2] and [3]. The main difference is that the characteristic function f is evaluated at the new grid points. Note also that the constraint f = 1 on I~Q [2] for a cubic domain Q is removed. 2. THE
DEFORMATION
TECHNIQUE
MAIN THEOREM. Let R be a bounded open set in R”, B be a ball (or an a.unuIus, a cube) in R”. Suppose there exists p : s2 -+ B, an orientation preserving CK+3 diffeomorphism (i.e., det VP > 0 in R ), IC 2 1. Let f > 0 be in CK(fi) and sn l/f(x) dx = ICI]. Then there exists 21E oiffK(b) satisfying
det Vu(x)
= f(u(x)),
PROOF. It suffices to show the theorem w : Q -+ R is a solution to
u(x) = x,x E an.
x E a;
for the case when
det VW(X) = g(x), x E R;
w(x)
n = B (cf. [3,4]).
by u(x) = w-l(x),
STEP
v solves the linearized
v E Ck+‘(R,
Fan) such that
div v(x) = g(x) - 1,x E R; STEP 2.
Solve, at fixed x E a, the evolution $(x,t)
Observe
=x,xEai2,
where g(x) = l/f(x), then u : Cl + 52, defined we follow the two steps below. 1. Construct
(2.1)
satisfies
that
if
(2.2) (2.1).
problem
To solve (2.2),
of (2.2)
v(x) = 0, x E 8R.
(2.3)
equation
= rl(P(x,t)),t
> 0;
cp(x,O) = x,
(2.4)
*This work is partially supported by a grant from the National Science Foundation. **The authors would like to thank Mrs. Leslie Wagner for the excellent typing of the manuscript. Typeset by A,&-T&X 27
G.
28
where the deformation
LIAO,
J. Su
vector field q : d ---i R” is determined
??(Y) =
by
V(Y) t + (1- MY) *
Then the solution to (2.4) at t = 1, W(Z) = cp(z, l), satisfies shows that, H, defined by
(2.2). In fact, the direct computation
(2.6)
H(t, ~1= 0 + Cl- Mcp(x, t))) detVv(+, t), is time indepenent We conclude
(cf. [4]). Thus,
this section
H( 1, z) = H(0, z), which is equivalent
by a derivation
to (2.2).
PROOF OF (2.5). In this case, detV+$z,t) = &p(z,t), div v(z) = v. with respect to t, we get from (2.3) and (2.4), setting Y = cp(z,t), that
dH dt = dA(y)(t dy Thus,
%
I
of (2.5) in the case n = 1.
+ (1 - My))
Differentiating
(2.6)
- V(Y)]*
(2.7)
- v(y) = 0, which gives rise to (2.5).
= 0 if ~(y)(t + (1 - t)g(y))
3. DETERMINATION
OF
Much of our work is devoted to the numerical algorithm typical domains for which v can be written explicitly.
v of solving
(2.3).
Here we list two
(a) Q = [0, 113. Let h(z) be an arbitrary smooth function with h(0) = h'(0)= h’(1) = h(1) = 1. Then v = (VI, VZ,2)~) can be determined by [2] Vi(Z) =
0=i7
- 1)V 1,22,23)&
- h(zl)
/
'Cs-
(9 - l)(h,t2,~3)dtl~tt
1 JJ
l)(h,~2,~3)dh,
J0
v2(z) = h'h)
- h(Z2)
0
v3(4
= h'(q)h'(z2)~='1lI'(l-
0,
1
(3.1)
(9 - l)(w2,~3)&%
0
I
l)(tl,t2,23)dtldt2dt3.
For an alternative (b) R = B(R2)
formula which works for all n, see [3]. an annulus in R2. Let (r, 0) be polar coordinates. \B(&),
Set
sl(r, 4) = (g(r,e) - 1)- p(r) JR: s(s(s,e)- l)ds, Ra
g2w =
J RI
where p(r) 2 0 is any smooth be determined by
4ds, 0) - 1)ds, function
such that
Jil’ rp(r) dr = 1. Thea
u = (Vi, va) CZJII
(in:)+v(~$z4a)d~ (12:). (3.2) J sgl(s,B)ds r
v(r,e) = 1 r RI For n 13,
see [3].
Grid generation
4. RELAXING
2S
OF g = 1 ON BR
For the balls in R”, the method in [3] does not require g = 1 on BR. Thus, we can treat other domains without the requirement g = 1 on 852 as well. For example, we deform the cube Q to a ball B with ]B] = 1 by a radial mapping \E : Q + B such that p E BQ is mapped to q(p) E 6B and the interior points are mapped proportionally in the radial direction. Next, construct a :B --+ B and p :B --) B by polar coordinates [3] such that det V&(z) = det V!P-l(z), detV/?(z)
= g(W’(z))
z E B;a(z)= z,z E 8B;
-detVW’(z),z
E B;P(x)
= 2,~ E aB.
(4.1) (4.2)
Then u = \E” o o-l o ,f3o 9 clearly satisfies det VU(Z) = g(z), t E 60;
u(x) =
x,xEac2.
(4.3)
For other methods in grid generation, we refer to [5] and [S]. For related work in fluid mechanics, we refer to [7] and [8]. REFERENCES 1. J. Moser, On the volume elements on a manifold, IPrcrnr. A.M.S. 120 (November), 288-294 (1966). 2. G. Liao and D. Andemon, A new approach to grid generation, Applicable Analyris (to appear). 3. G. Liao and J. Su, A direct method in Dacorogna-Moser’s approach of gxid generation, Applicable Analyrir 4. 5. 6. 7. 8.
(to appear). B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. JuJ~. Htnri Poincar, Analyre Non Lindire 7 (l), 1-26 (1990). J.F. Thornwon, Z.U.A. Warsi and C.W. Mastin, Numerical Grid Generation, North-Holland, New York, (1985). J. Cantillo, S. Steinberg and P.J. Boa&e, Parameter estimation in variational grid generation, Applied Mafh. and Computers 28,156177 (1988). D. Anderson, Grid cell vohune control with an adaptive grid generator, Applied Math. and Computers 98, l-9 (1990). P. E&man, Grid generation for fluid mechanics computation, Annual Review of Fluid Mechanicr 17, 487-522 (1985).